 Okay, good afternoon everyone, thank you for coming to this week's TSVP talk. So this week we're joined by Mark Wilden from Royal Holloway University of London. And Mark is going to be here at OIST until the 10th of November and has an office here in Lab 4 on floor F22D if you have lots of questions and want to track him down after today's talk. Mark received his PhD at the University of Oxford in 2004 and since 2018 has been a full professor at Royal Holloway. He's also spent the last year as an honorary senior visiting researcher at Harbrunn Institute for Mathematical Research at the University of Bristol and has also been awarded an ATIA Fellowship of the London Math Society and will be visiting the American University of Beirut in Lebanon I believe next year. So today Mark is going to tell us all about Primes, Partitions and Power Series. Alright, well thank you very much Learons for that very nice introduction. I'm delighted to be here at OIST, it's been a wonderful change of scene for me and a chance to catch up on my research and collaborate with all the strong mathematicians doing the subject-free lovers part of Learons group in representation theory and algebra combinatorics and that's what I want to give you some idea about today but first let me just explain briefly why I seem to have all these multiple affiliations. So my home university is indeed Royal Holloway part of the University of London but I've been on the comments for the previous academic year and the coming one at this place called the Hyderbron Institute which is attached to Bristol University and the Hyderbron Institute is a curious place it has two dual purposes one is to promote pure mathematics in Britain and it does that by sponsoring conferences workshops and most importantly postdoctoral fellowships and the postdocs fair and people like me that are seconded spend half their time in this wonderful state where we can do any kind of maths that we like but the Hyderbron Institute isn't just a charity it also exists to do cryptographic research on behalf of the British government and that's what we spend our other 50% of the time doing and that's why there isn't a picture of it there because I don't break the official Secrets Act when I'm on holiday and when I'm on important visits in Japan but instead there's a nice picture of Grunel's suspension bridge over the Avon Gorge anyway so enough of that what I want to tell you about is a part of a mathematical story which stretches from Euclid the guy on the left to the great Indian mathematician Ramanujan and beyond if time permits I will talk a bit about Turing and Gerdell and well this talk is intended really for a general audience not even perhaps for people as technically minded as you so if you find it all too basic well okay I apologize but at any rate you will have had the unique experience of having been at a mathematics seminar delivered by a professional mathematician wishing at the end that the speaker had gone into more technical detail well let's see how it goes so I took of this little snippet of Euclid from a picture which I think should be more famous than it is in my view it is vastly superior to the last supper and there's something controversial to start with it's Raphael's School of Athens and if you know any classical philosopher it's likely that they appear here somewhere but I can't promise to identify all of them for you the central figures are Plato and Aristotle as many others the badly dressed person there he's probably an academic no that's Diogenes famous for living in a barrel that one is Heraclitus famous for you can't step in the same river twice also famous for finding philosophy a rather depressing experience but still sticking with it all his life the one I admire most is the guy in the rather nondescript gray green cloak any guesses who that might be he's the only person I feel who actually looks like he might be interested in the people he's talking to everyone else very just sort of posing slightly oh yes I made it into this great fresco but Socrates that is he actually does seem to be talking to some people and of course that's entirely appropriate because Socrates is now famous for his dialogues Socrates of course would have made a very poor academic because he never published anything well everything we know about Socrates survives because of what his student Plato recorded Socrates was much more interested in dialogue in discussion with people he went around Athens pointing out to people in the politest possible way that they were thinking in a completely woolly way and that most of their ideas were wrong and you can imagine how popular this made him but I won't dwell on his well slightly unfortunate ending but anyway there's Socrates I'll come back to and the guy with the slate bending down that's Euclid and I want to start my story with part of Euclid's elements proposition nine from book 20 so it's all about prime numbers so I'll spot the prime and for bonus points spot the Gretchen dieg prime so I'm sure you all know that 31 is the unique prime number in this little list of people playing this curious game so 25 that's not prime because it's five times five everything else has a prime factor even 57 or there might have come as a surprise to Gretchen dieg who we believe lived at such a vast plane of abstraction even beyond the normal reaches of pure mathematics that when asked for example of a concrete instance of his theorem he said oh well just take a prime 57 for example but 57 is not a prime but it's perhaps more famous now because of Gretchen dieg okay just a little aside one is not a prime well says who well says mathematicians and it might come as a surprise to the Greeks because they thought one was a prime but we've now realized that it actually works best if you declare that one by definition is not a prime and the reason is that we want unique factorization