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*To*: fom@math.psu.edu*Subject*: FOM: defining ``mathematics''*From*: Stephen G Simpson <simpson@math.psu.edu>*Date*: Tue, 21 Dec 1999 19:07:24 -0500 (EST)*In-Reply-To*: <199912070051.QAA15441@herbrand.ucsd.edu>*Organization*: Department of Mathematics, Pennsylvania State University*References*: <199912070051.QAA15441@herbrand.ucsd.edu>*Reply-To*: simpson@math.psu.edu*Sender*: owner-fom@math.psu.edu

Having concurred with Sam Buss on the importance of the predicate calculus, let me take issue with Buss on another point, concerning the definition of mathematics. Buss Tue Dec 07 00:51:16 1999 defines ``mathematics'' as follows: > "Mathematics is the study of objects and constructions, or of > aspects of objects and constructions, which are capable of being > fully and completely defined. A defining characteristic of > mathematics is that once mathematical objects are sufficiently > well-specified then mathematical reasoning can be carried out with > a robust and objective standard of rigor." Now, I think this definition of mathematics is wrong on two counts. First, Buss hangs his definition of mathematics on a methodological issue: rigor. But this doesn't seem to fit with the history of the subject. Our current standard of mathematical rigor evolved only in the 19th and 20th centuries. Would Buss claim that there was no serious mathematics in the 17th and 18th centuries, prior to that evolution? Second, Buss's definition of mathematics in terms of rigor/objectivity would seem to deny the possibility of rigor/objectivity in all sciences other than mathematics. For example, if biologists were to come up with a rigorous definition of ``arachnid'' which permits reasoning about arachnids to ``be carried out with a robust and objective standard of rigor'', would that remove arachnids from the realm of biology and make them part of mathematics? If I were a biologist, shouldn't I take offense at Buss's implicit suggestion that the standards of rigor/objectivity in biology are somehow necessarily lower than in mathematics? The idea of identifying mathematics with rigor/objectivity per se, or the rigorous/objective part of our thinking, has a long pedigree going back to Descartes. Nevertheless, in my opinion, this idea is fundamentally flawed. An interesting study of this topic is David Lachterman's book ``The Ethics of Geometry'', Routledge, 1989, 255 pages. It seems to me that the right way to distinguish the various sciences from each other is not in terms of methodological issues, but in terms of subject matter. Thus mathematics, like every other science, is to be defined as the study of a specific subject matter. To delimit that subject matter may be difficult, but as a first attempt let's call it ``quantity''. In other words, I am suggesting to define mathematics as the science of quantity. Some people may consider this definition old-fashioned and outmoded, but I think it has a lot of merit. See also my short essay on the foundations of mathematics at <http://www.math.psu.edu/simpson/hierarchy.html>. -- Steve Name: Stephen G. Simpson Position: Professor of Mathematics Institution: Penn State University Research interest: foundations of mathematics More information: http://www.math.psu.edu/simpson/

**Follow-Ups**:**Vaughan Pratt**- Re: FOM: defining ``mathematics''

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