Extensions of the propositional logic

In addition to the considerations presented in the last chapter, some important extensions of the propositional logic must be mentioned here in any case, in order not to let the reader believe that he or she has already become acquainted with a large part of the logic through propositional logic. The possibility of expression of the propositional logic is still very limited and an extension in this or that direction will soon be desired, if one is occupied longer with it. However, the knowledge I have provided so far about the origin and meaning of the concluding rules is sufficient to understand the further considerations in this book.

Predicate Logic

With the propositional logic one can find rules of inference that lead from true statements necessarily to true statements. The modus ponens here is the prototype of such a concluding rule.  But we know from Aristotelian logic also other conclusions which necessarily lead from true to true, e.g. the two sentences “All men are mortal” and “Socrates is a man” lead to “Socrates is mortal”. 

Here in the proposition “All men are mortal” the word “All” appears.  One could continue to consider the proposition as a whole and its truth value and thus stick to the propositional logic. But you can also look into the “inside” of the sentence and notice that here the number of elements of a set comes into play. This is not always the case, but with a view to expressing oneself in as much detail as possible, this possibility should also be taken into account in a formal language. This has led to the extension of the propositional logic to the so-called predicate logic, in which a special structure is now provided for the propositions, namely a quantification with “quantifiers” such as “all” or “none”.

The decisive factor for such an extension was that one learned to give a more general meaning to the term “predicate”, which we know from the grammar of a natural language. In logic, a predicate is everything that can be meaningfully attributed to an object, i.e. “predicated”. So, in the sentence “Socrates thinks”, “thinks” is the predicate. Thus, in addition to properties, these can also be verbs. But also multi-digit relations, e.g. R(x,y) can be predicates; here now two objects x and y are assigned to a relation R.

The best translation of such sentences into character strings has proven to be, for example, that a one-digit predicate is seen as a function of a variable and the object to which the predicate is assigned is seen as an argument. The function s(x) then means that to the object “x” the predicate s≔ “is mortal” is predicted, i.e. assigned. One can then easily formulate that the predicate belongs to several or all objects from a given set or to none. This therefore gives the possibility of quantification.

This function s(…) has as function value the truth value of this statement; s(x) is therefore 1 or true, if x is mortal. If we now define the function M(x) with M(…):= “is a human being”, we can now formulate:

For all x applies: If x is a human being, then x is mortal, or

∀x (M(x) → s(x)).

where we have introduced the symbol ∀x for “For all x”.  The symbol “∀” is called a quantor. Of course, you have to determine beforehand to which quantity “all” refers.

Useful is also the symbol “∃x” for “There is an x”. In principle, however, the symbol “∀x” alone can be used for all quantifications.

The predicate logic therefore includes the propositional logic. Additional concluding rules in predicate logic are first:

∀x M(x) ∧ (x = c) ⊨ M(c),

i.e. “If a predicate M is assigned to all x from a given set, then also to a single element c of this set”, and second:

M(c) ⊨ ∃x M(x),

i.e. if an object c has the property M, then there exists an object x that has the property s.

After all, it does not work without the mode ponens, e.g. in the form

M(c), ∀x (M(x) → s(x)) ⊨ s(c),

where “c” stands for a specific element x from a predetermined set.

All syllogisms of Aristotelian logic can also be formulated in predicate logic. The syllogism “Barbara” reads here for example:

∀x (G(x) → M(x)), ∀x (M(x) → s(x)) ⊨ ∀x (G(x) → s(x))

where “G” stands for the predicate “is Greek”. Here a concluding rule is applied which can be derived from the modus ponens. But it can also be derived directly from the tautology

 (A → B) ∧ (B → C) → (A → C).

So far, quantification has always extended to object x, an element from a given set. One speaks here of a first-order logic. You can also introduce a second-order logic by extending the quantification to the predicates as well.

We want to study this from a famous example: Let us first consider the statement

P(x) → P(S(x)),

i.e. if the predicate is assigned to object x, then also to object S(x). This makes sense, for example, if x is a number and S(x) = x + 1, the successor of x. If, for example, an equation for x is fulfilled, then it should also be fulfilled for x + 1.

This one can be required for all natural numbers, and one gets

∀x (P(x) → P(S(x))).

This is one of the prerequisites for a particularly prominent concluding rule in number theory, the “mathematical induction”. One should not confuse the word “induction” with the logical induction. The danger of confusing it with physical induction is probably less. In order to arrive at a conclusion, there must still be an induction start, e.g. P(0), i.e. the property P is assigned to the number 0. Then, with the above implication we may conclude: ∀x P(x), i.e. the property P applies to all natural numbers x.

If, on the other hand, one wants to demand this statement ∀x (P(x) → P(S(x))) for all predicates P, then one writes:

∀P [∀x (P(x) → P(S(x))].

So, this is a statement in the language of second-order logic; the quantifier operates also on a predicate P.

You can also continue predicting, i.e. assigning a property, by assigning properties, i.e. predicates, to properties again. If F designates a property that can be assigned to a predicate P, you can, for example, first formulate:

F(P) → P(x),

That means: If a predicate P has the property F, then x has this predicate P. If you require this for all predicates P of a certain set, you can, for example, formulate:

∃x ∀P(F(P) → P(x))

i.e. for all predicates P applies: If they have the property F, then there is also an object x, which has this predicate P.

If the property F means “is good”, whatever that is, then the proposition says: “There is an object x that has all good properties.” Of course, this does not say anything about whether this statement is true or can be derived from any axioms. One can only formulate it.

About the differences of the first- and second-order logics with regard to their means of expression there are deeply rooted propositions in mathematical logic to which one can gain access only through intensive study. Here it can only be shown how to extend the language of propositional logic. It should be noted that the concluding rules of first-order logic are “sufficient as building blocks for all mathematical modes of argumentation” (Ebbinghaus & Thomas, 2018, p. 62). 

Modal and Deontic Logic

In the propositional and predicate logic, one always restricts oneself to a certain type of proposition, namely to those that can either be true or false. The law of the excluded third applies here, i.e. A ∨ ¬A is a tautology. 

But often we do not know for sure whether the statement A about a fact is really true. We now also want to consider that it is only possible that a proposition p is true. (In the following we describe propositions with small letters p, q, …, so that the notation remains clear later.) However, for statements p, which we have derived from accepted assumptions according to logically correct concluding rules, we would then say that the proposition is necessarily true.

Finally, we also know the situation in which we can only say about a fact that it can be possible but does not have to be necessary, that it is therefore “contingent”. If the language of predicate logic is extended by characters for such terms, one speaks of modal logic.

Analogously, another extension of the language of propositional logic can be developed, which deals with duties such as obligations or prohibitions. You then get a deontic logic (δέῖ, δέῖ gr. = you have to). But you can’t make proposition in this area, but you can make actions: The symbol “p” then stands for an action instead of a proposition. And one can also allow actions, i.e. neither forbid nor command them.

Here we see parallels between the terms “necessary, impossible, contingent” and ” obligatory, forbidden, permitted”. Fig. 1 also shows them clearly.

Fig.1 The three basic modal and deontological terms

It turns out that the modal terms “necessary”, “impossible” and “contingent” are disjunctive to the other two terms: What is not necessary can be impossible, but also contingent. What is not impossible may be contingent, but even necessary, and what is not contingent is either necessary or impossible. The same applies to deontological terms: What is not obligatory may be forbidden, but also permitted, etc.

With these three terms one must now expand the formal language of propositional logic by introducing new symbols. Actually, you only need a symbol for one term, because it shows that you can express all the others with the help of negations through this one symbol.

This new symbol shall be the sign “N” in modal logic, which can also be interpreted as an operator, which acts on a proposition p: “Np” shall mean: “It is necessary, that p is true”.

In modal logic we introduce the operator “O”, “Op” now means: “It is obligatory that the action p is performed”. 

All in all, you get this:

Np: It is necessary that p is true, i.e. it is impossible that p is false,
¬Np: It is not necessary for p to be true, i.e. it is not impossible for p to be false,
  N¬p: It is necessary that p is false, i.e. it is impossible that p is true,
¬N¬p: It is not necessary for p to be false, i.e. it is not impossible for p to be true,
¬Np ∧ ¬N¬p: The statement p is contingent true because it is not necessary and not impossible to be true.

One can formulate accordingly:

Op: It is obligatory to perform the action p,
¬Op: It is not obligatory to perform the action p,
O¬p: It is obligatory not to perform the action p, i.e. it is forbidden to perform the action p,
¬O¬p: It is not obligatory not to perform the action p. So, it is allowed to perform the action p.
¬Op ∧ ¬O¬p: It is neither obligatory nor forbidden to perform p.

A distinction must be made here between norms that are to be set with a law and propositions about norms. The norms are brought into effect by setting, the propositions about norms can be true or false.

Fig. 2 shows the different operators in the so-called modal or deontic hexagon. With these terms and their symbols, which are comprehensible for everyone, all specifications can be clearly represented.

Fig.2: The modal and deontic hexagon (after (Joerden, 2010), (Honerkamp, 2015)). The arrows indicate an implication (what is obligatory is also permitted), ” >-<” a contravalence or contradictory opposition (e.g. what is not forbidden is permitted), “—-“an exclusion or contrary opposition (e.g. being obligatory and being forbidden are mutually exclusive) and “…”” a disjunction or subcontrary opposition (e.g. contravalence >-< to “permitted” and to “not obligatory”).

Further explanations such as the discussion of possible axioms and a calculus in connection with the calculus of propositional logic go beyond the scope of this book.


However, the list of extensions to the propositional logic has not yet been completed in any way. There are also so-called non-classical logics in which, for example, the law of the excluded third does not apply. One then does not assume that propositions are only either true or false. In modal logic we have assumed this, only considering the possibility that one does not know it exactly. Here now it is to be renounced that there is “ontologically” only this alternative. It will no longer be adequate to map the Boolean association of statements to a binary Boolean association. There is a wealth of approaches to such multi-valued logics, and “intuitionist logic” is a prominent example of this.

The so-called fuzzy logic must also be mentioned here. It still permits nuances of the predicates such as “very” or “rather”, however treats them quantitatively again. It plays an important role in many control engineering applications.

But the most elaborated approach is “dealing with uncertain knowledge”, which is based on mathematical probability theory. Here a proposition is assigned a measure of the probability that it is true. This can also be seen as a measure of the credibility of a statement.

Thus, we can pick up the theme of Aristotle, which he hinted at in the classification of conclusions, when he spoke of a dialectical conclusion. Here already questions arose, which could not be answered at that time: What can be said about a conclusion in dialectical inferences? Can one even develop a calculus for propositions that are merely credible? How could conclusions be drawn as strictly as possible in such a calculus? What would “strict” even mean here? These questions will be addressed later.

Tautologies and Concluding Rules

In the excursus on formal languages in the last blog post, we already got to know the signs and more general expressions of propositional logic. We have seen that there are special character strings, so-called tautologies, which are always true, i.e. independent of the truth values of the individual characters. We had also already introduced such a tautology, namely (A ∧ (A → B)) → B.

Here we will show how logically correct reasoning rules can be formulated with the help of tautologies. But let’s first ask which simple tautologies still exist and how to generate tautologies in general. Interesting in this context are also character strings, which are always false, because from their negation one can also gain a tautology.


Let us first introduce prominent tautologies:

            A ∨¬A is in any case true,

because either A is true or ¬A. There is no such thing as a third according to our requirements. This statement is called the statement on the excluded third.

On the other hand, A ∧ ¬A is wrong in any case, because the statement A and the statement ¬A cannot be true at the same time. For example, it cannot be that it is raining and not raining at the same time. A and ¬A contradict each other. One generally calls a compound expression, which is false regardless of the truth values of the individual statements, a contradiction. Then, the following applies to the negation:

¬(A ∧ ¬A) is true in any case.

This statement is called the principle of contradiction.

Here we should now also list the tautology from the last blog post again.

(A ∧ (A → B)) → B is in any case true.

How do we find further tautologies in order to be able to form further rules of inference?

It can be shown that the propositional logic can be regarded as an axiomatic-deductive system. Axioms can be all tautologies of the form (by Kutschera & Breitkopf, Alfred, 2007, p. 69).

 A → (B → A),

(A → (B → C)) → ((A → B) → (A → C)),

(¬A → ¬B) → (B → A)

and as a rule of inference or concluding rule the modus ponens, which we already mentioned in an earlier blog post. In a moment, we will really introduce this concluding rule by deriving it from a tautology.

All logical expressions, that can now be derived from these axioms using the modus ponens, are tautologies again. So you can set up as many concluding rules as you want. Only a few will be needed.

Now we understand why Wittgenstein says: The propositions of logic are tautologies. So the propositions of logic say nothing (Wittgenstein, 2006, pp. No. 6.1, 6.11).

From Tautologies to Concluding Rules

Let’s have a look at the truth table for the tautology (A ∧ (A → B)) → B again:

A B A B A ∧ ( A B) (A(A B))B
1 1 1 1 1
1 0 0 0 1
0 1 1 0 1
0 0 1 0 1

Let us first consider the first line, in which both premises A and A → B are true. From the second column of this first row we then infer that B is true. The statement B must therefore necessarily be true if both A and A → B is true. Only in this way can the character of tautology be respected.

This is now a conclusion that results from inspection of the truth table of a tautology. No reasoning can be more elementary. This is also the conclusion with which all other concluding rules can be won. So here we have the origin of logical reasoning before us, the “mother” of all reasoning rules.

You write, with the sign “⊨” for a logical conclusion:

A ∧ (A → B) ⊨ B,

but frequently also in a form in which the individual premises are separated only by a comma:

A, (A → B) ⊨ B.

The symbol “⊨” is not a sign of propositional logic, but an abbreviation for the phrase “follows logically from this” in colloquial language. Otherwise we should be able to count with this symbol like with “∧” or “∨”. It only expresses the relationship between the statements A ∧ (A → B) and B in the meta language, our colloquial language:  In the case that the statement A is true, and in the case that the statement B follows from A, then B is true.

You may be confused at first and ask why all the effort.  They knew that a long time ago. It is indeed trivial, in the truest sense of the word, because in the Middle Ages the word “trivial” was coined from insights gained in trivium, the lowest level of education in a monastic school. This trivium, in turn, was named in ancient times for a place to which three paths lead, and where many who share the same opinion can gather.

Here, however, in a formal language, what Aristotle has already defined is very concretely realized: “A conclusion is thus a speech in which, with certain assumptions, something other than the presupposed follows with necessity on the basis of the presupposed”. (after Schupp, I, p.267). The emphasis is on “necessity”.

This is the modus ponens. It is the most prominent logical conclusion, was already known in ancient times by the philosophers of the Stoa and subject of many discussions in the Middle Ages (see “the Logic of the Stoics”).

We still want to investigate what this conclusion tells us if one of the premises or both are wrong. In any case, the entire premise is then wrong, because the individual premises are linked by a “∧”. We extract the two relevant columns from the table above and arrange them somewhat differently:

A (A B) B
1 1
0 0
0 1
0 0

So, if the total premise is false (2nd to 4th line), B can be true, but also false, i.e. nothing can be said about the truth value of B. Everything can be deduced from a false premise. At first this is surprising, and in the history of logic it has been discussed for a long time. But if one can derive both B and ¬B, then the conclusion is meaningless.

From the derivation of the modus ponens we can learn how to create a general final rule: From every tautology that can be found in the form

M → B, you can use the final rule

M ⊨ B

because then it follows immediately from the truth table for “M → B” that under the condition that M is true, also B must be true, because this implication M → B is a tautology after assumption and therefore true.

This is a statement on the meta level, not in the calculation of the syntactic level. However, we know that because M → B is a tautology, we can go from the character string M to the character (string) B on the syntactic level without leaving the realm of true statements on the semantic level. We write this in the form of

M ⊢ B,

and call this operation a derivation at the syntactic level. That what is a conclusion on the semantic level is called a derivation on the syntactic level.

At the syntactic level, we can therefore pass from the character string M to the character or character string B without leaving the realm of true statements at the semantic level.

But now one knows how to “calculate” on the syntactic level, namely according to the rules for the formation of character strings and according to the concluding rules, by which one may transform certain character strings into others, which are also usually shorter. Such a system of calculation rules is called a “calculus”.

The calculus of propositional logic therefore has nothing to do with any meaning of the statements in terms of content. To a certain extent, it only provides the tracks on which truth of statements can be safely transported from premises to conclusion. Wrong statements on such tracks lead to arbitrariness. Without the truth of the premises “everything is nothing”. That’ll still occupy us.

Two remarks are in place here:

Let us consider the premises:

A: = „2 + 2 = 4“,

B: ≔ “Freiburg is located in the south of Germany”. 

The statements A and B are true, thus also the implication A → B, i.e. if 2 + 2 = 4, then Freiburg lies in the south of Germany”. Thus, the expression A → B is well formed, but futile. Then the logical conclusion

A, A → B ⊨ B

is also meaningless. That doesn’t have to be irritating. Even in our colloquial language we can form grammatically correct sentences that are meaningless: “The moon babbles red suit.” The tracks are not responsible for what goes on them. Incorrectly formed sentences or not well formulated expressions are senseless anyway.

