Mathematical Transgressions (June 3-4, 2024)
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okay I think we got
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it may I start
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B yes please okay so good morning good evening everybody it's uh 6 PM in Poland
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in Caro that's the last session of the first day of mathematical
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transgressions uh we scheduled two lectures on this session the first one
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is by Norman wildberger uh Professor wildberger is a honorary professor of the University of
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New South WS Sydney Australia although currently he is in Canada so I guess
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there is Al in demonic in Canada uh so Professor please if I may say take the
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floor thank you so can you hear
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me yes perfectly okay great um all right well thank you for um the invitation to
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talk here thank you Barbara for organizing the uh this meeting uh it's
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um my intention to give a sort of an overview talk about the role of
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arithmetic in in mathematics um both in mathematics education and in research um so
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arithmetic is traditionally you know a very Elementary topic that students
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learn about in their first grades at school um but actually it's um it's a
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subject which ends up embedding itself in a lot of different areas in mathematics and especially sort of at
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the higher level uh there's actually some from my point of view logical
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issues uh that we need to think about especially with regard to the interface
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with computation and and and modern uh Computing so I'm hoping today to uh give
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you an overview of uh of arithmetic perhaps starting with a historical uh
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perspective and um a good place to start from a historical perspective is perhaps to go back to the Egyptians and the
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Babylonians and think about their arithmetic so um this is Maybe 3,00 BC
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so we're talking roughly 5,000 years ago these two ancient civilizations which
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are somehow the the pillars of the modern Western World at least um their
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arithmetic they both had very highly developed arithmetics um but they're noticeably very different and it's uh
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it's really interesting to compare the two the two arithmetics and the differences are manifest themselves even
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even today so the Egyptians um had a system where they could you
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know add and multiply they had a sort of a binary approach to to multiplication but um it's invariably
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division which is the sort of more problematic or challenging operation to
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get on top of so when students learn arithmetic they learn addition and multiplication first uh typically with
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natural numbers and then when subtraction is involved well it's a
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little bit more problematic because sometimes the difference is a negative number so to have a consistent
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arithmetic including subtraction you are sort of obliged to extend the natural
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numbers to integers and we should keep in mind that historically this was a very big step
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and that even dayart for example was highly suspicious of negative numbers
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and tried to avoid them so in any case um the J gyptians uh so they had uh
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addition and multiplication some kind of subtraction but it's really the approach to division which characterizes U it's
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really sort of the heart of computational problems in arithmetic is how do you
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divide and uh the Egyptians had uh a notion of something close to fractions
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but the fractions that they involved are rather special kind where you have one
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over a natural number so um these are sometimes called Egyptian
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fractions and these days uh this is not given enough uh credit and uh and it
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also I think it's a topic that quite interesting to students to to learn about the Egyptian point of view so we
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know not that much about Egyptian mathematics but what we do know is largely from a few papyri so there's one
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called the the Rin Papyrus um written by a scribe name is Ames around
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1,500 BC although it's probably a copy of an earlier um an earlier work but uh
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in that we we get a pretty good idea of how they they dealt with arithmetic and arithmetical problems and so for example
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let's say you want to subdivide uh let's say you were working on the pyramids and you have a work team of of eight people
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and you're given five loaves of bread and you want to subdivide these five loaves of bread into eight okay so this
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is like a standard kind of um question then and if you ask the primary school student today well what what do you get
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when you divide 5 by 8 well um they might say uh you get
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58s so then you ask them okay what does that how does that work but what would you do and they would say well you have
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these five loaves of bread I guess you should divide each of them into eight pieces each equal eight pieces and then
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after you've done that you give each of the eight people five of these U pieces but that's actually quite a lot
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of slicing and the Egyptians would have done something differently they would have expressed um 58s as a sum of
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Egyptian fractions as 1/2 plus 1/8 and with that in mind they would say
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okay the way you do this is you have to give everybody a half a loaf so take four loaves divide them into into halves
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so each person now has a half and then there's the fifth Loaf and you divide that into eight pieces and give each one
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1/8 and that way you've subdivided the loaves um in a more efficient
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fashion so what happens if you dividing by a number which is a little bit more challenging than eight for example seven
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so what's five divided by seven well the Egyptians would have said
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well what we want to do is we want to write five over s they didn't think about five over seven they they rather
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Express things in terms of the sum of these Egyptian fractions so they um am
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would have written 5 over 7 as 1/2 plus one4 + 1
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14th and uh and you might ask well how do you how do you get that subdivision it's an very interesting question so
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there's an algorithm to do this that I perhaps was first enunciated by uh by
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Fibonacci or Leonardo of Pisa which called the greedy algorithm so you
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divide you actually look at the reciprocals let's say 7 over5 um that's a number between one and two
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okay it's less than two so so two is kind of the number you want so the biggest Egyptian fraction which is less
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than 57s is2 right so what you do is you you subtract one2 from 57s and then you
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repeat the the process so if you do that the the greedy algorithm re shows that to 57th is 1 12 plus 1 15 plus 170th
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um that's not as good as the the am solution okay which is one half plus
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one4 plus 1 over4 because because you don't want to
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divide a loaf of bread into 70 and modern computer Studies have shown that U the the way am has subdivided things
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there was somewhat somewhat sophisticated uh tables U actually corresponds pretty pretty well with
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optimal um Optimal Solutions that computers are able to find so there's
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some pretty sophisticated stuff going on there with the Egyptian fractions now if we go to the
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Mesopotamians this is the the fertile cresant the land between the Tigris and
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the Euphrates rivers where we have the Mesopotamian civilization the samarians
