The Mathematical Truth | Enrico Bombieri

good evening ladies and gentlemen I'm Peter kadar director of the Institute for Advanced Study and it's my pleasure to welcome you here to this Institute lecture at the end of the lecture there'll be an opportunity for questions today our speaker is honoree Enrico Bambi re Enrico has been a professor in the School of Mathematics here at the Institute since 1977 and IBM von Neumann professor since 1984 he was born in Milan and completed his university studies there becoming a professor then in mathematics in Pisa in 1966 he was a member here in 1973 to 74 and then from 1974 he was professor at the squalor normally in Pisa in Rico's work ranges widely over many areas of mathematics from algebra and algebraic geometry to analytic number theory but it centers on number theory and analysis in 1974 he was awarded a Fields Medal for his work on the classical mathematical problem of the distribution of prime numbers in recent years his work has been mainly concerned with daya fantine approximation and daya Fantine geometry which is concerned with the solution of equations and inequalities using integers and apart from being one of the world's leading mathematicians Enrico as many other interests he's a talented artist and printmaker a distinguished collector of postage stamps and their seashells but today he's going to talk about the mathematical truth well the first thing can you hear me yes okay so the lecture will be on a topic which is really not my field because I'm not a logician or a philosopher but it's interesting to me because studying the question what is truth in mathematics helped me to some extent to understand what I'm doing so let's see first something which was said in 1653 in his book The Compleat angler Isaac Walton said that angling the art of fishing with with fly-fishing may be said to be so like the mathematics that it can never be fully learned at least not so fully but that there will still be more new experiments left for the trial other men that succeed us well this is still true today mathematics can never be fully learned so what is mathematical truth I will show now a couple of alternatives first one and absolute through this truth that's the end of the story there is no no choice about what is truth second well no no no no it's a relative notion through the pen only depends on the context in which in which you work in which you assert your truth other point of view well is a tautology I mean it just has no importance whatsoever for wel this the skeptical say does not exist never existed in fact and the some social scientists will say well is a product of culture and in studying what truth can possibly mean it turned out that maybe it's a mix of all these things so let me begin with the absolute which starts with the Platonic view that numbers in particular geometry and from their mathematics exist on in the Platonic world of ideas as absolutes well this is bit narrow the practice has shown that this view is a bit narrow and it's been some philosophers are proposed no alternative form of Platon is called plentiful platanus which allows for the existence of an objective set of distinct mathematics okay so the realist view is therefore that the mathematics exists independently of us but mathematician also believed that mathematics is not just formulas propositions and theorems and Hardy interview is quite explicitly we'll see in a moment the other view of mathematics is partially constructed yeah mathematics is there but we construct it at the same time so is a construction of the mind or perhaps the collective mind because really it comes the mathematics is the foreign body work of many people of all mathematicians from all times so the role of the mathematics the mathematician is like an architect and instead in platonic thing will be an explorer of the world which exists on its own the problem of formalism which we'll talk a little more in a moment is that infinite constructions quickly lead to paradoxes to contradictions to things which do not make sense at least in plain English so I'll give an example the rustle paradox is the the finest set set an aggregate of objects and you say I wanted to define from new set R with the elements of our are sets and the those sets which for which the set itself is not an element of this set so for example is the universe as a set also an element of universe well you decide but it's is a question of this type and the fact is if you take literally you see right away that it does not make sense so how you resolve these things well they push push it away and the just keep going and try to restrict the definition of set or the operations of set can you talk about this set of all sets is that a set or not and so that's a way mathematicians they keep changing the in the formalistic view they keep changing the rules excluding certain constructions or allowing other constructions and till things look more or less ok so mathematicians said they'd like to be to take the view the Platonic view that mathematics is there and we try to simply to understand it in practice mathematician is on the weekdays is a formalist on weekends is a platonic there are other views of mathematics the intuition is everything mathematics must be effectively defined mathematical entities do not exist until they have been constructed in Pearce's size mathematics the result empirical research it's like any other science like a church the quasi empiricist requesting the validity of mathematics that that's good postmodern istic view you start by destroying everything and then you put it together in a different way social constructivism is prodded mathematics a product culture and it does not exist until as being thought out mathematics is shaped by the fashions of the social group it well perhaps this may be true in some aspects there is a some fashion in mathematics at any given time because certain type of mathematical research maybe is more attractive there is a feeling that there are many many results new results to be discovered with a relatively simple effort and so everybody shifts into the field until has been explored completely nor nor becomes too difficult and so certainly there is some aspect of that on the other hand one can find many examples of mathematics done quite independently by different groups or different mathematicians and with essentially identical construction and I have an example in which I was part of that there were Japanese