You often hear people say: “It’s logical, isn’t it?” and the speaker means that the conclusion immediately makes sense to him. This feeling, however, probably does not refer to the concluding rule, but to the implication, which is of course true for the speaker. The speaker thus confuses the logical conclusion with his assumption that his premise is true. He should say, “It’s a plausible assumption for the modus ponens, isn’t it?” He would be met with venerable understanding.

Important Concluding Rules in Applications

The modus ponens is probably the most prominent concluding rule, even in generalized form, which reads:

(A1 ∨ A2 ∨ …   ∨ AnA) (¬A B1 ∨ B2 ∨ …   ∨ Bm)

  ⊨ A1 ∨ A2 ∨ …  ∨ AnB1 ∨ B2 ∨ …   ∨ Bm.

The underlined portion of the premise represents exactly the simple modus ponens. This term then also appears accordingly as B1 in the conclusion.

The premise here is based on a character string in normal or standard form, into which each character string can be brought by a systematic transformation process. In general, this is the normal form:

(A1 ∨ A2 ∨ …   ∨ An) ∧ (B1 ∨ B2 ∨ …   ∨ Bm).

Such procedures and applications of inference rules as the generalized modus ponens are implemented in so-called resolution algorithms. The length of the character string can be successively reduced. Within the “Prolog” program, for example, such an algorithm can be processed on a computer.

A particularly popular inference rule in mathematics, already known in ancient times, is proof by contradiction:

If one wants to prove that a statement A implies another statement B, one first assumes for the proof that besides a premise A also ¬B is true premise. If then one can deduce a contradiction from it, ¬B cannot be true. So B must be true, because there’s no such thing as a third. Here it is shown that the negation of an assumption can be reduced to a contradiction. In the Middle Ages this form of proof was therefore also called “reductio ad adsurdum”.

The contradiction can show itself e.g. in the fact that one can deduce from A ∧ ¬B a statement C, and in addition, the statement ¬C. In order to show that this strategy can also be represented as a concluding rule, one only needs to find the corresponding form M → B. This is

((A ∧ ¬B) → C) ∧ ((A ∧ ¬B) → ¬C) → (A → B).

This is indeed a tautology and therefore the concluding rule:

((A ∧ ¬B) → C) ∧ ((A ∧ ¬B) → ¬C) ⊨ (A → B).

In a somewhat different form one uses the proof by contradiction if one wants to know whether a statement B is contained in a knowledge base W and thus also true. So, you’re asking whether

W ⊨ B

applies. This is the case if W → B, i.e. ¬W ∨ B is a tautology. Since ¬W ∨ B can be transformed into ¬(W ∧ ¬B), we must therefore ask whether ¬(W ∧ ¬B) is a tautology and thus W ∧ ¬B is a contradiction. So, we can see that a statement B can be deduced from a knowledge base W if

W ∧ ¬B

is a contradiction. That is also plausible: If the information of B is contained in W, the contradiction B ∧¬B must show up somehow with the evaluation of the expression W ∧ ¬B.

In order to now show that W ∧ ¬B leads to a contradiction, the expression W ∧ ¬B is transformed into the disjunctive normal form within the framework of the calculus and then the receiving expression is successively reduced with the help of the generalized modus ponens until an expression is shown that represents a contradiction – or not, depending on whether the statement B is contained in the knowledge base W or not. 

The proof of the Pythagoreans that there are infinitely many prime numbers is e.g. of this form. The statement B is then: There are infinitely many prime numbers. The statement ¬B is: There are only finitely many prime numbers. The knowledge base consists of the rules of arithmetic for integers.

On this basis of ¬B and with the knowledge of W one shows then that one can always find a new prime number to every set of finitely many prime numbers, thus ¬B is false, in contradiction to the assumption that W ∧ ¬B is true.

The formal language of propositional logic

After briefly introducing Aristotle’s syllogistics in the last blog post, I should now actually explain how it were received and elaborated in antiquity, the Middle Ages and into modern times. In particular, the work of Gottfried Wilhelm Leibniz (1646 to 1716), in which important approaches to modern logic can already be found, should be honoured. It would also be his great project to describe a “Scientia generalis” based on the model of mathematics, in order to “bring about in all scients what Cartesius and others did by algebra and analysis in Arithmetica et Geometria”. (All writings II,1. p.160, after Schupp III, p.238). For this purpose Leibniz wanted to develop a “lingua characteristica” with suitable characters (characteristica universalis). But this and other developments in this direction can be better assessed in retrospect. Therefore, I want to skip this and talk about modern logic.

The mathematician and philosopher Gottlob Frege (1848-1925) founded modern logic with a book entitled “Begriffsschrift – Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens” (“Conceptual writing – one of the arithmetic formula languages of pure thought”). The foreword says: In this “small script I have now tried a rapprochement with the Leibniz thought of a lingua characteristicistica”. Leibniz is thus the godfather of the modern formal logic (Wille, 2018, p. 96).

Gottlob Frege was a philosopher and mathematician, his father had written a New High German grammar (Wikipedia: Gottlob Frege). So Frege probably knew the power of a formal language; with such a language he then raised logic to a new level, on which one could lead proofs as in mathematics. He thus discovered a “new science” for logic, as Galileo had done for natural research. And again one can see how formalization leads to an accelerated development of a field of knowledge, then “emigrating” from philosophy.

The final form of a “propositional logic” was then created somewhat later by Alfred North Whitehead (1981 to 1947) and Bertrand Russell (1872 to 1970). The “term logic”, as the philosophers had always practiced until then, was soon replaced by this propositional logic; logic developed into a predominantly mathematical discipline. Its mathematical form made it possible to transfer logical reasoning rules to machines, which has led, among other things, to the now flourishing field of “artificial intelligence” research.

Since one experiences again and again, which shyness most contemporaries have at mathematical formulas, one has to exercise great caution with the representation of a formal language for the logic. The best thing to do is to look at the structure of a formal language in analogy to the structure of our mother tongue and to demonstrate the similarity of a formal language with our colloquial language. It’s just all so much easier, and so simple, that at first you feel yourself as a stranger. The language of mathematics may serve as an intermediate station between the formal language of logic and our colloquial language, because it is already formal and the contemporary has at least some experience with it.

Thus, we now look at three languages: logic, mathematics and our colloquial language. In all three languages there is a certain set of characters with which you can form strings. In colloquial language this set represents the usual alphabet, strings are words and several strings represent sentences. So you learn to write the letters first, then the words and then the sentences.
Here the important difference between syntax and semantics has to be discussed. Syntax is what is called grammar in natural languages. It shows, for example, how words are declined or conjugated in a sentence regardless of their meaning, depending on their function and position, and in which order they can occur in a sentence. The syntax or grammar thus regulates the formal structure of a language. Semantics, on the other hand, deals with the meaning of words and sentences.

In formal languages, the distinction between syntax and semantics becomes even clearer. The syntax specifies exactly what should be allowed with regard to the formation of character strings. At the syntactic level, therefore, we are only dealing with strings of different complexity, without these having already been assigned a meaning.

Even a rudimentary knowledge of mathematics gives us an indication of the usefulness of a formal language. One can “calculate” in it, i.e. formally argue, in such a way that one does not have to know the meaning of the characters and character strings, which one changes, generates or removes there according to certain rules. The whole thing happens on the syntactic level and the rules are made in such a way that true statements are transformed into true statements – completely independent of the meaning of the signs and strings of signs. Only in this way is it understandable that we can transfer the calculating and logical closing to a machine, that we can develop algorithms to solve tasks that would require a certain intelligence from ourselves.

The syntax of the language for a propositional logic

In general, a construction of a formal language begins by first providing a character set, a certain set of symbols, i.e. defining an alphabet.
In propositional logic, the alphabet consists of a stock of

– Variables A, B, …, i.e. capital letters of our everyday language.
– the characters ‘¬’ and ‘→’, ‘∧’ and ‘∨’.
– the sign “≡”,
– the brackets “(“” and “)”; these should always be used, if you want to see a couple of strings as a unit, to keep an overview. Also, sometimes a space is used to increase the clarity, and finally
– the constants “0” and “1”.

With the help of these characters or “letters” you can form character strings. We will see that in the formal language of propositional logic there is no need to distinguish between letters, words and sentences. You only know characters and strings, and these are uniformly called expressions.
How character strings, i.e. more complex expressions, can be formed from the characters of the alphabet is regulated by certain laws, namely:

– If A is an expression, then ¬A is also an expression.
– If A and B are expressions, so are A ∧ B and A ∨ B.

You can always create new expressions in this way, e.g. ¬A ∨ B or A ∧ (¬A ∨ B). The expression ¬A ∨ B will later play a major role, so we will introduce the abbreviation A → B for him. This is another sign at our disposal. Any expression of the form A → B will always be understood as ¬A ∨ B.
With A and A → B and A ∧ (A → B) we can e.g. get also (A ∧ (A → B)) → B. We will deal with this expression later.

One calls expressions, which are formed according to these rules, well formed. So A → B and A ∧ B ∧ C C are well-formed, but ABA or AB→ are not.

Our natural languages also have an alphabet. We are most familiar e.g. with the alphabet of the German language. Other alphabets deviate more or less from it. These characters are also used to form strings, which we call words. These words are formed by the way people deal with the things of the environment. This is a historical process, and through the experience of always new things in this world, new words are always invented. Here there are hardly any rules, at most the condition that the word must be pronounceable. For example, the string mjk?a will not be regarded as a meaningful candidate for a word. So the word formation happens on the semantic level: “The thing must get a name”. A rule is then formed for how the accepted words must be written. Orthography is the teaching of “correct writing”, the correct representation of “accepted character strings”. Words are used to form sentences in colloquial language, which can then each represent a statement. The words are then modified, declined or conjugated depending on their position in the sentences.

In contrast, in a formal language “words” are formed at the syntactic level, and according to only a few rules. “Orthography” here means respecting the rules for the formation of strings. There’s no such thing as sentences.

– Mathematics is also a formal language. Here the character set usually consists of the set of symbols {0,1, … ,9}, i.e. the numbers from 0 to 9, and the characters {+, -,  ∙, :, =}. Depending on the mathematical theory, further characters are added. Furthermore, you still need symbols as placeholders for numbers that are not further specified at first. Often one takes letters of the alphabet of the colloquial language. There are also clear rules here for the formation of “words” and “sentences”. “Words” are then such strings as x + 0 or 5-4, which are also called “terms”. “Sentences” are equations like x + 2 = 0, and they are also formulated according to certain rules.

Herewith we have described the syntax of three languages. It regulates which character strings may be formed on the basis of a character set.

The semantics respectively the evaluation

In semantics we must now say what we want to understand by the characters we have introduced in syntax, what meaning we want to give them.

In propositional logic, the variables A, B, … should stand for statements in our colloquial language that are either true or false. Note that this is not about the content of the statement and its meaning, but only about an evaluation: true or false. In this respect one should actually only talk about evaluation instead of semantics. But we don’t have to become that exact in our colloquial language, in which we construct the formal language and in which we later talk about it. The colloquial language we call the “meta language”, the language of the propositional logic is then here the “object language”. We have to be very specific in this one. But in our colloquial language, however, we may have enough vague terms.

This condition that we only want to consider statements that are either true or false is constitutive for the entire propositional logic. So it’s not about interrogative sentences or exclamations. We also disregard the fact that there may be statements where you can never decide whether they are true or false. The fact that this limitation is not self-evident for some mathematicians does not need to be discussed here.

Let’s look at some simple statements. Be, for example.
A:= “It’s raining.”
B:= “The road’s wet.”
So e.g. A is true exactly when it rains. Truth is given when the statement is consistent with the facts.

In order to evaluate all the expressions that we can form with such statements according to the rules of syntax, we must now say what we intend to do with the other characters from the character set:

We interpret the “1” as true, the “0” as false.

The sign “¬” before a statement should negate it:
¬A = It is not the case that A, i.e. in our example: “It does not rain”.
It’s the same:
¬1 = 0, “not true” means “false”,
¬0 = 1, “not wrong” means “true”.

The expression A ∧ B represents a combination of two statements, reminiscent of 5 + 4 or 5 – 4. We determine the meaning of the sign “∧ ” by saying when this expression should be true depending on the truth values A and B and when it should be false. In principle, we have 16 possibilities there. The default choice is shown in the following truth table:

1 1 1
1 0 0
0 1 0
0 0 0

This assignment is done in such a way that the symbol “∧” corresponds to the colloquial “and”. We’ll test this against our example:
A ∧ B = “It’s raining” and “the road is wet”.
We obviously perceive this statement as true exactly when both statements A and B are true, as it is also stated in the table.

We define the meaning of the sign “∨” in such a way that in the expression A ∨ B it resembles the colloquial “or”:

1 1 1
1 0 1
0 1 1
0 0 0

The expression A ∨ B only has the rating 0, i.e. is only false, if both statements A and B are false. So with this “or” the statement “A or B” is true, if either A or B are true, but also if both, A like B, are true. This corresponds to the inclusive “or”. When we use the “or” in colloquial language, we sometimes mean, however, the exclusive “or”, i.e. the “either-or”. This is a typical example of the ambiguity of our colloquial language.

These three rules are actually sufficient to construct arbitrary strings.
The expression ¬A ∨ B with the truth tablet is particularly important and useful:

A ¬A B ¬A B
1 0 1 1
1 0 0 0
0 1 1 1
0 1 0 1

The link ¬A ∨ B, also written as A → B, is called implication. This applies to this:
If A, then B. The statement A is therefore a sufficient condition for B: Always, if A is true, B is also true.
The implication is only false if B is false.
If A is false, then the implication A → B according to the truth table is always true, no matter if B is true or false. That means: If the statement A is not sufficient, then one cannot say anything about the truth value of B in the case of a true implication.

In this way, the truth value can always be determined for longer character strings. We would like to give an example here of what will still concern us in the future. We’re looking at the string:

(A ∧ (A → B)) → B.

For these one receives the following truth table:

A B A B A (A B) (A (A B))B
1 1 1 1 1
1 0 0 0 1
0 1 1 0 1
0 0 1 0 1

So there are expressions in propositional logic that are true in every case, regardless of the truth value of the individual statements. Such expressions are called tautologies. We will see in the next blog post what important role such expressions play in the formation of conclusions.

The Syllogisms of Aristotle

In the last blog post, Aristotle presented the three types of conclusions: the logical conclusion, the dialectical conclusion and the false conclusion. Now we will have to deal with the logical conclusion, which derives a true conclusion from two true premises.

This is, of course, the most important conclusion, which ensures a safe “transport” of the truth of statements. That such a possibility exists is the good news, and you can’t overestimate it. The “bad news” would then be that truth is not gained in any way, it is only passed on. The premises must be true. The question as to how and where one can begin with true statements in practice will be of great concern to us later on.

With the logical conclusion one can distinguish now again several kinds. The individual proofs, as he also called them, differ in the nature of the premises and the conclusion, which then results from a “summation” of the premises. In the Greek “summation” means “συν-λογισμός” (syllogismos); thus, one also speaks of a syllogism, the doctrine of syllogisms is called syllogistics. 

Let us first look at an example of syllogism:

All humans are mortal.
All Greeks are humans.
Therefore: All Greeks are mortal.

So, there are three statements, two propositions from which one proceeds, and a conclusion in which the propositions are “added together”.

The form of sentences and their representation

In this example, all sentences are of the form “All A are B”.  The sentence “All A are B” can also be formulated as follows: “B applies to all A.”  This formulation suggests that B is a predicate of all A: B is attributed to all A (predicated), e.g.: To be mortal applies to all human beings.

The formulation is also the one that comes closest to the Greek text, so it is the more original form. In scholasticism however, this was rewritten then as “All A are B”. All A has the predicate B: All humans are mortal. 

The above syllogism is then in its original form:

To be mortal belongs to all human beings,
To be a human being belongs to all Greeks.
Therefore: To be mortal belongs to all Greeks.

Moreover, this formulation also suggests a method that was discovered much later to illustrate the relationship between two terms. One takes advantage of the fact that each term applies to a set of entities. “Human beings” can be Greeks but also Egyptians, Thracians or Asians or Europeans. If one represents the set of human beings by a circle or any closed curve (B), the set of Greeks as well (A), then the circle for the Greeks lies within the circle for the human beings.

Fig.1: The set of Greeks (A) is a subset of the set human beings (B). To be a human being (B) applies to all Greeks (A).

Such pictures are called Venn diagrams, after the mathematician John Venn (1834 to 1923), who introduced them following Leonard Euler (1707 to 1783). Actually, the philosopher Gottfried Wilhelm Leibniz (1646 to 1716) already used them. These Venn diagrams thus illustrate relationships between two sets in general, no matter of what kind the elements are. Within the framework of set theory, we would also write: A ⊂ B, i.e. A is a subset of B.

But this form of a sentence is not the only one at syllogisms. Aristotle obtained an overview of all possible forms of sentences and statements. This resulted in: (Prior Analytics, Book I, 1,  see

A premiss then is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite.
By universal I mean the statement that something belongs to all or none of something else;
by particular that it belongs to some or not to some or not to all;

by indefinite that it does or does not belong.