and if we made one a prime then 57 would factor in the normal way but it would also factor as one times three times 19 one times one times three times 19 and this will just muck up our whole beautiful fundamental theorem of arithmetic so that's why we eventually realized that it was best to declare one not to be a prime but if that guy was bearing down on me loudly insisting that one is a prime i don't think i would be arguing with him okay so here are some of the primes they start two three five seven eleven sure you know here is a slightly bigger spiral so if i tell you that the smallest gap on a spiral's arm is 40 you should be able to work out why there are 16 arms to the spine because the phi function of 40 is 60 but anyway this all strongly suggests that the sequence of primes goes on for some time we should be able to find 27 2017 there that's the most recent prime you're going to have to wait another five years for the next prime year so here is the sequence then it will be followed immediately well as soon as possible by another prime going on a bit more a billion and seven is prime a billion and nine is prime so the big question is does the sequence of primes go on forever or perhaps it stops perhaps there's some really big number which is the biggest prime number and beyond that everything has a proper prime factor all the numbers are composite so i expect you know the answer but a thing of beauty is a joy forever as Keats wrote about the greece and urn but i think it applies just as much to mathematical truths so i'd like to just revisit euclid's proof for you and if you read euclid's original proof apart from being phrased in terms of lengths of lines because full of greeks everything was geometric rather than a number in fact they didn't even really have a very good notation for numbers they repurposed their alphabet for it but it doesn't work hugely well but anyway if you reread euclid's original proof you'll find that what i'm going to give you more as an example of the proof is essentially what he considered a vigorous demonstration so we'll do it by a little example suppose someone is under mistaken belief that two three and five are all the primes well we multiply them together to get 30 and we add one there is 31 and we now consider that when we divide by this supposed complete list of primes two three and five every time we see a remainder we always see a remainder of one but every number is divisible by some prime number so there must be some prime either it's 31 itself well it is or some prime we've missed but it's not two three or five and happens to divide 31 but either way there's a prime that isn't in that list so we were mistaken in thinking we'd collected all the primes there is he's going through suppose we do it with the first six primes well we multiply them together you add one you observe that there's always a remainder of one so again we deduce that there must be some new prime and in this case I put this in because 30,000 and 31 is in fact not prime it has a prime factorization but still we managed to pick up some new prime so the argument presented by euclid as it's usually given now is essentially just a general version of that and I'd like to present it as a little dialogue between our friends Socrates and Euclid but just to turn the tables on Socrates for once he's going to be the keen student and Euclid will be the one with all the answers so here is Socrates with a mistaken belief that he's collected all the prime numbers and so Euclid says to him please multiply them together and add one and well Socrates knows he's part of a Socratic dialogue and no one ever gets bored or says they've had enough in a Socratic dialogue so he says okay and Euclid makes the same observation I've made just slightly more algebraically but it still applies in fact maybe it's even easier to see now because when you divide n by a p i there is rather obviously going to be this remainder one leftover Socrates admits he's correct many Socratic dialogues are essentially Socrates saying things and then his interlocutor saying gosh you are right you are correct how clever you are anyway Euclid n is divisible by some point and Socrates is therefore forced to admit that there is a prime not in his list and Euclid concludes that this shows there are more than any finite number of primes because we could have run this dialogue we could have run this argument of any finite collection of primes and we would have found that there was one more and Socrates admits he is correct okay and well it's a wonderful thing about mathematics but anytime we can revisit this argument in the laboratory of our minds and be convinced ourselves afresh that there are infinitely many primes or if we want to be a bit more ambitious perhaps infinitely many primes congruent to three mod four or five mod six or not congruent to one mod 10 there are increasing levels of difficulty but I suspect you'll be able to work out how to do it all refinements of this Socratic or Euclidean argument there's one feature I'd like to point out about the argument because people often get this wrong and it makes it seem more complicated than it needs to be so we've got this statement P there are finitely many primes this is incorrect and it's logical negation which I phrased like this there are more than any finite number of primes because that's how Euclid's conclusion is often translated nowadays the Greeks didn't like to talk about actual infinity they were quite rightly aware that talking about infinite numbers can cause problems so instead well we would say there are more than any finite number which is I think how most people if pushed would understand the word infinite infinite means not finite so in the Euclid proof we showed not P is true by assuming P and getting a contradiction so since P is false we are obliged to conclude that not P is true