The Omega Man: "Math is built on randomness"

If you ever wanted a use for the term "begs the question" here you go: "If we could live for 70,000,000 years, there’d be no theory of evolution, and certainly no creationism: we could watch speciation and adaptation with our eyes,"

He assumes that evolution is true, then therefor if we lived to 70,000,000 there would not be creationism. However if evolution wasn't real, but creationism was, and people lived to 70,000,000 year you could say the same thing: they saw creation happen, and no one would imagine such a thing as evolution.

Which makes his sentence effectively meaningless. And he "begged the question".

It's an interesting article, but it has a number of errors that severely diminish it. Plus he really needs to stop bashing on the bible and stuff (that myth that the bible says Pi is 3 isn't even true).

My biggest gripe with it is this:

"Take Goldbach’s conjecture, that every even number 4 or higher is a sum of two prime numbers: 10=7+3, 18=13+5. The conjecture has resisted proof since 1742. Yet we could design a Turing machine with, oh, let’s say 100 rules, that tests each even number to see whether it’s a sum of two primes, and halts when and if it finds a counterexample to the conjecture. Then knowing BB(100), we could in principle run this machine for BB(100) steps, decide whether it halts, and thereby resolve Goldbach’s conjecture."

That's WRONG! A turing machine that checks every single integer by definition never halts, so the entire exercise is pointless - you'll get no info. Yah, if it halts you know the answer, but if it doesn't you know nothing, so calculating BB(100) doesn't help you at all.

On top of that it's possible that BB's for large number don't exist - I don't mean that they can't be calculated, I mean that for whatever numbers of steps you chose you can make the machine halt in that number of steps, but there is no upper limit at all, and you can always create a machine that halts after more steps.

PS. If you want a really large number that ordinary students would understand try 9!!!!!!!!.... (repeated factorial).

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.

I see what you are saying, but think of it in reverse:

If the only way to prove or disprove Goldbach’s conjecture is to try every integer forever until you find a counter example (i.e. you can never stop - there is no point at which you can say, I checked enough numbers).

Then the same would apply to calculating BB(100) - you can never actually calculate the value of it - you never know if you have run it long enough.

That's my point: there might not actually be a value for large BB's. It is simply impossible to know if you need to keep running the machine or not. You can only set a lower bound. You can't, even in principle, ever calculate it.

So you can define the meaning of BB, but you can't calculate the value of it. If you can't calculate the value of it, then it has no defined value.

That's what I am trying to say - it is impossible, no matter how many (finite) resources you have, to calculate large BB's. There is never the point at which you can say: this program is not infinite.

Edit: I just read the wikipedia page on it, and I'm not saying something new: it's well known that BB is not computable, and point is that if it _was_ computable then you could use it to calculate Goldbach’s conjecture and others.

He really should have made the point a little clearer than you can't actually calculate the value of BB(100) - he made it seem like it was hard and needed massive resources, but not that it was impossible even in principle.

And finally, I say that since you can not calculate the value of large BB's - they are not actual numbers, and thus can not be used for his large number game.

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. "

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.