Here, in digested form, are six more responses related to the recent discussions involving "What is mathematics" and "What are the questions about the homotopy groups of spheres."............DMD _________________________________________________________- Date: Wed, 20 Jan 1999 12:54:51 -0500 From: rrb@math.wayne.edu (Robert Bruner) Subject: Re: htpy gps of sphs I have to agree with the criticism of 'well defined' or 'well definable' as the essential aspect of mathematics. I always answer this question along the lines that mathematics is the study of structure or pattern, and that many of the patterns are derived from our intuitions of number and space and that the most beautiful results often relate structures not heretofore seen as related. Tibor Beke's response says this much more beautifully. The main merit of the preceding paragraph is its brevity. I do find that getting the word 'beautiful' into the discussion right away intrigues most non- mathematicians. Bob Bruner ________________________________________________________ Date: Wed, 20 Jan 1999 21:35:30 -0600 From: Michael Spertus Subject: Re: htpy gps of sphs I think that the question of whether the problems are "well-defined" may not be relevant to the contention that "mathematics is the study of well-defined things." I think that the important distinction is that "the homotopy groups of spheres" is a "well-defined thing" even if we may ask many ill-defined questions about them and rely on brilliant and intuitive, rather than algorithmic, proofs. There is certainly a rigorous and elementary definition of pi_k(S^n) that makes it a well-defined thing, whether or not we can compute it or prove the sort of theorems we would like about it. I agree with the characterization that mathematics can be defined as the study of such well-defined objects, regardless of how well the open problems can be stated. As an example, I think that studying solutions of Maxwell's equations is an example of doing mathematics, while studying electromagnetism is an example of doing physics. A physicist may study electromagnetism, which is not well-defined. However, the (somewhat accurate) model given by Maxwell's equations is well-defined. As a result, when a physicist studies electromagnetism by means of Maxwell's equations, what s/he is doing is best characterized as "applying mathematics to physics", even if s/he is trying to solve a poorly-stated problem like "How can I build a better transistor?" This rings true to me, and I believe it would to most people. I think it is useful to consider a degenerate case of the above example. If physical reality is ever reduced conclusively to a set of equations (and so becomes well-defined), I believe that it would be fair to conclude that physics would become a branch of mathematics. Michael Spertus mps@geodesic.com Geodesic Systems (312) 832-2039 414 N Orleans, Suite 410 http://www.geodesic.com Chicago, IL 60610 ______________________________________________________________ Date: Wed, 20 Jan 1999 23:48:47 -0500 (EST) From: "Daniel H. Gottlieb" Subject: re well defined The definition of Mathematics as "the study of Well-Defined Things" may or may not be a public relation disaster, but it IS exactly what Mathematics is. I answer the key question:" Is the concept of Well-Defined itself Well-Defined?" My answer is YES! There are two subtleties with this answer, and one very important implication. First the implication: The definition I gave is a mathematical one, except that the word "study" may be ill-defined. Therefore, one would expect that every valid description of Mathematics, such as "Mathematics is a language" or "Mathematics is the study of patterns", should be a special case of, or derivable from, that primary definition. Also, the great trends in the history of Mathematics should be explainable in terms of that definition in the same way that the Hypothesis of Evolution explains the facts of Natural History. And finally, we should be able to make predictions as to what will happen in the future based on those explanations. For example: The ancient Greeks predicted that Mathematics was the tool by which the secrets of the World would be uncovered. Events have proved them correct. They did not know Algebra, they did not know Calculus. At the time of Plato, they didn't even know Euclid's axiom system. How could they have foreseen all that was to follow? Because by the process of eliminating ambiguity in their arguments, they arrived at the fact that there were incommensurable lengths. In modern language, with considerable increase in sophistication, we would say they discovered irrational numbers. Now in the presence of the Sun and the Moon, it would be difficult to see a Super Nova. But against the dark background of a few stars, it might be obvious. And the obvious Super Nova was this: By eliminating ambiguity in a systematic manner, it was possible to learn and to know a truth that was inaccessible to the only other way we have of finding knowledge; experience. And since what we can experience is so limited, if we are to transcend that limitation, it has to be via mathematics. And that is what happened and what will happen. The Western intellectuals believed this up