mathematician did something and I did exactly same thing quite independently and I was very surprised that turn out that way but if you think about it was not so surprising after all so fictional ISM but Mightiest meaningless in absolute is at best a useful fiction well maybe there is another type of mathematics applied mathematics the subject of study has its roots in the description of reality well let's see the opinion it's very hard to describe actual phenomena by simple mathematical models it's certainly century ago is really hard today mathematics has become rather sophisticated the models which were considered too difficult to study today maybe are approachable are the famous British mathematician in the book mathematicians apology took a dim view applied mathematics say mostly finest products of an applied mathematicians fancy must be rejected as soon as they have been created by the brutal but sufficient reason that they do not fit in fact well as an example sophisticated mathematical models investment finance turned out to be grossly insufficient to take into account the distinction between real wealth and paper profits as everybody knows by personal experience well this can be solved by introducing in the system what I call bonne biere is law not this bomb Gary but community father the banker bumpier lots of finance profits are on paper losses are in cash and perhaps if one will take into account such simple constraints things will be a little easier well mathematicians believe the mathematic objects are not just proposition theorems and what matters hardest view is the pattern in other words the whole construction is more important than the end result or a special statement like a theory and it says that in doing mathematics in the mathematicians works in a creative fashion like a painter or a poet so these ideas are very close to victim Stein for example aspects in which the aspect of things is very important and in the same way that a painter a paint a painting is not just a collection of molecules on a canvas and which produce certain vibrations or light which we where we see as colors and lines and shapes or even worse that is a collection of just atoms in a certain way is the aspect of the painting which is which matters which gives the substance to it so why aren't viewing this is the mathematics is designs relations what matter matters is the relation and in that respect it so studies how different things interact so patterns are aspects of relations and sometimes can be identified with relations but sometimes relations are not patterns so it's not quite the same thing anyway here is truth in the Renaissance here is a famous fresco of Raphael School of Athens in the Vatican and represents rational truth and there is the dual aspect of that theological truth is another fresco and I'm interested here in the rational truth and you will see here is Plato here Aristotle's this guy here is Euclid this one here is Pythagoras let's see here is Ptolemy and the roster and so many of the characters here represents the famous philosophers or scientists astronomers mathematicians of antiquity so let's see the next one - maybe this a little more clear the Tigers with and Euclid here the model for Euclid actually was a carpenter a friend of Raphael so the question about mathematics and truth is classical mathematics free from contradiction well that's not clear at all in fact my colleague Vladimir expressed some serious doubts that today is mathematics is free from contradiction and also studying ideas how to get around difficulties of this type it's obvious that some large part of mathematics today is certainly free from contradiction but how to get convinced about that this that's another question does mathematics deal with truth well maybe yes maybe no in fact sometimes there's a lot of mathematics which is made under assumptions which are not verified so even if the logic may be correct the conclusion may be wrong because the premise may be wrong so can true the truth truth is the same as proof or verification the answer is no we'll see in a moment truth or proof be achieved by consensus well that will be may be the social science point of view it's a question which is also you know you can give a whole lecture about that for example you start with the problem of refereeing the paper is how do you know that the paper is correct so you in the center referee and maybe if this turn out to be a bit difficulty center second referee maybe to the third one you try to get a consensus and then you accept it or maybe you go to some mathematician friend say you know this thing better than me is that correct he says yeah I think so the guy is pretty strong it's a good mathematician must be correct and then you keep going on so one has to be a little careful about that it's there it in notional probable truth that's a very interesting question we'll see at the very end and it is automatic verification by computer acceptable in mathematics some mathematicians say no way and I will say absolutely yes so the now talk a bit about truth in formalistic mathematics Hilbert proposed a program to obtain a complete axiomatization of mathematics and proof of consistency starting from the assumption of the consistency of a small number of intuitive basic axioms and it turns out that any sufficiently large model of mathematics cannot prove its consistency within itself and of the story so girdled brought down the whole Hilbert program and of course we don't know whether these are male Fraenkel set theory which is used today as foundational mathematics whether it really is sufficiently large but probably it is so I don't think that sir Melo Frankel theory proof can prove its consistency within itself so what we do well the formalization continued successfully with the Bourbaki group with the actualization large part of algebra analysis and geometry and some things were read flee this bit easier to formalize the other parts of mathematics were not so easy to formalize for example known nonlinear analysis and so and entire sectors were excluded at least initially from its program and it so the result was a bit of lopsided development at least in my view on the other hand the Bourbaki had a very good influence by unifying the language of mathematics the very positive influence now the formalization of truth in a formalistic model is possible how do you do that this was done by tarski in a famous paper it's not a long paper but the pretty deep thirsty