tag:blogger.com,1999:blog-3722233.post2041922047028548572..comments2024-09-11T19:08:31.467-05:00Comments on Computational Complexity: The Future of Education is PersonalLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger15125tag:blogger.com,1999:blog-3722233.post-29918864138017223452022-12-07T07:34:24.944-06:002022-12-07T07:34:24.944-06:00Sorry Pascal, I was changing the subject. Overly s...Sorry Pascal, I was changing the subject. Overly suddenly. Yes, using computers as a tool is a sensible idea. I don't think the ability to generate random, often dizzy, proof-like things is likely to be useful, but I'm a pessimist, and some of the people doing that sort of stuff are envisioning math without mathematicians. My understanding is that the current computer math tools are way more interensting (that is, way more capable of doing abstract math) than MACSYMA or even it's current derivitives (MATLAB, Malple, Wolfram, whatever). Whether that (modern abstract math) stuff combioned with LLM-like things will be interesting is a question.<br />DJLhttps://www.blogger.com/profile/04036156397398405817noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-33268745115744406242022-12-07T02:30:43.474-06:002022-12-07T02:30:43.474-06:00DJL: I was not suggesting to get rid of human math...DJL: I was not suggesting to get rid of human mathematicians, but to use AI as a tool. For instance, since you mention the claimed proof of the ABC conjecture: one could envision a (far away?) future where the authors could feed their human-written proof to the AI, and obtain a formal proof as output. If the AI gets stuck, it could ask for hints. If the authors still cannot obtain a formal proof, then they do not have a proof at all by the standards of year 2***.Pascalhttps://www.blogger.com/profile/14201150679841329835noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-30938963697245070952022-12-06T23:28:05.883-06:002022-12-06T23:28:05.883-06:00Are college students still taught math that MACSYM...Are college students still taught math that MACSYMA could do in 1975?<br /><br />(Just in case that's too obscure, (if memory serves) by the mid-70s, symbolic math programs could find the closed form for any integral that had a closed form.)DJLhttps://www.blogger.com/profile/04036156397398405817noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-44581378065100614852022-12-06T20:19:58.470-06:002022-12-06T20:19:58.470-06:00Presumably Shtetl-Optimized has been discussing th...Presumably Shtetl-Optimized has been discussing this, but there has been discussion in the math comminity about what automated proof generation and checking means for math as a human endeavor. The LLM-fans envision a world in which mathematicians are no longer useful/meaningful. It seems to me that progress in math is about finding and using relationships between previously unrelated things, e.g. the use of eliptic functions (was it?) in the proof of Fermat's last theorem, the step in Mochizuki's ABC Conjecture proof that some folks consider not proven. That is, a proof isn't just a random collection of statements that follow logically from the preceding one, but a thing with an overall structure. And what needs to be proved is a question not for machines, but for we humans.<br /><br />See: https://siliconreckoner.substack.com/<br />for deeper discussions of this.<br /><br /><br />DJLhttps://www.blogger.com/profile/04036156397398405817noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-36296592200591360982022-12-06T18:35:06.392-06:002022-12-06T18:35:06.392-06:00Do kids today still bother being taught math a cal...Do kids today still bother being taught math a calculator can solve?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-9757119301748338592022-12-06T05:58:13.479-06:002022-12-06T05:58:13.479-06:00One way to do something useful with this technolog...One way to do something useful with this technology might be to generate proofs that can be checked by formal proof verifiers such as Coq, Isabelle, etc... Has this been tried yet?Pascalhttps://www.blogger.com/profile/14201150679841329835noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-22415838997028341742022-12-06T05:07:30.497-06:002022-12-06T05:07:30.497-06:00The last I heard, MOOCS had the problem that the p...The last I heard, MOOCS had the problem that the percentage of people who succeeded in completing one was extremely low. Remote learning, even with a live instructor, did pretty badly during the pandemic. I have good intentions of taking the MIT linear algebra course, but haven't finished reviewing the prereqs. I'll fess up as to how well I do, if I ever get around to it. If.)<br /><br />Also, the Google certificates world is odd: a non-comp-sci programmer with 5 years of experience has written thousands of times as much code as a comp. sci. grad. But is concerned with proving his/her abilities. How do you motivate folks to make it through freshman calculus? That's a much harder problem.<br /><br />(The self-driving car has the problem that it doesn't solve the problem that the private car is a bad idea; it may even make the problems of the private car worse. Of course, most Americans think there's nothing wrong with the private car...)<br />DJLhttps://www.blogger.com/profile/04036156397398405817noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-82100086840039978842022-12-06T04:56:07.788-06:002022-12-06T04:56:07.788-06:00That second "proof" very much looks like...That second "proof" very much looks like it was written by an LLM. Oops, the first one does also. Anyway, both are bogus.<br /><br />Scott Aaronson embedded himself in an LLM group doing math and commented that it created some amazing stuff, but also made "freshman errors". I think that this is a fatal problem. The AI fans don't. YMMV, as they say.<br /><br />(I'm personally aghast that anyone takes the LLM stuff seriously. Just like the Markov Chain language models, LLMs generate stuff with no concern for (since they have no ability to have concern for) the real world. I don't get it. Lots of people think it's kewl. The world has gone completely nuts, it seems to me.)