Comments on "The Sequencing of the Mathematical Genome"TypePad2008-11-07T08:15:00ZMark Thomahttps://economistsview.typepad.com/economistsview/tag:typepad.com,2003:https://economistsview.typepad.com/economistsview/2008/11/the-sequencing/comments/atom.xml/Pitt commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535e7c318970c2008-11-10T14:22:33Z2008-11-10T14:22:33ZPitt1) is going back to fundamental axioms is going far enough back? There is literally no further back to go...<p>1) is going back to fundamental axioms is going far enough back?</p>
<p>There is literally no further back to go (without attempting to rewrite the fundamentals of arithmetic). These axioms are, for example, "there exists a 0 such that x + 0 = x"; they're not particularly controversial.</p>
<p>(As for your second question, don't be daft - this has nothing to do with economics, or with foolish risk-taking.)</p>
<p><br />
"We may be close to seeing how computers, rather than humans, would do mathematics."</p>
<p>Wow. Godel, Turing and Cook undone in one swipe of a hack journalists pen.</p>
<p>Idiocy. Pure idiocy.</p>
<p>This is idiocy why?</p>
<p>I suspect you don't understand the work of any of those people you've named, as none of it says computers cannot do mathematics. Godel's main result is on provability, and it applies equally well to humans and computers. Turing's most famous result (actually Church's) is on what can be computed, but there's no indication that mathematics cannot be done by a Turing Machine, and indeed it's never been shown that a human brain is strictly more powerful than a Turing Machine. Finally, Cook is known for the notion of NP, which is simply a way of categorizing algorithmic complexity.</p>
<p>None of these says computers cannot do math.</p>
<p>Indeed, the approach the article mentions - computers grinding through and mechanically checking loads of possible results, flagging potentially-interesting ones - sounds very much like computers already doing math, albeit not in exactly the same way humans do. There are certainly people in the field who would and have called this math (e.g., Shamos).</p>calmo commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535e235bf970c2008-11-08T06:53:57Z2008-11-08T06:53:58ZcalmoThomas Hales, one of the authors writing in the Notices, says that such a collection of proofs would be akin...<p> Thomas Hales, one of the authors writing in the Notices, says that such a collection of proofs would be akin to "the sequencing of the mathematical genome".</p>
<p>Dang! how did "akin to" get in there?<br />
<br />
Are we all that impressed with the dissection of the human genome? [This means you (buffalo-proof non-mathematicians who aren't going to be pushed around by some clown who couldn't manage the microbiology and defaulted to the foundations of mathematics) remain unswayed by the suspect rigor present in "akin to".]<br />
Or it could B that Hales is waiting for his comet (some of us miss our period) [You just wait, there will be a Calmo's Comet one day.]<br />
Or it could B that "akin to" is only the suspect English language so horribly inadequate to the task at hand which can only B expressed (you more rigorous philosophers out there will go on strike for "notated"...and of course I cave in immediately to any philosophical demand, you?) by the object language...not exactly a proprietary secret, but don't expect all the axioms to be written in a standardized notation.<br />
Ok, time to go sharpen my buffalo horns.</p>Walt commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535db13ad970b2008-11-08T01:21:53Z2008-11-08T01:21:53ZWalthttp://arsmathematica.netPatrick, to be fair, humans manage to do math despite Turing and Goedel, so it's certainly plausible that computers could...<p>Patrick, to be fair, humans manage to do math despite Turing and Goedel, so it's certainly plausible that computers could develop the ability to the level of humans. (Not that I think it's likely that it will happen soon.)</p>Patricia Shannon commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535db0dde970b2008-11-08T01:07:45Z2008-11-08T01:07:46ZPatricia Shannonhttp://www.patriciashannon.blogspot.comComputers Effective In Verifying Mathematical Proofs is the header of this story at http://www.sciencedaily.com/releases/2008/11/081106153638.htm<p>Computers Effective In Verifying Mathematical Proofs</p>
<p>is the header of this story at<br />
