"Pure Mathematics" seems remote from a lot of practical and "artful" techniques, but PM is usually how hidden mechanisms are discovered. Sometimes it's the other way around, for patterns of chaos in paintings and transmission of data. Focused interest attracts others simply by curiosity.
The title of this video (and of the talk) is misleading. The two cultures referred to by C. P. Snow are the sciences and the humanities, not science and art. There is a difference, there. Especially, the link between mathematics (patterns, mostly) and music has been acknowleged for ages - take, for instance, the famous book "Gödel, Escher, Bach", by Douglas Hofstadter (1979). This guy's central message is nothing new. Personally, I would say that the great devide is between science and theology (no, I don't agree with Stephen Jay Gould's "non overlaping magisteria"). Art is compatible with both. Within science, there is, perhaps, some tension beween physical sciences and social sciences (and postmodernism is bullshit - google the "Sokal affair"). I have the impression that the lecturer in this video didn't spend too much time thinking about the "two cultures". He didn't get the point. In any case, I see no problem at all between science and art. Hell, we even have "science fiction" as a literaly genre!
He made his point well. I don't know the context of the entire event, but he spoke directly to mathematics and art as the two cultures in the beginning. Then showed and spoke upon many ways that they do combine in positive ways. It may be nothing new but it helped me to understand some of what I've been feeling as I move into mathematics after almost a quarter century of writing poetry and stories.
Once again arts is described as theater or music or similar. It isn't, it's so much more than that. I agree that his starting point is flawed and biased especially with relation to the composer who applied mathematics to his work. The Beatles discredited and put to bed this deconstruction of the tonal scale. He then adds that an insect also sings in a mathematically pattern. The quote of Henri Poincare so depicts the mathematically minded view of creativity it is embarrassing, and we are living with its consequences today. Sorry, you might be correct, but you think in a way that many of us don't, like a labyrinth, and we all know what lurks here.
deadeaded Yeah, I see what you mean. Is it bad that I'm not overly impressed though? So, I read up about the one that re-proved a section of the principia mathematica, then I read a bit of the principia itself and, well I can see why Russell said that writing it deadened his mind! I don't know how to put this other than it's very symbolic and derivational, so it doesn't surprise me all that much that a computer could churn it out. If you told the computer to give a result when it derives an equation you already know is significant, and give it a preference to simplify when it can, it seems that you could just set it off to combine all the rules you start it off with in as many ways as possible. It feels a bit monkeys on typewriters. Could the program, on the other hand, find one off the more inventive proofs of something like Pythagoras' theorem though, without giving it some major hints that you know the computer can take its trial and error technique to eventually get there?
There are computer proofs now that humans find rather hard to read (and, quite to the point, classification of finite simple groups seems to be one of them - do you know the order of Monster Group btw.?)
Slawomir P Wojcik You have to be more specific: are you talking about finite examples that computers were required to be able to check, with humans proving why these finite examples being true would imply a more general result, i.e. the 4 colour problem? You mention finite simple groups, but what precisely did computers prove about their classification and can you link a source to the proof?
You might be right, in a sense. I probably wandered too far, but let me think, if I were to start the whole project again, how would I proceed. First of all, I would motivate myself. Why would I want to know what the simple groups are? Because they are the building blocks of the structure of this world. They show us how particles and atoms can combine and to what dimensionality this associations belong. They are limits of possible in this world. Even the simplest of them - cyclic groups of prime order - are being catalogued by computers - nobody searches for prime numbers by hand any more). Then, there comes a next cohort, the alternating groups and here, although the task is simple - just quotient Sn by Z/2Z - it is better done on matrix representations than by hand, or so I would imagine. Then, yes, there are groups, that have had been discovered by hand, like, mentioned in the video, Mathieu groups, but the process of checking of whether a group is simple is a negative one, so, I imagine (I - rather obviously, never done this) it must work as a sieve. First one excludes some group orders which determine that there must be at least one normal subgroup. Then one has to find a pair of generators of a group, that doesn't "split" (every Sn can be generated by a pair, but since one of these is a 2 cycle all the thing can be split into Z/2Z and An). So the question of fishing for a pair of generators stays (why a pair - if it were a triple than every pair would generate a normal subgroup within) This is simple, in a sense, because as the size of a group grows the probability that any pair of elements of a group are its generators grows as well. Then there is another helpful moment: every group is a subgroup of some permutation group and these have very simple matrix representations (in ever higher dimension). In other words, it can be seen without computer, that you need a lot of time (more than a lifetime, I guess) and luck without ... computer. If you find any fault in this argument, please, let me know. I might not be able to get much further in group theory, but, at least, some people reading it might. And why did I say simple groups, not "finite simple groups"? I don't believe in Ur elements. And it seems to be a question of belief (and convenience for others) So, here comes the most important remark, that I want to make. There is no such thing as a perfect AI. There are all the limitations of our thinking, and, they are pitfalls of every thinking AI included. There are questions of decidability (all this Red Queen race of axiom schemata or surreal numbers - if you choose only some infinitesimal numbers would they be inherently bent and always form one or another conic?) AI is a way to go further, but it isn't a way to absolute truth nor absolute happiness. Some derive great happiness from being better than others (criterion?), others are happy to let all of us move somewhere slightly better. This is a rather unpopular direction of late Here is a link to the one mentioned in the article brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/ The algorithms are not open source, and if you read the above you might (you might have had before) guess why - this is stuff with not only civil applications
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"Pure Mathematics" seems remote from a lot of practical and "artful" techniques, but PM is usually how hidden mechanisms are discovered. Sometimes it's the other way around, for patterns of chaos in paintings and transmission of data. Focused interest attracts others simply by curiosity.
