
Set Theory: Should You Believe? - ColinWright
http://web.maths.unsw.edu.au/~norman/views2.htm
======
ColinWright
Question for the people reading this:

I thought this was interesting, and the upvotes show that others here on HN
agree. But this item is now ranked below 850 and falling - clearly it's being
flagged.

I don't much like StackOverflow, MathOverflow and similar - I much prefer the
style of HN. If I provided an alternative to HN that overcomes this problem,
would you be interested?

~~~
patrickdowell
I certainly would be. I've noticed similar problems in the foundations of
mathematics and I'd be very interested in an alternate system (and a strong
defense of how this new system overcomes the problems found in other systems).

------
andrewcooke
[update: better use the wayback link]

cache:
[http://webcache.googleusercontent.com/search?q=cache:oyOudsA...](http://webcache.googleusercontent.com/search?q=cache:oyOudsA59NwJ:web.maths.unsw.edu.au/~norman/views2.htm+set+theory+should+you+believe&hl=en&gl=cl&strip=1)

(there's quite a few annoying typos with the text "MATH" - is that a problem
with the cache, or was the article converted from some other format, poorly?
ah, thanks to the wayback link - those are cache errors)

summary: it's a rant (i'm not being critical here - it's quite interesting,
but that seems the best description) against standard set theory as the basis
for mathematics (more generally, a rant against an axiomatic approach at all).
but it doesn't really offer much in the way of alternatives. no mention of
constructivism or intuitionism.

~~~
h_r
What about the following sentence?

"At no point do we or should we say, `Now that we have defined an abstract
group, let's assume they exist'. Either we can demonstrate they exist by
constructing some, or the theory becomes vacuous."

Doesn't that imply constructivism?

~~~
andrewcooke
well, yes, i think the article as a whole is heading in that direction. all i
meant was that it doesn't get into any details - it wasn't meant as criticism,
i was just trying to explain that it stopped before that point.

------
begriffs
So Mr. Wildberger has rediscovered mathematical constructivism. This article
would have been more newsworthy around, say, 1920.

------
lobo_tuerto
I don't buy this rant. I found his last argument a bit weak. Just because you
can't write a number because it is so big, it doesn't mean it can't be
simbolized by a handful of characters, or abstracted successfully.

Also it is more powerful to abstract than it is to enumerate.

~~~
ColinWright
Here's the thing.

Let's suppose you write down all the syntactically valid 1 byte programs (in
some language, say, Python). there aren't many of them, and you can certainly
write them all down.

Now write down all the syntactically valid 2 byte programs, then 3, then 4,
and so on. Now you have a list of all valid programs.

Some won't terminate, and some won't print anything, but any number that can
ever be computed by any program will be printed by (at least) one of these.

But there are only countably many of them. That means that most of the numbers
between 0 and 1 will not be printed by _any_ program. Ever.

Are you now still happy talking about the collection of all the reals? Most of
them (by which I mean uncountably many compared with countably many) can never
be printed or calculated.

~~~
lobo_tuerto
Well, I'd say then, that's what abstractions are for!

Seriously, I think that's part of the role of abstractions, and that's why we
_currently_ can talk about and use reals.

Even though we can't write the entire Pi number, and the program used to
calculate all its digits might never finish, we do have a symbol for it, and
put it to good use. What do you think?

~~~
ColinWright
But we _do_ have a symbol for _pi_ and we _can_ write a program which, for any
required N, can print the first N digits.

But there are only countably many numbers for which that is true. Uncountably
many numbers _cannot_ have symbols and _cannot_ have programs to represent
them. If we can never name them, not even in principle, does it make sense
even to say that they exist?

(Note: Don't think you can extrapolate my own personal view from the above, I
am merely pointing out that there are non-trivial questions here, questions
that most people don't seem to notice or understand.)

------
argaldo
archive.org's wayback machine has it ...
[http://web.archive.org/web/20110616020815/http://web.maths.u...](http://web.archive.org/web/20110616020815/http://web.maths.unsw.edu.au/~norman/views2.htm)

