

The Omega Man: "Math is built on randomness" - yters
http://www-2.dc.uba.ar/profesores/becher/ns.html

======
robertk
<http://en.wikipedia.org/wiki/Chaitin%27s_constant>

But seriously, don't upvote this story, it is written by a fifth grader (or at
least makes that its audience).

EDIT: In response to the comment below ("just because it's simplification
doesn't make it inaccurate"):

Because it's not "simplification." It's--pardon--bullshit. Chaitin's constant
has nothing to do with "randomness in mathematics."

Besides, number theory is certainly not "the foundation of pure mathematics."
Gauss would say that in the 18th century ("Number theory is the Queen of the
Sciences"), but obviously it's long been replaced by logic, model, and
axiomatic set theory.

Here, I'll condense the article to one sentence:

 _In other words, the randomness of the digits of Omega imposes limits on what
can be known from number theory--the most elementary of mathematical fields._

There. So what? There are limits on what can be known from _anything_ in
mathematics. Nothing to see here.

[http://en.wikipedia.org/wiki/Diophantine_equations#Typical_q...](http://en.wikipedia.org/wiki/Diophantine_equations#Typical_questions)

This is just saying that some of those five questions are answerable by
Chaitin's constant for that particular huge Diophantine equation.

EDIT2: Disclaimer: I am a graduate student in mathematics.

~~~
randomwalker
A big part of the problem is that Chaitin is a shameless self-promoter. He is
the Bruce Schneier of Algorithmic Complexity. Actually, no, Schneier is not
nearly as bad. Let me quote from the highest voted Amazon review of his book:

"Let the author speak for himself. From page 7, "Gödel's 1931 work on
incompleteness, Turing's 1936 work on uncomputability, and my own work on the
role of information, randomness and complexity have shown increasingly
emphatically that the role that Hilbert envisioned for formalism in
mathematics is best served by computer programming languages[.]"

Imagine if a working composer wrote, "Bach's preludes and fugues, Beethoven's
symphonies, and my own string quartets have shown increasingly
emphatically..." This man's reputation in his declared field is nowhere near
his apparent stature in his own mind. The ideas discussed in this book are
worthy of late-night musings over a nice brandy, or maybe a Scientific
American article, but only after extensive revision. They are not ready for
publication in a monograph. "

~~~
jhancock
I assume you weren't really picking on Schneier. He's a self-promotor in a
pretty healthy sense. He has written much content on the human aspects of
security useful to laymen. He does so without being arrogant or saying "I
invented this". All in all, a good educator that doesn't talk down to his
audience and can explain things so that his "grandmother could understand" (my
favorite Einstein principle). He has interwoven this practice into his
business, the beneficiary of his self-promotion, which seems to have done well
for him.

I might even recommend Schneier as the poster-boy of healthy entrepreneurial
self-promotion.

~~~
randomwalker
No, didn't mean to pick on Schneier. He's a self promoter, but the analogy
breaks down after that.

------
davido
The first day at University, my Maths Professor told us: "We will asssume you
never knew about Maths and we will start fron the beginning". We defined what
a number is, what addition is ... and from theorem to theorem we reached some
very high and complex domains in Maths a few years later. The process of
understanding each step, each demonstration and ultimatly each concept -
starting from scratch - made few students among the group Maths masters.

~~~
yters
That is really neat. I've never had that experience, and I definitely envy
you.

~~~
cool-RR
You can just attend calculus 101 when the next semester opens, in any
university.

~~~
ars
You mean audit a class? Not sure why you got downmodded, as far as I know you
most universities allow people to audit as much as they want.

Edit: Never mind, you go the downmod for saying calc 101 - it's certainly not
a 101 class.

~~~
wensing
Not 101 (technically it was labeled 161), but at the Univ. of Chicago (where I
went), Spivak was the first Calc class on the honors track--i.e. the honors
variant of 101. And that course did exactly what he's talking about.

------
emmett
Another introduction to this same topic, that goes much further and is better
written:

<http://www.scottaaronson.com/writings/bignumbers.html>

~~~
robertk
_You have fifteen seconds. Using standard math notation, English words, or
both, name a single whole number--not an infinity--on a blank index card. Be
precise enough for any reasonable modern mathematician to determine exactly
what number you’ve named, by consulting only your card and, if necessary, the
published literature._

Judging by the rest of the article, I would have won. I would have written
"Ackermann function on Graham's number and Graham's number." For added
enjoyment, add "Call this B_1. Define B_n = A(B_n-1,B_n-1) for n > 1\. Now
take B_(B_1)."

