[FOM] Wildberger on Foundations

[Timothy Y. Chow]

Recently, someone drew my attention to Norman Wildberger's views on the 
foundations of mathematics.  (I think it might have been an FOM reader 
emailing me off-list, but I don't remember who it was.)  Searching the FOM 
archives, I found Joe Shipman's post from July 2012:

https://cs.nyu.edu/pipermail/fom/2012-July/016563.html

Among other things, Shipman wrote:

> In his discussion with me, he asks for examples of texts where the
> modern framework of Analysis is developed completely rigorously from
> first principles.
> 
> Can anyone suggest some source books that might satisfy his request?

I don't have an answer to Shipman's question, but after listening to some 
of Wildberger's videos, I think there are some questions that arise from 
considering Wildberger's views---questions that are of interest even to 
those who don't share those views.

First let me try to summarize Wildberger's views.  I would characterize 
him as some flavor of constructivist.  To begin with, he rejects the 
concept of a completed infinity.  Thus he rejects the concept of the 
square root of 2 as a completed infinite decimal---it is okay to talk 
about rational approximations to sqrt(2) and about computations in the 
field Q[x]/<x^2-2> but not about a Dedekind cut or an equivalence class of 
Cauchy sequences.  One concept that I believe is original with Wildberger 
is what he calls "rational trigonometry," which roughly speaking is a 
development of trigonometry using only rational numbers---so for example 
by consistently working with sin^2(x) and cos^2(x) instead of sin(x) and 
cos(x), many traditional references to irrational numbers can be avoided.

So when he asks Shipman for texts which develop modern analysis from first 
principles, I think we can approximately parse his question as:

Question 1.  To what extent can we develop modern analysis on the basis of 
PA alone (or perhaps even Heyting arithmetic), without any reference to 
infinite sets?  Can this be done in an undergraduate-friendly way without 
a lot of extra baggage from logic?

Wildberger is more than just an anti-completed-infinity evangelist, 
however.  He is also anti-formalist.  He regards the enterprise of setting 
up any old formal system you like (so long as it is not manifestly 
inconsistent), and merrily deriving its consequences, as misguided. 
These formal systems are, in his view, castles in the sky, and represent a 
departure from the One True Mathematics.  So he would not approve of the 
above statement of Question 1, since it refers to PA; he does not 
necessarily regard theoremhood in PA as a sufficient condition for 
accepting a statement.  From the point of view of a conventional 
mathematician, this is unfortunate, because it means that it is hard to 
predict in advance exactly what kinds of mathematical assertions 
Wildberger will accept.

The final ingredient in Wildberger's brand of constructivism is his 
ultrafinitism.  Ultrafinitist views are notoriously hard to pin down 
precisely, and Wildberger's views are no exception, but here is an 
attempt.  Wildberger distinguishes between three types of notations for 
natural numbers; I'll call them unary, decimal, and big-number (but note 
that Wildberger doesn't use this exact terminology).  Unary and decimal 
are familiar; for "big-number" think something like Knuth's arrow notation 
or Ackerman or perhaps something even more exotic.  For Wildberger, a 
statement such as "for every n there is a prime p > n" or "for every even 
number 2n there exist primes p1 and p2 such that p1+p2=2n" has different 
meanings depending on what notation system you're using, because he does 
not acknowledge the existence of a natural number unless a notation for 
that number is *physically constructible*.

As an illustration, suppose we fix Knuth arrow notation as our big-number 
system (along with some small number of other symbols such as addition and 
subtraction and parentheses).  Given a natural number n (I am now speaking 
as a conventional mathematician), define its "notation complexity" c(n) to 
be the smallest number of symbols needed to express n using our 
Knuth-arrow notational system.  Then the "big-number" version of 
Goldbach's conjecture might look something like this:

Big-Number Goldbach.  For every even number 2n with c(2n) < 1000000, there 
exist primes p1 and p2 with p1+p2 = 2n and c(p1) < 1000000 and c(p1) < 
1000000.

Wildberger argues heuristically that Big-Number Goldbach is surely false.

Question 2.  Arguing conventionally, can we definitively disprove 
Big-Number Goldbach?  What if we modify Big-Number Goldbach by keeping the 
upper bound on c(2n) at 10^6 but relax the upper bounds on c(p1) and c(p2) 
to be, say, 10^100?

Tim