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and then the acadians and then later the the Babylonians we have a completely different arithmetic
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and this is a hugely important the arithmetic that we have there that was a base 60
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arithmetic sexal arithmetic um where they had digits from 0 to 59 essentially
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and uh they didn't use fractions they basically were working in a in a decimal
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environment but not decimal but rather base 60 so the modern Hindu Arabic
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system that we use a place value system that students learn um you know I used
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to think that this was it's we call it a Hindu Arabic system right um and and the
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Hindus made made great progress and the Arabs made great progress in developing this but in fact the basic idea really
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goes back to the Babylonians already their their sex adjustble system really contains the main features and it's such
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a powerful it's such a powerful uh technology that even in into Greek times
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when when tmy was making calculations in an astronomical fashion uh the Greek the
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Greek arithmetic was just not powerful enough I mean the the Greeks of course
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were very strong mathematically but perhaps a little bit more in the geometrical side arithmetically not so
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much um and soamy worked in a Babylonian system and even into modern times in
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Islamic scholars even in the 14 1500s when they were compiling trigonometric
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tables they still often resorted to the base 606 admal system because it's just
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more efficient so why is this more efficient well it's because the number
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60 uh has factors 2 three and five and that implies that when you're
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expanding numbers out um in base 60 then any number that let
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say any any fraction that has a denominator which is regular in the system meaning that its factors are only
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twos threes and fives will have a finite expansion in other words you can encode
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it completely in a decimal uh sort of place value system without any kind of
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uh truncations or or repeating decimals to to to to complicate the issue so that
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meant that the Babylonians were actually able to divide by more numbers than we
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are able to divide purely in a finite decimal world so you know we will know
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that a number like 1/8 in our system is has a finite decimal
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0.125 and the numbers that have that property are are exactly the numbers that are multiples of twos and fives
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because those are the prime factors of 10 but the Babylonians because they had base 60 also had the crucial three so
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not just not just two and five and and 10 but also six and then 12 and uh you
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know and uh 15 so these numbers they could all they could divide by these
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numbers exactly see if you went if you took an old Babylonian scribe and you took them
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into a primary school classroom these days and uh and and the student was
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asked okay what's what's 5 ided 7 let's let's let's see how division works what's five divided by seven well your
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primary school student is going to say uh well it's it's 57s it's 5 over
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seven that fraction yeah okay but the Babylonians
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say will say yeah you so you've just restated the problem you still haven't
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divided five by seven yet you know what do you actually get when you actually do that
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to a Babylonian 5 over s five divided by seven was actually problematic because
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seven was not regular there in their system so you couldn't write down an exact value for that that was not
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something that was really a a legitimate object in their
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arithmetic now our Primary School student if he's learned about decimals would say okay but I can I can do that
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division s of long hand perhaps not sure if students can do that these days but you can divide long hand and you get a
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decimal but it's a repeating decimal okay so that's that's a legitimate um
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kind of alternative to try to extend the arithmetic of finite decimals to repeating
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decimals and of course that's a step on the way to considering infinite decimals
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or so-called real numbers uh which is like a a Mainstay of modern analysis
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but even if we just restrict ourselves to repeating decimals for the moment
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which are just a decimal equivalence of fractions um it's a very interesting
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question to ask about the foundations of that arithmetic
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so if you ask a student okay let's work in the in the realm of repeating
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decimals if I have two repeating decimals which are you know completely finitely prescribable you you have you
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know maybe it's 0.53 and then 46 repeated so 46 46 46
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and you put a bar over the the 46 and that's a completely a finite expression so it's a legitimate question to ask
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well suppose you have two such things how do you define addition of
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those two things how do you define multiplication of those two
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things well one way is to convert both of them to fractions so students how to
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do that and then do the arithmetic over in the fraction land and then convert back but that's a sort of a a kind of
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cheating isn't it I mean it it you really require a consistent arithmetic if you're going to be working in the
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decimal framework you really ought to have consistent definitions of the
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operations of these objects and that turns out to be not impossible but
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challenging and in my understanding I'm I have never actually seen a proper
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account of that I have never seen a proper account of arithmetic with repeating decimals that's that that does
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the the following here are the objects that we're going to deal with okay that's pretty easy to do here are the
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operations here are the definitions of the operations here is how we going to Define operations of repeating decimals
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say and here are the laws of arithmetic that we want to have satisfied
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and proofs of those laws logically correct proofs that these
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laws actually hold so think about that I mean this is
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this is a foundational kind of thing right we're talking about repeating decimals how do you do arithmetic with repeating decimals so do you know of any
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account or maybe you do please share it with me where we have this all laid out in terms of what the objects are what
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exactly the operations are and proofs of the laws of arithmetic and I'll say again what we
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don't want is to say just go back to the fractions and and and just try to pull over whatever we've done in the fraction
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world no we want a consistent story with these fraction with these decimals because ostensibly there are such things
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as infinite decimals which are not repeating representing so-called irrational
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numbers now I have to you know admit at this stage that I am a bit of a heretic
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when it comes to my pure mathematician colleagues I don't believe in arithmetic with real numbers I think it's it's
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something of a fraud I think this is a a make-believe world that we've introduced you know
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because it's convenient for us in terms of statements that we're allowed to make
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in this context for example the intermediate value theorem or or the fundamental