Thus, beside the general sentences like “All Greeks are human beings” or “To be human belongs to all Greeks” there is also the negation (no-compliance) and the particular statement. Altogether you get the following types of sentences:

B belongs to all A, (all A are B),
B doesn’t belong to any A, (no A is B),
B belongs to some A, (Some A are B),
B doesn’t belong to some A, (some A are not B).

We can illustrate the other types of statements as follows:

 Fig. 2: Left: B doesn’t belong to any A. Middle: B belongs to some A. Right: B doesn’t belong to some A.

In scholastic circles these forms are abbreviated as (A a B), (A e B), (A i B) and (A o B). The letters “a” and “i” should remind us of “affirmo”, “e” and “o” of “nego”.

The structure of syllogisms

Two such forms of statement then form the premises, one such the conclusion. Each conclusion can thus be characterized by three letters from the set {a,e,i,o} and by the position of the three letters, which stand for the concepts in the respective sentence.

One of these terms, the so-called middle term, must occur in both conditions, it can be in first or second place, or in one condition in first, in the other in second. This results in four different shapes or figures. The first figure is the following (see example above)

A – B

B – C

A – C.

Now Aristotle selects in all forms that combination of two forms of statement that necessarily lead to a conclusion. He simply sorts out the conclusions for which he finds a counterexample.

The valid conclusions can then each be represented by the form and a specific combination of the letters a,e,i,o. And in order to be able to remember such combinations better, one has integrated them into corresponding peculiar words, e.g. one remembers the combination a a a with the word “Barbara”, and knows in addition that here the 1st figure is present. This conclusion corresponds exactly to the above example.

Fig.3: The Barbara syllogism

Another important conclusion, also of the 1st figure, is called “Celarent” in this way:

Fig.4: The syllogism Celarent

Aristotle demonstrated in this way that one could systematically formulate a system of conclusions in which from the truth of the premises necessarily follows the truth of the conclusion.

With the systematics of Aristotle, one now has a complete overview of all possible conclusions. Previously however, in syllogistics, “the whole art of syllogistics had consisted in searching around with great effort of time and effort”. as Aristotle wrote in his work On Sophistical Refutations (after Schupp, I 275).

But here one already has a system of statements that is reminiscent of the formal predicate logic that will be introduced later. Like there, you can “quantify” using the predicates, i.e. you can operate with quantities such as “all”, “none” and “some” for the predicates.

Most importantly, Aristotle must select the valid conclusions “by hand” from the set of all possible combinations, simply by discarding those that he recognizes as invalid with the help of an example. This “recognition” is an intuitive one, one with “common sense”. One does not doubt the correctness of the conclusion, but for a strict science in today’s sense this kind of knowledge is not sufficient.

Even if one describes the relations of terms mentioned above with the help of set theory, and thus justifies the conclusions within the framework of set theory, the insight is a mathematical one. Thus, the conclusions would be only indirectly logically justified, because mathematics only uses, as we know today, the rules of inference that are ultimately gained in modern predicate logic. Only in this predicate logic can the conclusions be strictly justified by deriving them from tautologies. We’ll see about that in a later blog post. Only in this logic is the bottom reached on which one can incontestably win a true sentence again from true sentences.

We still would have to comment on Aristotle’s remark on the case that the conditions are only likely to be true or credible. Then the conclusion could also only probably be true. Saying more here was not possible at that time.  It was not until the beginning of the 20th century that a theory of probability was developed with which one can become more precise in this case. This will be explained in more detail in a later blog post on the topic “How to deal with insecure knowledge”.

Syllogisms in an axiomatic-deductive system

But Aristotle’s logic has not only shown us how safe knowledge can be passed on safely. He has also shown that “It is possible also to reduce all syllogisms to the universal syllogisms in the first figure.”, as he writes in the Prior Analytics, 1st book, 7th chapter (see https://ebooks.adelaide.edu.au/a/aristotle/a8pra/book1.html).

Thus, we already have a concrete axiomatic-deductive system. This logical order of the statements in a field of knowledge represented for him the ideal of a science. In the Posterior Analytics, 1st book, 3rd chapter he writes:

On the other hand, I maintain that any science must be based on proofs, but that the knowledge of the unmediated principles is not provable. And it is clear that this must be necessary. For since a knowledge of the earlier propositions from which the proof is made is necessary, but one stops once at unmediated propositions, these must necessarily be unprovable. This is my view and I maintain that there are not only sciences, but also supreme principles of them through which we learn the concepts of conclusion. (Here I prefer the translation of a german version in http://www.zeno.org/Philosophie/M/Aristoteles/Organon/Zweite+Analytiken+oder+Lehre+vom+Erkennen/1.+Buch/3.+Kapitel).

The mathematician Euclid of Alexandria logically arranged the geometric knowledge of that time in this way in the time around -300. This organization of a scientific theory as an axiomatic-deductive system is still a model for any rigorous science today. One tried in all centuries to imitate this organization of a thought building, thus “more geometrico”, after kind of the Euclidean geometry, to arrange the knowledge of his science. In “Die Idee der Wissenschaft – Ihr Schicksal in Physik, Rechtswissenschaft und Theologie” I described how successful it has been to this day to realize this idea (Honerkamp, 2017).

Who first had this idea is not clear. The mathematical proof was already known to the Pythagoreans. If there are enough statements in an area, secured by proofs, one will probably at some point consider which statements could be regarded as axioms. This poses the question: Which statements do I need as a basis in order to be able to deduce all the others from them? Or also: How do I create a logical order?

Such specifications will then always be special statements, have special properties. They can be immediately insightful, i.e. “certain by themselves”, as in theories of mathematics, but also highly abstract and far from our idea, as in theories of physics.

Aristotle and the Stoa

If one examines the insights of the ancient thinkers as to whether they can still be regarded as generally valid today, one encounters Aristotelian logic, apart from many mathematical and some physical statements. Although at the end of the 19th century something “better” was found with modern logic, Aristotle’s statements on the laws of thought are still valid and will always remain so. It is highly admirable how clearly Aristotle saw the structure of an argumentation and how he worked out the decisive criteria for reliability.

The first step is a detailed analysis of a conclusion.  In a sentence from Topics, Book 1, Chapter 1 of Aristotle:

The conclusion is now a discourse in which some things are presupposed and then something different from them results from it with necessity mediated by those propositions.

The point is here: “Some” is assumed and “some of it different” results. The extent to which this “with necessity” arises will still have to be discussed.

Let’s have a look at a classic example:

The “some” that we presuppose are the two sentences, also called antecedents:

One: “All human beings are mortal.”

Second: “Socrates is a human being.” 

The “different” of it, the conclusion is the sentence:

   “Socrates is mortal.”

So here we have three terms: “Socrates”, ” human being” and “mortal”.  These are put into relation, “human being” is the generic term to “Socrates”, “mortal” to “human being”. If the relations of terms are correct, the antecedents are true. The conclusion then follows from this Dihairesis, i.e. a classification of terms.

The conclusion seems evident to us; to Aristotle it indeed follows “with necessity”. No one would deny that, everything else would be “unreasonable”. However, we still use our “common sense”, which is what we call reason.

We will leave it as it is and deal with this point again later. It should be noted, however, that the protagonists of this conclusion, i.e. those involved in the conclusion, are three terms, or to be more precise: two relationships for a total of three terms, in each case between two terms. For this reason, one speaks here of a term logic.

Logical inference, dialectical inference and false conclusion

However, something is also said about the sentences in which the terms appear, here e.g. that the antecedents are true. With regard to these, Aristotle now makes a decisive case discrimination (Aristotle, no date):

The conclusion provides a proof or a logical inference if it is derived from true and general superordinate sentences, or from such, which are based on true and superordinate sentences of the science concerned.

Dialectical, on the other hand, is the conclusion derived from credible sentences.

So, it depends on whether the first sentences are true or only “credible”. Often the Greek word translated here as “credible” is translated as “probable”. This suggests the idea that one could indicate here a degree of probability for whether the statement is true. However, only in the last century was it possible to develop a theory of probability in which one can calculate with different degrees of probabilities. These can be applied just as well if one assumes degrees of credibility. We will use that in later blog posts.

But in the case of a proof where one can speak of a logical inference, we are dealing with “certain knowledge”, whereas in the case of a dialectical inference we are dealing only with “uncertain knowledge”. In later blog posts we will deal in detail with what kind of knowledge can be gained from certain or uncertain knowledge respectively.

First it is important to distinguish between a “proof”, in which one must assume that the first sentences, the antecedents, are true, and a dialectical conclusion, in which one can only proceed from “credible” antecedents.

About the “true and general superordinate sentences” he says:

True and superordinate propositions are those which are not mediated by others but are certain by themselves. Because for the most fundamental principles of the sciences one must not demand a reason for them, but each of these principles must be certain by itself.

These are principles which were later called “axioms”. They play a major role in an axiomatic-deductive system by Euclid of Alexandria. However, we may assume that there are also sentences “certain by themselves” outside of special sciences, such as: “All human beings are mortal”.

He determines the “credibility” of sentences as follows (ibid.):

Sentences are credible if they are accepted by all, or by most, or by wise men, the latter by all, or by most, or by the most experienced and credible.

Here we are now “in the center of life”. It is almost always the case that we are dealing with sentences that we can only believe. Even as a scientist you will have to believe almost everything, e.g. the statements of the scientists of another subject – even those of your own subject, if you have not checked the statements yourself or cannot check them directly. “Credibility” is therefore a high good for a society. Today in particular, it is often not easy to decide who to believe.

Finally, Aristotle also deals with the fallacy:

A false conclusion is one which is derived from apparently credible sentences without them really being, or which is derived only apparently from credible sentences or from sentences that only seem so.

The error can therefore lie with the antecedents, that’s trivial. More interesting is the case that the rule of inference is not valid, that a inference is only “apparently”, not truly present.

One such false conclusion, which often remains hidden, is the “fallacy of four terms”. This is demonstrated particularly clearly in the following example (Wikipedia: Fehlschluss):

One: What’s got a beard can be shaved.
Second: Keys have a beard.
Conclusion: Keys can be shaved.

A shift in meaning has occurred here in the transition from the first to the second antecedent. A “beard” in the first sentence means something different than a “beard” in the second sentence. One should better speak of two terms “beard1” and “beard2”, and there would not be three, but four terms in the game – hence the name.

Since we constantly argue with unclear terms in our considerations and discussions, we are often undermined by such false conclusions.

The first major step in the analysis of an argumentation is thus taken: a discrimination between a rule of inference and the “presupposed”, a case discrimination between true and only credible antecedents as well as an investigation of the possibilities of a false conclusion. Here again the great systematist shines through. In the next blog post we will, again systematically, distinguish between different types of antecedents and rules of inference.

The Stoics’ logic

About 100 years later, a different approach to logic emerged in the philosophical school of the so-called Stoa. The philosopher Chrysippus (-276 to -204) of Soli (Cilicia) was probably the representative of this school who most successfully dealt with logic. According to Diogenes Laertius his extraordinarily numerous books were very famous at that time (Laertius, 2015, p. 415ff).

In the long run, however, Aristotle’s approach was far more effective. In all centuries up to the time of Gottlob Frege (1848 to 1925), who founded modern mathematical logic, logic was associated with the name Aristotle; stoic logic was almost forgotten in the Middle Ages, its significance was rediscovered only in 1950 by the American science theorist Benson Mattes. I think it is still underestimated.

The stoic logic was based on the findings that had already been gained by the Megarian school who referred to Euclid of Megara. Stiplon, Diodoros Kronos and Philon von Megara were the most prominent representatives of this group. Stoic logic was already a propositional logic in its approach, while Aristotelian logic, as already mentioned, was a term logic. Aristotle had grown up in the Platonic Academy and had therefore probably incorporated the Platonic terminology (Dihairesis). His logic thus became term logic. The Megarian school was free of such influence, and probably saw dialectics more directly as the problem of checking an argumentation for its conclusiveness. For them, the statements were thus in the foreground.

Aristotle had already seen that it depends on whether the antecedents were true, only credible or not. The rules of inference, however, had to be concerned with the relationships between terms. But now the rules of inference are focused on the “transport of truth” – from the antecedents to the conclusion. Then why shouldn’t the protagonists who are supposed to accomplish this be the sentences themselves? In a propositional logic then it must be only a matter of whether the antecedents are true or not. Terms no longer appear explicitly.

It is then also no longer of interest to “categorical judgements”, such as “All human beings are mortal”, in which a judgement is made about the categories, i.e. in which the category “human being” is set in relation to the category “mortal”, for example. Such a judgement, which corresponds to a division of terms, is to be distinguished from the “synthetic judgement”, which in today’s language corresponds to the connection “A and B” or “A or B” of two statements A and B respectively. So, here sentences A and B are connected in various ways.

A particularly important connection is the “implication”: If A, then B, e.g. “If it rains, then the road is wet”. Where A = “it rains”, B = “the road is wet”. So, if A is the case, then B is the case. An implication can also be true, credible or false. The Stoics were already familiar with this connection: “An implication is true, if the after-sentence is contained in the preceding sentence in the sense”.  (Sextus Empiricus: Pyrronic Skepticism II,112. p.181, after Schupp, I, p. 346). With this statement they excluded the case that the implication makes no sense, as in the example: “If 2 + 2 = 4, then my friend has birthday today”. Such a case led to difficulties in formulating certain rules of inference.

The Stoics formulated five “unprovable rules of inference”, and there are also said to have been rules on how more general final formulas can be traced back to these fundamental “unprovable conclusions”. One of these conclusions is identical with the “modus ponens”, a rule of inference in which implication plays an important role. That conclusion is:

Be the statement A true, and be also true:

If A is true, then B is true.

Then follows: Statement B is true.

This can be seen immediately by everyone, and so it is not surprising that this rule already belongs to the “unprovable conclusions” of the Stoics.

But there were always great discussions about the “modus ponens”. One always discussed all logical conclusions only by given meaning of sentences or terms. However, the “modus ponens” contains the implication as an antecedent. Because this could now also be senseless as in the above example, “If 2 + 2 = 4, then my friend has birthday today”, the logicians always doubted the general correctness of this rule of inference.

The “unprovable conclusions” long belonged to the school knowledge of late antiquity; writings by Cicero (-106 to -43) or Isodor of Seville (560 to 630) bear witness to this.

The difference between Aristotelian and Stoic logic was not seen by many even in late antiquity, as can be seen from works by Cicero or Galen. However, the neo-Platonist Porphyrios (234 to 305) still compared the stoic with the Aristotelian logic in terms of terminology and objective. Boethius (477 to 524) could then only report on the stoic logic. When one spoke later of logic, one always meant the Aristotelian logic (Schupp, I, p.349) and one referred thereby predominantly to the writings of Boethius. What remained of stoic logic was the distinction between categorical and synthetic judgments and the knowledge of “modus ponens” as a rule of inference, without knowing stoic logic.

Stoic logic, by being an early form of propositional logic, was much closer to modern logic as formulated by Gottlob Frege at the end of the 19th century. What it lacked was a decisive step: the discovery that a formal language of its own is necessary for logic, just as it is for mathematics, so that the laws of logical thought can be formulated independently of the meaning of the statements. Then you can “compute” like in mathematics. The correctness of conclusions can then be defined and checked at this level. With such a strict separation of syntax and semantics, i.e. of grammar and meaning, the “meaning problem” of the implication described above becomes irrelevant.

Thus, we will have to deal first with formal languages, before the modern propositional logic can be introduced. But first we should study explicitly the rules of Aristotelian logic for the sake of completeness.

