The truth (or falsity) of the Riemann Hypothesis is intimately related to the distribution of the primes. Specifically, if the RH is true, then the primes are distributed about as regularly as possible.[1]
I wonder if you have forgotten your original question -- "Sure, one can argue that if two theories are functionally equivalent, there's no downside to taking the simpler one. But has anyone demontrated this logically or mathematically?".
The answer is still yes, it has been demonstrated mathematically. If you're actually interested in a demonstration, you might want to pick up a book about Kolmogorov Complexity Theory sometime.
No, I understood your point. Kolmogorov Complexity Theory gives a mathematical proof that explains why simpler theories work better (and therefore should be preferred), just as I stated in the first sentence of my post ("Actually, yes, Occam's Razor can be considered to be a mathematical conlusion of Kolmogorov Complexity Theory.").
Most of the rest of the post is just to give a short introduction to the fundamental idea behind Kolmogorov Complexity Theory, not a proof of Occam's Razor (hence the second sentence: "Briefly, Kolmogorov Complexity Theory is the study of the compressibility of strings of symbols....")
The last sentence gives a possible intuition for how/why Occam's Razor is true (which it is, since it has a mathematical proof): "One way to think about why Occam's Razor is true: shorter theories are less likely to have arbitrary, extraneous features which imply incorrect conlusions (predictions)."
To see and understand an actual proof of Occam's Razor using Kolmogorov Complexity Theory, you'll have to do a little more work than reading three paragraphs on Slashdot!
I don't know of a proof online, but here is a reference to the definitive text An Introduction to Kolmogorov Complexity and Its Applications. The book is a very good introduction, and I find the whole subject to be extremely interesting and beautiful.
Actually, yes, Occam's Razor can be considered to be a mathematical conlusion of Kolmogorov Complexity Theory.
Briefly, Kolmogorov Complexity Theory is the study of the compressibility of strings of symbols. E.g., consider the three 10 digit strings "0123456789", "4294967296", and "5286354993". Which is most compressible (or, almost equivalently, easiest to remember)? Well, the first is obviously easy to remember (compress): count from 0 to 9. The second is (not as obviously) perhaps even easier to remember (compress): it is 2^32. I believe the third to be difficult to remember (because probably it has to be completely memorized - I typed it in "randomly").
Now suppose we consider infinite strings instead of finite strings, and we consider all computer programs that print out the first n symbols of a given infinite string. In Komogorov Complexity, Occam's Razor is equivalent to the idea that the shorter the program that prints out the first n symbols, the more likely it is to print out the correct (n+1)th symbol. This can be made completely precise, and then "Occam's Razor" is a provable conclusion.
One way to think about why Occam's Razor is true: shorter theories are less likely to have arbitrary, extraneous features which imply incorrect conlusions (predictions).
Slashdot Mirror
In 2006.
When Obama was secretly President.
God damn him and his time machine.
Operation Fast and Furious began in 2009. I believe Obama was president sans time machine.
The truth (or falsity) of the Riemann Hypothesis is intimately related to the distribution of the primes. Specifically, if the RH is true, then the primes are distributed about as regularly as possible.[1]
Carl
[1] See, for example, equation (2) of Riemann Hypothesis
Study Kolmogorov Complexity Theory if you really want to understand. It gets into the epistomological questions you're hung up on.
:-)
(And the answer is still yes.
I wonder if you have forgotten your original question -- "Sure, one can argue that if two theories are functionally equivalent, there's no downside to taking the simpler one. But has anyone demontrated this logically or mathematically?".
The answer is still yes, it has been demonstrated mathematically. If you're actually interested in a demonstration, you might want to pick up a book about Kolmogorov Complexity Theory sometime.
No, I understood your point. Kolmogorov Complexity Theory gives a mathematical proof that explains why simpler theories work better (and therefore should be preferred), just as I stated in the first sentence of my post ("Actually, yes, Occam's Razor can be considered to be a mathematical conlusion of Kolmogorov Complexity Theory.").
...")
Most of the rest of the post is just to give a short introduction to the fundamental idea behind Kolmogorov Complexity Theory, not a proof of Occam's Razor (hence the second sentence: "Briefly, Kolmogorov Complexity Theory is the study of the compressibility of strings of symbols.
The last sentence gives a possible intuition for how/why Occam's Razor is true (which it is, since it has a mathematical proof): "One way to think about why Occam's Razor is true: shorter theories are less likely to have arbitrary, extraneous features which imply incorrect conlusions (predictions)."
To see and understand an actual proof of Occam's Razor using Kolmogorov Complexity Theory, you'll have to do a little more work than reading three paragraphs on Slashdot!
Yes. "Provable" *means* "there exists a proof". (In this case the proof has even been written down and published.)
I don't know of a proof online, but here is a reference to the definitive text An Introduction to Kolmogorov Complexity and Its Applications. The book is a very good introduction, and I find the whole subject to be extremely interesting and beautiful.
This Usenet article gives more references, and Google searches for "Kolmogorov Complexity Theory" and "Kolmogorov Complexity Theory Occam's Razor" gives many hits. Check it out - it's very interesting stuff!
Actually, yes, Occam's Razor can be considered to be a mathematical conlusion of Kolmogorov Complexity Theory.
Briefly, Kolmogorov Complexity Theory is the study of the compressibility of strings of symbols. E.g., consider the three 10 digit strings "0123456789", "4294967296", and "5286354993". Which is most compressible (or, almost equivalently, easiest to remember)? Well, the first is obviously easy to remember (compress): count from 0 to 9. The second is (not as obviously) perhaps even easier to remember (compress): it is 2^32. I believe the third to be difficult to remember (because probably it has to be completely memorized - I typed it in "randomly").
Now suppose we consider infinite strings instead of finite strings, and we consider all computer programs that print out the first n symbols of a given infinite string. In Komogorov Complexity, Occam's Razor is equivalent to the idea that the shorter the program that prints out the first n symbols, the more likely it is to print out the correct (n+1)th symbol. This can be made completely precise, and then "Occam's Razor" is a provable conclusion.
One way to think about why Occam's Razor is true: shorter theories are less likely to have arbitrary, extraneous features which imply incorrect conlusions (predictions).