everything is either true or false so notice that we did not do a classical proof by contradiction this is a proof by negation and the difference does actually matter to some people if it was a proof by contradiction we would have started with Q we'd have got a contradiction from not Q and from that we would be entitled to infer not not Q which for most mathematicians is logically identical to Q but not for everyone but the thing is Euclid's proof does not need to raise these issues even intuitionists have to accept Euclid's proof and I'll come back to this whole business of well what do we mean by mathematical statements and what do we mean by by proofs towards the end of my talk okay so I want to head on to things which are a little more typical of what I work with day to day so I'll start with compositions so a composition of a natural number is the way to write it as a sum of natural numbers possibly itself I allow that four is a composition of four and the eight compositions of four are all written there so a little exercise how many compositions are very free well I'm going to do it for you there's three two plus one one plus two that counts as a different composition and one plus one plus one so there's four and there's two compositions of two and one composition of one so one two four eight so perhaps by now you're guessing that the number of compositions is going to be some power of two but of course we would like to to prove and I'm going to show you the kind of proof that I admire most a bijective proof and the idea is that we consider the partial sums in these compositions so say I take one plus two plus one well one is one one plus two is three and one plus two plus one is four so its partial sums make the set one three four and you can do this to all of them the only partial sum of four is four three is three you add one you get four you're always going to end four because everything is a composition four but I put it in any way because that's where we stop so there are the eight subsets corresponding to the compositions and if I get rid of four I've now got these eight subsets and I think you can see that these are every possible subset of one two three and because I've made a bijection I've made a one-to-one correspondence with these subsets we should be able to play the same game going back so pick your favorite subset of one two three I shall just pick one three find it here imagine that they are the partial sums once you've shoved in four that means we must have started one then we added two then we added one so it corresponds to the composition one plus two plus one so this is all just to convince you that there is a bijective correspondence between the compositions we wanted to count and subsets not of one up to n but one up to n minus one so to count the number of compositions we just need to count the number of subsets and while I claim that this is something easier to do I feel it should be obvious I feel if I was allowed to go up to someone on the street oh dear really the language barrier would immediately be a problem for me which I'm not proud of imagine I'm back in the UK when I go up to someone on the street or more likely since I live in central Bristol I'm asked by a homeless person for some money I say to them I shall be happy to oblige but first may I have your opinion on this how many subsets do you think there are of one up to n or n minus one well I hope I'll be able to convince them that it was two to the power n minus one and that this wasn't a terribly deep fact because after all for each element of one up to n minus one you've just got a binary yes no choice do I put it in so you've got two choices for one multiplied by your two independent choices of two and so on and eventually you will reach two to the power n minus one so by our bijection and by our ability to count subsets this gives us a perfect count of the number of compositions and well what we've really done here is show that these sets are isomorphic two sets are isomorphic in the category of sets if and only if they have the same number of elements but the isomorphism was chosen to be particularly memorable we didn't just pair them up in some random way we had some structure to it and I want to just try to explain or give you some flavour of what mathematicians mean by isomorphic because it is absolutely central to what I get up to as an algebraist but it would take rather long time to set up the algebra so I'm going to do it in another way by doing some juggling so juggling is a very mathematical thing it has a notation called sight swaps and so far everything I've done is an instance of a single sight swap 3 3 3 I'm just waving my hands around in a complicated way but if I depart from that and do something a bit more ambitious then you can probably tell that the rhythm has changed and that was a non isomorphic trick but the difference between this and that is nothing except five years of practice mathematical mathematically that they are completely identical in structure so that was isomorphism versus non isomorphism if you remember one thing let's make it that so I'll now go on to the partitions that were part of my title so partitions are a special kind of composition where the numbers have to be written in decreasing order weakly decreasing non increasing if you prefer because you're allowed to repeat apart so the partitions of four is five of them four three one two two two one one and one one one so the number of partitions p of four is five and here's a little table showing the number for small n and well there's no longer any particularly obvious pattern I hope you'll agree although there are many very beautiful non obvious patterns I'll just draw your attention to one take all the n which are congruent to four modulo five so like four nine and fourteen you'll find the number of partitions is always divisible by five five