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theorem of algebra so it's nice to have those theorems but they ultimately rest on this arithmetic which in my view is
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is dubious so to to go to go in that direction let's let's fast forward a
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little bit uh to the to the European era um where we have to talk about um
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arithmetic assuming a a much different role not just computation U Commerce uh
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measurement surveying you know astronomical calculations Etc but it it
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ended up playing a much more fundamental role having to do with
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geometry so in Greek times geometry was sort of
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the main subject and arithmetic was somehow secondary to Geometry so the
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Greeks thought about arithmetical operations in terms of line segments and operations that you do with line
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segments um then in the 17th
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century uh deart and FMA introduced a a a very novel new approach to Geometry
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which is we we call this the cartisian framework it's a really really a revolutionary uh shift in thinking where
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the foundational aspects of geometry are pinned to an underlying arithmetical
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structure so let me remind you how that works that let's say we're doing planer
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arithmetic and we're we're in you know we're looking at um the geometry of the
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plane a natural question is well what's the definition of a point what's the definition of a line what's the definition of a circle Etc so those
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questions are a little bit hard for uclid to answer really precisely dayart
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said look one way of doing this is to think about the points as being ordered
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pairs of numbers with respect to some coord we imagine having some coordinate axes which are suitably labeled and then
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by going this way this far in this direction this far in this direction Etc we can um you know assign a pair of
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numbers to a point but logically that means that we can then Define a point to
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be a pair of numbers so this has huge implications because then lines can be defined in
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terms of linear equations more complicated uh curves and surfaces can be defined in terms of the equations
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that that connect the various coordinates of of the points in question and it also becomes much more easier to
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to imagine higher Dimensions after all then a three-dimensional space is just what you get when you take triples of of
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of points those are your numbers they're triples of points okay so then going to four dimensions uh you you have
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quadruples of points Etc so um whereas
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previously people had hadn't really thought about Dimensions greater than three but with this cartisian or
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analytic approach to Geometry this becomes quite natural and reasonable and
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really is then the starting point of you know huge developments in terms of studying surfaces and varieties and
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algebraic geometry and and many other things but
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one you know so it's a great Advance it's a great Advance this cartisian
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revolution but we should be mindful that there's also a
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flip side which is rarely discussed okay which I want to bring to your attention
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that we lose something in some sense also or or maybe we should say there's a
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price to be paid for having this cartisian framework to the foundations
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of geometry you see let's say just in the planer case so if I mean what you what
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you're doing really is right you're you're taking this this flat plane maybe it's a piece of paper or or a tabletop
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or something which is a priori without any markings it's just a it's just a flat surface and what you're doing is
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you're imposing some structure on that you're saying let's draw some axes let's label the axes you know with with
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coordinates Etc so what that does actually implicitly is impose some
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distinguished directions in your plane it also imposes a distinguished
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point because after all there's an origin to your point to to the to the of the plane so now you've you've somehow
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um broken the Symmetry that's in you know classically there in the
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plane and a little bit more subtle is what you've done is you've uh introduced
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implicitly whether you like it or not number theoretical issues because the
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the the arithmetic that's underlying these numbers right I've said numbers but I haven't been very precise about
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what numbers we're talking about but it turns out that which numbers we're talking about ends up influencing the
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geometry so let let let me try to explain that so the simplest point of view would
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would be to say well let's use the number system that we're most familiar with say the rational number system
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fractions basically um so if if we start in in that fashion
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then and let's say we're working in a two-dimensional plane to begin with it turns out that there are no equilateral
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triangles in that framework in other words you cannot find three points with rational coordinates
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in the plane that form an equilateral triangle so that's disconcerting to a
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classical geometer like uclid because constructing an equilateral triangle with ruler ENC Compass was one of the
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very first things that uclid sort of accomplishes in book one of The
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Elements and if you have a a plane which is unadorned by anything and you just have a ruler andc Compass well you're
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able to make these constructions and now we're saying that well wait a minute in this cartisian framework that might be
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problematic but then what happens if we go to three dimensions and just consider
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a three-dimensional space with axes x y and Zed say um then it turns out that
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equilateral triangles do exist because you can take the here are the three axes you can take the plane which goes
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through the unit points on each of these axes what the point one 0 0 0 1 0 and 0
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01 you take that plane which is the plane with equation x + y + zal 1 and it
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does have an equilateral triangle namely those three points that we've just been talking about so now we find out that in
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three-dimensional space if our underlying number system is the
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natural numbers not all planes are geometrically the same this natural
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homogeneity that we sort of assume about the space that we're living in is lost
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in this mathematical representation of it because the plane x + y + z = 1 has
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equilateral triangles but the plane Z equals 0 or the XY plane does not have equilateral triang
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triangles so this became um even clearer in in the 19th century when um people
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started investigating more abstract algebraic structures like rings and
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Fields and groups and people uh eventually realized that the arithmetical structure that's basically
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encoded with the rational numbers has analoges has other analoges in
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particular it has finite analoges so there are things called finite fields and they are also in
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principle arithmetical domains which are potential building blocks for
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geometries so the 19th century saw a massive expansion in in people's thinking about geometry and arithmetic
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in in on the geometry side they discovered the non- ukian geometries
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like Boler levski Gus's discovery of hyperbolic geometry uh the projective