solution is you have to deal with a very well-known paradox when I say this sentence is false what if it falls then it must be true if it's true and it's false so which is which how you solve this paradox so if you have the axiom the symbol a or negation of a is true so it means any state of is either true or false one of the two must happen and that this is by the way classical philosophy the Aristotelian philosophy scholastic philosophy for all the Western civilization has been founded on that so suddenly you say you cannot define truth then you don't know what through this so I you solve the problem the solution e problem can be exemplified by this phrase it's knowing way what is white in commerce is true if and only if snow is white now this is an actual phrase tarski was having a discussion with the philosopher karna in a cafe I think in Prague not sure about the notion of truth and current cannot ask but what do you think through is and he said is phrase and became kind of understood that the first knows why it was only a sentence and his second is an effort affirms that the sentence is a proposition and that solves the problem in a certain sense and Carib then understood what what he meant so they said it's an anecdote but Carnap himself tells the story and so how do you do it I will not do the technicalities because that I will not be able to explain that but the definition of truth in a language l so alphabet collection of words phrases according to syntax must be given in another language called let's call meta language ml the meta language contains within itself a copy of the language and she'll be able to talk about sentences and the syntax of the smaller language the meta language will contain symbol true where true X means X is a true sentence of L so it's a predicate attached definition of true is a sentence deform for all X through XF and on live fire exercise a function in the meta language and true is not appear they otherwise you get a self referential definition and the equivalents if and only if must be provable using axiom in the meta language that do not contain again true well you want so conventional that what you think true should be should follow should be actually a property of T so D the language l the convention T makes the liar paradox inevitable so the already problem the meta language should be much larger than L and then tarski shows that in this case it is really much larger you can find a formula in ml in the meta language which defines the truth in this smaller language does not define truth in the whole thing just in this modern language and the intuitive truth the negation of a is true if and only a is not true and so on did the a or b is true if and only if a is true or B is true I mean this is playing plain English the second a special convention team okay so the advantage is so the mathematicians however he works it turns not tarski notion of truth for l will be say mathematically Zermelo Frankie Laxman and the and L the meta language MLS arrow elbow should be ml the meta language will be just playing English because we write sentences in plain English and well this would be satisfactory until someone finds the Mello Franco action which are contradictory and that may happen in that case we change the action will continue with a different language and and we'll find out some way around it now to show so quickly an example why this very famous example about notional truth is showing that something may be the inertial truth really changes according to the choice of the meta language so the Continuum Hypothesis is the following the simplest infinity our lives zero is the infinitely counting one two three infinity well some they said and Amazonian tribe counts one to infinity however you know what what I mean so the continuum is the set of all real numbers by the way cardinality means the totality how big it will not give a formal definition what is a real number well it would actions defined essentially real numbers by the following by considering all fractions less than something and old fraction greater that that's something if two things and this was actually formalized by daddy king and the e coincides in a sense with the notion that the real number it's thinking of some decimal number you write all the digits there's only one little rule you cannot have a number like zero point nine nine nine nine nine nine nine without because the number is one you're here to make that kind of identification so yeah that will be the continuum and Cantor showed the ID if you take all subsets of the natural integers that I said that as the as many element as the continuum so it's written that way because if we take for example set consisting one element the only subsets are the empty set and set itself say so there are two elements two subsets if I have two elements a and B these subsets are the empty element element a and then be and it be together there for them to squared if I had these subsets of a set with three elements there are 2 to the power 3 there are eight subsets again County can figure out so the subset of our set contains and elements are two to the N and that's the reason why he writes 2 to the power of 0 it's a consistent notation and Cantor then show tried to see whether show that the continuum has many many more elements than the than the set of than out of zero it's uncountable well suppose I started putting some numbers at random and like that well you take what what is called a diagonal marker for example the digit six digit zero this is zero digit 9 digit 5 this is 2 7 is on and obtain this number so the nth digit is the coincide with the nth digit of the nth element in the list and now you change it you change every digit for example you add one if the digit is less than 8 you have to avoid the 999 so if 8 and 9 you remove 1 okay so you get a number which cannot be in the list so you cannot put the the continuum in a list of this type so on the other hand ok let's see how it works that way this will be how you try to reach the continuum every time this was done is done with it digits a zero and one think of that is zero this is zero point zero this is zero point 1 this is 0 point 0 0 this point 0.01 this point 0 1 so this point 0 1 0 this is point 0 1 1 and so on that one is the continuing binary binary digits ok so by going this way the limit is the continuum ok so what is the continuing part it is well there's something you call the axiom of choice which allows you to from