Guess what, folks? Probability and cosmology are weird when they interact. (Big Brain Theory: Have Cosmologists Lost Theirs?, by Dennis Overbye, January 15, 2008 NY Times.)
Basically, it appears to be more likely that we are some sort of naked brain living in an illusion of a world than that we live in the actual world we perceive. Roughly speaking, this occurs if we assume that the universe is infinite -- and thus everything that can occur does occur -- because a naked brain is supposedly much more likely to form by chance than the reality we think surrounds us does.
The obvious rebuttal, if one is wedded to this particular model of cosmology, is an evolutionary one -- maybe naked brains aren't so likely after all, because brains are produced (or so we think) by evolutionary processes, so is one really so likely to find a brain just sitting there without the biology in which it evolved? Overbye's article only mentions physicists; I wonder what (if anything) the biologists have to say. And I don't think our probabilistic understanding of evolution is quite to the point where the first sentence of this paragraph can be made rigorous. (On this point, I'd love to be told I'm wrong!)
edit: Sean at Cosmic Variance has written about this much more insightfully than I, and with links to a lot of the relevant research.
Showing posts with label Overbye. Show all posts
Showing posts with label Overbye. Show all posts
14 January 2008
18 December 2007
Particularly numerically egregious typos
Language Log's Mark Liberman wrote this morning about The biggest typo in history, from this New York Times article, which referred to 10500 instead of 10500. (They've fixed the mistake now; I wonder if they found it independently or if someone pointed out the Language Log post to them?) The number here refers to the size of the string theory landscape, which I don't pretend to understand.
The Times made a similar mistake in The Death of Checkers, December 9, writing:
(James Kushner, a reader of this blog, pointed this out to me; I meant to mention it at the time but didn't.)
This particular mistake is actually so common that I barely notice it any more, especially because I usually have a decent idea of what the number should be. But to someone who doesn't know, it could make a big difference in how they perceive the article. For one thing, if you interpret the checkers quote literally, you get the idea that checkers is more complicated than chess: after all, 5 × 1020 =5010 5100.
Another paper that's been known to make these mistakes is The Tech, MIT's student newspaper. The difference here is that The Tech ought to know better; their entire staff is MIT students, who ought to be familiar with exponents. The Times at least has an excuse. (Disclaimer: I went to MIT as an undergraduate.)
Now if only I could get the Times to write about tilings of the plane, where they could say that an 80 by 80 rectangle has "10800" domino tilings. (That's "ten thousand eight hundred", as opposed to the correct number, which is around "ten to the eight hundredth".) Or just in general to write about statistical mechanics, which is where numbers like this come up.
(The Times article which provoked all this -- on the question of whether the laws of nature are inherently mathematical or whether we've just gotten lucky -- is interesting as well. The way I see it, even if a nonmathematical physics is possible, the mathematical physics is so far ahead that we might as well just act as if physics is ultimately governed by mathematics. But I don't think a nonmathematical physics that has the same explanatory power as the physics we have is possible. That might just be a matter of faith.)
The Times made a similar mistake in The Death of Checkers, December 9, writing:
The brute-force method is slow, which is its big limit. Schaeffer says he suspects you couldn’t use it to solve chess, because that game — with between 1040 and 1050 possible arrangements of pieces — is far more complicated than checkers, which has 5 × 1020 positions. “Chess won’t be solved in my lifetime,” he predicts. “We need some new breakthrough in technology to do that.
(James Kushner, a reader of this blog, pointed this out to me; I meant to mention it at the time but didn't.)
This particular mistake is actually so common that I barely notice it any more, especially because I usually have a decent idea of what the number should be. But to someone who doesn't know, it could make a big difference in how they perceive the article. For one thing, if you interpret the checkers quote literally, you get the idea that checkers is more complicated than chess: after all, 5 × 1020 =
Another paper that's been known to make these mistakes is The Tech, MIT's student newspaper. The difference here is that The Tech ought to know better; their entire staff is MIT students, who ought to be familiar with exponents. The Times at least has an excuse. (Disclaimer: I went to MIT as an undergraduate.)
Now if only I could get the Times to write about tilings of the plane, where they could say that an 80 by 80 rectangle has "10800" domino tilings. (That's "ten thousand eight hundred", as opposed to the correct number, which is around "ten to the eight hundredth".) Or just in general to write about statistical mechanics, which is where numbers like this come up.
