Do numbers exist? What a crazy question for a computer scientist to consider! I admit that my curiosity on this topic is not due to any application in Computer Science that I can see. Reading Feferman’s In the Light of Logic, though, has gotten me interested in some of the philosophical questions that some logicians — Feferman and Goedel are two notable examples — have devoted considerable attention to. This question regarding the ontological status of the objects of mathematics is considered by many very serious philosophers, certainly in the 20th century and no doubt before (though I know little about the history of the question). I think it is quite interesting to consider, since it sheds light on exactly what is happening when we do theoretical work.
The most straightforward answer to the question “Do numbers exist?” is simply, “yes (of course!)”. Numbers and more complex mathematical objects are indispensable to scientific practice, which W. V. Quine famously argued justified our acceptance (at least in some sense) of the existence of whatever objects mathematics requires, at least mathematics that is needed for scientific practice. But as Feferman argues in Chapter 14 of In the Light of Logic (and also in earlier chapters), the logical theories required in principle for the development of all the mathematics needed for modern science are quite weak in their ontological commitments. Indeed, Feferman puts forward theory W, a weak system of recursive operations on the natural numbers, and predicatively defined classes of such entities, and argues that it is sufficient for the needs of all of modern science. And theory W is a conservative extension of Peano Arithmetic, and proof-theoretically reducible to it. So enormously complex cardinal numbers, for example, need not be assumed to exist in order to carry out scientifically relevant mathematics. Thus if we follow Quine’s idea that we should accept mathematical objects that are needed for science, we need not accept such things. In this way Feferman rejects a strongly Platonist conception of mathematics, where the entire menagerie of higher set theory is accepted as actually existing.
But all this proof-theoretic fanciness is, in a sense, a distraction. After all, on Feferman’s approach we are still left with Peano Arithmetic at least. So just seeing that we do not need the more esoteric mathematical objects to do modern science does not let us avoid the question of whether or not (or how) numbers exist. And indeed, I personally am not motivated by Quinean concerns, which are ultimately rooted in naturalism. And naturalism, I suspect, ultimately represents a desire to banish the concept of God or other religious beliefs from intellectual discourse and indeed, from human life. There are very weighty reasons for thinking that that is a bad idea, which I will not discuss here: that topic is, of course, even more controversial than the existence of mathematical objects!
So: do numbers exist? Really I think we should consider two separate questions: do theoretical objects exist, and do abstract objects exist. While I have my doubts, it seems quite reasonable to believe that, at least in some sense, at least rather small finite numbers could be said to exist. I actually am open to persuasion on that point, but when I say the number of books on my desk is currently 4, maybe it is not just a figure of speech, but an actual statement about this thing called a number. I don’t know. But the question of whether or not theoretical objects exist — that is, objects that a particular theory is committed to — this seems clearer. Why on earth should we believe that just because a particular mathematical theory speaks of certain objects, that they definitely exist? I can easily write down a set of first-order formulas giving certain characteristics of unicorns, or ghosts, or whatever else you like. We would certainly not wish to say that because such a theory is committed to the existence of those things, they really exist. They might exist, of course. I could have a theory of cars, or other generally noncontroversially existing things. But just having a theory about them does not tell us anything about whether or not they exist. We would need some independent reason for believing in their existence.
Of course, in mathematics or other formal disciplines (Computer Science, too), people spend a lot of time working with the particular entities the theory considers. In Computer Science, one might work with Turing Machines, which are, of course, idealized computers with an infinite amount of memory. No computer has or perhaps physically could have an infinite amount of memory, so in that sense, Turing Machines do not exist. Working within a theory, it is natural to talk and act as though the theory were definitely true.
Perhaps psychologically, one must, in a sense, enter the world of the theory to be successful at deriving new theorems or reaching new insights into the theory. Indeed, this idea of entering the world of the theory suggests a modal interpretation of theoretical truth. A version of this interpretation is proposed by Hilary Putnam in “Mathematics without Foundations”, which can be found in his book Mathematics, Matter, and Method. The idea roughly is to say that when we assert a theorem of arithmetic, for example, as true, what we are really asserting is the necessary truth of the statement that the axioms of arithmetic imply the theorem. So any structure which satisfies the axioms of arithmetic will also satisfy the theorem. With this kind of perspective, “modalism” (which amusingly, Putnam says explicitly that he is not trying to start as an alternative foundational philosophy, though certainly it by now is) is a kind of structuralism, which asserts that structures exist, even if mathematical objects are identified only by their roles in such structures.
To me, structuralism’s structures are no better than the original mathematical objects. I would take a more proof-oriented version of modalism: to say that a theorem of a particular theory is true, is just to assert that it is provable from the axioms of the theory, using whatever logical axioms and rules one accepts. There is no ontological commitment at all there. Yes, we could understand this deductivist interpretation modally: if a structure exists that satisfies the axioms of the theory, then that structure will satisfy the theorem. But if we look at it that way, I think we must avoid phrasing this as above: “it is necessary that the axioms imply the theorem”. For this interpretation requires us to assume also that the axioms could be satisfied. If they absolutely could not be satisfied, then all formulas become theorems: if no structure models the axioms, then in every structure, the axioms are false, and the implication “axioms imply theorem” is true. I would not want to ground that modality in possibly existing structures — we’d just be back to where we started, because we would need a theory for those, and we’d be unsure of their status. It would be more acceptable to ground the modality in actually existing structures. So then the modal interpretation would amount to saying that for any structure that exists, if it satisfies the axioms, then it satisfies the theorem. But then arithmetic would become inconsistent if the universe is finite, for example. So we would not want to take this kind of more semantic modalism. We just need a more modal notion of implication, which we can find in constructive logic: if we were to enter a world where the axioms are true (regardless of whether or not there is such a world), then the theorem would be true. This has a bit of the air of the fictionalism I read attributed to Hartry Field. More simply, the theorem follows by the rules of logic from the axioms of the theory, and that is the end of the story from an ontological perspective.
Why one theory is of more use or interest than another is then a wholly separate question from whether or not use of the theory requires acceptance of the existence of the theory’s objects. Asserting a theorem, no matter how vivid the intuition for a world satisfying the axioms of a theory, is ultimately no more than asserting that the theorem follows deductively from those axioms.
Ok, after getting my attempts to learn some of this philosophy out of my system, I’ll be back to more technical topics next post.