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geometry was was revitalized and and and that was a a huge new new development
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there was also apine geometry and ultimately then relativistic geometries
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were not far behind so the geometrical world became much more variated there
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was no longer just one geometry there were many different possible
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geometries so a similar kind of thing really sort of
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was happening on the arithmetical side also when we start to realize that you
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know there's there's different arithmetical domains so I mean yes we already know
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that at some level because Primary School students are are taught arithmetic by starting with the natural
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numbers you know and then you and then you extend the natural numbers to the integers and then you extend the
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integers to the rational numbers and then if you do believe in infinite processes and you know such things maybe
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you can extend that then to the real numbers then maybe to the complex number so that we you know we have we have
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certainly a notion of of um arithmetical systems contained in bigger arithmetical
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systems okay but what I'm saying now is something a little bit different that actually there's actually almost
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completely separate arithmetical systems you know like a finite field doesn't fit inside the the rational numbers okay um
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and you have finite fields for every Prime in fact you have for every Prime power so there's lots of finite Fields
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there's also other things like pic fields and and and and in fact U you
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know polom Fields involving polinomial or transcendentals so the the world of
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of possible arithmetical systems becomes very rich and and
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varied and and because of this cartisian revolution we then have the possibility
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of doing geometry over basically any of these things right because we said that a geometry was something that we build
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by starting with an arithmetical foundation in terms of points being three tuples or whatever of of numbers
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so may if our numbers come from a finite field that means we could be doing geometry over over a finite
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field so um you know the subject becomes um
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really um much bigger and uh you know and and and so this is I think something
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that that maybe is not really appreciated by by students that this this seemingly very Elementary
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subject called arithmetic actually is um is is a much more varied landscape it's
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really comparable in its richness to the richness of the geometrical landscape and involving all possible all kinds of
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different geometries okay and so now now we have
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to sort of go up and and talk a little bit about um the the real numbers okay so this
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this is it's at this point here where unfortunately we start to run into very serious
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problems and this is a very sensitive topic in some sense because a belief in
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real numbers and real number arithmetic is at the core of modern analysis so you know any attempt to
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question this is is seriously threatening to a lot of existing
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theories of of analysis and not just analysis but related subjects differential geometry algebraic
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topology uh even algebraic geometry so you what's what's the issue
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the issue is that you know if we're going back to sort of Elementary point of view towards
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decimals we've talked about finite decimals and we've talked about repeating decimals so
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then there were these irrational numbers okay which popped up of course in in
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Antiquity and the there's sort of two prototypes in Antiquity one is the
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square root of two that baffled the the pythagoreans and the other one is pi
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which is ostensibly the area of a circle so we have to talk a little bit about irrational numbers
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so the the ancient Greeks knew that there was no rational number whose
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square was equal to two now currently that it's stated in another way
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currently that stated in the following way that the square root of two is not a
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rational number that way of stating it assumes that there is a square root of two and
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it just happens to not to have the property of not being a rational number so in my view that's a wrong logical
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thinking okay the existence of square root of two or even the definition of square root of
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two is is is a highly problematic um notion so let's talk about that all
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right so back in the Babylonian times solving
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quadratic equations was an important important thing the Babylonians knew how to solve quadratic equations they
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basically had the completing the square algorithm okay that's that's well established in there they also by the
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way had the um the Pythagorean theorem which they called the diagonal rule they refer to it uh in terms usually in terms
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of a rectangle so if you have a rectangle and a diagonal then the relationship between the sides of that
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rectangle and its diagonal that was the diagonal Rule and they knew they knew the relation
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and they Al you know I mean implicit in this is that if you're working with lengths you know you might have a
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rectangle which has you know integral lengths but that diagonal is not going to have integral diagonal it's not even
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going to have necessarily A a rational Diagon Al okay and that's in terms of
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its length and so the pythagoreans actually had had a very subtle um
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approach to to dealing with this problem so which is encoded in the remarkable tablet climton 322 which is the most
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famous of all the ancient Babylonian mathematical tablets which is a
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basically a tabulation of of U right triangles or or or rectangles with with
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Rel ations that allow you basically to to understand the relationship between the three sides of of of a right
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triangle if you know two of them you know you want to calculate the third or you want to calculate ratios involving
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the third and what they what ended up happening in in in Clon 322 is that they
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are often working with the squared quantities instead of the actual lengths so if you want to stay within a rational
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framework you want to um reframe the Pythagorean theorem to avoid
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irrationalities and the way you do that is is exactly the way ukl did it I mean
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uid if you have a right to Triangle like this you know what's Pythagoras's Theorem Pythagoras's Theorem is not that
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the square of this the square of this length plus the square of this length equals the square of length this length
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that's not the Pythagorean theorem the Pythagorean theorem according to ukl is that if you build a
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square on this side and you build a square on this side and you build a square on this side like a geometrical
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Square then the areas of these two squares will add up to the areas of these two squares precisely when the
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triangle is right now that way of saying it avoids the irrationalities because you don't
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have to take the square root to calculate the length the the way Pythagoras is usually
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discussed in all textbooks in terms of a S Plus b² equal c2 in in implies an
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implicit understanding of lengths and the lengths involve square roots and the
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square root operation is an unending U algorithm actually also going back to