Socrates, Plato and Aristotle

If I wanted to write a treatise on the pre-Socratics, I would now have to deal in this blog post with the so-called pluralists and atomists such as Anaxagoras, Empedocles, Leukipp and Democritus. But I do not want to give an outline of the philosophy of the pre-Socratics here. This has happened often enough elsewhere, and certainly after a much more detailed study of the literature of the past millennia than I can show. I am essentially interested in the thoughts of the pre-Socratics, in which one can discover the precursors of the way of thinking of modern physics, mathematics and logic.
There it is without doubt the three basic ideas which have been expressed so aptly and so early in the history of ideas by the school of Miletus, by the Pythagoreans and by Xenophanes: In nature there is such a thing as a causal connection, the regularities in nature can be formulated in the language of mathematics, and knowledge can be attained by “finding the better by searching”. But there can be no ultimate knowledge.
The first tentative steps of the pre-Socratics towards basic concepts such as ” motion ” are also remarkable. The long history of the clarification of this concept over the millennia clearly shows how much effort it takes to gradually move from the darkness of the first reflections to a clear idea with which one can argue reliably.
If I now follow the development of the “logos” in the history of philosophy, I will be even more selective. I will concentrate on the highlights, i.e. the works where substantial progress has been made. Of course, one often recognizes these only by looking back from today’s standpoint. The focus will be on such developments in logic, mathematics and physics that have become significant for the present state of the art in terms of both methods and the formation of concepts.
So, if one begins to look at the development in the time after the pre-Socratics, one must first speak of the three greats of Greek philosophy: Socrates (-469 to -399), Plato (-428 to -348) and Aristotle (-384 to -322). As already mentioned in the first blog post, Socrates introduced a new topic into philosophy: Ethics. But it was not only this new topic that led to a new era in philosophy. Anaxagoras from Kleizomenai ( Ionia ) had moved to Athens in the year -462 and had made the thoughts of the Milesians and other pre-Socratics known there.
Where more people can come together, it is also more likely that people with the same interests will meet and talk about them. In addition to teacher-pupil relationships, communities can now also emerge in which discussions can take place on an equal footing. Dialectics came into fashion. Already Zenon of Elea had been her great friend, Aristotle had later even praised him as their inventor (Mansfeld & Primavesi, 2011, pp. 361, No.4). Attic democracy flourished, it was an early type of democracy in which the “people of state” ruled, where the people of state consisted of the full male citizens of the city of Athens who had reached the age of 30. These full citizens could also turn to matters which were not directly necessary to daily life; for such work there were enough slaves, women or immigrants. Especially in political meetings and in court, eloquence and the art of dialectics were in demand, i.e. to find the right reasons for opinions and to spread those that could be applauded by listeners. Sophists taught for money to master this art, even to be able to skilfully involve the opponent in a contradiction in discussions.
About Socrates we know some about Diogenes Laertius (Laertius, 2015, pp. 67-90), in particular he is supposed to have said: ” I know nothing except that I know nothing” (DL 83). And “the good is nothing small, but start with small things” (DL 83). But we don’t know any of his writings. Plato, however, puts a lot into his mouth in his works. These works always represent dialogues that Socrates conducts with a wide variety of dialogue partners and in which it is always a matter of clarifying a concept or a question.


After the death of Socrates, Plato, in his years of teaching and travel, first stayed with a follower of the philosophy of Parmenides, “travelled to Cyrene to the mathematician Theodoros and from there to Italy to the Pythagoreans Philolaos and Eurytos; from there to Egypt to the prophets” (Laertius, 2015, p. 141). Thus he collected various impressions and combined them to a grandiose mythical thought building, which is still effective today and continues to inspire many people.
He must have been a great storyteller. In his works he described in the form of a dialogue the effort to find concepts and answers to the question of correct behaviour or a good life. Who first chose this literary form is controversial. “It is said that Zenon, the Eleate, was the first to write dialogues, but Aristotle names as such […] Alexamos from Styra […]. I, on the other hand, think that Plato, through his strict treatment and training […], has secured himself the right to first place […]”. Dialogue is “a question-and-answer execution of a philosophical or political theme”, but dialectics “is the art of persuasion, through which we prove something void or right on the basis of the question-and-answer procedure of the undermentioners” (all quotations: (Laertius, 2015, p. 159)).
Diogenes Laertius says of Plato’s works: “TIMAIOS belongs to the field of physics, to the logic belongs POLITIKOS, KRATYLOS, PARMENIDES and SOPHISTES” (Laertius, 2015, p. 160).
When he, as in TIMAIOS, came to talk about the ideas of the pre-Socratics about the cosmos, he did so in mythical form or tied to Pythagoras and saw the world built up from geometric forms. At that time, logic was still predominantly understood to mean the work on terms, i.e. the uncovering of the relationships between related terms, e.g. their classification as superordinate or subordinate terms. A clearer definition was expected from such a division of terms, called “dihairesis”. One could already see, however, that a definition often required definitions of the determinants, thus leading to an infinite regress.
Aristotle grew up in the Academy with this method. Thus, in his logic, which he was soon to develop, the concepts were also at the centre. This thus became a so-called term logic. Modern logic, on the other hand, is a propositional logic. In the next blog post I will work out the difference exactly.
Plato mainly used induction, i.e. the conclusion from special to general, for a demonstration of evidence, “which through some true cases opens up the same truth for other cases in an appropriate way”, as Laertius (Laertius, 2015, p. 161) says. For a long time this conclusion was considered “appropriate”, only David Hume stressed that this conclusion is not mandatory, thus not always “appropriate”. In modern logic there is no place for this conclusion either.
Plato’s central theme, however, was the human soul and how it can express itself in language, ethics, art and politics. He adorns existing myths, redesigns them or invents completely new stories.

The Platonic Academy

For the development of the “logos” all this does not give much. It is interesting in this context, however, that he also promoted training in analytical subjects, probably motivated by the impression that Pythagorean mathematicians made on him. He was not only a great narrator, but also a good organizer and designer. He founded a school where young people were taught philosophy and science. For this he acquired, probably in -387, a plot of land near a grove called Akademeia, because it had been dedicated to the Heros Akademos. In addition to metaphysics, ethics, dialectics and the doctrine of the soul, physics and especially mathematics were studied there, which soon became part of the basic education of every student of a philosopher. The academy remained, with interruptions, over many centuries, it was closed only around 530.
The Platonic Academy became a model: since Augustine (354 to 430) and Martianus Capella, a Roman encyclopedist from the 5th or 6th century, a canon of seven subjects had become established for the schools of late antiquity. This was divided into a trivium (grammar, rhetoric and dialectic or logic) and a quadrivium (arithmetic, geometry, astronomy, music theory). These subjects were called the seven liberal arts because they were “worthy of a free man,” and “free” was a man when he was free from the necessity of earning a living.
In the Middle Ages this study of the liberal arts was regarded as preparation for scientific studies in theology, jurisprudence and medicine. In the universities, the liberal arts were soon taught within the framework of a separate faculty, the Facultas Artium. The “Liberal Arts”, which we know from the USA e.g. as “undergraduate studies” and which are introduced nowadays also e.g. in Germany, try to continue this tradition. However, dialectics is more in the foreground here than any of the mathematical disciplines.


Aristotle, a pupil of Plato’s, had particularly fertile ground for his rigorous training at the Platonic Academy. Born into an educated and wealthy family in Stageira (Chaldidike) in the year -384, he was sent to Plato’s Academy in Athens in the year -367 at the age of 17. The mathematician Eudoxos von Knidos (ca. -395 to ca. -350) played an important role there.
Aristotle, on the one hand, loved to argue and discuss with dissenters. On the other hand, he showed – as with people who have not passed a mathematical education without consequences – a tendency to work systematically and to collect the doctrines of former philosophers such as Pythagoras or Democritus (Schupp, I 256).
Aristotle became such a great systematist, attentive to the methodical in a thought process and vigilant for connections. He was the first of this kind, and he was also to establish a completely new kind of philosophy, which is no longer a poiesis, a “production” of doctrines in a poetic manner. Instead, the focus is on analysis and methods, reasons are required and the results are checked for conclusiveness.
It is probably plausible that one feels the urge for such things when one has a large corpus of doctrines before one’s mind’s eye and at some point wonders why the respective philosophers can so firmly hold their views. But you also have to live in a stimulating environment and, above all, have the talent to make such a new start. So Aristotle soon went his own way in the academy. According to Diogenes Laertius, Plato is said to have said: “Aristotle beat me like young fillings do against their own mother” (Laertius, 2015, p. 225).
In the list of Aristotle’s writings, Laertius lists 146 titles and speaks of a total of 445 270 lines. According to him, the work on logic is “very clearly marked as a tool for all sub-areas.” These writings were later combined into a collection of six books entitled “Organon” (gr. ὄργανον = tool).

The Organon

The titles of the six books in the Organon Collection are as follows:
1. the categories,
2. on the interpretation (peri hermeneias),
3. the doctrine of the logical conclusion (analytika protera, first analysis),
4. the study of evidence (analytika hystera, second analysis),
5. the topics and
6. the Sophist Refutations
The books 3,4 and 5 are particularly important for the development of logical thinking, i.e. the doctrines of the logical conclusion and the proof as well as the topics. Here the so-called Aristotelian logic is developed. This is therefore the fruit of a time in which dialectics occupied a high position in society. Even in discussions about more fundamental questions such as those about a good life or “reasonable” morals, it was soon more a question of the power of the arguments than of authoritarian settlements, and finally it could not fail to be thought about directly which forms of argumentation possess such power that they are incontestable, i.e. must be accepted by every sensible person.
The new era of philosophy, with its emphasis on dialectics, has therefore by no means allowed the idea of the logos to recede into the background or hindered its dissemination. On the contrary, dialectics was precisely the field in which this idea could prove to be a particularly useful and valuable tool for thinking. The idea took shape in Aristotelian logic, which was to become a model for a science that today plays a significant role within the framework of artificial intelligence. In the next blog post I will deal explicitly with Aristotelian logic.

Aristotle and the System of Sciences

Aristotle’s talent for analytical thinking and systematic work, as well as his passion for collecting the doctrines of earlier philosophers, showed other fruits. He establishes a systematics of all sciences known at that time. It divides the sciences into practical and ethical sciences, into poetic (manufacturing) sciences such as medicine or craftsmanship, and finally into theoretical sciences. In the latter he distinguished mathematics, natural science and “first philosophy”, to which he added theology, ontology and logic. Theology at that time consisted essentially in a study of the unchangeable “divine” stars.
This division of Aristotle is, so to speak, the first “cashing down” of knowledge of a time and subsumes everything that creates knowledge under science. Probably he also counted the theoretical sciences of mathematics and natural science among philosophy, albeit not among the “first”. About 500 years later Diogenes writes of Laertius:
As for the parts of philosophy, there are three: physics, ethics, and dialectics.(Laertius, 2015, p. 10),
whereby dialectic here also stands for the doctrine of thought principles, i.e. for logic and more generally for epistemology. Ethics had thus immigrated into philosophy and mathematics had been eliminated. There could no longer be a thinker like Pythagoras, who practiced mathematics but also represented a certain world view with religious fervour. Important ancient mathematicians, such as Euclid of Alexandria or Archimedes of Syracuse, do not appear in the work of Diogenes Laertius on the lives and opinions of famous philosophers – except for Eudoxos of Knidos, who was a member of Plato’s Academy for a time.
But physics, too, was to emigrate from philosophy, albeit only about 2,000 years later, when Galileo Galilei overcame the Aristotelian theory of motion. By demonstrating how to describe regularities in nature in the language of mathematics, he discovered a “new science”. Natural science became a modern physics. The consequences of this discovery will be discussed in detail.
Finally, at the end of the 19th century, logic in the form of mathematical logic emigrated from philosophy. It is now a branch of mathematics and computer science. Nowadays it is even observed how epistemology is becoming an area of cognitive science that not only deals with our ability to think, but also with all conscious and unconscious processes in our brain such as perceiving, learning or remembering.
Cosmology, the theme of the pre-Socratics and the first theme of philosophy ever, is today an area of modern physics. A history of cosmology from the pre-Socratics to Hawking would be highly interesting: the “question of the whole” has arisen in mythical thinking and in this form is still present in all religions, and the pre-Socratics already began to look for physical reasons for their ideas. But only for about 100 years has there been a physical cosmology in which the logos rule strictly. This cosmology is a reconstruction of the history of the universe with consistent consideration of physical theories. The development of the cosmos could only be convincingly told after the phenomena of nature had been understood on the grounds of reliable theories. Because this knowledge was just the necessary guidelines for a history of the universe, which no longer wants to be a mythical but a logical one.

Zenon of Elea, the Movement or how to find seeking the better

A disciple of Parmenides, Zenon of Elea (around -490 to around -430) still attracts special attention today because of the paradoxes with which he annoyed his philosophically interested contemporaries. We now understand paradoxes to mean argumentations that lead to contradictions because an unclear or incorrect idea of a concept is at play. That was the case for both movement and infinity at the time. Today we have clear ideas about these terms and can resolve the paradoxes that have so unsettled his discussion partners. 

Zenon wanted to support the theses of his teacher Parmenides with his considerations. At least this is how we read it in Plato’s dialogue PARMENIDES, in which he lets Zenon say: (after (Mansfeld & Primavesi, 2011, pp. 365, No. 5):

In reality, my writing is something of a support for the Parmenides’ thesis, which is aimed at those who try to make him ridiculous.

His four movement paradoxes are particularly famous. We want to deal here with the third paradox, because it is based on an error, the enlightenment of which stood at the beginning of modern physics. This is about the apparent contradiction between the observation of a flying arrow and the Parmenides assertion that this movement of the arrow is only apparently present, since the being remains in absolute silence.

A flying arrow, Zenon argues, is at a given time in a certain place, which is always as big as the arrow itself. Since he was there in the “now”, he could not be in motion then. So, he is at rest in every moment, the arrow actually stands still. The motion we’re watching is a fake.

I don’t find that argument convincing. After all, Zenon already shows a scepticism towards our everyday perceptions, albeit in the most extreme form. In any case, Zenon seems to me to have the idea that in every “now” the state of an arrow is determined by a location alone, and he assumes that there is no motion. Aristotle says in contradiction: “In the “now” neither rest nor movement can take place” (Mansfeld & Primavesi, 2011, pp. 383, No. 23). So he already sees that Zenon only thinks of the most obvious when describing the state of a moving body. Zeno’s argument does not convince him either. But his counterargument does not go any further as well.

The arrow paradox dissolves when one knows that the state of a body in space is determined by a location and (!) by a velocity (or impulse) at any time, in any “now”.

This was Galilei’s discovery in the early 17th century. He studied the motion of a small ball as it rolls down an inclined long wooden channel. Not only did he find that the distance she travels on the wooden channel increases with the square of time. He also extended the channel beyond the inclined part and observed that the ball on the horizontal channel continues to run the longer the less its movement is affected by unevenness of the ground. He concluded that, with ideal ground, it would then have to continue to run forever. The motion remains as it is if there are no external influences on what is moving.

So movement is a state. In medieval “impetus theory” the movement was still a process: an “impetus” had to work constantly. This would be given to the body at the beginning, kept the movement upright, but was also slowly used up, so that it gradually came to a standstill. Galilei, on the other hand, attributed a slowing of a motion to external influences, e.g. friction. By refraining from external circumstances, he was able to discover a principle of nature that would prove extraordinarily fruitful for the further development of physics. I’ll come back to that soon. But first we have to look at how a velocity in the “now” can be grasped in concrete terms.

The instantaneous velocity

Galilei did not yet have the possibility to calculate an instantaneous velocity. In mathematics he was still at the level of the ancient Greeks, where geometry as a description of nature stood in the foreground. But his contemporary, the French philosopher and mathematician René Descartes, discovered how geometric problems could be converted into arithmetic problems. An “Analytical Geometry” was created, which represented a great advance compared to ancient mathematics and which for the first time enabled one to go beyond the status of the ancient Greeks.

One now learned to describe the location of a point in a coordinate system and to see such points as locations of material bodies, if one abstracted from their expansion. One could also represent the place x(t) of the body in dependence on the time t in a coordinate system.

An average velocity in a time period dt was easy to calculate by forming the ratio dx/dt, where dx may be the distance covered in the time dt. But it became difficult if you wanted to extrapolate to the instantaneous velocity, i.e. if you had to determine the velocity in the “Now”. The time span dt should actually be zero, the distance dx thus also, and the ratio 0 to 0 makes no sense. One had to choose somehow a very small time span dt, which should be however arbitrarily small, but still unequal 0. Somehow these quantities had to have something to do with the “infinitely small”. They called them infinitesimal. It was not a clear idea, but they succeeded in consistently calculating the ratio dx/dt in the “now”, the “differential quotient”. This “infinitesimal calculus” was developed independently by two great thinkers of the time for general functions f(x): Isaac Newton needed this knowledge for his reflections on motion. Gottfried Wilhelm Leibniz regarded it as a purely mathematical problem which had to be solved if one wanted to determine the tangent in a point of the curve of a function in a diagram.

For some time such calculations were very popular; they inspired many new ideas and questions. At the end of the 18th century, mathematicians were no longer satisfied with the justification of such calculations with infinitesimals. The Italian mathematician Lagrange found a method for calculating the differential quotient without having to use the term infinitesimal. In the 1960s, a new type of number could finally be defined using hyperreal numbers in a so-called non-standard analysis. A clear definition of the infinitesimal was now possible: they were certain hyperreal numbers.

Mathematics is sometimes like physics and actually every science: New concepts are not always clearly defined at first. But you can already use them and when you notice that they are “good for something”, you start to take care of the conceptual basics at some point. But it often takes some time for satisfactory clarity to be achieved.

The Evolution of Movement Theory: One Finds “Seeking the Better”

Knowledge of how to calculate the instantaneous velocity from a time-dependent spatial coordinate was a prerequisite for a theory of motion in the language of mathematics. While Galilei had discovered a relationship between distance and time in free fall, one theory was now concerned with describing the position and velocity of a body as a function of time.

The physicists and mathematicians of that time knew their ancient models very well. Especially the ELEMENTS of Euclid of Alexandria, in which he brought the then known laws of geometry into a “logical order”. Euclid has thus set a benchmark for what a mathematical or physical theory should look like. At the beginning there are definitions, conventions and axioms. According to this, all statements of the theory must be logically deducible from the axioms according to mathematical rules.

