Curry's Philosophy:

Formalism

This article is out of sequence: he assumes you know what Hilbert's Program is and what (Brouwer's) intuitionism is. We'll read those next, but Curry is so much more straightforward that even his off-hand remarks serve as useful orientation.

In addition, of all the reactions to the failure of logicism, Curry's formalism is the most straightforward.

The view should just be called formalism, which is what Curry calls it. But, unfortunately, Hilbert's view is called formalism, though the name applies better to Curry's view. Hilbert got there first, and so we're stuck.

I will always refer to views like that of Curry (even slightly different views held by others) as Curry's formalism, and I will avoid calling Hilbert's philosophy of mathematics formalism.

You need to be aware in reading the works of others that the term "formalism" is ambiguous. You need to check.

Curry calls certain views realism, Platonism, and so forth. His terminology is completely wrong. (He didn't make any mistakes, things just haven't gone his way terminologically.)

There is an obvious response to the problem with the logicist view; a response that was at one time advocated by Curry:

Russell, following Peano, and Frege, on his own, developed complete formal notations in which all of mathematics can be represented and all mathematical proofs carried out. They thought the notation was logical, and that didn't work out so well. However, we still have the notation, which is given quasi-concretely and precisely. So, there is an obvious proposal: give up on logic and retreat to the symbol system. What arithmetic is about is Peano's axioms. A "proof" is just a sequence of symbols obeying certain rules. A "theorem" is just the last line of a "proof."

This view was roundly criticized for many reasons:

  1. Mathematics is, on this view, just a game of symbol manipulation. It expresses no propositions, and so it doesn't even include any candidates for mathematical truth. "2+2=4" is, on this view, more like a legal position in a game than it is like a true statement. The short version of the criticism is that the view fails to distinguish mathematics from chess.
  2. The Löwenheim-Skolem theorem, which says that every theory that has an infinite model has infinitely many distinct infinite models, shows that, for example, the Peano axioms don't characterize the natural numbers, and so aren't really an adequate ersatz for them.
  3. Gödel's incompleteness theorem shows that there are truths about Peano arithmetic, even particular truths we can get our hands on, that don't follow from Peano's axioms. No other axiom system does any better.
  4. We are left with no way to pick out which axiom systems are interesting.
  5. Since mathematics expresses no propositions, we cannot explain how it can be applicable.

Curry's view in the paper we've read, empirical formalism, is intended to fix these problems: On the new view, mathematics isn't the use of formal systems, but the study of them, often from outside.

This way, we do have a formalism that acts like the natural numbers:

$|+\_=|$
$x+y|=(x+y)|$
$x|=y|\rightarrow x=y$

Consequences:
$||+||=||||$
$x+y=y+x$

This system is too weak to serve as a replacement for number theory, but number theory can be construed as the metatheory of this system.

This view has its attractions, and one can still find some mathematicians who, on Sundays, claim to subscribe to it.

What I take to be the knockdown criticism of this view is that the objects of study are now formal systems, and those are mathematical objects that are fully abstract, even more abstract, for example, than the number 2.

The point is not that the view is incoherent, but that it doesn't have the claimed advantages. The criticism is Quine's generic criticism of all attempts at nominalism: the elimination of abstract objects.

Now I want to stop talking about Curry's view and talk about his analysis of the landscape of views.

He divides philosophies of mathematics into "realism," "idealism," and "formalism." Since his use of these terms is completely nonstandard, I'll try to always type the scare quotes.

-"Idealism" is the view that mathematics deals with "mental objects" of some sort. This includes "Platonism" and "intuitionism." Platonism is just realism about mathematical objects: they exist and mathematics tells us the facts about them, in rather precise analogy to, say, biology. According to Platonists, mathematical objects are most certainly not mental. According to intuitionists, mathematics is not about objects at all, it is about possibilities for mental constructions. Thus, Curry's "idealism" is best construed as the view that mathematics concerns something nonphysical or perhaps something abstract.

-- ProfessorShaughanLavine? - 24 Jan 2005 -- ShaughanLavine - 24 Jan 2007