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ancient times people knew how to have algorithms to compute square roots but they they realized that these algorithm
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or these algorithms don't terminate you want to find the square root of three you're not able to do that you can run an algorithm which will which will
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produce numbers which have the approximate property that they their squares close to three but not equal to
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three the actual square root of three that we like to talk about is actually the result only of taking that algorithm
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and running it to Infinity okay and that's the crucial thing that that the the the number
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square root of three or whatever only comes into being once you run this
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algorithm to Infinity so you have to believe that you are able to run an
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algorithm to Infinity it's like a computer having a special red button at you on them back
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and you've got some algorithm and it's it's churning away and and and going going going going it's an unending
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algorithm and see you don't want to wait a trillion years even if you wait a trillion years it's not going to end so
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you press the go to Infinity button on your computer and suddenly boom it's
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finished well computationally you can't do that because an infinite amount of memories required to hold this this the
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result of this computation and so it clearly doesn't fit in our universe and um you know the bound of the universe
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gets overthrown long before you get to infin to Infinity so um but nevertheless it's
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become important in in many applications to pretend that
36:13
you know we do have things like square root of two and square root of three and and Pi Etc and and E and all and cosine
36:20
of seven and log of eight Etc so that we we take these U applied mathematical
36:28
Concepts and you know and sort of we sort of insist that they're actually
36:33
exact real arithmetical objects which which um
36:40
which correspond to them but which have where all the information is laid out so just in terms of Pi um in terms
36:48
of the area of course one approximation to Pi that's going back historically is three okay that was an approximation but
36:55
uh the Egyptians and the Babylonians had so um if you ask an Egyptian like Ames
37:02
in in the Rin Papyrus we find the following prescription if you have um a
37:07
circle and you want to calculate its area approximate area okay then what you
37:13
do is you take the diameter let's call the diameter D and you multiply it by 8
37:18
over9 multiply the diameter by 8 over9 and then you square it and that's the approximate area of
37:27
the the circle so that's amounting to effectively that Pi is approximately 256
37:35
ided 81 which is about 3.16 so it's not not too
37:42
bad we should also mention Archimedes Archimedes made the first sort of
37:48
serious estimates of the area of of a circle and he did that by inscribing and
37:54
circumscribing polygons poly of a special kind and he's able to calculate these areas of these
38:00
polygons at least approximately and so therefore so have a bound on on the area of the circle so it's not that that the
38:07
area of circle has a well- defined value it's that we can uh we can find a bound
38:13
and say well if the area did exist it would be somewhere in there so it might be disconcerting for
38:19
you to to hear a pure mathematician claim that you know a unit circle the unit circle does not have an exact area
38:26
but you know I'm s people that's the way the the mathematical world really is there's no exact area we might we might
38:34
want there to be we might pretend that there is but there isn't there are
38:39
algorithms sure there's algorithms that will will generate decimals but we can't run those algorithms to Infinity to
38:46
obtain the the results and and just using some Greek letter like Pi or whatever you know and
38:53
and and imagining that this this Greek letter encodes the infinite complexity of this this algorithm that run to
38:59
Infinity it's a you know it's a sort of a misrepresentation of mathematical
39:05
reality and it's and this is a topic that's so important for for Education
39:10
because young people should really be exposed to you know the way things
39:16
really are not the way we want them to be because it's convenient for
39:22
us let me give you another example right you all know the the fundamental theorem
39:27
of algebra right you write down a polinomial um and it has a if it's non
39:34
constant polinomial it has a complex zero so if I write down a polinomial like Z 7 plus Z minus 3 that there's a
39:43
complex number Zed that you plug that in there and you're going to get
39:48
zero okay now a student might well ask all right um so what is that complex
39:55
number what is the complex number Zed such that z 2 + z - 3al 0 and the
40:02
answer is you can't find such a pol such a number you can get your computer to
40:07
spit out something with a finite number of you know decimal digits and and plug it in and you're going to get something
40:13
that's close to zero but you're not going to get exactly zero you're not so it's like it's sort
40:21
of like demonstrably you know wrong to assert that there is a a a complex
40:26
number for when you plug it in you get exactly zero that's but that's what we do but that's it's it's very convenient
40:33
I I admit it's very convenient to to pretend that because then we can we can take that imaginary thing and and and
40:39
and ostensibly use it for further calculations ostensibly but actually we we can't actually do the computations
40:46
because we can't actually run uh our algorithms to Infinity to get to get what we
40:52
want so this all has um a lot of applications in the modern world because
40:59
our computers are becoming so powerful now okay and we have to start adjusting
41:05
our our understanding of mathematics to prepare ourselves for the AI machines
41:10
which are coming okay soon AI machines are going to be doing mathematics and you know currently the
41:16
AI machines are probably plugged into our way of thinking they've been programmed you
41:21
know with our assumptions and accumatics but like alpha alpha zer Alpha go um
41:29
they will you know very soon be a tasked with figuring out things on their own
41:35
like not with the assumptions that we would like to have built in and once they're doing that then you can probably
41:41
be pretty confident that they're not going to be asserting that there are things like Pi or the square root of
41:48
three as exact objects they're going to know you know and and they're not going
41:54
to have this religious um position that that we give our
42:00
students and so to prepare for that I think what we what we really ought to do
42:05
is is think much more clearly about a logical foundation for for mathematics which
42:11
is in accord with computational reality so that you know I I make a statement
42:17
and you say okay I'm going to check that statement I'm going to take that statement and put it in my computer and I'm going to see if the leftand side
42:24
actually does equal the right hand side like you're claiming okay and but if I'm making a statement
42:31
of a kind that you say well I I my computer's not able to take that statement and encode it because it it
42:36
involves I don't know some abstract thinking which a computer is not capable of then the rest of us should be
42:43
skeptical right we should be skeptical about such statements which are not encodable on a computable on a computer
42:49
and checkable by a computer I mean that's that's a reasonable Criterion for logical
42:55
correctness isn't it but of course it's uh hugely it's hugely
43:01
disruptive I understand you know um a lot of analysis has to be redone a lot of um mathematics has to be re
43:09
rethought and on my YouTube channels that's one of the things I'm I'm interested in doing so I have a lot of
43:16
videos on math foundations I have actually a series on box arithmetic if you like to think about the arithmetic
43:24
and its foundations box arithmetic is a sort of a point of view which lays out a new foundation for for basic arithmetic
43:32
so um I think that's maybe a good place to stop that with this this view that you know the AI the AI machines are
43:39
coming and they are going to straighten us out and maybe it's up to us to
43:46
preempt that and to start thinking clearly and logically about arithmetic on our
43:52
own and the way to do that is to have a a solid logical arithmetic which is in
43:59
accord with what our computers can do that's a litmus test for reasonable
44:08
mathematics all right so I I thank you for your patience and listening to my uh
44:13
my my story and my my little diet tribe and uh if you have any questions or whatever um be happy to try to answer
44:20
them okay thank you Norman so it's questions time h
44:29
happily in our community the next speaker is the expert
44:36
in doing geometry over finite Fields I deal with the extensions of real
44:43
numbers uh and in my recent book okay je
44:50
for Fran Mar if I'm correct sir yes yes absolutely hi yeah je FR
44:59
that's that's in French I'm based in Montreal um thank you very much for the talk that was extremely interesting and
45:08
um it's very fun to uh to listen to uh maybe I'll just I'll I'll just tease you
45:15
with a with a question um what would you say about about the
45:21
idea that that perhaps those those numbers irrational numbers
45:27
um are are just a way to um to give a name to some phenomenons you know like
45:35
uh like a limit like something that we're getting closer to like the name that we give to this uh to this segment
45:43
when we have a right triangle with side one and it because what is a number
45:48
that's a big question that that uh you know we've been discussing and and that's not well defined and I mean it's
45:55
been years that we're we're saying that you know we don't know what numbers are but we still work with them we still
46:00
talk about about them and do things and and so perhaps uh that could be kind of
46:06
a way around I'm just I'm just wondering what you would say about that and you know maybe the notion that uh that
46:13
perhaps there is something nice in the analogies that we can make between different uh reference or
