6 ERFLETT A. BISHOP clear. We represent the terms of the sequence by vertical marks marching to the right, but remaining to the left of the bound B. The classical intuition is that the sequence gets cramped, because there are infinitely many terms, but only a finite amount of space a ~ i l a b l e to the left of B. Thus it has to pile up somewhere. That somewhere is its limit L. The constructivist grants that some sequences behave in precisely this way. I call those sequences stupid. Let me tell you what a smart sequence will do. It will pretend to be stupid, piling up at a limit (in reality a false limit) L f ' Then when you have been convinced that it really is piling up at Lf , it will take a jump and land somewhere to the right! Let us postpone a serious discussion of this example until we have discussed the constructive real number system. The point I wish to make now is that under neither interpretation will there be any disagreement as to the truth of (*), once that interpretation has been fixed and made precise. Whenever a student asks me whether a proof he has given is correct, before answering his question I try to discover his concept of what constitutes a proof. Then I tell him my own concept, (C) above, and ask him whether he finds his argument completely convincing, and whether he thinks he has ex- pressed himself clearly enough so that other informed and intelligent people will also be completely convinced. Clearly it is impossible to accept (C) without accepting (B), because it doesn't make sense to be convinced that something is true unless you know what it means. The question often arises, whether a constructivist would accept a non- constructive proof of a numerical result involving no existential quantifiers, such as Goldbachls conjecture or Fermatts last theorem. My answer is sup- plied by (C): I would want to examine the proof to see whether I found it com- pletely convincing. Perhaps one should keep an open mind, but I find it hard to believe that I would find any proof that relied on the principle of the
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