thirty hundred and thirty five and well the mathematician I'll come to Ramon Nujan found a particularly spectacularly beautiful proof of this there are also somewhat bijective proofs where you take that set of partitions and partition it divided up into five sets which are visibly of the same size this is done using the famous rank or its successor the crank of partitions but I don't want to tell you all about that today it would be an entire talk said I'll tell you just a little bit about what concerns some of the early workers in the field of partitions so they wanted an asymptotic formula for how fast p of n grows and this is now something we can explore very neatly on the computer so here's a couple of graphs there's p of n and you can see it growing fairly quickly so your first guess might be perhaps it grows exponentially after all the compositions they grew exponentially with base two so perhaps partitions are like the compositions but they just grow a bit more slowly but when you take logs it doesn't really look like that because if it was exponential growth and you take logs it should be a straight line and this is really a bit more curved but if you mess around a bit more you might try thinking this looks a bit like the square root function so we'll try log p of n over root n and that looks like it might perhaps be heading to a limit slightly tendentious I know because of course I do you know what the answer is but you can demonstrate this if you do a log log plot then you can see the convergence quite neatly the time you've got to 10 to the 10 log p of n over root n is extremely close to what the eventual limit is so the Hardy Ramanujan theorem is the strongest version of this so what I try to demonstrate to you is that log p of n grows about like square root n so it has a sort of complicated kind of intermediate growth p of n is something like e to the root n and Hardy and Ramanujan proved this in a stronger form nailing down all the constants exactly so this little symbol here means that the ratio of the sides tends to 1 as n tends to infinity in fact to be completely honest Hardy and Ramanujan proved something even stronger they gave a divergent series for p of n and divergence divergent there isn't a typo but their series doesn't converge but it's still very useful for calculating p of n because although it diverges if you add up too many terms if you pick the right place to stop you will be within 0.5 of the correct value which is a really remarkable achievement and well I want to mention one of one of my papers which I'm quite proud of because it was my second published paper after a time I'd had a slightly fallow period and I was interested in partitions because of my research in representation theory and I used one of the tools from my work this this abacus which is a certain way of representing partitions to get asymptotic estimates for their number and I gave what mathematicians would call an elementary proof of this slightly weaker version of Hardy Ramanujan but here elementary should probably have a very heavy pair of inverted commas because what it means is I didn't use complex analysis I did build on what other people had done using real analysis and my proof involves a whole pile of epsilons and deltas but it was just a real analytic proof that even I could understand so yeah I was interested in partitions because they come up in my research and well I want to try to give you some idea of that and a good sort of linking point is this idea of generating functions so a generating function for a combinatorial sequence is a way of recording all the values of that sequence in a single object so we take p of zero p of one is the coefficient of x p of two is the coefficient of x squared and so on so what I've defined here by capital p of x is simply some kind of power series since we know the small values of p I can tell you how it starts all right it's one plus x plus 2x squared plus 3x cubed 5x to the 4, 7x to the 5, 11x to the 6, 15x to the 7, 22x to the 8 can utilize and a lot of my youth just writing out partitions I find it incredible this has actually been a source of employment for me so the thing is although there is no simple formula for p of n there is a very beautiful closed formula for its generating function and this is not actually stereotypical but it's still a little miracle one should enjoy so as a warm-up although I feel this audience probably doesn't need it we're going to do a sort of slightly easier special case so let q of n be those partitions where the parts are distinct so that means that you don't have any repetitions so for partitions of four four and three one are good distinct parts partitions but we don't count two plus two because that's got two as a part twice and we don't count two plus one plus one or one added up four times because they've all got repeated parts so we've now got another generating function q of x so that starts one plus x plus two x squared plus two x cubed plus two x to the four and so on and I claim that its generating function factorizes in this very simple way as an infinite product maybe you can see how this is going to work because say I want to count the distinct parts partitions of four well what's going to happen when I multiply this out how can I get a coefficient of x to the four well I could just take x to the four and one one one and that would be like the partition four or I could take three plus one and that would be it I can't take two plus two because I've only got one power of x squared so when I multiply this out I will get coefficient two attached to x to the four and you can play the same game with three and check that it will be three or two plus one so here is a complete proof when you multiply out the right hand side coefficient of x to the n is visibly number of ways to write x as a sum of natural numbers and that is what I mean by a