46:20
arithmetics or geometry and that because this exist in one maybe uh we we can we
46:27
can also want to have something similar in another one even if it's not as well defined or etc etc so I mean just just
46:34
to keep the the conversation going I would be curious to hear you on that
46:39
well thanks Jean Fran uh yeah that's those are um really good uh good questions and I think um it's it's
46:46
something we should naturally aspire to we should ask well you know how do we how do we create these irrational
46:52
numbers you know in some in some good way and you know along lines I think
46:57
perhaps of what you're hinting at I mean one natural thing that would occur a computer person is to is to think in
47:03
terms of the algorithms that uh that generate such things okay and and that's
47:10
certainly a profitable way of thinking is is to think well let's look at the algorithms but it turns out that that's
47:18
that's also not so easy because uh for example let's say Pi okay there's
47:24
there's hundreds of of algorithms formulas that that will will churn out
47:29
the digits of pi right so if you're going to go in that direction you have to be prepared for for this this
47:36
landscape of algorithms and how to deal with this landscape in particular you have to you
47:42
have to start wrestling with okay you've got an algorithm for pi ostensibly I've got an algorithm for pi
47:49
ostensibly how do we tell whether these algorithms actually are the same sure we
47:57
we can can run them a few a few million digits and say okay they agree up to 10 million but how do we know how how do we
48:03
prove that the algorithms are are are going to generate the same thing and that turns out to be U let's say um
48:12
difficult to Impossible um so this is like a sort of a standard computer science conundrum um
48:19
you know how do you tell whether one you know the two algorithms same and and I think most computer scientists sort of recognize that essentially this is an
48:25
impossible task because for example I I could build in some some num some
48:30
unsolved num theoretical problem um you know into my algorithm I could say that
48:35
okay I'm going to do the same thing as you do unless the the nth digit is a is an odd perfect number in which case I'm
48:42
going to change it okay so my algorithm is almost the same as yours except it has this little hiccup and if there are
48:49
no odd perfect numbers which is a famous unsolved conundrum in in number Theory then our algorithms are the same and if
48:56
and if if there is you know so no being able to answer the question is my algorithm the same as
49:02
yours involves being able to solve this this famous unsolved number theoretical problem so that's that's a big headache
49:09
you know that's and that's that's um I'm not saying that's not an interesting thing to to think about but it's not
49:16
easy at all to try to come up with an alternative way of accessing irrational
49:22
numbers in in that fashion but it's definitely worthwhile thinking but
49:28
absolutely okay so other questions so maybe I'll go back to my uh
49:36
to my question so so I studied the history of real numbers and history of
49:42
geometry in that 20th century in the 1922 something like that polish
49:49
mathematician korski provided an arithmetic for decimal numbers and as
49:55
far as as you know it was not implemented in the 60s of the past
50:01
Century John car Roa and his team also developed an arithmetic for real numbers
50:07
which are represented by sequences of 01 it's not decimal represent
50:13
representation it's very strange however he provides algorithm for multiplication especially for division so I still don't
50:22
know whether it was implemented and just recently I discovered
50:27
because you know in a Greek the irrational numbers were in fact
50:32
represented by a so-called algorithm anti phases so it was based on some
50:37
theory of similarity that Greeks could show that the algorithm does not end and
50:44
if it does not end the the to simplify it represents the irrational number and
50:50
for a long time there was an representation by continued fractions but however we do not know the AR
50:56
arithmetic of this numbers and just recently a Italian mathematician
51:02
developed an arithmetic for this representation of uh let's say real numbers the problem now is that all this
51:11
all this stuff is very involved very complicated and it cannot be provided
51:17
for the students as far as I know it students are not aware how the computer
51:22
works regarding the arithmetic of real numbers so it's I guess it's too
51:29
complicated what's your take concerning the complication lot of interesting
51:34
interesting points there um so one thing U is in terms of you know algorithms for
51:42
say dealing with decimals or or or binaries let me just say it in terms of decimals it might be more familiar to the audience so if you have two uh
51:49
two decimals right in front of you infinite decimals and you try to specify an algorithm for adding them you know
51:57
you're going to have to sort of take initial sections and and add them and
52:02
then and then modify that as you as you go um but so one one problem there is is
52:09
the issue of carries you know that if if you have if at some point you you you have some decimal and then you have a
52:14
whole series of nines then you have to keep a question mark about over the the
52:20
digit be that precedes the nines because maybe that will kick over and and and you'll have to
52:27
get a carry or or maybe not so then you've got this sequence of nines and maybe it's going on for millions of
52:32
years um so so at what point do at what point does your algorithm specify what
52:39
the actual value is you know for for that preceding digit because in in
52:44
principle you you might have to run this this algorithm to Infinity before you
52:49
actually know the value of that of that that 20th digit that predes all those
52:54
nines that you're getting okay I that that that's one issue the other issue I
53:00
wanted that that we should mention also is that um in the world of analysis right there's a little bit of a
53:06
subdivision actually most probably analysts are are believing in non so-called non-computable real numbers so
53:14
that what we're talking about is having having algorithms to specify the the the digits uh is not the whole story in fact
53:22
your standard analyst believes that there's a whole zoo of non-computable real numbers whose whose
53:29
decimal expansions are not obtainable by finite uh descriptions or
53:35
algorithms you know so so ostensibly this is part of our arithmetic although
53:41
it's it's impossible to really sort of demonstrate this in any kind of fashion because anything that we can write down is almost by definition U you know the
53:48
result of some finite process or algorithm but nevertheless it it's the reigning religious belief that no no
53:56
computable numbers are just a measure zero subset of the of the of the vast realm of uncomp of of real numbers most
54:03
of whom are uncomputable okay so to me this is is is little short of a religious position um
54:10
but you know um I I think we certainly need to look at this kind of kind of belief and and and ask ourselves is this
54:17
really what we want to be thinking you do we really want to be insisting that there's a there's an
54:22
arithmetic of uncomputable numbers out there even though it's it's almost by definition impossible to verify any of
54:31
that yeah yeah that's are there any other questions if there are not so uh to to
54:39
the audience Professor Norman will visit CCO in August I guess in August uh the
54:47
papers the studies I mentioned are parts of my book it's in Polish so I try to
54:53
translate into English before your com so so yes that the car is the basic
54:59
problem and the papers have mentioned address this problem anyway anyway it
55:05
seems it's too complicated for students even not to mention secondary students
55:11
students of mathematics but even as far as I know it students don't know how the
55:18
arithmetic of real numbers works really so yeah a few of of experts who who know
55:25
how it works anyway any other
55:31
questions so uh for this who are interested so please let you know that
55:37
Professor will visit us in August so we'll have a chance to to discuss his
55:42
topics in more details and once more thank you very much for your inspiring talk thank you thank you thank you for
55:51
listening okay okay okay
56:04
oops great uh in a moment there will be another speaker in five
56:13
minutes uh Whata will you present any slides or will just we'll just give a
56:21
talk uh yes I've got a SL the slides and I have checked
56:27
they work an hour go if you can wait four minutes because some other person would like to join you I will share the
56:35
screen I introduced you as an expert in doing geometry over finite fillets am I
56:44
right no I basically working overseas so sorry I'm uh working mainly oversea
56:53
overse fields okay however the next topic is strictly
57:01
philosophical and goes back to the let's say 70s of the past Century and the
57:06
philosophy of Latos who oppose that common attitude to
57:13
mathematics that consists in acatic Attitude acatic explanation of
57:19
mathematics it's a completely different view so let's wait three minutes with
57:55
okay
58:25
e
58:55
e e
59:47
I think we can start so the next speaker is from University of the National
59:54
Education Commission in k and she will present the talk Electic manifestations of lacos Quasi empirism
1:00:02
in contemporary algebraic geometry so go on thank
1:00:08
you first of all I would like to thank uh tobasa who organized the whole
1:00:14
conference um and um my talk is about um lato view
1:00:22
on mathematics and it signs in cont algebraic geometry I'm algebraic
1:00:28
geometer I working oversea uh basically um so um I will look for some signs of
1:00:36
uh Latos theories in what I'm doing so let me
1:00:43
start uh with uh the content of this quasi empiricism by
1:00:50
lat uh at the beginning of 20th century we had uh three big PR problem programs
1:00:57
the logicism by Russell whad the in intuitionism by Brower and the formalism
1:01:04
by Hilbert and um they attempted to build uh the
1:01:11
mathematics from breaks and to show the foundations and how uh theorems are one
1:01:17
on the built one on the other um and this
1:01:23
program the Hilbert Pro program failed with gle's incompleteness theorem from
1:01:30
1931 GLE um showed that any consistent formal system with certain
1:01:39
arithmetics uh in in any such system there exist statements which can be
1:01:44
neither proved nor disproved um so we cannot build a
1:01:50
consistent system um and if we assume that the system is
1:01:57
consistent that the consistency cannot be proved in the
