> I have posted an article at http://web.maths.unsw.edu.au/~norman/views.htm > that has caused a bit of discussion in some logic circles. > > My claims in short: 1) most of `elementary mathematics' is not > sufficiently well understood by the mathematical establishment, leading to > weaknesses in K12 and college curriculum, 2) the current theory of `real > numbers' is a joke, and sidesteps the crucial issue of understanding the > computational specification of the continuum, and 3) `infinite sets' are a > metaphysical concept, and unnecessary for correct mathematics. > > Analysts and set theorists are welcome to send me reasoned responses. > > Assoc Prof N J Wildberger > School of Maths > UNSW
I need to mull over what your paper says before I attempt to provide much in the way of analysis. I tend to agree with the spirit of the presentation, though I am not quite as dismissive of the established dogma.
Somewhere in the dust piles of my neglected papers is a one or two page derivation of Kepler's laws beginning with no other assumption than basic arithmetic and the belief that I could reasonably well communicate such concepts as the Pythagorean theorem through simple drawings. The presentation provides an illustration of what taking a derivative means; IIRC using epsilon delta arguments. It gives a definition of an ellipse, of Newton's laws of motion, and of the inverse square law of gravitation. When I consider that Russell and Whitehead took some 360 pages to get around to 1+1=2, I have to wonder if they really chose the best set of axioms. I acknowledge that their objective was "perfect" rigor, where mine was intuitive clarity, but I am inclined to believe the two objectives are far closer to one another than is suggested by the current formalisms of so-called foundational mathematics.
For now I will make only one comment about your paper. Regarding the proof that multiplication is associative; that fact is "proved" in the volume I am currently struggling through. This set of volumes attempts to outline the state of the art of foundational mathematics in its day. Perhaps it is nothing more than the formalization of that to which you object, but it was intended to address many of the concerns you appear to be expressing.
Fundamentals of Mathematics, Volume I Foundations of Mathematics: The Real Number System and Algebra Edited by H. Behnke, F. Bachmann, K. Fladt, W. Suss and H. Kunle Translated by S. H. Gould
Fundamentals of Mathematics represents a new kind of mathematical publication. While excellent technical treatises have been written about specialized fields, they provide little help for the nonspecialist; and other books, some of them semipopular in nature, give an overview of mathematics while omitting some necessary details. Fundamentals of Mathematics strikes a unique balance, presenting an irreproachable treatment of specialized fields and at the same time providing a very clear view of their interrelations, a feature of great value to students, instructors, and those who use mathematics in applied and scientific endeavors. Moreover, as noted in a review of the German edition in Mathematical Reviews, the work is ?designed to acquaint [the student] with modern viewpoints and developments. The articles are well illustrated and supplied with references to the literature, both current and ?classical.??
The outstanding pedagogical quality of this work was made possible only by the unique method by which it was written. There are, in general, two authors for each chapter: one a university researcher, the other a teacher of long experience in the German educational system. (In a few cases, more than two authors have collaborated.) And the whole book has been coordinated in repeated conferences, involving altogether about 150 authors and coordinators.
Volume I opens with a section on mathematical foundations. It covers such topics as axiomatization, the concept of an algorithm, proofs, the theory of sets, the theory of relations, Boolean algebra, and antinomies. The closing section, on the real number system and algebra, takes up natural numbers, groups, linear algebra, polynomials, rings and ideals, the theory of numbers, algebraic extensions of a fields, complex numbers and quaternions, lattices, the theory of structure, and Zorn?s lemma.
Volume II begins with eight chapters on the foundations of geometry, followed by eight others on its analytic treatment. The latter include discussions of affine and Euclidean geometry, algebraic geometry, the Erlanger Program and higher geometry, group theory approaches, differential geometry, convex figures, and aspects of topology.
Volume III, on analysis, covers convergence, functions, integral and measure, fundamental concepts of probability theory, alternating differential forms, complex numbers and variables, points at infinity, ordinary and partial differential equations, difference equations and definite integrals, functional analysis, real functions, and analytic number theory. An important concluding chapter examines ?The Changing Structure of Modern Mathematics.? </quote>
The part of the first volume specifically dedicated to foundational mathematics strikes me as fairly agnostic regarding which of several possible approaches are ideal in establishing the proper foundations of mathematics. It is so condensed as to be cryptic. Each subsection could fill a chapter, each section could fill a book, and each chapter could fill a book shelf if they were elaborated upon to the point of covering their topics comprehensively.