distinct parts partition so we'll just ramp this up slightly and run the same argument to do all general partitions so my proposition says that p of x is this infinite product here and the proof is why you expand each factor as a geometric series so it looks a bit more like what we had for a distinct parts partitions in fact that would be just like one plus x squared stopping a bit sooner but you see now the menu of options open to us includes more different powers of x and we'll be able to use these different powers to encode a completely general partition as a way of multiplying out the brackets and here's the rule so say I take x to the m1 from the first bracket x to the 2m2 notice that all the powers are even so I'll be taking an x to the 2m2 from the second bracket next to the 3m3 from the third bracket and so on when I multiply it in this way I'll get a contribution of one to the coefficient of the sum 4m4 and so on all being omitted and this is exactly counting the partition that has m1 parts of size 1 m2 parts of size 2 altogether contributing to m2 m3 parts of size 3 altogether contributing 3m3 and so on so the coefficient you'll get of x to the n is p of n let's just do a quick example it's something that undergraduates are never taught but you can always turn a proof into an example so let's do this one three plus three plus two plus three ones so the multiplicities there are three lots of the part one one lot of the part two two lots of a part three so I take three times one one times two two times three picking out these powers so I've got x cubed x to the five x to the six I multiply them together to get x to be 11 which is indeed the size of this partition three six eight nine ten eleven and again you you can go back so pick a way to expand this product say I take x to the nine there just one there and x squared there well that means I took three parts of size three and two parts of size one so that was encoding the partition three three three one one I can't do you say the partition with a the partition 10 plus one because I would need the brackets which are somewhere over here but well the idea still works okay so that that gives you some idea of generating functions and I'd like to show you one more example where we perhaps use and prove something a bit less obvious so here's a proposition the number of partitions of n into odd parts is equal to the number of partitions of n into distinct parts so we've already discussed distinct parts and odd parts well that just means what it says so as little example there are eight odd part partitions of nine I've listed them here and there are eight distinct parts partitions of nine so if you look at some other cases you'll find the numbers agree to so can we make a bijection between them can you find some simple rule that will map from one side to the other but if you can and you've never seen this before you've done something seriously impressive there are bijective proofs some very beautiful ones but none are completely obvious and instead I'd like to show you how to do it using generating functions so here is a generating functions proof we will take the generating function for the odd parts partitions which I have the same idea I showed you before is a product like this except now we're only doing odd parts so I only want to see the odd numbers here if you remember before one over one minus x squared became one plus x squared plus x to the four and so on and we used that to get parts of size two well now I don't want parts of size two they're off the menu so I just get rid of that factor so the left hand side has this infinite product as its generating function and now I'm going to just manipulate it algebraically so for no very obvious reason but it's something you could you could do I insert a whole pile of ones and then I rearrange the numerator by sliding all of these factors a bit to the left so now I've got one minus x squared over the one minus x one minus x to the four from there over one minus x squared and so on and when you do the product of two difference of two squares identity on it you get one plus x one plus x squared one plus x cubed which I hope you remember it was only one slide ago is the generating function for the distinct parts partitions so if you accept this argument you are forced to agree that these two sequences the first sequence counting the odd parts partitions and the second sequence counting the distinct parts partitions you are forced to agree that they have the same generating function it might look different but it is the same but since they've got the same generating function they are the same sequence QED so I quite like this because it's a it's a non-trivial application of generating functions but but not not not too hard and it's a kind of proof that well I really enjoy so as I said there are also bijective proofs but but they need a bit more work and I think I won't show you one now but you can ask me at the end and or come and see me you know where to find me I want to talk to you about your work if you think you've got any problems that might benefit from some pure mathematical input well I'm very happy to talk to you and find out something about your research but here is a little one liner for the experts here the brow character table of a symmetric grouping characteristic two is square and of course you will remember if you've seen all this otherwise it makes no sense to you whatsoever but the columns are labeled by the partitions with no even parts and the rows are labeled by the partitions with distinct parts so there's a little algebraic proof excruciatingly high growl but an honest proof that you can get to the point where you understand all these objects without having first proved the theorem so why do we care about all these different proofs what's the point of having proofs at all why do we have multiple proofs that