The title of this video (and of the talk) is misleading. The two cultures referred to by C. P. Snow are the sciences and the humanities, not science and art. There is a difference, there. Especially, the link between mathematics (patterns, mostly) and music has been acknowleged for ages - take, for instance, the famous book "Gödel, Escher, Bach", by Douglas Hofstadter (1979). This guy's central message is nothing new.
Personally, I would say that the great devide is between science and theology (no, I don't agree with Stephen Jay Gould's "non overlaping magisteria"). Art is compatible with both. Within science, there is, perhaps, some tension beween physical sciences and social sciences (and postmodernism is bullshit - google the "Sokal affair"). I have the impression that the lecturer in this video didn't spend too much time thinking about the "two cultures". He didn't get the point. In any case, I see no problem at all between science and art. Hell, we even have "science fiction" as a literaly genre!
He made his point well. I don't know the context of the entire event, but he spoke directly to mathematics and art as the two cultures in the beginning. Then showed and spoke upon many ways that they do combine in positive ways.
It may be nothing new but it helped me to understand some of what I've been feeling as I move into mathematics after almost a quarter century of writing poetry and stories.
A literary genre? As in literature? As in the humanities?
Once again arts is described as theater or music or similar. It isn't, it's so much more than that. I agree that his starting point is flawed and biased especially with relation to the composer who applied mathematics to his work. The Beatles discredited and put to bed this deconstruction of the tonal scale. He then adds that an insect also sings in a mathematically pattern. The quote of Henri Poincare so depicts the mathematically minded view of creativity it is embarrassing, and we are living with its consequences today. Sorry, you might be correct, but you think in a way that many of us don't, like a labyrinth, and we all know what lurks here.
Why this man getting so much hate
I wonder, what is the density of true statements in the set of all statements?
is the set of all statements finite
Computers can't churn out proofs. I they could we would have perfected AI
deadeaded Evidence?
deadeaded Yeah, I see what you mean. Is it bad that I'm not overly impressed though? So, I read up about the one that re-proved a section of the principia mathematica, then I read a bit of the principia itself and, well I can see why Russell said that writing it deadened his mind! I don't know how to put this other than it's very symbolic and derivational, so it doesn't surprise me all that much that a computer could churn it out. If you told the computer to give a result when it derives an equation you already know is significant, and give it a preference to simplify when it can, it seems that you could just set it off to combine all the rules you start it off with in as many ways as possible. It feels a bit monkeys on typewriters. Could the program, on the other hand, find one off the more inventive proofs of something like Pythagoras' theorem though, without giving it some major hints that you know the computer can take its trial and error technique to eventually get there?
There are computer proofs now that humans find rather hard to read (and, quite to the point, classification of finite simple groups seems to be one of them - do you know the order of Monster Group btw.?)
Slawomir P Wojcik You have to be more specific: are you talking about finite examples that computers were required to be able to check, with humans proving why these finite examples being true would imply a more general result, i.e. the 4 colour problem? You mention finite simple groups, but what precisely did computers prove about their classification and can you link a source to the proof?
You might be right, in a sense. I probably wandered too far, but let me think, if I were to start the whole project again, how would I proceed. First of all, I would motivate myself. Why would I want to know what the simple groups are? Because they are the building blocks of the structure of this world. They show us how particles and atoms can combine and to what dimensionality this associations belong. They are limits of possible in this world. Even the simplest of them - cyclic groups of prime order - are being catalogued by computers - nobody searches for prime numbers by hand any more). Then, there comes a next cohort, the alternating groups and here, although the task is simple - just quotient Sn by Z/2Z - it is better done on matrix representations than by hand, or so I would imagine. Then, yes, there are groups, that have had been discovered by hand, like, mentioned in the video, Mathieu groups, but the process of checking of whether a group is simple is a negative one, so, I imagine (I - rather obviously, never done this) it must work as a sieve. First one excludes some group orders which determine that there must be at least one normal subgroup. Then one has to find a pair of generators of a group, that doesn't "split" (every Sn can be generated by a pair, but since one of these is a 2 cycle all the thing can be split into Z/2Z and An). So the question of fishing for a pair of generators stays (why a pair - if it were a triple than every pair would generate a normal subgroup within) This is simple, in a sense, because as the size of a group grows the probability that any pair of elements of a group are its generators grows as well. Then there is another helpful moment: every group is a subgroup of some permutation group and these have very simple matrix representations (in ever higher dimension). In other words, it can be seen without computer, that you need a lot of time (more than a lifetime, I guess) and luck without ... computer. If you find any fault in this argument, please, let me know. I might not be able to get much further in group theory, but, at least, some people reading it might.
And why did I say simple groups, not "finite simple groups"? I don't believe in Ur elements. And it seems to be a question of belief (and convenience for others)
So, here comes the most important remark, that I want to make. There is no such thing as a perfect AI. There are all the limitations of our thinking, and, they are pitfalls of every thinking AI included. There are questions of decidability (all this Red Queen race of axiom schemata or surreal numbers - if you choose only some infinitesimal numbers would they be inherently bent and always form one or another conic?) AI is a way to go further, but it isn't a way to absolute truth nor absolute happiness. Some derive great happiness from being better than others (criterion?), others are happy to let all of us move somewhere slightly better. This is a rather unpopular direction of late
Here is a link to the one mentioned in the article brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/ The algorithms are not open source, and if you read the above you might (you might have had before) guess why - this is stuff with not only civil applications