[FOM] R: Wildberger on Foundations

Antonino Drago drago at unina.it
Thu Dec 6 17:34:33 EST 2018

Timothy Y. Chow wrote:

When I was an undergraduate, the "rigorous" introduction to analysis 
defined the real numbers as a complete ordered field with the least upper 
bound property, and constructed it using Dedekind cuts (or maybe Cauchy 
sequences---I don't remember).  Dedekind cuts are of course infinite sets.

The question is whether we can dispense entirely with all the traditional 
constructions of real numbers, and still write a textbook for a "rigorous 
first course in analysis."  Formal systems such as PA or EFA or whatever 
should not intrude into the foreground---current undergraduate texts on 
analysis don't mention formal systems, after all.

I don't think it's possible to talk about analysis without functions, so 
functions have to be introduced somehow, presumably as "rules" rather than 
as infinite sets of ordered pairs.  One of the main challenges is to 
figure out how to avoid talking about the set of real numbers or even an 
"arbitrary real number," without introducing unnatural circumlocutions.

It is an old question how teach analysis in a modern way. The review Am.
Math. Monthly devoted several papers on the subject. 
  There exists a simple and natural way to introduce calculus in a formal
and yet easy way. 
Feferman proved that Weyl's elementary mathematics corresponds to introduce
one quantifier about a constructive set on numbers. 
This result suggests a new educative technique for teaching analysis: to
apply Weyl-Feferman theory of calculus. 
One more easy way is to remark that both Cavalieri's method of indivisibles
(i.e. to give reality to the word 'omnes' (all) and  Torricelli's method to
consider the 'elementi finali' (ending elements) of a series (or even the
geometrical intuition of a single point as either a tangent point of a
straight line to a curve, or as being the common point of two curves, etc.)
correspond exactly to assume one quantifier on calculable elements, i.e.
Weyl's mathemaics. 
A bonus is thus to reiterate grosso modo the historical path leading to the
birth of the first kind of calculus, occurred before the invention of
Leibniz's infinitesimal analysis. 
As one more bonus, a teacher can add the first historical version of inertia
principle (by Cavalieri, before Descartes). 
All that is presented by my paper: The introduction of actual infinity in
modern science: mathematics and physics in both Cavalieri and Torricelli,
Ganita Bharati, Bn/I. Soc. Math. India, 25 (2003), pp. 79-98;  Una nuova
proposta didattica per l'analisi matematica, Rivista di Epistemologia
didattica, 1, (2006) pp. 253-262 (with R. Vella).
Best regards
Antonino Drago 

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