Mathematical science being a tautology inquiry by Poincare

Question

I was reading a quote from Poincare from his "On the Nature of Mathematical Reasoning" that states:

THE very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour 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?

I don't understand this statement.
I think it should mean something like:
$A \implies B \space \And B \implies A$
but I don't really understand how that is logically meaningful because
$A \implies B$ means that $B$ is a form of $A$ in "disguise" but the direction of the arrow can not be reversed i.e. $B \not\implies A$

So what is the actual meaning of the statement about tautology here?

Answer 1

Alongside his well-known mathematical and scientific personality, Poincaré was a philosopher of mathematics and science with shrewd observations. The chapter the quote is from, On the Nature of Mathematical Reasoning, focuses, though not exclusively, on a thought called logicism in philosophy. Logicism is the collective name of several views considerably differing from each other, but none of them claims a naive continuity from logic to mathematics. Carnap gives us a concise overview of what it comprises (see "the Logicist Foundations of Mathematics" in Philosophy of Mathematics: Selected Readings, edited by P. Benacerraf and H. Putnam (Cambridge University Press, 1983, p. 41):

We will split the logicist thesis into two parts for separate discussion:

1. The concepts of mathematics can be derived from logical concepts through explicit definitions.

2. The theorems of mathematics can be derived from logical axioms through purely logical deduction.

Logicism is recurrent theme that actually acts as a motivational force for the advancement of logic. For the interested, Tennant tidies up its past and current course in his article Logicism and Neologicism. I shall try to spell out further the concept of tautology, since the present issues hinge on it. It would be an unfair anachronistic expectation from Poincaré to take it up in comprehensiveness it has gained, specifically through Wittgenstein's elaboration, since then.

In propositional calculus, a proposition is said to be a tautology if it is true under all possible assignments of truth-values (a rare, but a nice counterpart for a formula of predicate logic is 'tautologically valid'; similarly derived phrases are '$\phi$ tautologically implies $\psi$' and '$\phi$ is a tautological consequence of $\psi$'). That is the narrow definition. There is also a broader definition, handed from the Greek root word, tautologia, which means "repetition of what has been said; telling the same idea again in different words." Mostly, it was and still is a pejorative expression, for it has been regarded as useless. Poincaré seems to refer to both narrow and broad definitions, when he asks ". . ., how is it that mathematics is not reduced to a gigantic tautology?".

We see that tautology is essentially a semantic notion of logic. The compound proposition $$((P\rightarrow Q)\rightarrow P)\rightarrow P$$ has got the sense of tautology, because, in modal terminology, it turns out to be true in all possible worlds. Being thus, a tautology seems to carry no information, that is, no semantic significance that could make a difference. As Wittgenstein states in his Tractatus Logico-Philosophicus

4.461 Propositions show what they say: tautologies and contradictions show that they say nothing.

A tautology has no truth-conditions, since it is unconditionally true: and the contradiction is true on no condition.

Tautologies and contradictions lack sense.

(Like a point from which two arrows go out in opposite directions to one another.)

(For example, I know nothing about the weather when I know that it is either raining or not raining.)

Obviously, there is an inherent circularity in tautology (I shall use the term in generical sense). However, this circularity is hardly vicious. Being congruous with all possible states of affairs, it manifests a fundamental characteristic of the world:

4.4611 Tautologies and contradictions are not, however, nonsensical. They are part of the symbolism, much as '$0$' is part of the symbolism of arithmetic.

I would give the identity element (just for the sake of explanation, not to suggest a correspondence) in general as an example. Consider

$$5.1 = 5$$

The unity is there with all numbers and seems ineffective. However, it is an outstanding element and informative about the system in which it occurs. In a similar vein, tautology points to a way that the world is. Suppose $$P\vee\neg P$$ is not a tautology. Then, it would be a different view of the world, or, if one is strongly realist about the correspondence between logic and world, it would point to a different world than the one in which $P\vee\neg P$ is a tautology.

Every proposition is to imply tautology, hence, tautology is an invariant of the system (world, etc.). We know from mathematics and science that invariants tell us so many things about the matters they are related. It is no less for tautology. Being logical truths, they signify the principles of our logical framework.

Here, we can return to Poincaré. We may restate what he argues as that there is no mystery about conformance of mathematics to logic, just as jurisprudence or biology do, but it is wrongheaded to assert that there is genealogical connection mathematics bears to logic, for mathematics offers too much creative and rich content than that could be encompassed by a logical system.