<br /><br /><br />DJLhttps://www.blogger.com/profile/04036156397398405817noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-72323974121950364102022-12-06T03:50:05.688-06:002022-12-06T03:50:05.688-06:00Was the previous comment written by an AI? Is this...Was the previous comment written by an AI? Is this comment written by an AI? I know the answer to exactly one of these two questions - RQRVLVQQAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-50097852263691416272022-12-05T20:10:37.567-06:002022-12-05T20:10:37.567-06:00What is a proof that there are an infinite number ...What is a proof that there are an infinite number of primes?<br /><br />One way to prove that there are an infinite number of prime numbers is through a proof by contradiction. Suppose that there are only a finite number of prime numbers, and let's call that number P. We can then create a new number N by multiplying all of the prime numbers together and adding 1 to the result. For example, if P = 3, and the prime numbers are 2, 3, and 5, then N would be equal to (2 * 3 * 5) + 1 = 31. Because N is a whole number greater than 1, it must be either prime or composite.<br /><br />If N is composite, then it must be divisible by at least one of the prime numbers, because those are the only numbers that N can be divided by (since it is the product of all the prime numbers and 1). But if N is divisible by any of the prime numbers, then the remainder would be 1, which is not possible because N is a whole number. Therefore, N cannot be composite.<br /><br />If N is not composite, then it must be prime. But this is a contradiction, because we assumed that there are only P prime numbers, and N is a new prime number that is not in our original list of P prime numbers. Therefore, our assumption that there are only a finite number of prime numbers must be false, and there must be an infinite number of prime numbers. This is just one way to prove that there are an infinite number of prime numbers; there are many other ways to prove this as well.<br /><br /><br /><br /><br /><br />What is another proof?<br /><br />One way to prove that there are an infinite number of prime numbers is to use the fact that there are an infinite number of whole numbers. Because prime numbers are a subset of the whole numbers (i.e. all prime numbers are whole numbers, but not all whole numbers are prime numbers), it follows that there must also be an infinite number of prime numbers.<br /><br />Here's how this proof works: suppose that we have a list of all the prime numbers, and let's call that list P. We can create a new number by adding 1 to the largest number in P. For example, if the largest number in P is 19, then the new number would be 20. Because 20 is a whole number, it must either be prime or composite.<br /><br />If 20 is composite, then it must be divisible by at least one of the prime numbers in P. But if 20 is divisible by any of the prime numbers in P, then the remainder would be 0, which is not possible because 20 is a whole number. Therefore, 20 cannot be composite.<br /><br />If 20 is not composite, then it must be prime. But this is a contradiction, because we assumed that the list P contains all of the prime numbers, and 20 is a new prime number that is not in our original list P. Therefore, our assumption that there is a finite number of prime numbers must be false, and there must be an infinite number of prime numbers.<br /><br />This proof uses the fact that there are an infinite number of whole numbers to show that there must also be an infinite number of prime numbers. However, it is important to note that this proof relies on the fact that we can always add 1 to the largest number in our list of prime numbers to get a new whole number. This fact is not always true, because there may be some largest whole number that cannot be created in this way (e.g. if the largest whole number is the number "infinity"). In that case, this proof would not work.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-90780354952188104792022-12-05T18:06:14.919-06:002022-12-05T18:06:14.919-06:00I was inspired my a robotics talk recently -- usin...I was inspired my a robotics talk recently -- using robots to get kids to relax at hospitals -- to reread Neal Stephenson's "Diamond Age". Interestingly and also to advance plot points, the tutor in the Diamond Age was an AI, but because text to speech still sounded too mechanical, it was performed by actors in the cloud. My colleague who works in audio says that this tracks. Jason Hartlinehttps://www.blogger.com/profile/14571631306936100699noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-1495202677693557422022-12-05T16:33:13.030-06:002022-12-05T16:33:13.030-06:00A Young Lady's Illustrated PrimerA Young Lady's Illustrated PrimerAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-48238270411378240202022-12-05T16:00:26.737-06:002022-12-05T16:00:26.737-06:00The state of self-driving cars is pretty impressiv...The state of self-driving cars is pretty impressive although it will take longer to get social acceptance than I thought. MOOCS are disrupting higher ed, look at Southern New Hampshire or the Google certificates, each with about 150K students. ML will be disruptive as well.Lance Fortnowhttps://www.blogger.com/profile/06752030912874378610noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-73553289329635171862022-12-05T10:20:10.931-06:002022-12-05T10:20:10.931-06:00Before the current AI craze, education was suppose...Before the current AI craze, education was supposed to be "disrupted" by MOOCs. So far, it looks like the impact has been rather limited.Pascalhttps://www.blogger.com/profile/14201150679841329835noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-83262895997233187522022-12-05T08:01:41.622-06:002022-12-05T08:01:41.622-06:001) Having other people in the class bring up point...1) Having other people in the class bring up points, ask questions, is valuable and that will be lost. However, in a 1000-person Calc I or Discrete Math class, thats lost anyway.<br />2) Lance - you also thought that my now we would have self-driving cars. And I'm still waiting for my Jet Pack! There may still be both tech and sociological obstacles to your prediction for the future. This may mean it takes MORE time than you think, or it may mean that it never happens. gasarchhttps://www.blogger.com/profile/03004932739846901628noreply@blogger.com