**Note:**Edited to correct some egregious mistakes in grammar.

Here is Steven Landburg arguing that if we believe rocks exist, then we should also believe that mathematical objects exist. After all, rocks are mostly empty spaces.

Now here’s what genuinely baffles me: Apparently there are people in this world (and even, occasionally, in the comments section of this blog) who haven’t the slightest doubt about the existence of rocks, galaxies, squirrels, and the rest of the physical universe, but who suddenly turn into hardcore skeptics re the existence of mathematical objects like the natural numbers.and also

Why on earth would, say, a scientist, commit to the belief that there’s a physical universe out there but not a mathematical one, when we know that our perceptions of the physical universe demand constant revision, whereas our perceptions of the mathematical universe are largely eternal. My conception of the natural numbers is very close to Euclid’s; my conception of an atom bears almost no resemblance to Demosthenes’s.He then goes on to discuss what this implies for the existence of God. Now, Steven Landsburg is far more intelligent than I am, and also incomparably more knowledgeable about Mathematics. In this matter, though, I think he is wrong.

I really doubt it is true that his "conception of the natural numbers is very close to Euclid's." We now think of the natural numbers as being embedded in the real number continuum, a concept which Euclid would have found incoherent or even abhorrent. Why stop at the real numbers? We've gone and created Non-standard Analysis, with its infinitesimal and infinite numbers. Whether this changes our conception of the natural numbers depends on what you mean by "conception." We can now subtract any natural number from any other, which Euclid would have said is nonsense. I think that is a big change in our conception of the natural numbers. And we can divide any natural number by any non-zero natural number, and we believe the result is meaningful. Again, a big change in our conception of the natural numbers.

None of this, however, is my point. I just think that the very word "exists" is too baggy and commodious to be useful by itself. When we say something exists, we should say (or it should be commonly agreed) in what sense we mean to use the word. Rocks exist in the sense that when I kick one with my foot, it

*hurts*. Numbers exist in the sense that I can use to them to count and measure, and they form a coherent system. Words are conventional things, and we are being lazy when we simply assert that something "Does exist!" "Doesn't!"

When I was in college, I went to visit a friend of a friend, and he was reading a book or article titled "Do holes exist?" I don't remember anything about it except the title, and the impression it made on me: "is this what philosophers spend their time on?" I guess it must have been one of these. These days, I wouldn't be so dismissive: to remind us that meaning is often a matter of convention is a valuable service.

I think Reuben Hersh may have thought much more deeply about these matters than Landsburg. I would love to see a debate between them.

## 3 comments:

Basically Quine, "On What there Is" argued for the same position that you argue here. What matters are the conditions under which we can make sense that something exists. Though I must say I'm with Landsburg here - these conditions are broader for numbers than for rocks.

There is also this interesting paper by physicist Max Tegmark: http://arxiv.org/abs/0704.0646. The university is most of all a computable mathematical object!

Basically Quine, "On What there Is" argued for the same position that you argue here. What matters are the conditions under which we can make sense that something exists. Though I must say I'm with Landsburg here - these conditions are broader for numbers than for rocks.

There is also this interesting paper by physicist Max Tegmark: http://arxiv.org/abs/0704.0646. The university is most of all a computable mathematical object!

I've got to read Quine. However, re rocks vs numbers, I think rocks are actually *more* real. If that makes sense.

I used to be sort of platonist, but Hersh changed that. The way I see it now is that my cupboard contains 23 socks, 8 T-shirts, 2 short-sleeved shirts, 10 red clothes (some shirts, some socks), 3 items made of linen, 2 key-rings, 10 keys, etc. I mean: there are things in there which can be counted, but it requires an intelligent (if I can be called that) creature to do the counting, and so bring numbers into existence.

Hersh is very good on this: do give his "What is Mathematics, Really?" a try.

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