I think you're mistaken in dismissing this as a "theorem". Once when I TA'd in a course "math for elementary school teachers" taught by Leon Henkin, he went over a proof of this theorem---I thought it was pretty revealing, (although I have to say it wasn't very appropriate for the future elementary school teachers, who didn't understand the point.) But for me, after going through the proof and trying to explain it to my students, it made a lot of sense why there is something that needs proof and that can be proven, our early indoctrination to the contrary notwithstanding. In fact, there's a good proof reducing it to on more primitive intuition than counting. It's very close to the standard proof that finite-dimensional vector spaces over a given field are classified by their dimension---which few people think is obvious without proof! I.e., when you count a finite set in two different orders, why do you get the same number? You can transform one order of counting to the other step-by-step using a simple exchange. I kind of like the point of view that proofs as a dialectical construct, as in Lakatos' book "Proofs and their refutations", rather than part of a formal system (which in practice they definitely are not). Many things people accept for a time, or for ever, until they are challenged, when their beliefs can be shaken. Did you ever tabulate a big set of data by hand and find the column sums and the row sums did not reconcile? At times like that, my belief in the commutative law for addition can become shaken --- thinking back through why sums, or counts, should reconcile is a non-obvious task. Bill On Apr 28, 2006, at 5:51 PM, James Propp wrote:
Let me say, only half in jest, that my favorite theorem in math is the classification theorem for finite sets, i.e., the fact that finite sets are classified up to isomorphism by the counting numbers (including zero)!
And I'll respond, half-seriously: How would you prove that?
I wouldn't call it a theorem in the theorem-proof sense. If a proof is defined (epistemo-sociologically) as a text that makes you believe and/or understand a proposition more than you did before you read the proof (leaving aside the question of what "believe" and "understand" mean), then I don't think this theorem *has* a proof! You can undoubtedly set up a formal system in which this claim is proved by induction, but nobody who didn't already accept the claim would be able to understand the formal system, let alone follow the proof.
Instead, I am using the word "theorem" to mean any mathematical fact that makes you say "Well, will you look at that!"
And this proposition is a real looker. We all encountered it when we were kids and noticed that a set doesn't change its cardinality when its elements are shifted around, and this agreeable invariance probably fuels many kids' enthusiasm for counting (even though their knowledge of this invariance is intuitive and most of them couldn't begin to articulate it as a general proposition).
My serious point is that some mathematical facts may simultaneously be so profound and yet so commonplace that we don't recognize them as "theorems" as such.
But I think I want to change my mind about my favorite mathematical fact. How about the fact that there are infinitely many counting numbers? The realization that you can keep on counting forever and never run out of numbers (even if you run out of names for them) was such a mind-blowing experience for my young self that I retained absolutely, positively NO memory of the experience. :-) But my mind remained permanently blown, so there was nothing for me to do with my life but become a mathematician...
Is it a "theorem"? Well, I doubt that there are any formal bases for mathematics in which the axiom of infinity can be proved. Either you see it, or you don't. And once you've seen it, you can try to doubt it, but it's really hard to do.
Jim
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