that's what I want to to end my talk with and well I claim that we want proofs partly I have to admit because we want to know what's true but this really isn't I think the only reason and particularly say when you're lecturing to undergraduates it can't be the only reason because how many undergraduate students well of course I'm sure you're all exceptional but take the typical undergraduate student I don't think they believe things like the fundamental theorem of calculus because they can retreat to their their mental gym and mentally rehearse the proof I think they believe it because they've been told it and they found it in the textbooks and perhaps because they have experience of knowing that it works so the point of proofs isn't just to decide what is true or false I think proofs are important to undergraduate teaching principally because they are repositories of ideas because proofs are correct arguments they are memorable arguments this is one reason why so much philosophy philosophical discussion I find hard to remember because I'm never completely convinced whether it's right but I find mathematical proofs particularly the elegant proofs I like so much highly memorable because they are convincing and they help to tie together all the different strands of mathematical thought in my mind so that would be my answer to what's the point of having all these multiple proofs because they they really give you genuinely different perspectives you know here's the representation for a proof for experts here is a hint there's a bijective proof earlier I gave you regenerating functions proof okay so proofs are clearly something important and perhaps surprisingly the relationship between proofs and truths was not completely clear I mean if it was for many many years people thought that they were the same and then people started to have their suspicions and realized they might not be so that's what I want to end my my talk with and these are tricky things to discuss and I find that perhaps the clearest way to present a lot of it is to use something even more modern than Gerdel who was writing in the 30s to use the language of modern computing so that that's where I want to finish my talk so computing as I expect you know began with this guy Alan Schuring here is an extract from his report it's slightly illegible but well I've read it more clearly he must realize the ability to put down a proof neatly on paper so it is intelligible is important for a first-rate mathematician and well his schoolteacher had a point because after all mathematics papers are mostly words I was reminded of this example by Professor Yuga Abdullah's very nice talk yesterday where he talked about PDEs and harmonic functions well this is a complete proof of Neuville's theorem in any number of dimensions notice that it is mostly words but anyway going back to Schuring and the foundations of computing this is a bomb the special purpose device you wouldn't really call it a computer because it doesn't fit the von Neumann definition of a stored program computer that came later in the war with Colossus and Schuring's later work at the University of Manchester but it was certainly a fiendishly important device for its time which he used to to crack enigma and now while Schuring is probably known thanks to Andrew Hodges wonderful biography and films like The Imitation Game mainly for his work at Bletchley Park but probably intellectually he deserved most to be remembered for this theorem which stated slightly and precisely but not so imprecisely that I feel was an issue there is no algorithm that will decide the truth or falsity of a mathematical statement so here are some examples of mathematical statements infinitely many primes that's true and we proved it one about partitions true improved infinitely many primes ending one well that's true and it can be proved but I haven't given you a proof today infinitely many primes ending two that's of course false there's only one in fact a real function is equal to its Taylor series when that Taylor series converges I'm sure you're all far too knowledgeable about this to be tricked very tempting but completely false and here are a few more statements where we just don't know one I quite like is is the partition function equally likely to be even or odd I think almost everyone is convinced that it is equally likely in some sharp asymptotic sense it's equally likely to be even as odd but we have no idea how to prove this and the strongest results in this direction are quite weak but notice that all these statements they are all statements where we strongly believe that we will one day find a proof could there be statements that are simply unprovable well that's something that in fact follows from Turing's theorem here and was already known at the time when Turing was writing it's a version of Goerdel's first incompleteness theorem when I feel Goerdel's theorem is often misunderstood I remember once I had a very long discussion with someone who's sort of perhaps been to some public lecture I'm sure it was a wonderful public lecture but his takeaway was essentially oh you mathematicians you think you're so clever but you can't prove everything which is sort of what Goerdel says but it's sort of not as well and I'd like to be a little bit more precise about it so here is Goerdel's theorem stated with I feel only justified in precision so we fix a formal proof system and there exists a mathematically true statement everything is either true or false but has no formal proof in that system I'm not going to take 20 extra minutes to set up what I mean by a formal proof system but but here's a representative example now a somewhat old-fashioned one no one uses this anymore but it was very important at its time Russell and Wright heads formal system they used in Principia Mathematica this is a formal proof that one plus one is two and you can