Science and Hypothesis

Notes

Book is divided into four parts:

Part 1 - Number and Magnitude (Arithmetic)

Part 2 - Space (Geometry)

Part 3 - Force (Physics)

Part 4 - Nature (Engineering/Physics)

Part 1 - Number and Magnitude

The contradictions of mathematical proof

Poincaré begins the book by pointing out the contradictions at the heart of mathematics, and asking a series of challenging questions about why mathematics works, and how we can really prove anything or trust the results of proofs.

The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigor which is challenged by none? If on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology?

- Henri Poincaré, Science and Hypothesis (p. 1)

Poincaré spends Part 1 trying to resolve the tension between, on the one hand, the need to resort to direct experience, while on the other hand, the need to exclude an appeal to the senses to prove things. The way he resolves it is by introducing inductive (or what he calls "recursive") logic, and the technique of proof by induction. He says that this provides an entirely separate, third method of proving statements, one that is not simply substituting definitions but that genuinely leads to novel insight.

(Note that the book covers induction, but in a rather informal way. But we should keep in mind this lack of formality was not a product of the time in which it was written, or due to a lack of appreciation of the importance of rigor - the lack of formality is because Poincaré was writing this book for the layperson.)

Poincaré begins the discussion by sweeping much of mathematics off the table, so that he can focus on the most fundamental operations, concepts, and definitions. Accordingly, he starts with arithmetic, and starts with the "trivial" problem of proving that 2 + 2 = 4.

This "trivial" problem turns out to be quite complicated. Poincaré uses it to show the difference between verification (tautological reasoning that "leads to nothing") and proof (which reaches a conclusion more general than the premises).

Inductive Proofs

Poincaré uses an inductive proof method, which he calls "proof by recurrence", to state a hypothesis; he then demonstrates the base case to be true, and then he demonstrates that if the hypothesis is true for n, it is true for n + 1 as well.

Poincaré proves the associative and commutative properties of addition, then proves the distributive and commutative properties of multiplication, using induction.

The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinite number of syllogisms.

- Henri Poincaré, Science and Hypothesis (p. 9)

However great the number may be we shall always reach it, and the analytical verification will always be possible. But however far we went we should never reach the general theorem applicable to all numbers, which alone is the object of science. To reach it we should require an infinite number of syllogisms, and we should have to cross an abyss...

- Henri Poincaré, Science and Hypothesis (p. 10)

I asked at the outset why we cannot conceive of a mind powerful enough to see at a glance the whole body of mathematical truth. The answer is now easy. A chess-player can combine for four or five moves ahead; but, however extraordinary a player they may be, they cannot prepare for more than a finite number of moves.

- Henri Poincaré, Science and Hypothesis (p. 10)

In this domain of Arithmetic we may think ourselves very far from the infinitesimal analysis, but the idea of mathematical infinity is already playing a prepondering part, and without it there would be no science at all, because there would be nothing general.

- Henri Poincaré, Science and Hypothesis (p. 11)

Induction as a separate type of proof

Poincaré says that induction is such a special concept, it stands alone as a whole different kind of proof, different from proof by contradiction or proof through experimentation (verification?).

To put it a different way: Poincaré says that a priori synthetic intuition is infinite all at once, whereas inductive reasoning is infinite one step at a time.

...the power of the mind, which knows it can conceive of the indefinite repetition of the same act, when the act is once possible. The mind has a direct intuition of this power, and experiment can only be for it an opportunity of using it...

- Henri Poincaré, Science and Hypothesis (p. 13)

In this way, you could say that inductive reasoning is a kind of thought experiment, or Gedankenexperiment.

Wittgenstein tie-in

In Wittgenstein's book Foundations of Mathematics, Lecture 4, Wittgenstein says something that has echoes of what Poincaré is saying here.

The point is that the proposition 25 x 25 equals 625 may be true in two senses. If I calculate a weight with it, I can use it in two different ways.

First, when used as a prediction of what something will weigh - in this case it may be true or false, and is an experiential proposition. I will call it wrong if the object in question is not found to weigh 625 grams when put in the balance.

In another sense, the proposition is correct if calculation shows this - if it can be proved - if multiplication of 25 by 25 gives 625 according to certain rules.

...It is of course in the second way that we ordinarily use the statement that 25 x 25 equals 625. We make its correctness or incorrectness independent of experience. In one sense it is independent of experience, in one sense not.

Independent of experience because nothing which happens will ever make us call it false or give it up.

Dependent on experience because you wouldn't use this calculation if things were different. The proof of it is only called a proof because it gives results which are useful in experience.

- Wittgenstein, Foundations of Mathematics (Lecture 4)