Newton formulated his theory according to this model. Galileo’s idea that motion can be a state was the first axiom he incorporated into his theory of motion, which is now known as “Newtonian mechanics”: “A body remains at rest or in straight-line, uniform motion if no forces act on it.

Here, of course, something must have been said about space and time in the definitions beforehand, so that one knows what uniform motion means. So you have to know what a straight line is, and you have to say something about the course of time before you can speak of a uniform velocity, one that is constant in direction and size.  Only then can one speak of this particular motion in the axiom and postulate that this motion remains if there is no external influence on the moving body.

In a second axiom, Newton then logically describes a procedure for formulating a mathematical equation for the case that an external force now acts on the body. With a suitable mathematical expression for the force, one can then calculate all motions in the sky and on earth from such an equation of motion, taking into account existing circumstances.

This Newtonian theory of motion, briefly outlined here, was regarded as the only ideal of a scientific theory for over 200 years from the end of the 17th century, and its structure as an axiomatic-deductive system represented a model for future sciences.

Here is an opportunity to speak of two other theories of movement: on the one hand a theory that Aristotle had formulated about 2,000 years earlier, and on the other a theory that Albert Einstein developed some 300 years later and was soon called “special relativity theory”. One can very beautifully demonstrate with these three theories how indeed in the course of time people found “seeking the better”. This story doesn’t describe a special case. One can find many such examples. But let us first characterize the other two theories:

Aristotle was a great systematist, and so he first of all distinguished the motions in the sky from the motions on earth. He divided the earthly motions again into movements of living beings, into natural and finally into forced motions. He gave a different reason for each type of movement. The motions in the sky showed the eternal order. In natural motions the “disturbed order” was restored, e.g. smoke rises to heaven and a stone falls to earth, because light has its place above and heavy has its place below. In a forced motion, a force must always act, otherwise the motion would come to a standstill.

This Aristotelian movement theory had more than 2,000 years of existence. It was still taught in the academies during Galilei’s lifetime, and Galilei had studied it intensively himself until he finally overcame it.

Newton’s theory finally completely replaced it, for it is obviously “better”. From the equations of motion of Newton’s theory, it was possible, with a certain expression for the force which a body exerts on another body due to its mass, to derive the three Kepler laws for the motion of the planets around the sun, even to accurately predict the return of a comet. So with fewer assumptions one can explain more phenomena. The theory also makes predictions that can be tested and indeed have been confirmed. To this day, it is indispensable for calculations in everyday motions.

The theory of motion developed by Albert Einstein, the special theory of relativity, is in turn better than Newton’s theory. The motivation for the development were problems with the idea that the whole universe is filled with a subtle substance, an “ether”, which has prevailed since ancient times. It should also be the carrier of the electromagnetic waves, which at that time had only been known for about two decades. Ether should also mark absolute rest and one wanted to measure motion of earth versus the ether. Whenever and however you did it, you couldn’t see such motion.

Einstein took the bull by the horns, so to speak. He made this negative result the principle of his new theory: “The speed of light in any inertial system is independent of the speed of the light source”. So, this means that no matter how I move relative to the light source, I always measure the same speed for the light.

This theory is also structured as an axiomatic-deductive system. As such, it is even particularly “elegant”, because it is based only on this principle, as on a “principle of relativity” already known from Maxwell’s theory of electromagnetic phenomena. A wealth of phenomena could then be predicted; some of them differ strikingly from our everyday experiences, but in the mean time they have all been proven experimentally.

The hypothesis that there should be something like an ether was dropped. He was no longer needed. There is no absolute rest, but instead an absolute speed: the speed of light as measured in a vacuum. It is an upper limit for the transmission of effects.

When comparing these two theories, one finds that the special theory of relativity is an extension of Newton’s mechanics, in the sense that the smaller the speeds to be considered are in relation to the speed of light, the better the agreement between the statements of both theories. For velocities close to the speed of light, however, the astonishing phenomena already mentioned above are predicted, which have all been confirmed in the meantime.

A useful yardstick for the “goodness” of a theory is its scope of validity. Newton’s theory already had a very large validity range, because with it one can explain all motions which are “non-relativistic”, i.e. sufficiently small compared to the speed of light of approx. 300,000 km/sec. Here one could also try the theory of relativity. That wouldn’t be necessary, it’d just be harder. If one now considers increasingly higher speeds, the statements of the two theories will differ more and more. One leaves the scope of Newton’s theory but remains within the scope of the special theory of relativity. In this sense it is therefore an extension of Newton’s mechanics and thus the better theory.

If we look at the image of an evolution of theory, then we can say that the Aristotelian theory survived 2,000 years because there was no other theory that could become dangerous to it. But Newton’s theory was such a dominant competitor that Aristotle’s theory became extinct. The special theory of relativity is then a further development of Newton’s theory, so that there are now two theories that both have their own habitats. Where these overlap any theory can come into its own.  

Aristotle – Newton – Einstein: Aristotle has submitted, Newton and Einstein “found seeking the better”. Who knows when someone will come and find something better, and what further insights we will gain about motion and thus about space and time. Only one thing seems clear to me after 2,500 years: The way of Xenophanes to find “seeking the better” is also the better way to knowledge.

Xenophanes and Parmenides

The search for the “One” led the philosophers of the School of Miletus and the Pythagoreans to various principles and basic statements. The specific answers they gave are not so important to us today. It is much more interesting that they departed one step from mythological thinking and began to formulate rationales for their statements. Authorities didn’t have to be respected, doubts were allowed. What is even more interesting, however, is that in this effort they have already come across a number of problem areas that should prove to be fundamental for future sciences. These fields can be described by the concepts of “infinity” and “motion”.

The “Eleates”, those philosophers of the 6th and 5th centuries BC who lived in a Greek settlement Elea on the west coast of southern Italy, were to raise awareness of a third problem area of such importance. It was the question: How can we obtain reliable knowledge?

Thus, three fundamental questions are already under discussion in such an early period of our intellectual history. And each of these questions or problem areas stood for an area of philosophy at that time: “infinity” for mathematics, “motion” for physics and the question of reliable knowledge for what we call today epistemology. We know it by now: When you tackle big tasks, other big tasks also come into focus. And even if the original task cannot be mastered, there is usually fruitful insight in the treatment of the subsequent problems. In the technological field today, this is called “spin-off”. I will always keep an eye on these fundamental questions in the course of the later blog posts.

Who were the Eleates and how did they answer the question of reliable knowledge? Their most important representatives were Xenophanes (about -570 to about -470), Parmenides (about -515 to about -455) and Zenon of Elea (-490 to about -430).  Let us take a look, which possibilities of knowledge they saw and how the concept of motion came to the fore again, and in the wake of it the concept of infinity. 


Xenophanes was at first probably a Rhapsode, someone who recited Epen Hesiods and Homers, but also appreciated and recited elegies or mocking poems. He only settled in Elea after a long life in which he had got to know many a foreign culture.

Xenophanes takes a completely new view on the question of the possibilities of knowledge.  In a fragment (Mansfeld & Primavesi, 2011, pp. 229, No.32) it says:

The gods did not reveal everything to man at the very beginning, but in the course of time they seek and find what is better.

That is now strikingly topical. Here, already 2,500 years before our time, the knowledge of modern science theory is formulated that knowledge grows in a kind of evolution. We see this particularly clearly in the development of modern physics: the theories are adapted to the ever-increasing number of observations and experimental results. The range of validity of a theory thus becomes ever larger; it becomes “better”. A theory that does not work well enough in such a process of adaptation will at most become a topic in the history of physics. Using the example of motion, we will soon see clearly how such better theories will look in the course of the development of physics.

 Xenophanes thus formulates a path to knowledge in which people search and find something “better”, quite in contrast to mythological thinking, which “seeks nothing, always speaks apodictically and claims to be simply true” (Schupp, 2003a, p. 87).

But there is no mention of the possibility of once being in a definitive possession of the truth. On the contrary – according to Sextus Empiricus (160 to 210) Xenophanes (Mansfeld & Primavesi, 2011, pp. 231, No.39) says it:

Of course, no man has seen the clear, and there will be no man who has seen it with regard to the gods and all the things I declare. For even if one had succeeded to an extraordinary degree in saying perfection, he would still not be aware of it: In all things, there is only assumption.

The sentence “In all things there is only assumption” even presupposes that all our reliable knowledge of the world of all things, even if it exists in the form of physical theories, is ultimately based on certain basic thoughts, principles or equations.

Xenophanes therefore only calls our findings “opinions”, whereby this term does not mean arbitrariness, but the idea that our findings can still be outbid by a better opinion. The opinion could, however, be regarded as “similar to truth”, since it was as well founded as possible in each case. Xenophanes does not believe in the possibility of a revelation of a final truth, as Parmenides later accepts it. 

One can thus call Xenophanes the first decided metaphysical agnostic. There is nothing definitive to say about a God and the world as a whole. The theses of the Ionian philosophers may have seemed too ambitious to him, too far removed from empiricism, which he regarded as more important than the embellishment of myths.

Thus, he contrasted the idea of the Greeks that the goddess Iris appears in a rainbow with a natural explanation: “What they call Iris is also a cloud, one that looks purple, light red and yellow-green (Mansfeld & Primavesi, 2011, pp. 223, No. 21). He also relied more on common sense: “For all is of earth, and all ends as earth” (ibid. No. 4) and “The sea is the source of water, the source of wind” (ibid. No. 23a). 

Xenophanes also discovered that every culture formed its gods in its image: “The Ethiopians claim that their gods are blunt-nosed and black, the Thracians blue-eyed and redheaded” (Mansfeld & Primavesi, 2011, pp. 227, No. 28)
“But if the oxen and horses and lions had hands or could paint with their hands and works, as men do, the horses would paint like horses, the oxen like oxen like gods” (ibid. No. 30).

 He, on the other hand, sets a “single God, the greatest among gods and men, neither in form similar to mortals nor in thought” (ibid. No. 35).

Here we also see in ancient Greece the idea of a single god appearing. Whether we have a separate root of monotheism in front of us here is probably difficult to decide. In other fragments, however, we read how he also speaks of “gods”. The development of monotheism in the early religions is a highly exciting topic, but we do not want to address it here. We are more interested here in how Xenophanes imagines the only God (Mansfeld & Primavesi, 2011, pp. 231, No. 37):

He always stays in the same place, without any motion.

So, the only God distinguishes a place and represents absolute rest. Also, Anaximander had already spoken of an absolute state of rest, but it was the Earth that stood in perfect rest in the centre of the world.  So here it is the only God.


Parmenides from Elea is regarded by some as a pupil of Xenophane, but in any case he is said to have known his works. He is probably regarded as the most prominent philosopher from the time before Socrates. Relatively many fragments of his writing ABOUT NATURE (Περι φύσεωζ = Peri Physeos) have been preserved. But his speeches are considered “dark” by all. This has earned him a special prominence among the pre-Socratics, for his work invites a wide variety of interpretations. There has also been much discussion about the right interpretation of the fragments.

I don’t want to get involved. On the one hand one should have penetrated deeply into the language of that time, on the other hand it does not seem so important to me to know the thoughts of a thinker as exactly as possible. It is much more interesting to know which new thoughts a thinker has brought into the discussion with which motives and how important these are for the treatment of important questions. Parmenides is also primarily concerned with the question of how to acquire knowledge, and his merit is at least that he makes an alternative quite clear with regard to this question. He speaks of two very different ways of knowledge.

The first way is like a revelation. In this way the knowledge of the being is given to one. For Parmenides, it is a goddess who leads a human being to an absolute certainty about what exists by means of a “destiny”. He dresses these thoughts in the story of a journey to the goddess Dike. In a fragment we read (Mansfeld & Primavesi, 2011, pp. 321, No.4):

So, the goddess, […] received me confidently and spoke the following words: “Young man, […]. It is not an evil destiny that has led you away through this path to reach your goal […], but divine providence and justice. So, it is appropriate that you experience everything: on the one hand the unshakable heart of the well-rounded truth, ….

To the well-rounded truth belong then the statements about the “being”: This is timeless, there is no origin and decay. It is “because unborn also imperishable”, it is “present together in the present as a whole, one, connected”. It is also an indivisible continuous, because “… it is not divisible, because it is quite similar. And there is not a stronger being here or there that could hinder its connection, nor a lesser one. […] That’s why it’s quite coherent, because being is close to being.” And:

As the same and persisting in the same and on itself it is, and in this manner remains firmly in the same place. (Mansfeld & Primavesi, 2011, pp. 327, No.11).

Here again we encounter the topos “absolute rest”, i.e. no motion, no change of place. The extraordinary, the imperishable and the comprehensive had to have a motion which is exceptional compared to other motion. According to the view of the world at that time, such a motion could only be rest. Even today it seems to us humans that this is often the case. However, we are not always aware that motion is not a property, but a two-digit relation, i.e. a motion in relation to another object, in everyday life. This unconscious and erroneous assumption that there is absolute rest should allow Zenon of Elea his so-called movement paradoxes, with which we want to deal in the next blog post.

But let us first consider this first path of the Parmenides to true knowledge. While according to Xenophanes the knowledge about the “perfect”, about the “gods and all things” is denied to us, according to Parmenides man can only attain this knowledge if it is revealed to him.

This is now water to the mills of mythical thought. Here an unshakable truth is promised. There is no uncertainty from possible criticism and constant changes in knowledge, as was the case with the Milesians and the Pythagoreans.

Now, every human being will probably never get along completely without a mythical component in his world view, also he may stand firmly to his convictions of such a kind. But a person who believes that the deepest secrets of the world are revealed to him personally as reliable knowledge easily risks seeing this knowledge as obligatory for all other people. It does not always have to be a goddess or a god, one can also feel a strong sense of the evidence of one’s ideas, which one may regard as a profound unshakable truth.

But now we know that different people can also experience very different revelations and feelings of evidence. When power and mission are added to an unshakeable conviction, even from a single source of revelation, different faith communities can gradually develop that divide society. Intolerance, contempt, oppression and annihilation of dissenters was indeed often the result. Violence has been perpetrated in every form.

So, what is the second way to knowledge for Parmenides? According to the above fragment, the young man, on the other hand, should also experience: the opinions of mortals who do not possess true reliability.

The second way is thus the way of the usual mortals, who can only arrive at “opinions”. According to Parmenides, however, this knowledge is at most truth-like and uncertain when measured against the abundance of the existing. Here the influence of Xenophanes shines through: We mortals have insights that can only be regarded as “opinions” in view of our knowledge of what exists. We only find “names” for what we explore:

Therefore, all the name that mortals have set is in trust that it is true. Emergence and decay, being and non-being, changing place and changing the bright colour. (Mansfeld & Primavesi, 2011, pp. 329, No. 11).

To give something a “name” does not yet mean to recognize it: Thus, according to Parmenides, the existing is in reality motionless. But we see motion. They’re just “names,” they just seem to exist. On our own, we can only develop delusions that cannot be reliable. This thesis has of course attracted a lot of attention, as it has the charm of the extraordinary, the anti-intuitive.

Both paths to knowledge were taken in the following millennia. In an effort to understand the world in which we live, the thinkers have, more or less consciously, adopted one side or the other, with sometimes dramatic consequences for the living together of people and the further development of humanity.

With concern to the reliability of the acquired knowledge, the thoughts of Parmenides about the two ways of knowledge are in stark contrast to what has been shown in the course of time. The “opinions” about nature proved to be highly reliable, because their fundamentals were directly inquired by nature – through observations and experiments. The fact that we can use this knowledge in the form of the development of technical devices is evidence of this. But the situation with revelations is quite different. There can be no question of reliability here.

The Pythagorean Crisis

After having studied the mathematics of the Pythagoreans and other Greek philosophers of the time in the last blog post, we now have to deal with the conclusions the Pythagoreans drew from the results of their studies. For this we must first have a look at the Pythagorean community in order to understand the fate of Pythagorean philosophy.

The Pythagorean Community

The group around Pythagoras consisted by no means of predominantly unworldly philosophers who enjoyed mathematical problems alone. From later ancient sources (Iamblichos, 3./4. century) one learns that the Pythagoreans later had two groups: “’Mathematicians’ were called those who had been taught more in the special teaching of his science, cultivated with regard to accuracy, ‘acousmatics’, those who had heard only the brief regulations without exact justification”. (Mansfeld & Primavesi, 2011, pp. 137, No.20). 

This has already shown how social divisions arise:  For the acousmatics (ἄκουσμα = Akousma = auditory perception) the mathematicians were not real Pythagoreans at all. On the other hand, the mathematicians saw themselves as a kind of “higher” Pythagoreans. Franz Schupp mentions in this context that later there had been a similar distinction between Gnostics and early Christians: There were “Pistiker”, the simple believers, and “Gnostics”, who have a higher insight or only claim such (Schupp, 2003a, p. 67). In our time they are in the religions the “common people” and on the other side the group of priests and theologians.

But the comparison does not fit in all aspects. For one could not call it unreasonable if Pistikers or believers were to cast doubt on the teachings of the Gnostics or fall away completely from the faith. They then only turn away from the decisions of certain authorities. It would, however, be unreasonable, i.e. acting against reason, if acousmatics experts would not recognize mathematical proofs in principle. They could be informed only to the extent that they would be able to verify the accuracy of the evidence.  

