Characteristic numbers

Gottfried Wilhelm Leibniz developed the idea of characteristic numbers in 1679 . He used it as a model of Aristotelian syllogistic logic and hoped to have found a general method of solving all logical problems with the help of a calculus , namely calculating with whole numbers.

The idea

In 1679 Gottfried Wilhelm Leibniz wrote a small number of unpublished manuscripts. These texts belong in the context of his ambitious project of a Calculus Universalis , which he himself describes as follows:

“If one could find characters or signs that could express all of our thoughts just as clearly and purely as arithmetic expresses numbers or analytical geometry expresses lines, then in all matters, insofar as they are accessible to rational thought, one could do what one could in arithmetic and geometry does. "

- Leibniz

“In order to establish a general calculus, characters are to be invented for any expressions from which, after they have been connected with one another, the truth of the sentences composed of the expressions can be recognized immediately. As the most comfortable characters so far, I've found the numbers. They are easy to handle and can adapt to all objects, and they also provide certainty. ( Ad calculum universalem constituendum inveniendi sunt characteres pro terminis quibusque, ex quibus postea inter se junctis statim cognosci queat propositum ex terminis conflatorum veritas. Commodissimos characterum hactenus invenio ess Numeros. Sunt enim facile tractabiles omnibusque rebus accomodarient. ) “

- Leibniz

Leibniz's idea was to use prime numbers to represent the building blocks of his logic, the simple or elementary ideas. Compound ideas or concepts should then be represented by the product of prime numbers, just as all integers can be represented as products of prime numbers according to the fundamental theorem of algebra . The following simple example comes from him:

“For example, if it is assumed that the term 'animal' is expressed by the number 2 (or generally a), the term 'rational' by the number 3 (or generally r), then the term 'human' is expressed by the number Expressed 2 × 3, i.e. 6 as the result of multiplying 2 and 3 (or in general by the number a × r). ( Exempli causa, si fingeretur terminus animalis exprimi per numerum aliquem 2 (vel generalis a) terminus rationalis per numerum 3 (vel generalis r) terminus hominus exprimetur per numerum 2 × 3, id est 6, seu productum ex multiplicatis in vicem 2 et 3 (vel generalius per numerum a × r) ) "

- Leibniz

After some effort, Leibniz had to realize that he could not realize his idea with this simple method (the problems that arise are described in detail in). Leibniz solved these problems by moving from the representation by whole numbers to the representation by number pairs.

The number pairs

As with his simple approach outlined above, Leibniz assumes that there are simple, non-compound ideas or terms that can be used to compound all other terms. These basic concepts are again assigned to the prime numbers; d. H. each basic concept corresponds exactly to one prime number.

Compound concepts are determined by specifying which elementary concepts are contained in them ("positive" basic concepts) and which basic concepts are not contained in them ("negative" basic concepts).

For a given composite concept, let the product of those prime numbers that belong to the "positive" concepts and be the product of those prime numbers that belong to the "negative" concepts. Then you assign the pair of numbers to the term : ${\ displaystyle S}$ ${\ displaystyle s}$ ${\ displaystyle \ sigma}$ ${\ displaystyle S}$ ${\ displaystyle (s, \ sigma)}$

{\ displaystyle S \ rightarrow (s, \ sigma)}

.

Leibniz himself, however, did not use the modern spelling as a pair of numbers, but a different one: instead of he wrote . ${\ displaystyle (s, \ sigma)}$ ${\ displaystyle + s- \ sigma}$

The judgments

The aim of mapping concepts to numbers (pairs) was to interpret the categorical forms of judgment that go back to Aristotle using arithmetic formulas. Leibniz succeeds in doing this as follows.

Let us assume that the subject S and the predicate P are assigned the characteristic number pairs and respectively . Then the categorical judgments are interpreted as follows: ${\ displaystyle (s, \ sigma)}$ ${\ displaystyle (p, \ pi)}$

judgment	designation	interpretation
"All S are P"	A judgment	s divides p and divides ${\ displaystyle \ sigma}$ ${\ displaystyle \ pi}$
"No S is P"	E judgment	gcd or gcd ${\ displaystyle (s, \ pi)> 1}$ ${\ displaystyle (p, \ sigma)> 1}$
"Some S are P"	I judgment	gcd = gcd ${\ displaystyle (s, \ pi)}$ ${\ displaystyle (p, \ sigma) = 1}$
"Some S are not P"	O judgment	s does not divide or divide p ${\ displaystyle \ sigma}$ ${\ displaystyle \ pi}$

As usual, GCF (a, b) denotes the greatest common divisor of the whole numbers a and b.

Inferences

The laws of Aristotelian logic can be understood from the arithmetic interpretations of the judgments. Note that in the number pairs the numbers s and are always prime (as products of different prime numbers). ${\ displaystyle (s, \ sigma)}$ ${\ displaystyle \ sigma}$

Example From A (S, P) follows I (S, P).

Proof: Assume that all prime factors contained in s are also contained in p, and all prime factors contained in are also contained in . If one of the prime factors of s, say m, were contained in, then p and this prime number would have as a common factor, which is excluded according to the definition of the number pairs. In the same way, the assumption that p and have a common factor leads to the contradiction . ${\ displaystyle \ sigma}$ ${\ displaystyle \ pi}$ ${\ displaystyle \ pi}$ ${\ displaystyle \ pi}$ ${\ displaystyle \ sigma}$

Similarly, Leibniz confirmed all the other fundamental laws of Aristotelian logic (especially the syllogisms) in his arithmetic interpretation.

With his arithmetic interpretation Leibniz succeeded in setting up an intensional model of the logic of Aristotle. This model gained importance through the Polish logician Jan Łukasiewicz , who used Leibniz's characteristic numbers in his standard work to prove the completeness of his axiom system of Aristotelian logic.

Individual evidence

^ Translation: Fragments for logic, p. 90
↑ Academy edition, 6th series, 4th volume, p. 217
↑ Academy edition, 6th series, 4th volume, p. 182
^ Klaus Glashoff , On Leibniz 'characteristic numbers, Studia Leibnitiana Volume 43/2002, page 161
↑ Jan Łukasiewicz: Aristotle's syllogistic. From the standpoint of modern formal logic . Oxford: Clarendon Press, 1951.

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Gottfried Wilhelm Leibniz: “Complete Writings and Letters”, Academy Edition, 6th series, 4th volume, Part A, manuscripts No. 56–64 Berlin 1999
Fragments of logic. Selected, translated and explained by Dr. phil. habil. Franz Schmidt. Akademieverlag, Berlin 1960

[1] Translation: Fragments for logic, p. 90

[2] Academy edition, 6th series, 4th volume, p. 217

[3] Academy edition, 6th series, 4th volume, p. 182

[4] Klaus Glashoff , On Leibniz 'characteristic numbers, Studia Leibnitiana Volume 43/2002, page 161

[5] Jan Łukasiewicz: Aristotle's syllogistic. From the standpoint of modern formal logic . Oxford: Clarendon Press, 1951.