any aggregate of set any collection of sets to choose one element from each set it does not tell you how it tells you that you can do that then you can compare cardinalities so there is a first Carlin number greater if one greater than a left 0 and the continual policies is LF one equaled the continued canter try to find something between you cannot and then formulate a hypothesis that the first uncountable Cardinal is to continue there's a common mistake which is repeated again again in popular writings you know mathematics that the C equal to LF 0 is the continual polish no that is the definition of C so be careful if you read this type you know popular expositions so girdle proved in 1940 that the Continuum Hypothesis is consistent with the Zermelo Franco an axiom of choice in 1963 Paul Cohen here the Institute proved that the negation of the continual parties is also consistent so is that a contradiction no because the consistency is requires to amplify the Zermelo Frankel choice actions and and you have two different ways of amplifying the system in a coherent fashion and in one system then you can assume the continued policy as an axiom in the other system you can assume the negation of the continual policy as an action so this tells you that truth is depends at least in the formalistic models depends on how the extension that you that you make the extension of the mathematical language you have accepted up to a given moment so it's one model better than the other well some better is subjective so mathematicians and that by getting guided by aesthetic considerations the intuition simplicity of arguments linearity of patterns and and there is no definitely rule for that and and views differ on on this from there is no unanimous choice I mean what is better so it's through the same as proof answer no and there is a theorem of combinatorics I will not explain Adam don't want to be too technical but is related to properties of coloring of graphs of think of the following take for example five points join them in all possible ways what you get is a star Pentagon in style inside the Pentagon now color the edges with two colors red and blue then you will discover pretty easily that if you color the star painter in blue and the outside Pentagon in red there is no triangle or the same color with edges of the same color if you do it with six points then you discover no matter you do the coloring you will find a triangle with that property now what happens if we try to do the same thing with say six points nobody knows what is known that you can find the complete say you want to resell triangle say the next case will be a tetrahedron in width and you want to find the tetrahedron with the same color so there will be four vertices well if the set the point is big enough you can find such an option if with five yeah it is sided big enough you can find the five points in which all all the segments joining them would have the same color how many vertices they need is some number between 45 and 49 and nobody knows how to compute that because the number of possible cases to examine is so big the question is still open now the this is the very special case of the theory of Ramsay on computing combinatorics there is a little twist in which you put just a little extra condition which looks very not yours well if you user male Frankl axioms you can do an infinite version of the problem and once you have done the infinite version you can do the finite version and then you know that affirmative answer that you can get this extra coloring with the extra condition now in each case each final specialization is a final calculation and theoretically you can shall be able to check all cases so the fact is that this the file that you have an affirmative the final version as an affirmative solution is not provable in this standard arithmetic that's kind of sobering sobering fact so here we have an example of truth which you cannot prove in this system in which you work if you expand the system a bit then you can prove it so truth in other models well I will say something with the heart reveals fictional ISM mathematics dispensable you can tell with everything without mathematics well well maybe statements cannot talk about realities at best an useful fiction maybe that's why this name is such a one plus one equal to is meaningless in absolute and through only in the fiction of the world of mathematics if you go only company on the web you find the following statement the statement like two plus two equal four it just as false as Sherlock Holmes leave at twenty to be Baker Street but both are true according to the relevant fiction mathematician answer two plus two is true in primitive recursive arithmetic and also for the layman and is false in characteristic two or three the second stem is false for the brutal but sufficient reason that Sherlock Holmes live at 20 to 1 capital B Baker Street and here is a reference in fairness to Wikipedia to say this mistake was corrected few months later but a copy of that empiricism is my trial truth contingent to observation quasi empiricism the postmodern postmodern is applied mathematics questions the validity of mathematics because not only truth can be the validity can be proved from the beginning to the end but in fact the falsity from the conclusion to premises can also occur in other words this is related to the question of the logical system being noncontradictory so well in purses got support from string theory but general relativity got support from mathematics the relevant geometry differential geometry of space in high dimension was devil by riemann fifty years and the tensor calculus needed was ready for Einstein and provided to be very right an important tool so it to be a little careful with when we go to popular journalist otherwise we talk about things we have names black holes and quarks and quanta the quants are something else and I've seen on TV some imaginary depictions of black holes which are really like like a rotating funnel and so on and it's simply just it's not that's not it's not the reality of black holes intuition is okay objects cannot be considered unless obtained by experienced construction so the axiom that something's either true or false you can use that if you decide separately whether something is true or false you cannot say one of the two is correct so this is restricted but the girdle incompleteness theorem still holds in intuitionistic mathematics does not help to to