<a href="http://www.sciencedaily.com/releases/2008/11/081106153638.htm" rel="nofollow">http://www.sciencedaily.com/releases/2008/11/081106153638.htm</a></p>CBBB commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535dacf60970b2008-11-07T22:51:07Z2008-11-07T22:51:09ZCBBBWhat's described in the article sounds more like proof verification rather then computers devising their own proofs.<p>What's described in the article sounds more like proof verification rather then computers devising their own proofs. </p>Patrick commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535da7510970b2008-11-07T20:31:33Z2008-11-07T20:31:34ZPatrickcalmo: I have a bachelor degree in math and I work as a software developer. Having worked with computers for...<p>calmo: I have a bachelor degree in math and I work as a software developer. Having worked with computers for many years, I have a healthy respect for their limitations. Granted, the bit about verification is interesting and makes sense (i.e. verification is typically much easier than finding solutions/proofs). The statement about computers doing math on their own is, in my opinion, rubbish.</p>
<p>a: Perhaps ... in the sense that the incompleteness theorem applies equally to humans as it does to computers. Fortunately, humans have the capacity to be creative, while computers do not.</p>Patricia Shannon commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535da4747970b2008-11-07T19:13:09Z2008-11-07T19:13:09ZPatricia Shannonhttp://www.patriciashannon.blogspot.comI have an M.A. in math. Every branch of mathematics is based on a few definitions. The same symbol, eg.,...<p>I have an M.A. in math. Every branch of mathematics is based on a few definitions. The same symbol, eg., "+", is defined differently depending on the branch of mathematics. The proofs ultimately depend on the particular definitions of the appropriate branch of mathematics.<br />
</p>Eric Dewey, Portland, Oregon commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535e12bb8970c2008-11-07T18:27:49Z2008-11-07T18:27:49ZEric Dewey, Portland, Oregon"To get around these problems, computer scientists and mathematicians began to develop the field of formal proof. A formal proof...<p>"To get around these problems, computer scientists and mathematicians began to develop the field of formal proof. A formal proof is one in which every logical inference has been checked all the way back to the fundamental axioms of mathematics."</p>
<p>Two questions come to mind: </p>
<p>1) is going back to fundamental axioms is going far enough back?</p>
<p>2) does achieving the perfect purity of a formal proof prevent human beings using the "proven" mathematics, or models derived from them (complex derivatives!) from blowing up the economy again? </p>
<p>My sense is that, while these proofing programs are definitely advancing human knowledge overall, the answers to both these questions is "No"; in the latter case particularly, because it will simply make it more likely that decision-makers will be over-confident in the math used to support foolish decisions.<br />
</p>calmo commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535da02e7970b2008-11-07T17:31:12Z2008-11-07T17:31:12ZcalmoIssued from the American Mathematics Society, but is it about math, or its foundations...which might make it about logic, a...<p> Issued from the American Mathematics Society, but is it about math, or its foundations...which might make it about logic, a branch of philosophy, yes?<br />
Do you have a math background? a mathematical hobby? an active membership in a math club? a voracious reading practice in mathematics? the foundations of mathematics?<br />
Me neither.<br />
But I am somewhat numerate.<br />
And I do know what a formal proof looks like, what a metalanguage is, who Godel was, ...basically I might be able to write an article like this popularizing the use of computers and how they might be used in "validating" (there is another technical (unarguable and precise) mathematical use of the word "valid") mathematical proofs.<br />
I wonder how a meeting of the string theorists ("we don't need your 'reality'") and these authors ("we have not yet exhausted all the possible exceptions to your claims") would go? </p>a commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535e10bc6970c2008-11-07T17:26:12Z2008-11-07T17:26:12ZaIMHO Godel says nothing about with whether computers can do math or not.<p>IMHO Godel says nothing about with whether computers can do math or not.</p>Organic George commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535d9de48970b2008-11-07T16:29:20Z2008-11-07T16:29:20ZOrganic GeorgeProof? Well it all depends on the baseline assumptions, Garbage in Garbage out.<p>Proof? </p>
