The Nature Of Mathematical Reasoningby H. Poincare
1. Number And Magnitude
Mathematical science is an insoluble contradiction.
- If math is deductive, then where does its perfect rigour come from?
- If all the propositions of math can be derived by formal logic, why is math not a gigantic tautology?
If everything springs from the principle of identity, then everything should be capable of being reduced to the principle of identity.
- If follows that all the theorems merely mean that A is A.
- We may classify mathematical theorems as à priori synthetic views [judgements made from active thought confined within the mind].
But this is not a solution.
- It is merely giving it a name.
- The contradiction still remains.
No theorem can be new unless a new axiom intervenes in its demonstration.
Reasoning can only give us immediately evident truths borrowed from direct intuition.
- Reasoning would only be an intermediary parasite.
The contradiction between reason and intuition will strike us the more if we open any mathematics book.
- On every page, the author announces his intention of generalising some proposition already known.
If the mathematical method goes from the particular to the general, then how can it be called deductive?
Finally, if the science of number were merely analytical, or could be analytically derived from a few synthetic intuitions, it means that:
- a sufficiently powerful mind could with a single glance perceive all its truths.
- some day, a language would be invented simple enough for these truths to be made evident to any person of ordinary intelligence.
Mathematical reasoning has of itself a kind of creative virtue which distinguishes it from the syllogism.
All these modes of reasoning, whether or not reducible to the syllogism:
- retain the analytical character
- lose their power.
2. Mathematical Reasoning
Let us see how Leibnitz tried to show that
2 + 2 = 4.
I assume the number one to be defined, and also the operation x + 1
i.e., the adding of unity to a given number x. These definitions do not enter into the subsequent reasoning.
I next define the numbers 2, 3, 4 by the equalities
(1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4;
In the same way I define the operation x + 2 by the relation
(4) x + 2 = (x + 1) + 1.
Thus, we have
2 + 2 = (2 + 1) + 1, (def. 4);
(2 + 1) + 1 = 3 + 1, (def. 2);
3 + 1 = 4, (def. 3);
whence 2 + 2 = 4,
This reasoning is purely analytical. But the mathematician will say that “This is not a proper demonstration. it is a verification of mathematical reasoning."
We have confined ourselves to:
- bringing together one or other of two purely conventional definitions
- verifying their identity.
Nothing new has been learned.
Verification differs from proof precisely because it is analytical, and because it leads to nothing.
It leads to nothing because the conclusion is nothing but the premisses translated into another language.
A real proof, on the other hand, is fruitful, because the conclusion is more general than the premisses.
2 + 2 = 4 can be verified because it is particular.
Each individual enunciation in mathematics may be always verified in the same way.
But if mathematics could be reduced to a series of such verifications it would not be a science.
- A chess-player does not create a science by winning a piece.
- There is no science but the science of the general.
The object of the exact sciences is to dispense with these direct verifications.