to this century, and most educated people from the Age of Enlightenment until this century believed this also. But the devices we have developed and the science we have learned have increased our range of experience so much that modern people are too distracted by the bright light to note that it is only by Mathematics that we can learn beyond our experience. Now let us turn to the two subtleties of the statement that the concept of Well-Defined is itself well-defined. I thought about that statement for five years before I accepted it. I couldn't find anything ambiguous with it. Still, I have to admit that there maybe something can be ambiguous about it. But how is that different from the concept of a line or a plane, or any other primitive mathematical concept? So the first subtlety is that we may not know that our Well-Defined Concept is really well-defined. The second subtlety involves whether these concepts really exist. Plato was so impressed with the mathematical ideal that he went overboard and said that only the ideal really exists. This invites the counter opinion that things that you can't experience really do not exist: Like, for example, the irrational numbers. What do we do when we are confronted by a non-Platonist attack. We retreat, because we really don't know what "exist" means outside of its strictly mathematical manifestations. So then the rational numbers, or Well-Defined concepts, become just things we invent for our own amusement. But the concepts are so important that we expect our opponents to agree that they are more than nothings, even if we can't defend them mathematically. The fallback definition then is that Mathematics is a method whereby we act as if we are studying Well-Defined Concepts. This should answer both Martin Crossley's and John Greenlees comments that what they are really doing is studying concepts they hope will turn out to be well-defined. And it is the classical maneuver that mathematicians engage in when they meet a non-Platonist, so this addresses Andrzej Kozlowski's comments about Plato. As far as public relations is concerned, we will rarely meet a non-Platonist, and only a mathematician would know that he is really trying to make things well defined, so I think we can take the original position that Mathematics is the study of Well-Defined Things. Then, instead of describing Mathematics by its experiences, successes, and analogies, as a botanist describes a plant, we could derive the successes and experiences via a majestic argument. "Of course mathematicians win Nobel prizes in Economics and Chemistry, how could it be otherwise". "Naturally the computer came about from mathematician's ideas, who else could do it"? If you adopt this strategy, you will have to agree that Chess Problems form part of Mathematics contrary to all your experience; but then, isn't that the power of Mathematics? Dan Gottlieb ______________________________________________________ Date: Wed, 20 Jan 1999 18:23:25 +0000 (GMT) From: Ronnie Brown Subject: Re: well-definable Can we borrow from a description of the role of the poet? `To give to airy nothing a local habitation and a name.' (Midsummer Nights' Dream) This seems to me related to what John is saying. Grothendieck writes in a letter of the difficulty of `bringing concepts out of the dark'. Ronnie > >Date: Wed, 20 Jan 1999 13:22:04 GMT > >From: J.Greenlees@sheffield.ac.uk (John Greenlees) > >Subject: Why are we here? > > > > > > I've been hoping someone else would deal with this, but > >I can't restrain myself any longer. > > > > The definition of mathematics as the study > >of well defined things is a public relations disaster. > >I also believe that it is wrong, because it doesn't > >describe what I do. Most of my mathematical effort > >(certainly the most interesting bit) is expended on > >things which are not (yet) well defined. The answer > >sounds like the revisionist account of work we usually > >use when we publish. > > > > > > > > John Greenlees > > 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 _______________________________________________________- Date: Wed, 20 Jan 1999 10:21:51 -0500 (EST) From: James Stasheff Subject: Re: Why are we here? Greenlees makes a very good point a mthematical defintion of what we do maybe relevant to us but it is of NO help in getting the `real' worlkd to appreciate us cf. a marvelous German journalists article in re: the Berlin ICM available in a bilingual editon for Klaus Peters Verlag .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: Wed, 20 Jan 1999 10:43:34 -0500 (EST) From: James Stasheff Subject: Re: well-definable literally accurate? no way accurate in conveying a correct impression - that's what we should aim for .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 > From: Martin Crossley > Subject: Re: Why are we here? > > i agree that the "study of well defined things" > definition is wrong and disastrous for public relations. > it seems to me that a more accurate answer would be that > it's the study of "potentially well defined things", but > this is surely even more disastrous in terms of public > relations. is there no way to be accurate without sounding > crazy to a layman ? > > martin crossley > > >