Date: Tue, 26 Jan 1999 23:56:54 +0900 From: Andrzej Kozlowski Subject: Re: 3 responses to Wilson Only a short while ago I promised myself (and John Klein) never to write about this topic again and here I am unable to resist it yet another time. But resist it I can't for I still want to say why I like Done Gottlieb's description of mathematics better than anything else that has been offered in this discussion. First of all, the word "defined". I think it is essential. The things we deal with are defined by us. We do not take them from the real world, we have the freedom to be guided by our own imagination or any inspiration, however irrational, that comes to us. This is what makes the subject unique, no other science has this kind of freedom. Secondly, "well defined". Although this has a primary "logical" meaning, it also suggests that no all "defined" things are equally good. Of course not well defined object (i.e. self contradictory) are nonsense, but as we know even among the well defined objects some are much more worthy of study than others. Perhaps we can call them "beautifully defined". I think this definition, although at first rather surprising, is accessible to laymen and can be used for popularizing mathematics. For example, it describes very well the difference between a Turing machine and a computer. The first is a "well defined" (or a "beautifully defined") object, and thus belongs to mathematics. In the second case the word "defined" is not appropriate, and thus it belongs to a different area altogether. These two objects, a Turing machine and a computer, will appear to many non-mathematicians so close as to be the same, just as is the case with a mathematical geometrical object and a physical one, but I think almost every one is capable of grasping this difference and through this the nature of mathematics while at the same realizing its power. By contrast I find the approach via pattern and structure much less satisfactory. First of all, lots of people study patterns and structures and they are not all mathematicians. There are structures and patterns in society, in human psychology etc. and it's by no means obvious that studying them either is mathematics or really should be mathematics. I think nothing has done more damage to the reputation of mathematics than cavalier excursions into social sciences ala Christopher Zeeman. The study of structure and pattern sounds far too much like this sort of approach. Besides, people tend to think that patterns and structures are not objects by themselves but for most people patterns and structures involve some other things, which are not themselves patterns or structures. Far from being more "down to earth", this sort of definition is more abstract and difficult to grasp than "study of well defined things". Thus one might say that a Turing machine describes the structure of a computer, but to say that it is just a "structure" sounds strange. Again, in geometry transformation groups can be thought of as structures on well defined objects, e.g. manifolds. Abstract groups may be thought as "structures" by themselves, but how many people will see the point of such an abstract idea? The more I think about it the more I am convinced that "well defined objects" are preferable in every respect. On the other hand, a definition which refers only to numbers and geometry seems too restrictive, for example I am not sure if something like a Turing machine can be fitted into it. I think people who do not like the idea that mathematics is "a study of well defined object" mainly object to the fact that it does not describe the psychological process by which mathematical discoveries are made, the role of intuition, imagination, trial and error etc. But all this is true in all sciences, and does not differentiate mathematics from them. I think most of the things that were said by the opponents of "well defined objects" concern the psychology of mathematical discovery and not the nature of mathematics itself. Andrzej Andrzej Kozlowski Toyama International University JAPAN http://sigma.tuins.ac.jp/ http://eri2.tuins.ac.jp/