Both ways of thinking, the ones close to the myth and the one close to the Logos, existed in the community and of course also in individual minds. And for later purposes it is advisable to make another distinction in the myth, namely in a “purely philosophical” direction, which saw a similarity between structures of nature and the structure of mathematical concepts, that is, between the “principles of mathematics and the principles of things in existence”, and in a strongly religious direction, which interpreted this view of the world to such an extent that one believed one could deduce from it strict rules for life and coexistence. To this religious direction I also want to count the doctrine of the soul, the assignment of numbers to any virtues as well as all the rules for life, which one finds e.g. in the list of “Akusmata” of Iamblichos (Mansfeld & Primavesi, 2011, pp. 193, No.102).

What I have said about the doctrine of the soul should, of course, apply generally to the religious direction of Pythagorean thought. This is not meant to play a role here.  

The Principles of Mathematics and the Principles of Being Things

If one wants to get an overview of the mathematics of the early Greeks, then one must take the trouble to study the “Elements” of Euclid of Alexandria. Euclid must have written this textbook in the years around -300, and according to Proklos Diadochos (412 to 485) he “used much of Eudoxos, brought to a close much of Theaitetos’ treated, and what was represented by earlier only superficially, he supported by incontestable evidence”. We don’t know if that’s a fair judgement.

In any case, we must note that among the mathematicians who later appear in any lists of early Greek mathematicians, there have not been too many Pythagoreans. From the Pythagorean communities, however, we know two groups whose leaders Philolaos (-470 to -399) and Architas of Taranto (-428 to -347) were famous mathematicians, and whose mathematical results are still remarkable today.

From Philolaos we know about the collector of ancient philosophical writings of Stobaios (5th century), which world view the Pythagoreans derived from their mathematical studies. In one of the fragments that we have of him:

 “And it is truly all that can be recognized, number, for it is not possible to understand or recognize anything without it” (Mansfeld & Primavesi, 2011, pp. p. 147, No.28).

The concept of number is therefore the basis of every knowledge. If you want to recognize something, it must be possible to formulate it quantitatively. And in the more quantitative knowledge one discovers the order and harmony of the world. Another fragment says:

“However, it would be impossible that one of the things recognized by those who exist and by us humans would have come into being if it had not already existed the essence of the things from which the world is composed: the essence of the limiting and unlimited. But since these principles […] are not equal, […] it must necessarily be united by such harmony if it is to be contained in the world order.” (Mansfeld & Primavesi, 2011, pp. 145, 27).

With the concept of the number also the unlimited comes into view. From the fact that there are being things, one must therefore conclude that this Unlimited exists in harmony with the Limited. It is obvious that it was especially the discovery of the rational frequency ratios of the strings, e.g. of the tetrachord, which suggested the connection of numbers with a harmony. (Aristotle, no date) says:

“Since they became aware of the fact that the relationships and laws of musical harmony can be represented in numbers, and since all other phenomena also showed a natural relationship to numbers, but numbers are the first in all nature, they came to the conclusion that the elements of numbers are the elements of everything that exists and the entire universe is a harmony and a number.

Once one has conceived such a thought, one also associates the beauty of the regular polyhedra with the numerical proportions readable there. 

But the Pythagoreans went further. Aristotle mocks it: “What could only somehow be found in agreement between the numbers and harmonies on the one hand and the processes and parts of the vault of heaven and the entire structure of the world on the other hand, they collected and sought to establish a connection; but where they missed the opportunity to do so, they were not afraid of artificial assumptions, only to make their systematic procedure appear as strictly uniformly carried out.

He gave an example of this:

“Since they consider the ten to be the perfect number, and believe that these is concerned with the entire nature of numbers within itself, they make the assertion that even the bodies which turn in the sky are ten in number, and since we know only nine in real experience, they invent a tenth body in the form of the opposing earth.

Philolaos had invented this opposing earth. One also had an argument ready for the fact that one could never see these: It should always be right behind the sun when seen from Earth.

One remembers involuntarily some hypotheses of physics, e.g. the assumption of Wolfgang Pauli that there must be a certain particle, that a certain energy carries with it at the decay of a neutron, so that the preservation of the energy could also be confirmed here. But the difference is quite substantial: In Pauli’s day, the hypothesis was a mandate to examine, in Philolaos it was unthinkable to verify. So, it was pure metaphysics.  

The crisis: Incommensurable variables instead of numbers

But not from these and other unverifiable statements did the Pythagoreans threaten disaster with their philosophy. This came with a discovery that at its core shook her philosophy. To understand that, you need to make a little extra effort:

The Pythagoreans knew the natural numbers {1,2,3,…} and the positive rational numbers, i.e. ratios of natural numbers like 3/4 or 2/5. Since these numbers should reflect the condition of the world, also all things had to have a measure which can be expressed by these numbers. The length of a route, for example, had to be expressed by a multiple of a unit length and for two routes there had to always be a common unit length. Expressed in numbers: there had to be a common divisor g for two numbers, so that the two numbers m and n could be represented as integer multiples of g. The largest of these divisors is then called the “largest common divisor”. For the Pythagoreans, therefore, the length of all routes had to be “commensurable” in this sense, yes, all things in the world had to be commensurable, i.e. have a common measure. For numbers rule the nature of the world.

The ancient Greeks had even found an algorithm with which they could easily calculate the greatest common divisor of two natural numbers. This is demonstrated in Fig. 1:

Fig. 1: The algorithm of alternating removal to determine the largest common divisor of 44 and 12: One subtracts from the larger number, here 44, so often the smaller number, (here 12), until the remainder is smaller than 12. Then 12 is the larger number and you repeat the game – until you get zero. The last number to be subtracted for this, here 4, is the largest common divisor or common measure. https://de.wikipedia.org/wiki/Datei:Euklidischer_Algorithm.png

This procedure is found in Euclid’s “Elements”; but already the Pythagoreans are said to have known this algorithm.

But now the Greeks, and even the Pythagoreans, knew the so-called Pythagorean theorem. In a square of side length 1, the diagonal has a length whose square equals 12 + 12 = 2 according to this theorem. But they didn’t know a number with a square equal to 2. Side length and diagonal can therefore not be commensurable.

If someone is still looking for any way out, one can convince him with a strict proof:

Let us call the quantity whose square equals 2 already √2, as we do today, and make the assertion that this quantity can be represented as a number in the sense of the ancient Greeks, i.e. as a ratio of natural numbers. So then

√2 = m/n.

The numbers m and n can be chosen without loss of generality in such a way that the statement

A:= “m and n are relatively prime”

is true. Squaring the equation results in

2 = m2/n2, therefore also m2 = 2∙n2

It follows that m is an even number, so that m = 2∙k can be written, so m2 = 4∙k2. Thus, with the help of the previous equation 4∙k2 = 2∙n2 is also valid, i.e. n2 = 2∙k2, which finally means that n like n2 is divisible by 2. So: m and n are divisible by 2.

Altogether we conclude that m and n are not relatively prime, i.e. statement A is false, although it was assumed to be true. Then A can’t be true. Because one can never logically conclude from a true statement that it is wrong. That’s plausible. In a later blog post I will show this logical conclusion strictly formal within the framework of a so-called propositional logic.

The ratio √2:1 can therefore not be represented as a ratio of natural numbers m/n.  So, a natural number and a quantity like √2 have no common measure, they are incommensurable.

The quantity √2 was not accepted as a number until modern times. Such non-rational, irrational numbers were seen at most as “impossible” or “imaginary” numbers. You could count with them, their square was equal to 2, but you could never write down these quantities completely, as it was “appropriate for a number”. Only at the end of the 19th century did people learn to define the concept of numbers in such a way that even irrational quantities could be accepted as a special class of numbers.

This discovery that in nature there can be distances whose length cannot be represented as a rational number has greatly shaken the Pythagoreans. It was believed that this knowledge must remain secret. Iamblichos (245 to 325), in his book On Pythagorean Life, tells the story that someone is said to have divulged this discovery. He was then excluded from the common circle and later even perished in the sea (Mansfeld & Primavesi, 2011, pp. 171, No.61,62). Like all anecdotes from this time, you can sprinkle them in for entertainment. You don’t have to believe them.

The legacy of the Pythagoreans

The ancient Greeks were the first to turn the mathematical knowledge, handed down to them by earlier peoples, into a science. They not only found interesting relationships between several numbers and between numbers and geometric figures, bodies and sounds. Even more significant is the fact that they discovered an argumentation that is incontestable, that is, what we call mathematical proof today.

Euclid of Alexandria collected this knowledge and brought it into a logical order. In this order definitions and axioms are placed at the beginning, and all knowledge is derived from them in the form of mathematical proofs. An “axiomatic deductive system” was thus created. Thus, the idea of a strict science was not only born in the time around the year -300, but was also already realized once. This idea still inspires everyone who thinks about what a science actually is. In Die Idee einer Wissenschaft – Ihr Schicksal in Physik, Rechtwissenschaft und Theologie (The Idea of a Science – Your Fate in Physics, Jurisprudence and Theology) I have elaborated on this further (Honerkamp, 2017).

Also, in the years after Euclid the mathematics of the Greeks progressed further. With Archimedes (ca. -287 to -212) it has reached a temporary climax. Historians of science, however, speak of the first signs of the disappearance of creative forces in the coming centuries (Russo, 2005).

Pythagoras and the Pythagoreans gave in ancient Greece the impulse to this first bloom of a science. Though, with their doctrine of harmony they exaggerated it, their religious zeal including their rules for the way of life today seems to us highly strange, sometimes bizarre. They failed with their idea of numbers as a basic pattern for nature. 

Yet they were on the right track. It was not until the second attempt, 2,000 years later, that a combination of mathematics and natural science was to emerge, which then led to an understanding of nature from which people could develop machines rich in both blessings and horrors.

The Presocratics: The Mathematics of the Pythagoreans

At the time when Anaximenes in Miletus represented the tradition of the Ionic school, Pythagoras was active in another part of the Greek culture (ca. -570 to -490).  Born and raised on the island of Samos, he is said to have spent his youth in Babylon and Egypt. There he got to know the religious beliefs and also the mathematical knowledge of these cultures. Around -530 he settled in Kroton, a town rich through trade in a southern Italian Greek settlement, and he founded a school there. This became a conspired community; he himself became an unquestioned authority for his pupils. He must also have been a charismatic speaker, and he must have impressed them greatly with the ideas he took up in Babylon and Egypt. With the sentence “He said it himself”, the Pythagoreans are said not to have even had any discussions (Schupp I, p.63). So today we see strong religious traits in this school.

Much has been written about Pythagoras, much attributed to him. None of this is considered certain today. There are no writings by him, not even by his pupils, the Pythagoreans. From the ancient sources such as those of Heraclitus, Empedocles, Aristotle and from the late antiquity of Diogenes Laertios or Jamblichos we learn contradictions and highly different assessments. 

Two themes were at the forefront of Pythagoras and his pupils, the Pythagoreans, but in any case: the transmigration of souls and mathematics. Pythagoras was probably inspired by his travels on both topics. 

I will not go into the ideas that the Pythagoreans had of a “soul” and a “transmigration of souls”. The Pythagorean theory of the soul had a great effect and influenced later philosophers like Plato in particular. It was the dominant theme of Neuplatonism in the first centuries AD and has also flowed into Christianity through this philosophical current. However, this doctrine did not contribute to the strengthening of the Logos in people’s thinking.

The situation was quite different in this respect, however, with the study of mathematical problems. A wealth of mathematical insights was obtained. A completely new field for philosophical considerations was discovered. Aristotle reports in his Metaphysics:

“At the same time as those philosophers [Parmenides, Empedocles, Anaxagoras, Leukipp, Democritus] were active, but even before that, those called “Pythagoreans” began to pursue mathematical studies. They were the first to advance these studies; and when they were educated in them, they were convinced that the constitutive principles of mathematics were also the constitutive principles of things being. (Mansfeld & Primavesi, 2011, pp. 147, No.31).

The Pythagoreans were, after Aristotle, the first to increase mathematical knowledge, and Pythagoras had given the impetus. This was the second great impetus in the history of ideas. The first impetus was given by the Ionic philosophers when they began to explain the world from within, namely with arguments based on regularity in nature. Here now, in the second step, a language was discovered in which one can note regularities. But not only this; in this language one could also provide arguments that are undeniable.  And for the relationship between the “constitutive principles of mathematics” and the “constitutive principles of things in existence”, the Pythagoreans, according to Aristotle, provided a first answer. We will see in the next blog post how they felt about it. Later, the question of such a relationship will become very topical.

But let us first have a look at the studies that the Pythagoreans and other philosophers, infected by them, have carried out:


Most people associate the name Pythagoras with the SATZ OF PYTHAGORAS via the relationship between the squares of the sides of a right-angled triangle: If one designates the cathets of such a triangle with a and b, the hypotenuse with c, then a2 + b2 = c2 applies. The Babylonians already knew this relationship and Egyptians and Pythagoras, like Thales, had brought such knowledge from his travels. Legend has it that Pythagoras was inspired to prove this theorem by looking at an Egyptian tile (Fig.1).

Fig. 1: Left: Illustration of the theorem of Pythagoras: right: The tile could have looked like this (but with the same strokes everywhere), from which Pythagoras was inspired, according to the legend, to prove the proposition. The squares above the hypotenuse are marked by dash dots. The squares above a, b, c contain two or four triangles of equal size. (from Honerkamp, 2018).

Here we are dealing with a geometric figure from which we can derive a statement about the relation of three numbers, i.e. an arithmetic relation. This is probably the first connection between geometry and arithmetic. About 2,000 years later Descartes was to strengthen this connection with “Analytical Geometry” to such an extent that arithmetic became the focus of mathematics from then on.

The Greeks represented their numbers at that time still by letters, i.e. there were prominent numbers, i.e. those, which we today 1, 2, …, 9, 10, 100,… any other number was seen as the sum of these celebrities, and the summands were then noted in size in the form of letters. We still know such an addition system from the Roman numbers.

In the context of such a number writing, calculating is very difficult. This may have led Pythagoras to invent a geometric number representation in which each number is seen as a sum of ones. For each one a point is noted, whereby the points are arranged to a certain geometrical figure (fig.2). For example, there are squares, triangles, cubes, etc.

Fig.2: A square number (16=42), a triangle number (10 = 1+2+3+4) and a cube number (8= 23).

From such figures one can already read many other relationships between different numbers, e.g. the statement that the sum of successive odd numbers, if it begins with 1, is always a square number (Fig.3).

Fig. 3: A deduction from square numbers (from Schupp,2000a, p.74)

If you still write the last line on the right side of this illustration as :

(1 + 3 + 5 + 7) + 9 = 25, so 16 + 9 = 25, or 42 + 32 = 52,

one obtains a so-called Pythagorean number triple, i.e. three numbers, where the square of the largest can be represented as the sum of the squares of the two smaller numbers. In this way one discovers how easy it is to form such number triplets:  Each sum of such odd numbers ending with a square number yields a triple. So take only an odd square number, say 25 = 52, and form

1 + 3 + … + 23 + 25 = 169, i.e. 144 + 25 = 169, or 122 + 52 = 132.

The Babylonians already knew such triplets of numbers, and since a large number of such triples were found on their cuneiform tablets, methods for the formation of such triples must have been known for much longer.  Even in the megalithic culture of southern England the knowledge of such triples is said to have been used for constructions (van der Waerden, 1983, p. 9).

Inspired by such experiences, the Pythagoreans discovered many relationships between two, three or four numbers, e.g. the arithmetic and geometric mean, including the harmonic mean. Probably they already knew the “golden ratio”. Today one would book such studies under the title “number theory”.


Geometry has not been neglected in all these studies of ratios of numbers. Especially in Egypt, Thales and Pythagoras learned a lot about geometry. After Nile floods the Egyptians often had to reconstruct their fields. It was very useful to know how to construct rectangles and, above all, how to calculate the area of a field.

Not only right angles and rectangles were important. Also circles, triangles and other polygons wanted to be constructed, in practice – and with the Greeks then also on the papyrus with compass and ruler, completely without regard to a use. And one also wanted to be able to calculate the area contents of such areas.

A notorious problem for the Babylonians and Egyptians had been the calculation of the area F of a circle. It was well known that F must be proportional to the square of the radius, but the proportional constant, which we call today π, could not be determined exactly. The Babylonians had used the value 3 for this constant, the Egyptians (16/9)2 = 3.1604… . The Pythagorean antiphon of the Sophist (5th century B.C.) took advantage of the fact that one already knew how to determine the area of regular polygons, approached the circle by such polygons, and thus obtained an approximate value, which is the better the more corners this polygon has. This method was to become the norm. Bryson of Herakleia (-450 to -390) looked not only at regular polygons inscribed in the circle, but also at polygons circumscribing the circle. He thus received two approximations, one smaller and one larger than the value he was looking for. Archimedes was then to drive the method to a perfection that can only be admired today (Honerkamp, 2018, p. 82ff).

