
Intuitionistic mathematics for physics - DiabloD3
http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/
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granite_scones
I'm curious to see the smooth infinitesimal analysis approach to complex
analysis. Using the limit definition (the dreaded epsilon-deltas), you get the
correct definition of continuity (for the usual topology).

How (if at all) do you do the same thing with dz's?

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pash
One wonders whether most mathematicians and physicists reject the author's
point of view because to accept it requires either (1) a deep understanding of
set theory and its limitations or (2) an attitude towards truth that's rather
more cavalier than you'll usually find in the faculty lounge.

I suspect that the author himself fails (1) and so must believe what he does
due to (2). I will not pretend to understand the details or even the general
background of all the concepts needed to get at the heart of the issue, but I
will single out one example to illustrate what I mean (hopefully).

The author credits synthetic differential geometry [1] as an intuitionistic-
logical basis for infinitesimals (because of its rejection of the principal of
the excluded middle) and in the next sentence calls Abraham Robinson's non-
standard analysis a theory based in "classical logic". Now, this must seem
perfectly reasonable to anyone who has sweated through Robinson's process of
constructing the hyperreals [2]. But the author seems unaware of Edward
Nelson's alternative approach, called internal set theory [3], which takes an
axiomatic approach.

Nelson (who, by the way, is one of those ultrafinists whose existence the
author questions in his second paragraph) simply adds a few axioms to Zermelo-
Fraenkel set theory and calls it a day [4]. He can do this because those
axioms _must_ be consistent if ZFC itself is consistent. Which seems
unobjectionable enough, if you're not a mathematician; but if you are, you
probably don't like playing with axioms. (As Bertrand Russell wrote, "The
method of 'postulating' what we want has many advantages; they are the same as
the advantages of theft over honest toil." He then rejects that approach,
unlike Nelson, who admirably pilfers what he needs and proceeds to take a
nap.)

And then you _also_ probably don't like being reminded that ZFC's consistency
(or not) is unprovable [5]. You see, Nelson's axiomatic construction of the
hyperreals relies on the very same logical rabbit-hole as the author's
intuitionistic logic: some things are neither provable nor disprovable in
mathematics (or logic generally).

And that's why I say accepting the author's point of view requires either (1),
if you realize that math is on shaky ground from the start, or (2), if don't
much care and just want to get to the answer as quickly as possible (which is
why I love Nelson's approach to infintesimals).

1\. <http://en.wikipedia.org/wiki/Synthetic_differential_geometry>

2\.
[http://mathforum.org/dr.math/faq/analysis_hyperreals.html#co...](http://mathforum.org/dr.math/faq/analysis_hyperreals.html#construction)

3\. <http://en.wikipedia.org/wiki/Internal_set_theory>

4\. See Edward Nelson, _Radically Elementary Probability theory_ , available
at <http://www.math.princeton.edu/~nelson/books/rept.pdf> [PDF]

5\. Yes, this is Gödel's first theorem:
[http://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theore...](http://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorem#First_incompleteness_theorem)

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chalst
The condescending tone towards Andrej Bauer is unwarranted, who is from the
Dana Scott PhD stable and who has a well-thought out attitude towards
constructivism and realist foundations of mathematics.

Nelson is respected but not what I would call very influential. I took this to
be the point of Bauer's jibe about ultrafinistists. Note that, AFAICS, Nelson
does not call himself an ultrafinitist, but rather a (radical) predicativist.

I don't get the point about the cavalier attitude to truth: constructivist
rejection of PEM comes from worries that some propositions are not sharp
enough to be determinately true or false, the classical Brouwerian examples
coming from topology. This stance more deserves to be called pernickety than
cavalier.

~~~
pash
In my defense, that's a written-at-five-in-the-morning tone, not a
condescending tone. I'm not sure what I was trying to write with the "I
imagine the author himself fails (1)" bit, as I am sure Bauer is far beyond me
on the subject, as I admitted in my comment. Apologies to Bauer and to all for
coming off wrong.

Writing 'cavalier attitude to truth' was a poor choice of words, but let me
explain what I meant. I agree that intuitionistic math is _logically_ more
rigorous, and it sets a higher bar for what can be considered true. But
because it rejects the validity of tools like proof by contradiction and the
law of the excluded middle—tools that classical mathematicians use to
_disprove_ many concepts—intuitionistic math contains a much _wider_ universe
of possible mathematical objects. It sets a _lower_ bar for what _should_ be
considered _untrue_.

So when you use mathematical objects that only _potentially_ exist—that is,
objects whose existence you can neither prove nor disprove—it seems to most
classical mathematicians as more than a bit dishonest, because they work only
with objects that _must_ exist. (I brought up Nelson's axiomatic approach to
infinitesimals because it is analogous in this regard.) Thus Russell's
likening postulates to theft.

Intutionists get to play with toys that other mathematicians don't because the
former are, in a sense, more broad-minded about what toys are (could be) out
there. That's all I meant by 'cavalier attitude to truth'.

Like I said, I admire this approach, and have tried to pick up some axiomatic
approaches myself when they're helpful, but I still do think that Russell's
bit of pith is quite apt.

~~~
chalst
OK, that is clearer.

I guess we could call the revised argument the "cavalier attitude to
refutation". I don't agree about the "much wider universe": Kripke semantics
draws a line around what is possible, and tells us that intuitionistic
structures can be embedded in families of partial classical structures that
evolve towards (but need not reach) a classical structure. Structures
inconsistent with this view can be intuitionistically refuted.

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phren0logy
Does anyone else look at the word "intuitionistic" and want to scream?

~~~
phektus
That reaction is intuitive but nobody wants to be branded as grammar n*zis, I
guess

