3 responses to Steve Wilson's discussion of "What is mathematics." See http://www.lehigh.edu/~dmd1/postings.html for Wilson's original posting...DMD Date: Mon, 25 Jan 1999 15:47:24 -0500 (EST) From: "Daniel H. Gottlieb" Subject: RE What is Mathematics Steve Wilson said "Several structuralists put forth answers of the type 'the study of well defined objects' to which my gut response is YUCK!". But then he goes on to say "Math is the SEARCH for PRECISE language to discuss spatial and quantitative relations. (an emphasis on both search and precise)". PRECISE language can stand for Well-Defined; it means no ambiguity. So does Descarte's "Clear and Distinct Ideas". SEARCH is just a subconcept of study. Why limit yourself just to SEARCH? And why limit yourself to spatial and quantitative relations? What about motion and time? Why drum out Galileo from mathematics? He thought of himself as a mathematician. Why not ask your Kindergarten teachers to think about Aristotle's largly true law that heavier objects fall faster than lighter objects? Then ask them to make the concept of objects PRECISE, which with a little guidence should lead them to connectivity which may suggest a thought experiment which could lead to the mathematical statement that objects fall at the same rate in vacuum. Then give them homework to try to make the word 'fall' PRECISE. It is no accident that all the actors leading to the Law of Gravitation were mathematicians, except possibly for Aristotle. (By mathematicians I mean those who can perfect and contemplate well-defined concepts, and not those practitioners who can only use the algorithms we have discovered correctly and sometimes incorrectly to obtain useful results.) And it is no accident that Maxwell's Equations or General Relativity was conceived by mathematicians. And why should society support our arcane research, which is another of Ronnie Brown's questions? Because it is very difficult to discover these important concepts, even when they are very simple. History shows clearly that mathematicians discover simple all embracing ideas and methods and language by working on very definite, very DIFFICULT, questions. And why waste your time saying mathematics has beauty when beauty can only be know through experience? Especially when so much of beauty in mathematics derives from obtaining truths indepently of experience. Dan Gottlieb ____________________________________________________- Date: Mon, 25 Jan 1999 10:04:42 -0500 (EST) From: James Stasheff Subject: Re: Wilson & Koslowski I agree with a lot of what Steve has written but I think it is a mistake to emphasize arithmetic and geometry prefer pattern and structure with numbrs and shapes as major examples mathematics is the active study of pattern and structure in an attempt to achieve a precise (and useful?) understanding remarkably this leads to undrstanding patterns which superficially appear quite different but have an underlying structural similarity .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds ______________________________________________________- Date: Tue, 26 Jan 1999 10:58:40 +0000 (GMT) From: Ronnie Brown Subject: Re: Wilson & Koslowski Is discussion of foundations worthwhile? This is not a new debate! Here is an excerpt from a 1916 quotation from Einstein, not then exactly a case for early retirement (I have been unable to trace the origin of the quotation, but I got it from Math. Intell.). -------------------------------------------------------------------- How does a normally talented research scientist come to concern himself with the theory of knowledge? Is there not more valuable work to be done in his field? I hear this from many of my professional colleagues; or rather, I sense in the case of many more of them that this is what they feel. ........................................... For when I turn to science not for some superficial reason .... then the following questions must burningly interest me as a disciple of science: What goal will be reached by the science to which I am dedicating myself? To what extent are its general results `true'? What is essential and what is based only on the accidents of development? ...It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little... __________________________________________________________________ I should say that we have been working on the matters under discussion for some time, partly as a necessity in connection with teaching and popularisation, but also in relation to research (see Einstein!). So I won't rehearse matters gone over in articles available on our Centre for the Popularisation of Mathematics home page: http://www.bangor.ac.uk/ma/CPM/welcome.html (including a fuller version of the above quotation.) Here is a warning: Steve's teacher friend may well have a further question once she has some kind of answer on the nature of mathematics: "That is very interesting. I like it. How do you implement that view in your teaching?" As my eldest son said to me in 1988 or so: `Yes, Dad, but when are you going to do something about it?' Good luck! Ronnie Prof R. Brown, School of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382475 fax: +44 1248 383663 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ New article: Higher dimensional group theory Symbolic Sculpture and Mathematics: http://www.bangor.ac.uk/SculMath/ Mathematics and Knots: http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm