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INFINITE_WILDBERGER

                                             October 18, 2019


https://njwildberger.com/2015/11/26/infinity-religion-for-pure-mathematicians/

   "... I hope some of these quotes strike you as
   little more than religious doggerel. Is this
   what you, a critical thinking person, really
   want to buy into??"

That's a feeling I often have, reading
the platonist-oriented philosophy of math.

   "From the initial set-up by Bolzano, Cantor and
   Dedekind, the twentieth century has gone on to
   enshrine the existence of 'infinity' as a
   fundamental aspect of the mathematical world.
   Mathematical objects, even simple ones such as
   lines and circles, are defined in terms of
   'infinite sets of points'. Fundamental
   concepts of calculus, such as continuity, the
   derivative and the integral, rest on the idea
   of 'completing infinite processes' and/or
   'performing an infinite number of tasks'."

I have to say, the concept of "taking
a limit" has always left me feeling a
little queasy.  You're supposed to be     A comment from Tom Holroyd:
able to see where a curve is going
without actually taking it there...         "Fermat already knew how to get
                                            rid of the infinities in
There's something awfully hand-wavey        calculus, before calculus was
about this for it to be the basis           invented. Nilpotent infinitesimals."
of calculus.
                                             https://plus.google.com/108269652526642085924/posts/EcrwP9cDBz3
   "What would mathematics be like
   if we accepted it as it really            "Calculus without limits.
   is?  Without wishful thinking,            Automatic differentiation.
   imprecise definitions and                 Adequality."
   reliance on belief systems?"
                                             http://arxiv.org/abs/1210.7750
   "What would pure mathematics be
   like if it actually lined up
   with what our computers can do,    CEREBROUSUS_MATHEMATICA
   rather than with what we can
   talk about?"

That seems to be Norman Wildberger's central approach:
start with what's actually computable, with what we
can actually know.

In this video presentation:

  "Infinity: does it exist?? A debate with
  James Franklin and N J Wildberger"

     https://www.youtube.com/watch?v=WabHm1QWVCA

Wildberger lays out some more details of his
objections, making the point that the idea of
reasoning about the infinite set of natural
numbers looks a little strange once you get
up to the level of:


                                    10
                                 10
                              10
                           10
                        10
                     10
                  10
               10
            10
   N  =   10                             + 23

He makes the point that there's not a lot of
information in that number, as evidenced by how
easy it is to write down, but most of the numbers
between 1 and that number are actually physically
impossible to express, even with hard drives the
size of galaxies and bits written as quarks.



From the usual platonist point of view, that seems
like a shallow objection, a "practical" objection
that seems willfully obtuse about comprehending the
idealized concept--

It shows a lot about Wildberger's style of thinking:
what does it mean to have a "number" that you literally
can't do anything with?

Is it possible to develop an understanding of math
in more concrete terms, basing it in things we can actually
know about without taking them on faith?

   Wildberger objects that a typical math book
   just skates past a lot of dubiously founded
   concepts, just "finessing" the details,
   pretending that someone else has dealt with them.


Wildberger's debate partner, has a new book out 
(with a really great title):
                                                    
    "An Aristotelian Realist Philosophy of
    Mathematics: Mathematics as the Science of
    Quantity and Structure" by James Franklin
                                                            
                                                           


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