
 How To (really) Trust A Mathematical Proof - nickb
http://www.sciencenews.org/view/generic/id/38623/title/Math_Trek__How_to_(really)_trust_a_mathematical_proof
======
randomwalker
The mention of the Russell-Whitehead proof brought back memories. It's from
Principia Mathematica (same title as Newton's book, but it's a different
work.) Here's a representative page:

[http://quod.lib.umich.edu/cache/a/a/t/aat3201.0001.001/00000...](http://quod.lib.umich.edu/cache/a/a/t/aat3201.0001.001/00000401.tifs.gif)

Look at the last line of 54:43. Then look at the page number. This isn't a
joke. It's really how bad axiomatic mathematics is.

During my last year of high school and first year of college, I was fascinated
by this stuff to the point that I couldn't think about anything else. After
digesting Peano arithmetic, I moved on to Zermelo-Frenkel set theory (the "set
theory" that we normally use is self-contradictory and therefore meaningless
to a mathematician.) I worked through Godel's incompleteness theorem. I tried
my best to understand non-Euclidean geometries. And a bunch of other stuff
that I don't even care to remember now.

Soon, predictably, it started affecting my health. At that point, I broke down
and gave up that shit for good. The stuff I do today is positively quotidian
by comparison. But it makes me happy. Even now, I'm occasionally uneasy about
living in a universe where we don't know (actually, _can't know_ ) if the
continuum hypothesis is true or false. But I shove the thought quickly from my
head.

The point of all this is to tell you guys that unless you really know what
you're getting into, don't think too deeply about the meaning of proof :-)

~~~
jerf
In our actual, factual physical universe, the Continuum Hypothesis isn't so
much "true" or "false" as "irrelevant". It's possible the universe is
fundamentally discrete, in which case it just doesn't apply to anything
physical, and even if it turns out to be continuous in some sense, it's likely
to be some hybrid continuous/discrete combo where the hypothesis still doesn't
apply to anything physical.

(One of the points made in Reflections on Relativity (
<http://www.mathpages.com/rr/rrtoc.htm> ) is that both a continuous universe
and a discrete universe, taking the traditional senses of the term, are
logically contradictory things to build a universe out of, and Einsteinian
relativity calls for an odd mixture of both. No clean link to a single page
as, IIRC, the point is made over a series of sections, culminating in a unique
(AFAIK) discussion of Zeno's Paradoxes, which I _can_ link to:
<http://www.mathpages.com/rr/s3-07/3-07.htm> although you really need to read
the sections before that for full impact.)

So, sleep easy. It is nothing more and nothing less than a choice of axiom.
The axiom of choice is similarly irrelevant in the real universe; since use of
the axiom calls for the selection of an infinite set from another infinite
set, possibly uncountable, it has no particular connection to the real, finite
universe.

~~~
randomwalker
That makes sense, and it's likely that if I read through the entire book that
you linked to, I'd grasp it pretty clearly. But it was certainly out of my
mental reach as an 18-year old. And believe me, I tried.

In general, it appears that our ability to ask philosophical questions,
especially of the mathematical variety, far outpaces our ability to find
answers or even _comprehend_ answers that others have come up with. It is
highly plausible that this is linked to the disturbingly high occurrence of
depression, bordering on mental illness, among mathematicians. (I know enough
mathematicians personally for that observation to have some statistical
validity, but anyone who has read many biographies of mathematicians should
get the sense of it.)

What I'm trying to say is that while many of the questions of mathematical
philosophy pique our curiosity greatly, and it can certainly be a rewarding
experience to get a taste of the beauty of the field, everyone should have a
mental threshold for how much they are willing to get involved. Perhaps this
is obvious to other people, but since it was a painful lesson for me, I feel
obliged to share it :-)

------
kurtosis
"There is no permanent place for ugly mathematics"

-G.H.Hardy

I loathe these automatic proof systems (at least the state of the art today) -
I think they may be useful for verifying that a complex CPU correctly
implements integer arithmetic - but these proofs have little to offer a human
mathematician. Let's follow Paul Erdos who said "one need not believe in god,
but a mathematician should believe in The Book" The Book is where god (the
supreme fascist) has collected the shortest and most elegant proofs in
mathematics. Here's a great book written in this spirit (dedicated to Erdos) -

<http://books.google.com/books?id=KvQr9l0wgf8C>

when a theorem prover can produce a proof as elegant as as the one given in
this book of say sperner's lemma then I'll give them another look. (No
disrespect to Russell or the creators of these programs is intended, of
course)

~~~
yummyfajitas
I don't agree.

Lots of proofs are irreducibly ugly, even when the idea behind them is simple.
Just too many edge cases and details. Automatic proof systems can avoid this.

I think of it much like garbage collection and higher order functions. The
machine code of a garbage collected program might be messier than a manually
managed one. But python code is prettier than c code.

~~~
kurtosis
you have strong point - i hadn't thought about it this way before. Just the
other day someone called me out on a derivation and i spent a couple of hours
writing out a very tedious proof. there were lots of special cases and i
really resented the bastard when i was done. the experience was very much like
filling out IRS forms. it would have been very nice to have a proof assistant
to back up the points where i would have much preferred to handwave. I really
like the comparison to manual memory management that's exactly what this proof
felt like - freeing the mallocs.

but think about the following two situations: (1) you have serious doubt as to
whether a statement is true. presenting a proof removes all doubt. Computer
proofs, no matter how ugly, are a win for this situation (2) you have
overwhelming experimental evidence that something is true and you want a
mathematical proof to help you understand _why_ it is true. Computerized
proofs are less helpful here - this is where you want an elegant and simple
proof. it should be very intuitive how the result depends on the assumptions.
it should be something you'll be able to remember when you're 70 years old.

~~~
yummyfajitas
I actually suspect the second situation won't happen for a long time.

My guess is that we will instead have proof _assistants_. Basically, what will
happen is that a human will come up with the big idea which motivates the
proof, and will direct the proof assistant through the details.

I'm thinking something along the lines of Isabelle or Coq + ProofGeneral,
except of course much better.

~~~
ars
Or more along the lines of Computer Assisted Chess.

A person assisted by a computer can play better chess than any human, and also
better than any computer.

Maybe it will be the same here.

------
ars
“Most of what you learn from a textbook is in the exercises”

That explains a lot ....

Too bad no one told me that back when it mattered.

------
ced
"A proof is a repeatable exercise in persuasion"

\- Couldn't track the source. Shucks.

Computer proofs have their value, but so far it is mostly in determining
whether a proposition is true or false. One still needs a mathematician-
written proof to get insight into the problem. This is unlikely to go away
anytime soon.

Furthermore, for most important unsettled problems in maths, we already have a
very strong inkling that the statement will turn out true/false. Dramatic
surprises in maths are very, very rare. Gödel's incompleteness theorem is the
only one that comes to mind.

------
13ren
The "library of foundational proofs" sounds a little like this graph of
hyperlinked theorems online:
<http://us.metamath.org/mpegif/mmset.html#overview>

