Trevor

I would like to know the philosophical explanation of what is a number.

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A very fascinating and difficult question, and I like the way you put it: the "explanation" of what is a
number. Now, in logic, we might take Russell's (in the Principia) definition, which is, I believe: "the
class of all classes similar to the given class". If you think about it, he's saying that if we abstract from
everything that is similar to groups of, say, two members, we get the number two. Of course the
sticker here is the word "similar", which you have to define in terms of numerical correspondence. I've
never seen, really, how you get out of a kind of circularity there, unless you're just concerned with a
concatenation, in effect, of particular numbers. But enough of that.

I actually do not think that theexplanationof number is a philosophical question, but a cognitive, i.e., a
psychological, one. That is, we can show, for example, that animals are aware of numerical
differences between groups of things, up to two or three objects. Are they therefore aware of
"number"? Well, no. But it's a start. George Gamov wrote a book a while back titled "One, two, three,
infinity" in which he talks about our human comprehension of number, and that we cannot actually
comprehend more than three entities simultaneously, although we can talk about any number, in the
abstract. If you're really interested in this issue, that's a good place to start. In the paper, "The magical
number seven", Miller gets this up to 5-7 objects, but recently, Nelson Cowan seems to show that it
actually is three.

So the question then becomes, what is the nature of the abstraction process by which wesymbolize
entities that we do not fully comprehend? That is, how is it that we can speak of, write down, and
meaningfully manipulate numbers like 10003 and 10011, which we cannot possibly understand as
totalities in the way that wecanunderstand the numbers 2 and 3? No one knows. This is the general
problem of the nature of the abstractive and symbolic processes, and the nearest we've come to
solving them is knowing the neural basis for some types of sensory abstractions. That is, wedoknow
how it is we abstract rectangles, for example, from lines meeting at various angles on the retina. Now,
this isnotthe same as knowing how weseethose as rectangles, mind you, merely how (to a
reasonable extent — we don't know all there is to know about it by any means) the CNS codes them
as neural firings. If you're interested, there's amassiveliterature on that; any intro cognitive science
book will start you on it ("The mind's new science" by Gardner is good).

When you think about it, whydowe have any concept of number at all; why not just be aware of
differences between groups with different numbers? I mean, we're aware of separate colors, but not
as groupings, really, like numbers. We're aware of sound, in some sense, as groups, i.e., chords;
we're aware of shapes, as I said, as groups. But those latter things arethings,which we encounter as
objects, bonk ourselves on, fall over, etc.; we don't fall over 3s and 4s. An interesting question, isn't
it? We seem to have evolved this capability almost accidentally, perhaps as a consequence of our
evolution of linguistic abstractions. Lakoff and Nunez have a similar explanation in "Where
mathematics comes from", an interesting but poorly reviewed book.

So you've touched on a huge, difficult, and unsolved problem, and there are literally volumes on its
various aspects, from perceptual to linguistic to numerical.

Steven Ravett Brown