solve the problem of non contradiction the earlier statement I put three objects into boxes so one box contains at least two elements it is not acceptable in intuition is unless you tell which box so it's quite restrictive but mathematicians have tried to see how far you can go with that and it turns out a large part of calculus almost all calculus and almost all of analysis can be done in intuitionistic mathematics with some contortions that so I'll give an example and then I will conclude consider the function accounts number of primes up to X it's integral of log but it's very computable you can put X in number and it's not difficult to compute a very good approximation and the prime number theorem says that the number of primes up to X and this number are very close okay of course I put this this thing because I really like prime numbers it's so I have to promote myself a little bit so the Riemann hypothesis is the statement that the gap the approximation is not much more than this root of Publix in fact is equivalent that the statement 4x graded 2657 okay now Riemann notice already that the number of primes up to wakes up to 3 million is less than this quantity and commented on that the women had a formula for pi of x and by the formula had Li of X was the first term but there were other terms and those terms are oscillatory and really commented that will be interesting in a subsequent research to see the effect of these assimilatory terms on the on PI of X now calculation the computer show that PI ace is less than Li of X for X less than 10 2023 that's a pretty big number for a computer but in 1955 Stanley skills proved there's a number X less than 10 to the 10 2010 to 8,000 for which PI X is greater than L Phi of X so so numerical evidence by computer maybe is not always to be so so evident the excuse argument two parts first was done in 1933 and shows that holes in the Assumption the Riemann hypothesis the number was a little smaller the thousand was 34 hardy commented this certainly is the biggest number that ever served specific purpose in mathematics well in 1955 we came up with bigger number because that was a teeny assuming the failure the Riemann hypothesis and then by the principle the excluded middle you can say there is a sign change within the larger of the two this is not valid in interesting intuitionistic mathematics at least this proof but still you come up with a very specific number and well can you ever compute where the changes on the ring apology has been narrowed to something very precise one point three nine seven nine one point three nine eight four times ten to twenty sixteen this is likely to be the very first point I mean it's somewhere in between it's very large interrelation but then where in between it will be the first point where you get a sign change well my time is up so I will only mention that today there are many other aspects of truth especially in question of proof and verification or by computers proof by consensus if ever proof say of ten thousand pages which is the case for solution of problem of finding all finite simple groups nobody can read ten thousand pages of condense mathematics without burning his brain so the thing has been done in pieces and eventually goes through revision and revision and eventually consensus will say yeah it is a theory we know this is this loop we are getting close to that but it will take another several years to before to be totally convinced computer-assisted proofs are getting better and better and my prediction is that the computer will play a bigger and bigger role in the future of mathematics thank we had time for one or two questions yeah that's very interesting question the probabilistic proof is obtained by taking say conventional proof and it has been shown that you can list theoretically rewrite it in a formal way in just a bit longer not excessively longer and then you take a little sample just three three clauses and check whether it's correct or not well if it says false you just take everything put in the wastebasket say this is true it'd still be false but your sample the give me a wrong answer will be probably probability is less than they just say around fifty percent so you take another random sample it says false because the sample is independent the sorry that this happens is 25 percent well if you do this say 50 times the probability is that the computer gives a false answer is less than the number of atoms in the universe so it's you say okay it's a after all these all these things that by humans it's a human proof so that eventually this thing will be put in practice maybe so no more referring this that's not as clearly all the way of looking at proofs chopped dating interesting pieces and check and see how it's connected and so on intuition mathematics or constructive mathematics anymore is that been abandoned well yes no it's the the many models have been proposed in fact even doing away completely with these are metal franco actions in the difficulty is Romero Frankel action is there are too many sets innocent so there if you start putting serious constraints on what is constructible what is acceptable what max will be a little different some things cannot be done anymore but in practice you know instead of proving a general theorem which encompasses everything you prove something less but what you are really interested in still will still be proved English is suitable as a meta language in the tarski construction that's a good question well to say there's a after refers to the study of the language the Amazonian tribe became second studying that says they do not count 1 to infinity so I don't know the the fact say plain English or plain Italian for example since that was was my native language that's the way we communicate maybe Sunday we'll start communicate by telepathy and that will be different language just communicate by feelings that's another language so so we do what we can do the important thing is to keep it you know the mind open and if we have to change we change already they're pretty strong restrictions on the use of plain language in plain English in mathematics you to be very careful you cannot be sloppy yeah questions let's thank and reco gain from fine you you

Channel: Institute for Advanced Study

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Keywords: Enrico, Bombieri, Math, Lecture, Truth, Proof, Verification

Id: J-A_6CGY9dM

Length: 60min 24sec (3624 seconds)

Published: Thu Apr 26 2012