Leibniz's stand on axioms and definitions

Question

Recently, while discussing, a friend claimed that Leibniz was fond of proceeding axiomatically and from definitions, which I find hard to believe. My conception is that Leibniz was more interested in intuition and the process of construction rather than endproducts. This is exemplified by these two quotes:

Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves.

This quote I found in Polya's How to Solve It.

I am convinced that the unwritten knowledge scattered among men of different callings surpasses in quantity and in importance anything we find in books, and that the greater part of our wealth has yet to be recorded.

This is from Wikiquote, alledgedly from here.

Am I wrong?

Answer 1

In a letter to Hermann Conring 1678, Leibniz writes in latin (my asterisk):

Axiomata ego non ut ais, αναπόδεικτα*, sed tamen plerumque non necessaria demonstratu esse arbitror. Demonstrabilia vero esse pro certo habeo. Unde enim constat nobis de eorum veritate? non, opinor ex inductione, ita enim omnes scientiae redderentur empiricae, ergo ex ipsismet, id est ex earum terminis: quod fit vel quando idem dicitur de seipso (ex. causa, A est A, unumquodque sibi ipsi aequale est, et similes identicae) vel quando sola terminorum significatione sive quod idem est definitione, intellecta statim apparet propositionis veritas. Omnes ergo propositiones certae demonstrari possunt praeter identicas et empiricas.

My poor attempt at an translation. The Axiomata ego non ut ais, αναπόδεικτα part:

The axioms is not as you say, αναπόδεικτα (unsubstantiated/proofless),

The letter in its entirety can be found here. 162. LEIBNIZ AN HERMANN CONRING (Latin).

I noticed that the letter in Marcelo Dascal's Leibniz: What Kind of Rationalist? is similar in some regards, that translation is as follows (page 162):

I regard axioms not, as you say, as something apodictic but only as something that in most cases does not require a proof. Yet that they are demonstrable, I believe to be certain. Whence does it come that we are certain about their truth? As I believe, not from induction, since in that way all sciences would be rendered empirical; thus, from themselves, i.e. from their terms, which happens when the same is said of the same (e.g., A is A, everything is equal to itself, and similar identical propositions) or when only from the signification of terms or, what is the same, from the understood definition the truth of the proposition is apparent.

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*I am not entirely sure this is the greek word. It looks a bit different in the text.

Answer 2

This is a long comment...

I'm not able to give a complete answer "in general".

But if we consider specifically his logical works, we can see in :

Gottfried Wilhelm Leibniz, Logical Papers : A Selection (edited by G.H.R.Parkinson, 1966), at least :

• A Specimen of the Universal Calculus (1679-86 ?), page 33-on

• Addenda to the Specimen of the Universal Calculus (1679-86 ?), page 40-on.

In the last one [page 42] we can find :

[Six] Propositions true in themselves :

(1) a is a. An animal is an animal.

<p>(2) <em>ab</em> is <em>a</em>. A rational animal is an animal.</p>

[...]

An inference true in itself : a is b and b is c, therefore a is c.

Thus, he has a clear understanding of axioms and rules in logic.

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If we instead consider his mathematical works, the "failure" to axiomatize the infinitesimal calculus must not be considered a failure : we have to consider that it takes a couple of centuries only to attain a viable axiomatization of the real number system.

Answer 3

To elaborate on Leibniz's comment that "The axioms are not, as you say, indemonstrable" (see the answer https://hsm.stackexchange.com/a/369/604): Leibniz was rather successful in choosing the appropriate axioms and/or heuristic principles for the infinitesimal calculus he developed. I would note four particularly important items.

(1) Law of Continuity. There are several formulations of this (as distinct from the Principle of Continuity). One of them is the following: The laws of the finite succeed in the infinite, and vice versa." This is described by Abraham Robinson as closely related to the Transfer Principle of infinitesimal analysis (for example, a relation such as $\sin^2 x+\cos^2 x=1$, being true for all standard inputs, would necessarily be true by the Transfer Principle for all inputs, including for example infinitesimal ones).

(2) Relation of infinite proximity. Leibniz emphasized on numerous occasions that in the calculus he worked, not with the relation of exact equality, but rather with the relation of equality up to negligible terms. This is closely related to the Standard Part principle of modern infinitesimal analysis.

(3) The dichotomy of assignable vs inassignable number. While there is clearly no construction of this anywhere in Leibniz, he postulates the existence of such a distinction, which enables him to work both with infinitesimals and (their inverses) infinite numbers.

(4) Leibniz made it clear in a pair of texts from 1695 that his infinitesimals violate the principle of comparability found in Euclid's Elements, volume 5, Definition 4 (Leibniz referred to Definition 5 following contemporary editions of Euclid). This is closely parallel to the violation of the Archimedean property in modern number systems containing infinitesimals.

Thus all in all this leads us to suggest that Leibniz was rather successful in creating the germs of an axiomatisation of infinitesimal analysis -- something that has been clearly realized only recently.