EDIT: Or I guess just:

 _Let G be Graham's number. Let B_n be G if n = 1 and A(B_n-1,B_n-1) if n > 1
(where A is Ackermann's function). Take B_B_2._

~~~
kylec
I think your number is the XKCD number:

<http://www.xkcd.com/207/>

~~~
robertk
Yep!

Only my second example was doing that recursively...oh the horror...

------
davo11
Doesn't this mean that

1) There exist mathematics (systems of logic) for which there is no way to
reach logically from what we know now to how this system works, it must be
intuited

2) But does this mean that reality is embedded in one of these mathematics?
This is not necessary as far as I can see, so this still leaves the door open
for a theory of everything in the physical world (string theory or some such).

3) But it does seem to imply that no infinite system can be described by a
finite system ( a restatement of Godel ). So we can't set up a computer
program to run the universe - no real surprise there.

or are the implications bigger?

from <http://plus.maths.org/issue37/features/omega/feat.pdf>

"To put it bluntly, if the incompleteness phenomenon discovered by Gödel in
1931 is really serious and I believe that Turing's work and my own work
suggest that incompleteness is much more serious than people think then
perhaps mathematics should be pursued somewhat more in the spirit of
experimental science rather than always demanding proofs for everything.
Maybe, rather than attempting to prove results such as the celebrated Riemann
hypothesis, mathematicians should accept that they may not be provable and
simply accept them as an axiom"

So, for example, if we have the axioms of set theory, then for any theorem, it
may not be possible to prove this theorem from the axioms as a set of linear
deductions, somewhere along the line we may find a new theorem that requires
to be stated as axiomatic, i.e.

we have axioms A,B we have theorems C,D,E provable from A,B then we find F
which seems to be true, but we can't prove F from A,B, F must be stated as an
axiom, probably not that surprising really.

Again just because systems like this exist, doesn't mean we live in one.

------
gibsonf1
Call me uneducated, but isn't the rock solid foundation for math the real
world? To the extent that math corresponds to the real world, it is correct.
The "halting problem" sounds like an attempt to predict the future with math,
which seems like a futile effort. Maybe the conclusion from the "randomness"
in trying to predict the future is that such predictions are simply not
possible and a waste of effort?

~~~
emmett
You are Not Even Wrong, in this instance.

Math is not a simulation of the real world; to take a simple example, both
Euclidean and hyperbolic geometry are self-consistent and useful, and yet
clearly different. So "the rock solid foundation" of math is not the real
world, but rather sets of self-consistent axioms.

The halting problem is not an attempt to predict the future with math; it's a
description of a property of algorithms which is inherently non-computable.
You cannot, in the general case, prove whether an algorithm halts or not.

So as to your the question of whether the "randomness" comes from trying to
predict the future...the answer is not yes, nor is it no. The question is not
sensible.

The article was very poorly written though, so I'm not surprised it causes
confusion. If you are interested in actually learning about Omega, I recommend
<http://www.scottaaronson.com/writings/bignumbers.html> which is an amazing
introduction to all these ideas.

~~~
cousin_it
_So "the rock solid foundation" of math is not the real world, but rather sets
of self-consistent axioms._

You're mostly right, but there's a fun twist. The self-consistency of e.g.
Peano arithmetic (natural numbers) ultimately rests on our intuitions about
the real world - it can't be proven starting from nothingness.

------
jrp
> Chaitin's discovery implies there can never be a reliable "theory of
> everything"

Just because gravity works every time we check, there's no certainty that it
will work the next time.

Physics disproved - I'll take my Nobel please.

------
pixcavator
Let me try to summarize. Out of all possible mathematics, there is something
mathematicians care about and also there is a lot of crap. This is really not
that surprising...