see why we don't bother with them now why we prefer to present mathematics slightly informally but what I want you to observe is that this formal proof it's just a long string of symbols and in principle you can feed this into a computer and it will verify that it is syntactically correct okay so Goerdel's theorem says that there are correct statements that cannot be proved by any formal proof in the system and it follows from what Turing proved about the Enschiedung's problem so what Turing really showed is that there is no algorithm that will decide whether a Turing machine will halt and when you apply this to the statement Turing machine M halts you then find that there is no algorithm that will decide the truthful falsity of even this special class mathematical statements hence the formulation that I gave you earlier so we can now understand Goerdel's theorem as a corollary of Turing's theorem so here is a proof so what I'm going to show you first of all is that there is a form of incompleteness so suppose for a contradiction that our system is complete that every P is either provable or it's negation not P is provable and then we take our Turing machine and we let P M be the statement M halts and then we can spend week one trying to prove P M we can spend week two trying to prove not P M this is terribly like my life we can spend week three trying again to prove P M we can spend week four thinking you know that didn't work so well I'll go back to not P M but the critical difference is that because we're now enumerating four more proofs and these four more proofs are just finite strings of symbols and moreover we have assumed that every statement is either provable or its negation is provable we are going to win at some point we will find a proof but that means we can detect whether or not Turing machine M halts which according to Turing is something we cannot do so therefore there must be statements P such that neither P nor its negation is true sorry is provable take one of these statements either Q or its negation is true everything is either true or false but some things are not provable which is precisely what we wanted to show okay so that's a deduction of Gerdel's view in a way that can be made rigorous out of Turing's work on the halting problem and well as we approach an age where there will be working large-scale quantum computers we are constantly testing the limits of what it means to compute and well I find it very interesting that by understanding better and better what it means to compute we can still understand better and better what it means for things to be true or false or provable quite besides the joy of using computers like I did with those graphs to explore our wonderful mathematical world okay well thank you very much any questions for Mark do you have any examples of statements that are expected but not known to be unprovable that's a good question none leap to mind because many of the common statements of mathematics that we don't know about like perhaps the parity of p of n or the Riemann hypothesis in some sense they must be provable because the argument is that if they're not provable it can only be because they're true because if they're false you can just write down a counter example like for the Riemann hypothesis I can just write down non-trivial zero if it's false so a proof that the Riemann hypothesis is unprovable in a particular logical system has to be a proof that in fact it's true and it would moreover be a proof that would convince professional mathematicians or I think it would come as a surprise to them that you needed a detour through a particular logical system there are many interesting mathematical statements which are not of the artificial nature of the Gerdel sentence that would have come out of the kind of proof I showed you but are known to be unprovable you could look up say Hercules and the Hydra for one of my favorite examples but yeah these statements are the exception on the whole we believe that the axiomatic system we work in day to day is powerful enough to prove the things we care about any questions if there are no more questions for Mark excellent questions should I care about convergence arguably yes because I was manipulating with power series however it would be very easy to see that the radius of convergence was at least a half but I would claim the more important answer is no you do not have to care about convergence and I can justify this from two different points of view one is the way I set up generating functions simply as a device for storing all the numbers in the sequence at once and from that perspective a generating function is really just a formal object it's like one of those formal strings of symbols that made up the formal proof and you can manipulate these formal strings of symbols so that they form a ring with addition multiplication even division when you're not dividing by something in the unique maximal ideal and it will all work you don't need to care about convergence the ring of formal power series is a complete discrete valuation ring with the non-archimedial norm so whatever that means take it from me you don't have to care and I'm not I'm not completely wild in in having this view because well you could say we will wonderful book generating functionality where we discuss these issues and you know he says that for many purposes like the proof I showed you in the end we were just showing two sequences were equal you don't have to care about conversions but of course we might then want to get asymptotic results from our power series that that's what party and ramanujan did after all and then we have to care very much but you know just for getting things set up and understanding why the power series is what it is you can relax you have my permission for you needed all right if there are no further questions then let's thank mark once again