Poincaré On the Nature of Mathematical Reasoning

1894

Poincaré did not know about what we now think of as logic. When he is referring to logic as syllogisms, this is not just outmoded terminology: He is really thinking of Aristotelian logic.

Aristotelian logic is much weaker than what we think of as logic, and so Poincaré's claim that 'logic couldn't be all there is to mathematics' must be taken with a grain of salt.

Nonetheless, he emphasizes the importance for mathematics of recursive definition and proof by induction, aspects that we have heretofore ignored.

Induction and recursive definition: The term "inductive reasoning" is used by philosophers of science to refer to methods of reasoning from examples to generalizations: that ranges from the idea that if something has, so far as we know, always happened, that gives us reason to believe that it always has and always will happen to sophisticated statistical methods. It is not what Poincaré is talking about. What he is talking about is the method of proof by induction for the natural numbers:
To show that every natural number has a property $P$, it is enough to show that 1 has the property and that whenever a number $n$ has the property, then so does $n+1$.
To define a function on the natural numbers, it suffices to define it on 0, and, given the values of the function on numbers less than $n$, to define it for $n$. That is known as recursive definition.

Example: $n+0=n$
$n+(m+1)=(n+m)+1$.

Those two equations suffice to define addition.

These methods involve "loops": the same process or method is invoked over and over.
The slowest machine instruction on a computer is the floating point division instruction. If you filled the entire RAM of a typical desktop computer with nothing but copies of that instruction, the whole program would execute in a matter of seconds. So, why do computers (except those using Windows) run for weeks at time? Answer: all of that time is spent in loops, reusing procedures. Everything we do on a computer makes use of recursive definitions. Every area of mathematics involves at least a few critical recursive definitions, and mathematical logic and elementary number theory make use of nothing else.

So, whether or not logic can make a nontrivial contribution to mathematics (that is, logical reasoning from the axioms), induction is absolutely central to mathematics.

We will see that first-order logic is not adequate for many mathematical purposes and that second-order logic just builds in many assumptions that should be justified. There is an intermediate form of logic---first-order logic plus inductive definitions---that does much better.

Why is induction called "induction"? It has something in common with the scientific method of induction: it starts from information about individuals and leads to a general conclusion.

Without induction, about the only thing you can do with the numbers is particular computations and that, Poincaré says, is not mathematics any more than chess is mathematics.

That brings me to the second aspect of Poincaré's article worth emphasizing: computations and chess, he says, are not mathematics. That is in sharp contrast to the logicist, who doesn't care what mathematics is, since it is to be eliminated in favor of a logic that can handle chess and mathematics. The formalist has no reason to be any happier about a formal system of arithmetic than about a formal system of chess, though admittedly one may be more useful than the other. For the intuitionist, "mere computations" are mathematics par excellence. And so forth.

We have a clear intuitive sense that some things are "really" mathematics, others mere computation or symbol pushing, or at any rate not mathematics. (What about programming? Chess? Algorithms for computing primes or for multiplying efficiently?) Most philosophies of mathematics say nothing about this issue.

Poincaré's proposal is that what is characteristic of mathematics is the discovery of general conclusions about recursive definitions by use of induction. Whether or not that is right, the "demarcation problem"---the problem of deciding what is mathematics and what isn't---is a worthwhile one that has, for the most part, been ignored.

Mathematicians in fact have a very strict criterion for what counts as mathematics that is universally accepted. Here it is: you can make up any cockamamie theory you want, and it is legitimate mathematics if and only if it can be used to solve a problem that was posed prior to the introduction of the theory.

-- ShaughanLavine - 16 Feb 2005