Another introduction to this same topic, that goes much further and is better written:
The Omega Man: "Math is built on randomness"
21–30 of 47 posts
Re: The Omega Man: "Math is built on randomness"
#22Another introduction to this same topic, that goes much further and is better written: http://www.scottaaronson.com/writings/bignumbers.html
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.
Re: The Omega Man: "Math is built on randomness"
#23Re: The Omega Man: "Math is built on randomness"
#24Another introduction to this same topic, that goes much further and is better written: http://www.scottaaronson.com/writings/bignumbers.html
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 functio…
I think your number is the XKCD number:
Re: The Omega Man: "Math is built on randomness"
#25Earlier quoted context omitted.
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 functio…
I think your number is the XKCD number: http://www.xkcd.com/207/
Yep!
Only my second example was doing that recursively...oh the horror...
Re: The Omega Man: "Math is built on randomness"
#26The 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 student…
which university was that?
Re: The Omega Man: "Math is built on randomness"
#27The 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 student…
Sounds like my first lecture in Fluid Dynamics. 2 hours later, we all agreed that yes, fluids flow :-) It's very important before undertaking advanced study to know not only how to do something (which is just cranking through the machinery by rote) but why it works and what that implies.
Re: The Omega Man: "Math is built on randomness"
#28Another introduction to this same topic, that goes much further and is better written: http://www.scottaaronson.com/writings/bignumbers.html
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 functio…
Sounds like you didn't read all of the rest of the article! (Unless you mean you'd have won against the mathematically naive people he mentions at the start who wrote down things like long sequences of 9s.) Aaronson goes on to describe short names for big numbers that get much bigger with fewer symbols than things like your iterated Ackermann.
Specifically, by defining a sequence of numbers in terms of how much a program of given length can do, given that it halts, it turns out that you get a sequence that provably grows faster than any computable sequence. (It's actually traditional to do this with Turing machines rather than conventional programs.) This is in fact rather closely related to Chaitin's stuff at the start of this discussion.
Re: The Omega Man: "Math is built on randomness"
#29Call 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?
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 o…
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.
Re: The Omega Man: "Math is built on randomness"
#30http://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…
What's wrong with a simplification if it's accurate? We can't all be geniuses, I know I'm not. Here's Chaitin's layman's description of his theory: http://plus.maths.org/issue37/features/omega/feat.pdf His book about getting there: http://arxiv.org/PS_cache/math/pdf/0404/0404335v7.pdf And the halting problem thrown in for good measure: http://en.wikipedia.org/wiki/Halting_problem BTW, if someone wants karma, I haven'…
Wow, thanks for posting those links! I read the first two chapters of the book and I must say that Chaitin is a brilliantly lucid and entertaining writer.
I am pretty much mathematically illiterate but I am still enjoying his book immensely.