Review: Philosophy of Mathematics, ed. Paul Benacerraf & Hilary Putnam

Very early on in my study of philosophy, probably when I was still in high school or maybe freshmen year of college, I sat down to read Hobbes’ Leviathan and had a reaction that surprised me, and may surprise just about everyone else reading this: I thought Hobbes sounded really similar to John Lock in his Second Treatise, which I had read not too long before.

If you know anything about the history of philosophy, this may sound a little crazy to you. Hobbes was known as a defender of absolute monarchy, while Locke was a defender of individual rights and limited government. But the similarities are real: for example, they both approach political philosophy from a social-contract point of view. That there would be such similarities shouldn’t be surprising, as they were both Brits writing within a couple decades of each other.

For me, the take-away lesson was that philosophers are often defined not so much by the views that become associated with their names, but by the assumptions they make on their way to arriving at their views, and the questions that they think are worth having views on. This is something I was reminded of when reading an anthology on philosophy of mathematics that a friend lent to me, asking for a review.

The anthology was co-edited by Hilary Putnam, and while I’m reasonably familiar with Putnam’s work on mind and language, but I hadn’t previously encountered his work on philosophy of mathematics. Furthermore, I was reading the second edition of an anthology originally published in 1964, meaning it had an early 20th century focus, meaning the content was stuff I’d be most likely to encounter in a history of philosophy course (at least, and the two universities where I’ve studied philosophy).

And boy, was it alien territory to me. Much of the book was focused around the debate between three main views on philosophy of mathematics, intuitionism, formalism, and logicism. I understand logicism well enough, it’s basically the view that mathematics can be reduced to logic, and apparently it’s now widely accepted that the logicist project got very far but doesn’t quite work in its original form. Though regardless of whether it works in its original form, it’s always felt to me like logicism would just turn all our old questions about mathematics into questions about logic, so it’s not really a candidate for answering all our questions about mathematics. Or so it seems to me, anyways.

As for intuitionism and formalism, well, here’s a quote from L. E. J. Brouwer’s essay “Intuitionism and formalism” (included in the anthology):

On what grounds the conviction of the unsassailable exactness of mathematical laws is based has for centuries been an object of philosophical research, and two points of view may here be distinguished, intuitionism (largely French) and formalism (largely German). In many respects these two viewpoints have become more and more definitely opposed to each other; but during recent years they have reached agreement as to this, that the exact validity of mathematical laws as laws of nature is out of he question. The question where mathematical exactness does exist, is answered differently by the two sides; the intuitionist says: inthe human intellect, the formalist says: on paper. (p. 78)

I look at this, and I think that the phrasing of the question, “where does mathematical exactness exist?” looks just bizarre to me. There are lots of things in the anthology I had trouble wrapping my head around, but this one stands out especially. Just being written in the last century is, in this case, enough for the point of view to seem totally alien to me.

On the other hand, this passage from one of Putnam’s own contributions to the anthology, “Mathematics without foundations,” resonated strongly with me:

Philosophers and logicians have been so busy trying to provide mathematics with a ‘foundation’ in the past half-century that only rarely have a few timid voices dared to voice the suggestion that it does not need one. I wish here to urge with some seriousness the view of the timid voices. I don’t think mathematics is unclear; I don’t think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs ‘foundations’. The much touted problems in the philosophy of mathematics seem to me, without exception, to be problems internal to the thought of various system builders. THe systems are doubtless interesting as intellectual exercises; debate between the systems and research within the systems doubtless will and should continue; but I would like to convince you (of course I won’t, but one can always hope) that the various systems of matehmatical philosophy, without exception, need not be taken seriously.

By way of comparison, it may be salutory to consider the various ‘crises’ that philosophy has pretended to discover in the past. It is impressive to remember that at the turn of the century there was a large measure of agreement among philosophers – far more than there is now – on certain fundamentals. Virtually all philosophers were idealists of one sort or another. But even the nonidealists were in a large measure of agreement with the idealists…

Anyone maintaining at the turn of the century that the notions of ‘red’ and ‘hard’ (or, more abstractly ‘material object’) were reasonably clear notions; that redness and hardness are nondispositional properties of objects; that we see red things and see that they are red; and that of course we can imagine red objects, know what a red object is, etc., would have seemed unutterably foolish. After all, the most brilliant philosophers in the world all found difficulties with these notions. Clearly, the man [as in 'the man on the street' - Hallquist] is just too stupid to see the difficulties. Yet today this ‘stupid’ view is the view of many sophisticated philosophers, and the increasingly prevalent opinion is that it was the arguments purporting to show a contradiction in the view, and not the view itself, that were profoundly wrong. Moral: not everything that passes – in philosophy anyway – as a difficulty with a concept is one. And second moral: the fact that philosophers all agree that a notion is ‘unclear’ doesn’t mean that it is unclear. (pp. 295-296)

  • MNb

    “the take-away lesson”

    is also that philosophers are a product of their time and place, ie their social circumstances (Russell).

  • Richard_Wein

    Interesting. Putnam’s point about foundations resonates with my Wittgensteinian views. But I have a reservation with regard to clarity. Mathematics is clear in the sense that I don’t need any philosophy of mathematics in order to do mathematics (at least not at my level of mathematics, but I hesitate to speak for professional mathematicians). However, useful or not, philosophers are in the business of understanding ordinary things like mathematics in some deeper sense. As someone who takes a naturalised approach to philosophy, I would say that what I’m interested in is understanding what’s going on when we do mathematics. That’s deliberately vague, but hopefully it suggests a different emphasis from the more traditional philosophical one(s). Rather a lot of philosophers seem to be concerned with (to me) meaningless questions like whether numbers are objects, and whether they exist. I agree that the question, “where does mathematical exactness exist?” is weird. But I think it’s meaningful and interesting to ask why mathematics is exact in a way that most language is not.

    Similarly, the notions (words) “red” and “hard” are reasonably clear in the sense that we ordinarily use them without difficulty. They’re clear relative to some other words that we have more difficulty using. I suppose that makes them clear in the ordinary way that words are clear. On the other hand, maybe we don’t just want to use these words. Maybe (since we’re doing philosophy) we want to understand what’s going on when we use them. And that’s not quite so clear.

    I think that understanding what’s going on when we use language is not just of philosophical interest. It can help deflate a lot of pointless arguments that are mired in semantic confusion, especially in philosophy, but not only in philosophy. That said, “red” and “hard” are words which are not likely to cause such confusion, and perhaps that was Putnam’s point.

    (I don’t understand Putnam’s point about “non-dispositional properties”. The philosophical concept of dispositions was new to me, and after looking it up, I’m still not much the wiser.)

  • One Brow

    I’m surprised there’s nothing about mathematical realism in there.

    My amateurish take on the issue:

    Formalism treats mathematics like a game. If the rules are inconvenient, change the rules and see what happens. If the rules are working, don’t change them.

    Intuitionism treat mathematics as a construction to explain reality and experiences. You need to make changes to reflect reality differently, but there is some underlying reality and thus some mathematical models have something closer to truth in them.

    Realism is the notion mathematical concepts like “0″ and “set” represent some actual properties of reality, and that we are discovering mathematics, as opposed to inventing it.

    I’ve heard at least one mathematician (I think more than one) say that most mathematicians claim to be realists but publish works as if they were formalists.

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