(The Times article which provoked all this -- on the question of whether the laws of nature are inherently mathematical or whether we've just gotten lucky -- is interesting as well. The way I see it, even if a nonmathematical physics is possible, the mathematical physics is so far ahead that we might as well just act as if physics is ultimately governed by mathematics. But I don't think a nonmathematical physics that has the same explanatory power as the physics we have is possible. That might just be a matter of faith.)
27 June 2007
secret messages in human DNA?
In yesterday's New York Times, Dennis Overbye writes about the possibility of hiding secret messages in human DNA.
This seems vaguely plausible. Each strand of DNA is composed of a sequence of the four bases adenine, cytosine, guanine, and thymine. One could use these like the digits 0, 1, 2, and 3 in a base-4 number system; equivalently, they could be used as 00, 01, 10, 11 in a binary number system, so each base represents two bits.
Humans have done things like this. Freshly allocated memory in certain computing environments is filled with the repeated string (in hexadecimal notation) DEADBEEF; also ABADBABE, BAADF00D, CAFEBABE have been used. (CAFEBABE is apparently used in Java-related contexts; see this archive of a thread "why CAFEBABE?" on comp.lang.java.) It is of course quite unlikely that any of these strings would be found repeatedly in a computer's memory, if the memory is filled at random; the chance of getting, say, ten DEADBEEFs in a row (assuming there's not a process that's just copying some string over and over again) is one in 2320, which is more than the number of subatomic particles in the universe. As you may know, the Central Dogma of molecular biology says that DNA is transcribed into RNA, which is translated into proteins; each triplet of DNA bases maps to a single amino acid, of which there are twenty. There's a code that assigns a letter to each amino acid; the letters B, X, and Z are "special" letters; U, O, and J aren't used. It's possible to spell things with the remaining twenty letters, though, and I've heard that some genetically engineered food includes the name of the company doing the engineering in the junk DNA.
So what if someone designed us? Maybe they'd hide a message in the DNA? (For the record, I don't believe in intelligent design; however, if we were intelligently designed, that leads inexorably to the question of "who designed the designer"?) But how would they hide that message? They don't know what language we speak, and they certainly don't know that we'll invent this twenty-letter way of describing protein sequences concisely. And unlike in the DEADBEEF example, there appear to be reasons why you'd want stretches of DNA to be the same thing over and over again; these occur in the so-called junk DNA. Like many mathematicians, I'm inclined to believe that they'd hide the prime numbers. The idea behind this is that the primes should never occur due to a natural process, but any culture which is the least bit mathematically sophisticated should have them. (The idea comes from people who are searching for extraterrestrial intelligence; they assume that both us and the other species involved have radio astronomy, and inventing radios without mathematics is Hard.) The sequence
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...
which in base-4 is
2, 3, 11, 13, 23, 31, 101, 103, 113, 131, 133, 211, 221, 223, 233, ...
(Note that a very large number of these base-4 numbers, when read in base 10, are also prime! This is just a coincidence, although the fact that 4 and 10 are both even -- and therefore numbers which are odd remain odd under this transformation -- helps.) Replacing 0, 1, 2, 3 with A, C, G, T, we get
GTCCCGGTTACACCATCCTCTCCTTGCCGGCGGTGTT
and so if we see this string in DNA, perhaps we should be suspicious? Well, it's 37 base-pairs long; thus we expect it to occur once in every 437 base pairs. The human genome is about 3,000,000,000 base-pairs long, so if the genome were random, the probability of this string occuring is 3,000,000,000/437 = 1.6 × 10-13.
So if we find it? Then yes, there's probably a Designer. But this doesn't mean that creationists should go fishing for hidden patterns in the genome. First, my choice of how to encode the primes was entirely random. We could reorder A, C, G, and T. We could have encoded the primes in base 3, using the fourth base to separate them. We could have encoded the primes as
CAACAAACAAAAACAAAAAAAC...
where the number of A's between each pair of C's is prime. And so on. Creationists looking in DNA would, I suspect, take a Bible code-like approach to the search. And if there were slight errors? They'd blame it on mutations, which are inevitable (the Times article points out that there are certain "ultraconserved" segments of the genome -- but those sections also appear to be functional, so it would be harder to hide a message in them -- but then if these hypothetical designers are so smart, maybe they can make those sections be functional and hide messages...)