1:02:03
system so this was before far before lcat and then um in 1934 there was the
1:02:12
paper by K poer the um um he was previously in Vienna and so
1:02:19
the paper was published first in German it was logic that forun but then it was
1:02:25
translated and become uh known um as the logic of scientific discovery when poer
1:02:32
says how to how to um how the science works and also the other paper by poer
1:02:39
conjectures and reputations uh which developed the role
1:02:45
of conjecturing and refuting results in natural in in Sciences um was the basis for lcat to
1:02:53
think whether um such rules work in mathematics also in this area there was
1:03:00
the paper by the structure of scientific revolutions but I'm going to refer to it and also we may uh mention
1:03:09
the group of mathematicians uh who published under the uh nickname of
1:03:15
Nicola Baki the elements of mathematics um who also wanted to systematize the
1:03:21
mathematics and published books on certain areas of mathematics like set theory algebra and so
1:03:29
on so what is lat view on mathematics which he presents in proofs
1:03:35
on and reputations so proofs and reputations are lcat PhD thesis the
1:03:43
extended version of his thesis uh which he obtained at the University of Cambridge in 1961 under the super
1:03:51
supervision of poer who is a philosopher uh and it was on the faculty of
1:03:56
philosophy because at the time philosophy the mathematics were were in one faculty at the
1:04:03
University um so the um the paper of Latos was
1:04:10
published um Twice first in years 1963 64 in four parts in the British
1:04:19
Journal of philosophy of science and then after lcat death Latos died in 19
1:04:25
1974 and two years after the extended version was published uh which was
1:04:32
edited by oral and zahar and what's the idea of proofs and
1:04:39
reputation it is an attempt to show that uh the growth of uh in
1:04:45
mathematics is ruled by the similar rules to the rules of the growth of
1:04:52
knowledge in empirical Sciences so here is the first page of the uh of
1:05:00
the paper uh and as you see it is dedicated to paa who who was famous of his heavy
1:05:09
sticks um and he's also [Music] um Hungarian as Latos and to Kar poer so
1:05:18
the supervisor of laat and also um the original version of the PhD seases um is
1:05:26
not it is not digitalized but it is available in the library of the Cambridge University and it has another
1:05:33
also uh one more dedication um it refers to hegel's dialectics so these are The
1:05:40
Inspirations for Latos for for his view on mathematics and how Latos um see
1:05:47
mathematics let me show you this by showing the table of contents which more
1:05:54
or less will um clarify what lato wants to uh want wants to say so um first
1:06:02
there is a problem and a conjecture and then approve and from our school
1:06:07
mathematics we may say okay so we are done a problem a conjecture and we
1:06:13
proved so what else should we do but lcat said says it's not the end of the
1:06:20
story it's the beginning of the story now we have to criticize the proof by by
1:06:26
counter examples of different types by global local counter examples so the C
1:06:31
counter examples that uh that are dealing with a whole theorem or just
1:06:38
with some local LMA and then we come back to the
1:06:45
theorem um and we revisit the content and we um redefine sometimes the Notions
1:06:53
and then at the end we get um that
1:06:59
uh we can by by logical [Music]
1:07:06
um then we can build the mathematical true but the proof is not the end of the
1:07:14
story and the whole book um or the articles is built in the form of a
1:07:20
dialogue between teacher and students and this is how Latos explains his method he doesn't write a book on
1:07:28
methodology of mathematics but he writes a dialogue so there is a teacher there
1:07:33
are students named Alpha Beta and so on with Greek letters of alphabit uh and
1:07:39
then the students and the teacher try to um try to prove the oiler formul so um they do
1:07:47
some examples and they discover then for the polyhedron the number of vertices minus
1:07:54
the number of eight edges plus the number of faces is equal to two and they try to prove it and
1:08:03
succeed but then they discuss whether all the steps are correct and they prove
1:08:11
they find counter examples one of them is um the counter example given by a
1:08:20
cube cut from a cube so if we compute the number um which appears uh on the
1:08:29
left hand side of the Oilers formula we have the number of vertices uh in the example below 16
1:08:37
minus the number of edges is TW 24 plus 12 faces we get four and then we have to
1:08:46
rethink the notion of the polyhedron which polyhedrons are
1:08:51
allowed um so um the first proof was not
1:08:57
the end of the Hoster and also this method uh one may think that it it works
1:09:04
for Visible objects like empirical objects so mathematical may be quas
1:09:09
empirical because you can make a model of a polyhedron and you can touch it you
1:09:16
can really see the the number of faces and so on but Latos says um that the
1:09:22
same method appear um also works for for example for ki's proof for the theorem
1:09:30
that the limit of any convergent series of a continuous functions is continuous
1:09:37
so for theorem which is a theorem of mathematical analysis and for
1:09:43
sure it's not so visible so in every theorem of mathematics Latos
1:09:53
proposed that one can one should State a theorem prove it and then
1:10:00
criticize every step so every LMA and also the whole theorem and make sure
1:10:05
that the Notions are are correctly stated and so
1:10:11
on and latus points out that mathematics is a living growing
1:10:17
organism that acquires the certain autonomy from the human activity which
1:10:22
has produced but what is uh important here is this uh
1:10:29
growing um he he writes about growth not about development of
1:10:39
mathematics and also another quotation from laat book in formal quasy empirical
1:10:46
mathematics does not go it does not grow through a monotonous increase of the
1:10:53
theorems so we just we don't just add new theorems but through the Improvement
1:11:01
of guesses speculations and quases so also this human attitude human um factor
1:11:11
is taken into account but it is done in a formal way by the logic of proofs and
1:11:21
reputations so as we as we have seen it is not like we are taught at schools or
1:11:28
even at the universities that we have one example and from this example there's a definition or first the
1:11:35
definition then the example then there is some theorem and then there is a proof and we are
1:11:41
done um so it is not a building like that it's not like having solid
1:11:48
foundations and then building putting next bricks on the others
1:11:58
but mathematics uh in the view of latov is an output of a process and the proofs
1:12:04
are not the last stage of the process but there are somewhere in
1:12:11
between and once more the growth of mathematical knowledge is important it's
1:12:17
not what Hilbert wanted to do to build the whole building Define the axioms Define what
1:12:25
the axioms imply and so on but to see the mathematics as a growing
1:12:33
organism and um I see some signs of this
1:12:39
quasi empirical character in what I'm doing as a mathematician as an algebraic
1:12:44
geometry so I would like to show you um what signs of Quasi empiricism I
1:12:53
see in my work so first algebraic geometry is somewhere in between algebra
1:12:59
and and geometry uh so this area
1:13:06
here uh it doesn't matter what field are we working over what field are we we so
1:13:13
in practice we uh start usually when we are doing some new theorems we start
1:13:19
with examples and with a conjecture and then we try to uh I would
1:13:26
say test the conjecture so um we are doing more examples and this is often um
1:13:34
with computer use so we often use computer to test the
1:13:39
conjecture and often we modify the conjecture on this stage on this stage
1:13:44
because um the conjecture find um yeah some part of the conjecture seems to be
1:13:52
not true some times also the modification of
1:13:57
the definition of Concepts is needed if the concept was defined in for the
1:14:04
purpose of the conjecture sometimes it must be redefined for the conjecture to
1:14:11
work and also um we often modify the assumptions of the
1:14:16
conjecture in the way that is analogous to the Natural Sciences so like we add
1:14:21
different temperatures different um other conditions we change the conditions in the way to um for the
1:14:29
conjecture to work and also I would like to mention that this quasi empirical
1:14:35
character it is visible in the actual work of mathemati but it is not visible
1:14:43
in the F final version of the publication in the publication you will see examples conjecture or theorems
1:14:50
proofs and so on and everything is clear but um the the whole way how it appeared
1:15:00
uh it is hidden so only working in the on the problem you can feel how it is
1:15:07
really done so the area I'm working um in
1:15:14
started with the Hilbert's 14th problem which came from uh Hilbert's talk from
1:15:20
the Congress of mathematics in n in the one
1:15:26
2019 uh and it is the it was the problem if some algebra was finally
1:15:35
generated um of course there were some partial results and the results were
1:15:42
positive then they came there came some other results also confirming that this
1:15:48
algebra is finally generated I'm not going to discuss it but but uh there was um a feeling that
1:15:58
the question has positive answer and then um in
1:16:03
1958 Nagata showed a counter example so um we
1:16:10
may think that then the story stops but no the new chapter was
1:16:16
opened um in the paper when Nagata proved that uh algebra doesn't
1:16:23
have to be final gener at he had a theorem which looked almost like this
1:16:29
conjecture except for one moment when R was the square of uh a positive number
1:16:39
so if R is a square of a positive number then uh that the statement on the screen is
1:16:47
uh true and then we have that the degree of some curve is Li is uh limited by
1:16:55
number um which involves the number of points and the multiplicity of the
1:17:01
points but uh we may ask what if the number of point if if the number of
1:17:07
points is not a square of a number of natural
1:17:13
number and it opened a new Pro area of problems in algebraic
1:17:20
geometry and what algebraic geometer does when he has a problem and he cannot
1:17:26
solve it he generalized it so there are other generalization which um oh maybe
1:17:34
there are other conjectures which uh turned out to be um that to
1:17:41
be equivalent to Nagata conjecture in some case
1:17:48
um I'm not going to say what's the the other conjecture about