Not only surfaces, but also three-dimensional bodies in space were studied, besides the sphere the polyhedrons (polyhedron, after ἔδρα gr. = seat, καδ-έδρα gr. = catheter) were of particular interest. These are bodies that are bounded by several regular surfaces. There is a tetrahedron (quadhedron), the cube is a hexahedron, there is an octahedron (octahedron), a dodecahedron (dodecahedron) and an ikosahedron (twentyhedron) (see Fig. 4).

Fig.4: The five platonic bodies: tetrader, hexahedron, octahedron, dodecahedron, ikosahedron.

As the mathematician Theaitetos (-415 to -369) was able to show, they all have a certain point inside, which is equally far away from all corners, so that all these polyhedrons can be inscribed in a sphere. He also proved that there can be no other polyhedra of this kind. The regularity of these bodies and their uniqueness caused Plato later to assign tetrahedra, hexahedron, octahedron and ikosahedron to the elements fire, earth, air and water, in the sense of the Pythagoreans, who see everything in nature dominated by numbers. Plato thus gave these bodies a mythical shine; this led to the fact that they were later called “Platonic bodies”.

The French historian of science André Pichot lists in his work THE BIRTH OF SCIENCE twenty-one Greek mathematicians of the 6th and 5th centuries BC (Pichot, 2000, p. 371ff). Many mathematical theorems in geometry and arithmetic have been found by these. Also textbooks, called “elements”, must have existed in pre-Socratic times to introduce newcomers to mathematics. None of them has been handed down to us. Only from the time around -300 such a textbook is available to us. But it is probably the most complete and mature work from which we can take the state of mathematics at that time. They are the “elements” of Euclid of Alexandria. This work consists of three chapters (“books”) on arithmetic and 10 chapters on geometry. It has had a hardly surpassable influence in the intellectual history of the western world, above all because it could represent geometry in the form of an axiomatic deductive system and thus made the logical order of the knowledge represented transparent. I will come back to this at the appropriate time.


There was another, even more concrete, field where the Pythagoreans discovered that relations in space can be mapped to relations between numbers. This field was the music they played on the lyre, and here you could even experience relationships between nature and numbers with your senses.

The “tetrachord” had four strings, which could be plucked, struck or beaten to produce a certain tone, depending on length and tension. If one plucked two strings at the same time, one could hear a sound which one felt to be harmonious if the lengths of the two strings were in an integer ratio such as 1:2, 2:3, 3:4. Especially with a ratio of 2:1, the sound sounded almost like a single note. Let us consider here the simplest form of a tetrachord (Fig.5, see also Wikipedia: Tetrachord).

Fig. 5: The strings of a tetrachord and the ratios of the lengths of adjacent strings (after (Pichot, 2000)). Attention: Here the tones get lower from left to right!

Be the first string on the far left the shortest. The second, third and fourth string may then be 4/3, 3/2 or 2/1 times the length of the first. We then call the notes C’, G, F and C, the frequency of the notes G, F, C are 3/4, 2/3 and 1/2 times the frequency of the note C’ of the left string. The Pythagorean Philolaos gave the intervals certain names, later they were called fourth (from C’ to G), fifth (from C to F) and octave (from C’ to C) (fig.6)

Fig.6: The strings of an octochord and the ratios of the frequencies of neighbouring strings in the Pythagorean tuning (according to Pichot,200, p.385).

For an “octochord” with eight strings a finer subdivision was needed. Philolaos first introduced a “second” as the distance between fourth and fifth (from G to F, from Mese to Paramese). For this, the ratio of the frequencies is (3/4):(2/3) = 9/8. With two consecutive seconds (whole tones), the lowest note C (nete) can be used to reach the note F (paramese), but not yet. The remaining “distance” is (4/3):(9/8)2= 256/243. This interval is called a “semitone”, the Greeks called it Diёsis (from δί-ειμι, gr. = pass through).

The interval from G to C’, from the mese to the hypate, is also a fourth, and the result is the same as from C to F.

So, such a “semitone” is an interval consisting of two tones with a frequency ratio of 256/243. But it is not really something like half the interval of a whole tone. If you put two such intervals on top of each other, you get an interval with a frequency ratio of (256/243)2, which is not equal to 9/8. A very small interval with frequency ratio is still missing.

(9/8) : (256/243)2 = 531441/524288.

This interval was today called the Pythagorean comma (from gr. κόπτειν->κόμμα = something like “section”). One would have to call the “semitone” more exactly a small semitone, and then there is also a large semitone, an interval that is larger by the Pythagorean comma. 

As long as you play on the octochord, and everyone who plays at the same time uses the same fundamental C, this Pythagorean comma does not matter. 

This determination of a stock of six tones between the two tones of an octave for a given fundamental leads to the so-called Pythagorean tuning. It is therefore realized with such an octochord.

The six tones (now starting from “below”: D, E, F, G, A, H) can also be determined in this way: From a given tone you can find the pure fifth (frequency ratio 3/2). If this results in a tone above the interval under consideration, set the tone an octave lower, i.e. halve the frequency. So if you start from C, you get one tone at a time: G, D’->D, A, E’->E, H. Now the F is still missing. Then you should start from F, so now you get: F, C, G, D, A, E, H.

If you now want to increase the stock of notes, you can continue to use the fifth rule for this, so you get notes that we could call F#, C#, G#, D#, A#, because their frequencies are one Diёsis higher than F, C, … in each case. The next note would then be ice, and that would have to be F again. But that’s not true, because there are two small semitones (H-C and E-F) between H and the F sharp of this construction, instead of just one like the other such intervals. So the interval H-F sharp is too small, not a pure fifth, a “wolf’s fifth” (because it “howls like a wolf”). So, if you want to strictly follow the rule, you have to increase the frequency of the F sharp by the Pythagorean comma and so do all following notes. The frequency of the ice finally reached would thus be higher by this decimal point than F. This can also be seen if this construction is computed. It is

(3/2)12: 27 = 531441/524288.

If one were to construct the notes B, E flat and A flat from F “downwards” to the same extent, one would obtain the circle of fifths in the usual form. Then the notes G#/As show that the “circle” does not close completely. The tone G sharp is higher by the Pythagorean comma than the As.

The Pythagorean tuning system was fashionable until the 16th century. After a long period of experimentation, the equal tuning was introduced in the 16th/17th century. In this the octave is divided into twelve intervals of equal frequency ratios f. So it must be f12 = 2, hence f = 1.05946…. > 1,05349… = (256/243).

The Pre-Socratics: The school of Miletus

The city of Miletus, situated on the coast of a Greek settlement of the landscape of Ionia (Asia Minor), gained economic importance in the 6th century B.C. through extensive trade relations. A social class developed for which “just about everything was available for comfort and a higher standard of living”.

The cultural milieu was marked by Orphic or Dionysian cults, stories of Homer and epics of Hesiod (born before -700); influences from Mesopotamia and Egypt, Phoenicians and Chaldeans must have been noticeable as well. But time and leisure also brought with them the attempt “to escape ignorance without expecting any benefit beyond knowledge” (Aristotle, after (Schupp, 2003a, p. 43)). Philosophy can be born in such an atmosphere.


Thales lived in this town from about -625 to -547. At first he was a merchant and must have been around a lot in the world at that time. He is said to have brought geometrical and astronomical knowledge from Egypt. In his later years he also interfered in politics. But we only know of all this from later sources, whose credibility is not always given.

What is certain, however, is that he must have lived around 585 BC. He had predicted an eclipse of the sun, which according to modern astronomical calculations took place just in 585 BC. Babylonian astronomers already knew the cycles with which solar and lunar eclipses were repeated. Sun eclipses are not visible from every place equally well; but one knew thus when one could count on such at all. Thales had probably benefited from this knowledge, either through his earlier travels or through more recent cultural contacts. But he was also someone himself who studied the stars. Diogenes Laertius knew that Thales was said to have written two writings about solstices and equinoxes. Because of his knowledge he was also admired by later philosophers (Laertius, 2015, p. 13).

Many of us can surely remember that in school they learned the SATZ OF THALES in mathematics lessons. We do not know whether Thales himself discovered this sentence. Diogenes Laertius writes only 800 years later: “In geometry, a pupil of the Egyptians, he first entered the right-angled triangle into the semicircle, as Pamphile reports. (Laertius, 2015, p. 13). There it is also mentioned that Thales in Egypt should have measured the height of the pyramids, “giving their shadow, which he measured exactly at the time when our shadow and our body have the same length” (Laertius, 2015, p. 13). (Laertius, 2015, p. 15). Thales must have been highly intelligent in any case and he knew how to use his new knowledge independently.

The philosophers regard Thales of Miletus as the first of their guild. We do not know how Thales came to such thoughts, which today are called philosophical thoughts. The processes “emergence and decay” were in the foreground in the mythical narratives and on his travels Thales will not only have taken up geometrical and astronomical knowledge, but also become acquainted with the most diverse of such narratives. Since one constantly observes changes in nature, including emergence and decay, the question may arise as to whether there must not be something lasting and eternal “behind” all this. This must then be something causal and “underlying”, something like a “primordial principle” or a “primordial substance”, that is, something that could also be called “the One”.

Aristotle later wrote about the ideas Thales had developed for this:

“Thales, the first representative of this direction of philosophical investigation, describes water as such a principle. He therefore taught that the land, too, rests on water. The reason for this view was probably the observation that the food of all beings is moist, that the heat itself arises from it and lives from it. But what everything becomes of, that is the principle of everything. If this was one of the reasons for his opinion, then another was probably the fact that the seeds of all beings are of moist nature, but that water constitutes the principle for the nature of wetness.

It was not far-fetched that water should be the primordial cause of all things, for the Sumerians and Egyptians also spoke of an “primordial sea”, and in Homer’s case the inhabited world was surrounded by the Okeanos, a mighty river, which was also considered the father of the gods and the origin of the world.

It is important, however, that Thales argues this way here and elsewhere: Water causes moisture, food needs water, otherwise it dries up and becomes inedible. Water can form fog and clouds, and in the last instance even air (water vapour) and fire were formed from it. Sedimentation on the coast shows how land can be removed from the water.


Thales brought a whole new way of thinking into the world. Anaximander (ca. -610 to -546), who also lived in Miletus and is regarded by many as a pupil of Thales, took up this way of thinking, but immediately showed that one can come to a different conclusion in it. For him, water was perhaps too concrete as primordial matter, and he concentrated more on one particular aspect: the One which should be a primordial reason had to be unlimited, an apeiron (from Greek ἀπέραντος=aperantos, unlimited, infinite). The Greeks had the same word for “unlimited” and “infinite”, but could not yet imagine a curved space. Today we know spaces of different dimensions; a vivid example for an unlimited, but not infinite space is the surface of a sphere, a two-dimensional curved space.

This is where the term “infinity” came into play, a term which subsequently occupied almost all philosophers and mathematicians and which was only learned to be dealt with consistently and formally at the end of the 19th century within the framework of the Cantor set theory. According to Anaximander, what people do not know  nor what they can imagine, should apply to the Apeiron: “The Apeiron is without age” and the “Apeiron is without death and decay” (according to Schupp, 2003a, p. 53).

These were qualities that were granted only to the gods. The One was thus placed on the same level with the gods. Thus Anaximander came suspiciously close to the world of myth, but he answered the question of how the One would create multiplicity, more like a physicist: He spoke of opposing elemental forces, dry hot fire and humid cold steam, that could lie in battle with each other.

In this way he had an idea of what the sun, moon and stars were made of, and even knew how to explain how solar or lunar eclipses occur. According to him, the earth was at the centre of the circles on which the celestial bodies were moving, and it was in perfect equilibrium with all the stars.  In the shape of a cylinder it was similar to a “stone column segment”. Humans lived on the top of the cylinder; according to the geographer Agathemeros, Anaximander even “dared to be the first to draw the map of the inhabited world” (Mansfeld & Primavesi, 2011, pp. 65, No.2). The distances of the celestial bodies from the earth were in certain proportions to each other. Thus, Hyppolyt of Rome reports in the 3rd century that according to Anaximander the circle of the moon and the sun is 19 and 27 times the diameter of the earth cylinder, respectively (Mansfeld & Primavesi, 2011, pp. 75, No.20).

Even a weakening of the struggle between the hot and the humid a passing of the “existing” as well as a new creation from the Apeiron was planned. For Anaximander the Apeiron is therefore not only infinite; it can also create infinite many worlds and let them pass away again. One inevitably thinks of the quantum vacuum in Stephen Hawking’s M-theory and of the constant emergence of new universes from it through quantum fluctuations.

For the interaction between the opposing pairs he had an answer, which again fits more to the myth and the heaven of gods: “From which things the things in existence have their origin, in these also their decay takes place, as it must be, because they do each other justice and punishment for the injustice, according to the temporal order” (Mansfeld & Primavesi, 2011, pp. 71,Nr.15).

The whole picture already contains astonishingly many modern aspects such as the description of spatial relations in quantitative form and the concept of infinity.


At Anaximenes (approx. -586 to -527) the primary substance is now something you know from everyday life: the air. Instead of an abstract, it is now something concrete again. From the Doxographen Aёtios from the 1. century we experience: “Anaximenes set as principle of the being things the air, because from this everything develops and into this everything dissolves again. Just as our soul, which is air, holds us together through its power, so also the whole cosmos includes “breath and air”. (Mansfeld & Primavesi, 2011, pp. 87,Nr.3).

Anaximenes presumably took the air as the primary substance because he observed opposite properties in the air; it could be warm or cold, compressed or diluted. From the observation of a breath he believed he could deduce a relationship between these pairs of opposites: “For when the breath is compressed and solidified by the lips, it becomes cold, while when the mouth is open it escapes, it becomes warm through dilution”. (Mansfeld & Primavesi, 2011, pp. 89, No.5).

So he saw compression or dilution as the basic principle for the difference in things: Clouds consist of weakly compressed air, rain of more compressed air and ice like earth of even more compressed air. The solidification of matter generally has its origin in the cold air, the thinness and looseness in the warm air. Wind was moving air. The primordial matter was now something that could also produce multiplicity. Thus, he had an idea of how the One could become the Many. The air was a primordial substance which, as a breath of life, absorbed all living beings into the unity of all things.

So here we find the connection between breath and life, a thought that one encounters again and again in the intellectual history – e.g. also in the idea of Christianity that the Holy Spirit originated from the Father and Son through breathing (see Wikipedia: Hauchung).

In this earliest school of the pre-Socratics, the Ionian school or the school of Miletus, the concept of “being” or “One” as a primordial substance or principle is in focus. In later schools also the “existing”  or the “Many” will become more strongly in the view and the question about which possibilities of the knowledge we can have about the being like the existing.

Paradigm shift

With the Ionic School, a world view was created that was derived from observations of nature and not from stories about interventions from a “supernatural”. This was a big step for mankind. But it was only a first step in a new direction. Of course, there were no clear answers. The reasons for such statements as “Everything is water” or “Everything is air” were only more or less plausible, necessarily not at all. Later Empedocles (-490 to -430) should still claim that “everything is fire” applies. Generally one spoke of the four basic elements water, air, earth and fire.

It was to take more than 2,000 years before a second step was taken in natural science, from which followed what we now call modern physics. Two new thoughts had to emerge:

First: One does not immediately try to understand the world as a whole, but one has to “begin it in the small”, thus with a simple and most clear phenomenon.

Secondly, one should try to understand the phenomenon not only qualitatively, but also quantitatively, so that an as exact as possible examination of the reasoning becomes possible. This requires sufficiently developed mathematics.

Galileo Galilei was the one to whom these thoughts came in the early 17th century and who was fully aware that he had founded a “new science” with a demonstration of the fertility of such thoughts. If one loves the term “paradigm shift”, which was so unnecessarily strained by Thomas Kuhn (Kuhn, 1973), then one can say that Galileo caused a paradigm shift. It would be the second – after the first paradigm shift by Thales and the Ionian school. Perhaps today we are experiencing a third paradigm shift in artificial intelligence with data-driven machine learning.

I will come back to all this in later blog posts. First of all, we must follow the path that has created all the conditions for this second and perhaps third paradigm shift. This includes, in particular, mathematics in which one learns quantitative and truth-preserving reasoning. Pythagoras (ca. -570 to after -510) is at the beginning of such a mathematics. In the next two blog posts we have to deal with him and his students.

The Pre-Socratics: A Brief Overview

“The greatness of the pre-Socratics […] lies not only in the fact that philosophy began with them. For this can be argued about and has actually been argued about. Rather, it lies in the fact that many essential questions, themes and conditions of science and philosophy are to be found for the first time in the statements of these pioneers that we have received.” (Mansfeld & Primavesi, 2011, p. 9).

With these sentences, Mansfeld and Primavesi introduce their reworking of a collection of texts handed down to us by the Pre-Socratics themselves and the most important secondary testimonies of their doctrines and work. And in a somewhat later sentence, they characterize these questions and themes as follows:

“If there is an exemplariness of the pre-Socratics, it is above all founded in a critical and rational attitude, which should not be a mere cultural-historical fact, but which today hardly has to be achieved less than at that time”.