Sequencing the human genome is good for lots of reasons. But the search for messages from the past probably isn't one of them. They might be there, but we'd be searching for a needle in a haystack. And there would be lots of shiny things that aren't needles there, too.
This seems vaguely plausible. Each strand of DNA is composed of a sequence of the four bases adenine, cytosine, guanine, and thymine. One could use these like the digits 0, 1, 2, and 3 in a base-4 number system; equivalently, they could be used as 00, 01, 10, 11 in a binary number system, so each base represents two bits.
Humans have done things like this. Freshly allocated memory in certain computing environments is filled with the repeated string (in hexadecimal notation) DEADBEEF; also ABADBABE, BAADF00D, CAFEBABE have been used. (CAFEBABE is apparently used in Java-related contexts; see this archive of a thread "why CAFEBABE?" on comp.lang.java.) It is of course quite unlikely that any of these strings would be found repeatedly in a computer's memory, if the memory is filled at random; the chance of getting, say, ten DEADBEEFs in a row (assuming there's not a process that's just copying some string over and over again) is one in 2320, which is more than the number of subatomic particles in the universe. As you may know, the Central Dogma of molecular biology says that DNA is transcribed into RNA, which is translated into proteins; each triplet of DNA bases maps to a single amino acid, of which there are twenty. There's a code that assigns a letter to each amino acid; the letters B, X, and Z are "special" letters; U, O, and J aren't used. It's possible to spell things with the remaining twenty letters, though, and I've heard that some genetically engineered food includes the name of the company doing the engineering in the junk DNA.
So what if someone designed us? Maybe they'd hide a message in the DNA? (For the record, I don't believe in intelligent design; however, if we were intelligently designed, that leads inexorably to the question of "who designed the designer"?) But how would they hide that message? They don't know what language we speak, and they certainly don't know that we'll invent this twenty-letter way of describing protein sequences concisely. And unlike in the DEADBEEF example, there appear to be reasons why you'd want stretches of DNA to be the same thing over and over again; these occur in the so-called junk DNA. Like many mathematicians, I'm inclined to believe that they'd hide the prime numbers. The idea behind this is that the primes should never occur due to a natural process, but any culture which is the least bit mathematically sophisticated should have them. (The idea comes from people who are searching for extraterrestrial intelligence; they assume that both us and the other species involved have radio astronomy, and inventing radios without mathematics is Hard.) The sequence
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...
which in base-4 is
2, 3, 11, 13, 23, 31, 101, 103, 113, 131, 133, 211, 221, 223, 233, ...
(Note that a very large number of these base-4 numbers, when read in base 10, are also prime! This is just a coincidence, although the fact that 4 and 10 are both even -- and therefore numbers which are odd remain odd under this transformation -- helps.) Replacing 0, 1, 2, 3 with A, C, G, T, we get
GTCCCGGTTACACCATCCTCTCCTTGCCGGCGGTGTT
and so if we see this string in DNA, perhaps we should be suspicious? Well, it's 37 base-pairs long; thus we expect it to occur once in every 437 base pairs. The human genome is about 3,000,000,000 base-pairs long, so if the genome were random, the probability of this string occuring is 3,000,000,000/437 = 1.6 × 10-13.
So if we find it? Then yes, there's probably a Designer. But this doesn't mean that creationists should go fishing for hidden patterns in the genome. First, my choice of how to encode the primes was entirely random. We could reorder A, C, G, and T. We could have encoded the primes in base 3, using the fourth base to separate them. We could have encoded the primes as
CAACAAACAAAAACAAAAAAAC...
where the number of A's between each pair of C's is prime. And so on. Creationists looking in DNA would, I suspect, take a Bible code-like approach to the search. And if there were slight errors? They'd blame it on mutations, which are inevitable (the Times article points out that there are certain "ultraconserved" segments of the genome -- but those sections also appear to be functional, so it would be harder to hide a message in them -- but then if these hypothetical designers are so smart, maybe they can make those sections be functional and hide messages...)
Sequencing the human genome is good for lots of reasons. But the search for messages from the past probably isn't one of them. They might be there, but we'd be searching for a needle in a haystack. And there would be lots of shiny things that aren't needles there, too.
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