From owner-qed Wed Nov 9 09:57:03 1994 Received: from localhost (listserv@localhost) by antares.mcs.anl.gov (8.6.4/8.6.4) id JAA21775 for qed-out; Wed, 9 Nov 1994 09:52:33 -0600 Received: from svin07.win.tue.nl (root@svin07.win.tue.nl [131.155.70.232]) by antares.mcs.anl.gov (8.6.4/8.6.4) with ESMTP id JAA21769 for ; Wed, 9 Nov 1994 09:52:19 -0600 Received: from pc8012.win.tue.nl by svin07.win.tue.nl (8.6.8/1.45) id QAA19309; Wed, 9 Nov 1994 16:38:43 +0100 Date: Wed, 09 Nov 1994 16:36:18 From: debruijn@win.tue.nl (N.G. de Bruijn) To: qed@mcs.anl.gov Subject: Platonism Message-ID: Sender: owner-qed@mcs.anl.gov Precedence: bulk Eindhoven, 9 November 1994. Dear QED, The interest in philosophical problems arrising in the QED discussion is quite remarkable, and enjoyable. Some of the items are very essential for the structure of QED, but on top of that there are items which are psychologically valuable without having consequences for the way we do our work. For example, two people can write a paper together, where the first one believes that the results aleady existed and are just now being discovered, whereas the other one believes that everything is created on the spot for the first time. Or, the first one may be working in the spirit of Love for Nature and Absolute Truth, and the other one may just be driven by the desire to get a better rating in the Citation Index in a scientifically honest way. In spite of the different basic attitudes, the collaboration can be fine, and it should not be tried to push the two into a common philosophical straightjacket. It would be quite something else if the two authors would disagree on the mathematical basis, on the derivation rules or on the style in which the paper has to be written, on what to consider important, what as interesting, what as beautiful. I think that the larger part of the discussion on the role of intuition belongs in the category of "emotionally valuable but irrelevant for QED". There appears to be quite some interest in Mathematical Platonism. In the discussion I note some different ideas about this, quite far apart. Let me give them names. A-Platonism is the opinion that our mathematical objects have a real existence in some real world. B-platonism is the idea that there is, or there should be, a fixed basis for mathematics, and that such a basis will be satisfactory for ever. C-platonism is the idea that mathematics should be described in a fixed object language. Metalanguage (called "syntactic") should not be allowed in the bag of proof tools. The idea to label this point of view as platonism was expressed by David McAllester in his email of 27 Oct 94; I had never seen it before. There may be many more definitions of platonism. One definition I've seen is that a Platonist is someone who accepts the natural numbers plus power set axiom plus a general comprehension axiom. I suppose that we all agree that A-platonism is irrelevant for QED. And I think that many people will admit that A-platonism is a bit dangerous as well. It makes believers believe that there is no need for an absolute level of accuracy. Their idea may be that for the time being they can get away with sloppy definitions. They might believe that whenever any hard question comes up they can inspect the objects a bit closer and adjust the definitions a bit. The claim has been made that A-platonism agrees with the psychology of the mathematicians: during their work they imagine that the things they are talking about are real objects. Indeed, there are such emotions, but these are essentially the same feelings that a novelist has when writing imaginary stories about imaginary people and imaginary situations. An essential difference is that the mathematician may even be cheating: he starts with some assumptions that he wants to disprove, in the course of the argument he constructs objects, treats them psychologically as existing things, until at the end a contradiction is reached and the whole edifice falls into pieces. (This cheating is a point I miss in the Podnieks' chapter.) There is a stronger form of A-platonism: the idea that in the real world of mathematical objects there is an absolute notion of truth, even for cases which will always remain undecidable for human beings. A-platonism is the idea that when we are talking mathematical language, we talk ABOUT something. That "something" is called real by A-platonists, imaginary by others. With metalanguage this is different, since metalanguage talks about the texts in the language, and those texts can have an actual existence in a physical sense. They can be recorded in the physical world (like on a magnetic disk); retreiving them we get them back in the original form. So when we talk about the mathematical texts instead of about the mathematical objects, it seems quite reasonable to be platonistic. This is so trivial that it is hardly worth while mentioning. B-platonism is by no means irrelevant for QED. It calls for action: if we believe in the possibility of a fixed basis for mathematics, what should it be? The word possibility has both a mathematical and a social meaning here. And even if we believe in the possibility, should we take a particular fixed system as the base for the QED system? A serious alternative is the idea of the very liberal Automath Restaurant. Believers in B-platonism can eat there without any trouble. Is it possible to do mathematics without believing in B-platonism? Could one really turn oneself into a Relativistic Mathematician? I think one can. One can get into any kind of mood by just trying long enough. The matter of C-platonism seems to become more and more important these days. I think it has to be a key item in the discussion about what a proof is. Using pieces of metalanguage or metatheory can work wonders. Nevertheless it should be admitted that in many cases it is not essential, just a matter of making proofs faster and shorter. Quite often the ideas used in the metalanguage can be pushed back into the language, but even if it might be true that this reduction is always possible one might ask whether it will always be manageable. I think that during the 20th century most logicians and mathematicians (including myself) have been in favour of C-platonism, but this may change. If methods different from our traditional proof rules will still give satisfactory confidence, why fight rear-guard actions? For non-mathematical customers it is confidence that counts, not formal proof. Related to the matter of C-platonism is the use of computer programs. If a mathematical fact is established by the execution of a computer program for which a correctness proof is available, we may have a very strong confidence in the correctness of the result, and yet we have to admit that we have not produced a proof according to our proof rules. This is different in those cases where there is a program whose execution actually produces a formal mathematical proof in terms of our own proof system, a proof that can be computer-checked in the next stage. In such cases the correctness proof of the program plays no role any more. N.G. de Bruijn.