<p>Well it all depends on the baseline assumptions,</p>
<p>Garbage in Garbage out.</p>Patrick commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535e0c9f9970c2008-11-07T15:28:31Z2008-11-07T15:28:32ZPatrickI was with the author until he let fly with this rubbish: "We may be close to seeing how computers,...<p>I was with the author until he let fly with this rubbish:</p>
<p>"We may be close to seeing how computers, rather than humans, would do mathematics."</p>
<p>Wow. Godel, Turing and Cook undone in one swipe of a hack journalists pen. </p>
<p>Idiocy. Pure idiocy.</p>a commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535e0a8ee970c2008-11-07T14:26:40Z2008-11-07T14:26:40ZaAs the article says, you still need the mathematician to translate his informal proof into a computer proof.<p>As the article says, you still need the mathematician to translate his informal proof into a computer proof.</p>Alex Tolley commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535d98254970b2008-11-07T13:57:28Z2008-11-07T13:57:28ZAlex Tolleyhttp://www.mymeemz.comSerlin: "Another great way computers can basically do proofs, although not formally, is in calculating probabilities by simulation." You can...<p>Serlin: "Another great way computers can basically do proofs, although not formally, is in calculating probabilities by simulation."</p>
<p>You can do this in some cases, but for conjectures a single false case negates it. This would mean testing an extremely, possibly intractable, space of cases to test the conjecture, where just one case, a black swan, in this space that fails to work would be a disproof. If the space of cases is finite, then a computational approach along those lines is useful.</p>ken melvin commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535e08db5970c2008-11-07T13:34:56Z2008-11-07T13:34:56Zken melvinBeen to string theory lectures, back when, that reminded me of the old joke of the prisoners who numbered the...<p>Been to string theory lectures, back when, that reminded me of the old joke of the prisoners who numbered the known jokes then told them by just saying the number.</p>Richard H. Serlin commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535d95941970b2008-11-07T10:59:59Z2008-11-07T10:59:59ZRichard H. Serlinhttp://richardhserlin.blogspot.com/"It is sobering to realize that the means by which mathematical results are verified is essentially a social process and...<p>"It is sobering to realize that the means by which mathematical results are verified is essentially a social process and is thus fallible.", yes, but as he later alludes, this is not really a problem with short, uncomplicated proofs. There, you can really check carefully, line by line. The potential problem is with the many important very long, very complicated proofs, some dozens, or even hundreds of pages long.</p>
<p>Another great way computers can basically do proofs, although not formally, is in calculating probabilities by simulation. For example, suppose I wanted to find the probability of getting two heads in a row flipping a fair coin. I could prove this the old fashioned way writing out a proof essentially saying that there are 4 possible outcomes, HH, TT, HT, and TH, each with equal probability, so the chance of HH is 1 in 4. Or, I could just have the computer spend a few seconds to flip 50-50 virtual coins 1 million times, and count up the number that end up HH. It will almost surely be extremely close to one-fourth.</p>
<p>In this case it wouldn't be worth running the simulation. The proof is easy to do the old fashioned way, but many important probabilities in economics, finance, and elsewhere are very long and hard to deduce with standard analytical mathematics, but relatively quick and easy to find by just having the computer see what happens when the events occur virtually 1 million times. <br />
</p>Zipf commented on '"The Sequencing of the Mathematical Genome"'tag:typepad.com,2003:6a00d83451b33869e2010535d953da970b2008-11-07T10:36:42Z2008-11-07T10:36:42ZZipfI recall there have been some efforts along these lines. Notably, the Mizar Project. However, that seems to be built...<p>I recall there have been some efforts along these lines. Notably, the Mizar Project. However, that seems to be built around a library of human-entered and computer validated proofs. Sadly, its markup language appears antiquated.</p>