The pre-Socratics showed this new attitude in dealing with questions about the origin and nature of the world. Instead of inventing or embellishing stories with supernatural actors, they used observations of nature to find answers to their questions. In doing so, they relied on regularities of nature and used analogies and generalizations. To a certain extent, they invented rational reasoning, discovered logos as a tool of thought in the search for the truth about the constitution and order of nature. Equipped with today’s knowledge, we recognize here the first beginnings of our scientific age.

Things in common

One will first ask oneself why it is precisely those philosophers who worked before Socrates that are grouped together. In particular, this question arises when one notices that Zenon (-490 to -430) and Democritus (-440 to -370), for example, can already be regarded as contemporaries of Socrates (-469 to -399).

It is probably decisive that with Socrates a completely new topic arose in this so young philosophy. The philosophers before Socrates were natural philosophers, they were concerned about the order in the world and its beginning, about the “physis” – they were, so to speak, the first physicists. Socrates, on the other hand, “was the first to call philosophy from heaven to earth”, as Cicero (-106 to -34) said in his conversations in Tusculum (Cicero, 2008, pp. V, 10-11), and Diogenes of Laertius learned through several intermediaries that Socrates had recognized that natural philosophy was no good for “us” [by which he probably meant himself and his discussion partners]. So, he turned to the moral doctrine (Laertius, 2015, p. 77). Ethics became the new topic, questions about the best rules for the togetherness of people and for a “good” and happy life. One speaks of a Socratic turning point: Away from physics – towards ethics. This marked the beginning of a new era in philosophy.

Legacy and life data

Only a few fragments of the works of most pre-Socratics have survived. Our knowledge of their doctrines is often based on equally fragmentary works by Plato, Aristotle, Theophrastos and many later doxographers.

Much has been written and puzzled about the pre-Socratics. I found the books by Schupp (Schupp, 2003a) and Pichot (Pichot, 2000) particularly illuminating, in addition to the Mansfeld and Primavesi collections. The work “Lives and opinions of famous philosophers” by Diogenes Laertius, written around 220, is also worth reading, alone because of the many anecdotes. One gets an impression of how much fantasy must have been involved in such reports.

Fig.1 shows the life data of prominent pre-Socratics. With the ellipses the philosophers are grouped together who are assigned to a certain school.

Fig. 1: Chronology of prominent Greek philosophers from the 7th to the 2nd century BC (© J. Honerkamp)

Thales of Miletus created the earliest of these schools, the School of Miletus. The Pythagorean school around Pythagoras and the Pythagoreans played as influential a role as the school of Elea, whose most important representatives are Xenophanes and Parmenides. Elea was a coastal city of an Greek settlement in southern Italy. The philosopher Heraclitus, who is also listed here, cannot actually be assigned to any school. The philosophers Anaxagoras, Zeno of Elea, Empedocles and Democritus are the most prominent representatives of the pluralists and atomists. These were mainly active in Athens.

Areas of activity

As can be seen in Fig.2, most of the important philosophers and mathematicians of Greek antiquity did not live and worked in the Greek motherland, but in the coastal cities of Greek settlements, whether in southern Italy, Sicily, present-day Turkey or Egypt. It was not until Anaxagoras that Athens became the center of Greek philosophy.

Fig.2: Areas of activity of ancient Greek philosophers (after (Symonyi, 1990, p. 59)).

You can understand that. The basis of every civilization was already at that time trade, administration and building activity. Trade, in particular, attracted people who were courageous enough to take on the dangers of a journey and who could also get used to the conventions and customs of other countries. From the coastal towns there must have also been a lively cultural exchange with the respective hinterland, e.g. with Babylonia or Egypt. All this may cause a certain alertness and open-mindedness and favor unconventional thoughts. Moreover, the coastal cities had become rich precisely because of trade and allowed lifestyle that even nonconformists could endure or even appreciate.

This was an atmosphere in which philosophical thoughts could flourish and which attracted philosophers. Thus, we know from some pre-Socratics that in their young years they had travelled the world a lot, whether as traders, as refugees from political persecution or in search of a teacher. Here you can also see that cultural exchange can bear fruit in the long run. The ascent of an overseas settlement of Europeans in the 16/17th century comes to mind at this point: The United States of America soon attracted many intellectuals from Europe and has been culturally and economically dominant since the early 20th century to the present day.

The cultural situation of the time

The pre-Socratics could not know that the questions of the origin and nature of the world as a whole were also being discussed in other important cultural areas of the world at that time. There had probably always been a lively exchange of ideas about gods between all the cultures around the Mediterranean. But independently of this, Confucius (ca. -551 to -479) appeared in China and taught that the world had an order and that man’s highest goal was to live in harmony with this order. In India, between -800 and -600, a collection of philosophical writings, the Upanishads, was created. In our context the Vaisheshika is particularly interesting, a system of a natural philosophy in which five basic elements are mentioned; besides the four, which were also named by the Greeks, there was also the ether here. In Persia, Zoroastrianism spread from -800 to -300, and in Palestine the biblical prophets worked.  The philosopher Karl Jaspers (1883-1969) therefore coined the term “Achsenzeit” (axis time) for the period from -800 to -200. In the meantime, however, many historians see this as not particularly meaningful.

There was a parallel development not only in the field of myth and a beginning natural philosophy. People learned to carry out elementary calculations with numbers and to find solutions for simple mathematical tasks independently of practical problems. The Dutch mathematician B.L. van der Waerden studied the early mathematics of the Chinese and Babylonians and found astonishing parallels in problems and proposed solutions. Likewise, strong similarities with the mathematics of the Hindus were noticeable. Since the knowledge about Pythagorean triangles was also used in the construction of the megalithic monuments in southern England (Stonehedge) (cf. a later blog post), van der Warden sees the source of all this knowledge in a megalithic culture of the period from -3,000 to -2,500, and thus assumes a very early cultural exchange in the field of mathematics (van der Waerden, 1983, p. XI).

But I rather believe that humans were always roughly at the same evolutionary stage and therefore had to solve the same problems in their world. Trade, administration and construction require planning and therefore some skills in arithmetic and geometry. This phenomenon of parallel development shows at most the universality of mathematical thinking.

Be that as it may. The fact is that the ancient Greeks decisively developed the mathematics they received from the Babylonians and Egyptians and made it into a first science. In a blog post about the Pythagorean school I will address this. With this step, which was initiated by this school, the Logos has thus become “alive”, for mathematics acquires its rigor and infallibility by using only logical rules of inference in its deductions (cf. a later blog post). The Greeks discovered the mathematical proof. This is the uniqueness of ancient Greek culture.

With the rediscovery of this culture at the time of the Renaissance in Western Europe, it was possible to build on it and thus set in motion a development that for a certain time led to the cultural and economic dominance of the Western world. However, this soon seems to be a matter of the past.

Cicero, 2008. Tusculanae disputationes/Gespräche in Tusculum. Lateinisch/Deutsch. Stuttgart: Reclam.

Laertius, D., 2015. Leben und Meinungen berühmter Philosophen. Hamburg: Felix Meiner.

Mansfeld, J. & Primavesi, O., 2011. Die Vorsokratiker. Stuttgart: Philipp Reclam jun..

Pichot, A., 2000. Die Geburt der Wissenschaft – Vone den Babyloniern zu den frühen Griechen. Wissenschaftliche Buchgesellschaft Hrsg. Frankfurt, New York: Campus.

Schupp, F., 2003a. Geschichte der Philosophie im Überblick – Bd.1 Antike. Hamburg: Felix Meiner.

Symonyi, K., 1990. Kulturgeschichte der Physik. Thun: Harri Deutsch.

van der Waerden, B., 1983. Geometry and Algebra in Ancient Civilizations. Berlin: Springer.

The Oneness and the Unified Theory of Fundamental Interactions

The European nuclear research centre CERN is planning to build a new particle accelerator. In a ring-shaped tunnel 500 m below Lake Geneva and with a circumference of 100 km, the physicists want to study the interactions of electrons, positrons or protons at far higher energies than before. The particle accelerator LHC, which exists today, was used to test the standard theory, i.e. the unified theory of the so-called strong, weak and electromagnetic interaction. The result was a view of our nature on a length scale of about 10-15 m, in which on the one hand there was a good agreement between theory and experiment, but on the other hand new questions arose which the theory could not answer and for which the experiments so far could not yet give any indication. One would therefore have to look “more closely”, with even higher resolution, i.e. let the particles collide with each other with even higher energy, so that one can uncover a further substructure in today’s picture of nature through the analysis of the decay products.

The plan for such an accelerator represents a completely new dimension in the search for a unified theory for all fundamental interactions of nature. The search seems to be a project of the more recent history of modern physics, but basically it is a human project that was already started 2,500 years ago, by the “pre-Socratics”, those ancient Greek philosophers who mainly worked in the time before Socrates, more precisely in the years from -600 to -400.

People probably always felt the urge to reduce the complexity of their impressions of the world and thus always tried to make a sense of the world. At first this happened in the form of stories about the work of gods and spirits who ruled the world and people. We know such myths from the Gilgamesh epic or from the works of Homer and Hesiod. Thales of Miletus (ca. -624 to ca. -555) is regarded as the first of the pre-Socratics, as the first philosopher of Greek antiquity at all, who went a completely different way to explain the world. In the stories “about God and the world” reasons appeared now: For statements about the world as a whole, one now used everyday observations. The gods were no longer superhumans who could switch and rule as they wished. One could even think of them quite differently than in human categories, and one actually no longer needed their stories. Now there counted arguments with which one could justify his statements.

Two fundamental Questions

One could also call the pre-Socratics the first physicists, even the first cosmologists, because with their questions they went straight to the “whole”, as today only the theologians hold it in this directness. Nevertheless, they were “physicists” because they invoked nature (ἡ φύσις = physis, gr.: nature) in their justifications and no longer put forward mythical ideas.

Two questions crystallized, which became decisive for the whole history of the mind. The first question arose from the feeling that there must be something “behind” the plurality of things that we discover in our everyday world, something of which this “variety” consists or has arisen. The pre-Socratics called it the “primordial matter” of all things or the “being”. In the world in which we can live and gain experience, however, “being” shows itself as the “plurality”.  Hence there are always two levels of being, the “higher”, where being is a “oneness”, and the “lower”, where people live and encounter the “many”, the plurality. But the “oneness” could also be a “principle” or that which was called “God” in religions.

The question then was: What is the primordial matter, what does it have to do with the matter that we perceive with our senses? Or: What is the basic principle, the underlying.

A second question then inevitably arises: What can I even know about it? Or: How can I gain knowledge about it?

The first question concerns the nature of the world, the second our cognitive faculties. These are questions that still arise today.

In the first question we now see the connection with the physicists’ search for a unified theory of all the fundamental forces of nature. In this theory there should then be only one fundamental force, from which all other forces such as the gravitational force and the forces of the standard theory mentioned at the beginning result under certain circumstances. Obviously, people have made some progress in the question of the “oneness”, although the question is now posed on a completely different level, namely not on a “material” but on an “intellectual” level. One has good hope to find sometime the “one” theory which contains the “many” other theories about the individual fundamental forces.

This hope can indeed be had today because concrete progress has been made on the second question, even in ancient Greece. There mathematics came to its first bloom and it had been learned that mathematical conclusions were “irrefutable”. In the field of mathematics one can therefore gain reliable knowledge. This led to the search for necessary rules of inference outside mathematics. Aristotle (-384 to -322), in a sense an early “post-Socratic”, was probably the first to formulate rules of inference of this kind and thus to establish a doctrine of the laws of thought, a logic.

“Logical” for us today, however, in colloquial language stands for something like this that goes without saying in the context of our world of experience and thought. One sometimes speaks of women’s or men’s logic. We often hear “It’s logical – isn’t it?” when a friend tells us about a decision that he has made and found to be highly reasonable because it obviously corresponded to his goals or principles. However, this has only a limited connection with real logic. We will see this in detail in later blog posts.


The term “logic” developed in ancient Greece from the word “logos”. This initially meant something like “word” or “speech”.  In the context of speech, however, the scope of meaning expanded to include reasonable speech and that which makes a speech intelligible and convincing: Conclusions which prove to be “necessary”, irrefutable, i.e. which cannot be rejected, if one does not want to be regarded as unreasonable. “logos” thus stood for understanding and reason, and one discovered that one could find rules for such reasonable reasoning.

Of course, this development could only exist in a community in which much had to be said and discussed in public. This was the case in the early democracies of ancient Greece. Rhetorical arts and dialectical skills offered a great advantage in court and in political assemblies, and so-called sophists taught these arts. At some point one had to realize that argumentations could be incontestable, others not; yet others had to be seen as attempts at persuasion or as truly misleading.

Thus “logos” initially stood for logically respected thinking and the doctrine for this thinking became “logic”.

But the term “logos” itself made a career as well, with the second question mentioned above, to what extent we as humans can have access to the “higher level”. Of course, the “logos” had to play a role, understood as reason or cognitive ability. What role exactly this should play differs from the various philosophical schools that had emerged in the course of history. But the “logos” became the cause of a cognitive capacity, could even become a being of its own, a mediator between the two levels and later also played an important role in Christianity. But that is another story.

On this occasion it should only be mentioned that the role of the “logos” as mediator between two levels is also today found in the concept of “logistics”, a field of knowledge in which it is about the transport of goods, people and information. And it is no coincidence that a large German logistics company is called “Hermes”, after the messenger of the gods who, in Greek mythology, passed on the decisions of Zeus, the father of the gods.

Looking back and with this term “logos” in our luggage, we can say that with the pre-Socratics the logos came into a world in which until then myth had explained the world to mankind. This new line of thought was to become stronger and stronger in the course of the intellectual history, and on many further questions it also competed with the myth. This led to a division of philosophy into two fundamentally different schools of thought: logos versus myth. The logos only gained ground in thrusts, with long periods of strong predominance of myth in between. Today – in the globalized world as well as within individual cultures – we are dealing with a variety of intellectual situations in which the relationship between logos and myth is very different, and it is not clear which relationship could be particularly beneficial for humanity.

I will not report here on the struggle between myth and logos for supremacy in people’s minds, however exciting it may be. We will focus here on logic, which in another way is also very exciting and will lead us to the most modern developments in physics, artificial intelligence and cognitive science.


Therefore, “logic” is a knowledge of how to use reason, an answer to the question of what rigour in thinking actually means. Aristotle and representatives of the Stoa took the first steps to answer this question. They discovered in the first place that there are rules of inference in argumentation and that they can be very different in terms of convincing power. In particular, they discussed how to construct irrefutable rules of inference.

For over 2,000 years it was believed that there was no need to add anything to these logical doctrine until the philosophers and mathematicians Frege, Whitehead and Russel developed a mathematical logic at the end of the 19th century which was more general and meaningful, and which seemed to have been made just for digitalisation. Not to forget, however, Gottfried Wilhelm Leibniz, who in the 17th century had already foreseen important aspects of modern logic such as the necessity of a formal language and the meaning of a calculus at the level of this language.

With modern mathematical logic, logic has become a science in its own right.  Today is pursued at faculties of mathematics or computer science. It has, in a sense, emigrated from philosophy, as mathematics did already in ancient Greece.

Logic and Mathematics

Logic had actually been in people’s minds even before the pre-Socratics, in the form of a rudimentary form of mathematics. In the early communities there was already trade and administration, so one had to be able to count, measure, add, multiply and divide, and one will have already known that this kind of thinking leads to indisputable results. In my book ” Denken in Strukturen und seine Geschichte – Von der Kraft des mathematischen Beweises” (“Thinking in Structures and its History – The Power of Mathematical Proof”) I reported, among other things, on such first mathematical approaches in early Egypt and Mesopotamia and also on how this mathematics also reached Greek culture through trade.

The Greeks then brought mathematics to its first bloom. The preoccupation with mathematics and above all the discovery of mathematical proof may have inspired Aristotle to develop logic. Only much later should one understand the connection between logic and mathematics.

Mathematics and Physics in modern Times

During the Renaissance, when the scholars of Europe got to know and receive the culture and science of the ancient Greeks, the logos entered a phase of accelerated growth. Galileo Galilei discovered a “new science” in which mathematics, including logic, was the language of communication. He demonstrated the power of this new science in a phenomenon that is highly unimpressive for us today, the fall of a sphere. Two generations later, Isaac Newton developed a general theory of motion within the framework of this new science and also showed how motions in the sky and on earth can be calculated and reliably predicted under the influence of gravity. Modern physics had emerged.

In the following centuries new fundamental forces of physics were to be discovered and theories based on Newton’s theory were to be established for their effects. The idea of a unified theory always stood in the foreground, and indeed it was often possible to find a common, superordinate theory for two theories. Today we believe we know all the fundamental forces and are on the verge of a final step towards unification. The goal is a single theory for all fundamental forces. Then the programm of the pre-Socratics would have come to a conclusion in a way, however in a way that is quite different from what one could have imagined at the time.

I will deal with this important topic of our intellectual history in more detail in future blog posts.