1:17:53
but what's interesting in that it was F found by four scientists completely
1:18:00
independently so there are there is Nagata conjecture there is a conjecture
1:18:06
and also Nagata conjecture can be expressed in the language of s constants which are some other constants coming
1:18:14
from uh mathematical analysis uh so there are different
1:18:20
conjectures and also there are generalizations of the open
1:18:26
conjectures for example unexpected curves that's uh that's the stuff that's
1:18:32
um explored now and um they generalized the SAG conjecture which hasn't been
1:18:40
proved but um so so we are building something that doesn't have the solid
1:18:49
foundations and um let me mention this three conjecture so three conjectures so
1:18:54
there was a conjecture or uh on existence of irrational multipoint s
1:19:01
constants the other conjection on on rationality of one point s constant and
1:19:07
S a conjecture and um there was a common belief that all these conjectures called
1:19:15
but then uh in 2013 uh the four authors DKI coronia M
1:19:21
and shenberg proved that if one conjecture holds then the other cannot
1:19:27
hold so we have also in algebraic geometry the theorems that have a
1:19:32
conjecture in an assumption which um
1:19:37
doesn't look like a school M or even University mathematics um so it is not building on
1:19:45
something that is proved it is making implication from an open conjecture
1:19:55
and what I I'm would like also like to mention is the use of computer support
1:20:01
in algebraic geometry so we often use computers because the computations are
1:20:07
very complex so the computer does it quicker and it's you may we may say that
1:20:14
it supports the process of proofs and reputations in the sense of
1:20:19
lacat uh I would like to give two examples so one example
1:20:25
uh from the area of the Shad constants we could prove theoretically a
1:20:32
Bound for S constant of some line bundle and at at a
1:20:37
point but this bound is inefficient so we may have a bound but
1:20:46
then what can we really say we we cannot
1:20:51
um so this number involves M which is the uh multiplicity at the point and we
1:20:56
should take all the multiplicities and like we cannot say anything more so we can use maple to
1:21:05
analyze a few first let's say first few hundred cases and say what we get if we
1:21:13
say that L square is equal to the following numbers and then we
1:21:18
take the natural numbers and M is from 2 to let's say 10 or or 20 or whatever and
1:21:26
then we can make a table and we um discovered that uh the biggest number M
1:21:33
that appears is m equal to 7 which is completely not visible from the
1:21:39
formula um but the computer showed us what we can prove
1:21:46
theoretically and we are we were able to prove theoretically that um if we want
1:21:52
to have a Bound for S con stand in this
1:21:58
um in the area I I wanted to say um then um we have to take in uh into
1:22:06
consideration only M's from 2 to 7 but it works for Big s Square from
1:22:14
like 1,000 some something so the remaining cases of course were done by Maple or any other program but still by
1:22:23
the computer we could have done it by hand but it would take a lot of time and
1:22:31
we trust the computer at this stage that uh it gave um the right
1:22:39
result but um this is not the only um case when we use computers so the second
1:22:47
example is the cutting diagram method which was used by Martin tki um for proving
1:22:54
some version of the SAG conjecture in this Ty in
1:23:00
this in this proof the computer computes the grer basis and um uses um Lots the
1:23:08
the the computations are so complex that um we uh we don't even know what the
1:23:16
computer does at every step so um the computer is used in two different uh
1:23:24
ways one is just to speed the computation that in principle could be done by hand and the other is it does
1:23:31
something and it gives a result that we believe that is true because we have the
1:23:37
algorithm we have checked theoretically the algorithm the algorithm is okay but
1:23:42
um of course there may be some problem with um implementing the the
1:23:50
algorithm uh so let me come to the methodology iCal
1:23:57
conclusions so I would like to ask does the methodology of algebraic geometry take into account this quasy empirical
1:24:07
character so we use computer AED computations and we know the computer
1:24:15
are not perfect but we believe in the
1:24:20
results so of course there are some errors that might be in Hardware or in
1:24:27
software and um it's a good practice that mathematicians report the errors
1:24:32
and then the the libraries are uh improved but still there might be some
1:24:37
errors and um even the proof theorems might not be really true because they
1:24:43
use the computer also algebraic in algebraic geometry uh we use various system of
1:24:50
symbolic computation so it's not just one program we have singular we have M
1:24:56
we have CA we have other programs and uh even if the paper is sent for
1:25:02
review then the reviewer may be using the other system so it is he is he may
1:25:10
usually he doesn't check the algorithm and this may uh
1:25:15
cause the problem that um there appears um theorems they they
1:25:24
appear theorem who are which are not true and the system in algebraic
1:25:31
geometry and also in other areas of mathematics promotes working on new
1:25:36
problems and doesn't reward checking or redoing using other methods usually if
1:25:42
you do the same then the publication is not worth as many points as many credits
1:25:48
as the original one so uh the effort is put to solve new problem s and push new
1:25:56
areas of re research but uh the errors so we may see sooner or later the
1:26:05
replication crisis also in mathematics which um so which is not not
1:26:13
unexpected in like psychology and other Sciences like this but in mathematics it
1:26:19
was completely unexpected and now I think we may see it in in years that
1:26:28
um there there are some errors they replicate they people site and build
1:26:34
other theorems on the papers which have some failures because for example because of the use of of computers or
1:26:41
because of the um non ideal way of reviewing that we have so unfortunately
1:26:49
I don't have more time so thank you for uh for your attention
1:26:55
uh thank youa so it's questions time you're waiting for some questions
1:27:03
so maybe in the meantime uh you take for granted the idea of a
1:27:09
proof uh in mathematical logic especially as the idea goes back to to
1:27:15
hbert a proof is a sequence of formula starting with axum and maybe basic
1:27:20
definitions however it is a sequence of formula do however in uh currently when
1:27:28
we study AI mathema mathematics based on an i for
1:27:33
example in our seminar we studed Alpha geometry and some other system a proof
1:27:40
is represented as a tree so it's not it's not a linear
1:27:46
sequence how do you address this problem does Latos has any idea what a proof is
1:27:57
okay like um so I didn't come across to uh with the distinction whether it's linear
1:28:04
or not linear just um he points out that at every point of the proof every step
1:28:11
of the proof we have to be cautious so maybe it is still linear
1:28:18
because we have this Lamas sequences of Lamas and so on and then every Lama
1:28:23
should be careful Yul looked at um to find whether the Notions
1:28:30
are really correspond to to what there should be and so
1:28:36
on so I didn't see any um idea of um
1:28:42
nonlinear proofs okay
1:28:47
okay let's me check any other questions are there any other questions
1:28:54
as far as I see the none I guess the students maybe a bit
1:29:02
far shy so you will have a chance to to ask wsia of our corridors but for
1:29:10
the moment thank you very much and thank you for maybe if I can I can just ask
1:29:16
maybe for you to comment on something uh for me a lot of uh of lakatos's work is
1:29:23
is also to uh show us the historical process of of doing mathematic and the
1:29:30
importance of those conversations on how one mathematician respond to another and
1:29:35
so on and I think a part of that is also to uh show us how what is uh relevant in
1:29:42
mathematic is is socially constructed so it's not just that you you can prove
1:29:47
anything and everything that you can prove is worth proving there's there's a part of of of that that comes from
1:29:55
mathematicians interacting and responding to one another and when you mention AI so this is something that
1:30:01
might be interesting to think about so those computers will be able to prove a lot of things but uh will they be able
1:30:08
to know what is worth proving what is interesting or not and I think perhaps
1:30:14
this is a this is something to keep considering when we're looking at this at this process and it's just not the
1:30:20
proof that is that that is important it's also the context and why we want to
1:30:25
have a proof for that and so on thank you okay W do you like to respond thank
1:30:32
you no the comment was perfect uh in the spirit of Latos because he also points
1:30:38
this social um aspect of mathematics um but I also uh studied books um some
1:30:47
articles on social constructivism and um just social aspect is not in enough uh
1:30:55
and that's why I think lato view corresponds better to um to the research
1:31:02
in mathematics than just social explanations but of course this common sense um which is the area that we
1:31:10
should look at look at now um the discussions the the importance of the
1:31:16
problems which um come from some some type of intuition I would even say but
1:31:22
not intuition in the sense sense of Brower but uh the intuition um in the
1:31:28
sense of the common sense of mathematicians I would say okay yes yes and and maybe just to
1:31:36
complete I think it's it's interesting also to situate this in in the perspective of uh fire aband because uh
1:31:44
as as you know fire aband wrote his his book as a response to latash and they were supposed to have a conversation and
1:31:52
and I think that this aspect of uh of understanding why we do math and why we
1:31:57
solve these problems and the way that we do versus other kinds of methods of
1:32:03
approach or interest that we have is also something interesting to uh to have in mind there thank you so much thank
1:32:11
you okay so I cannot see any other questions so once more thank you wsa
1:32:18
thank you for your talk and cheers so I guess
1:32:24
Barbara it's at the end of the of this session is it it yes yes it is and
1:32:30
tomorrow we start at 10 a.m. um according to polish time so today
1:32:37
tomorrow the session starts at could you repeat 10 AM 10 AM 10 a.m so thank you
1:32:43
and see you tomorrow bye goodbye thank you
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