Category Theory for Neuroscience (pure math to combat scientific stagnation)

I actually feel like I owe everybody a little bit of an explanation so I jump right in the first thing is what do I mean by scientific impasse and so I'll keep this very brief so the question is are we stuck is there problem in science is there an impasse and I would just let somebody else do the argument for me so this was published in nature not too long ago and based on a lot of data the conclusion was that there's indeed a problem in science overall if you look at the past and what happens now things seem to have slowed the Neuroscience in particular I could point out for example this time period in the 1950s to 1960s and show you various findings that happened I took this from this excellent book in neuroscience and I think it's quite striking that we went from basically inventing the intracellular electrodes to finding what DNA is what Gaba is serotonin dopamine the full genetic code we found Action potentials epsps ipsps we're explaining all of this all the way to orientation tuning was still in the 1960s where all of this got discovered REM sleep reward hippocampus and memory and on the theoretical side we went from cybernetics to the Hodgkin huxe model to artificial intelligence which started in the 60s and this whole brain is a computer metaphor so what has changed not much has changed since this since this time it seems and yet if we look at the number of Publications starting in the 1950s in neuroscience and neurology they have exploded exponent eventually if we look at the number of phds that have graduated in the last 30 years they have gone exponentially so there seems to be a mismatch here perhaps I suggest between the effort the money and what is going into science and what's coming out of it and it seems that that pertains to Neuroscience as well now how could this be fixed well as a biologist who learned that the biggest theory in biology the theory of evolution did not come about by people sitting in an armchair my first instinct would be to say we need more data or better techniques and again I'm not alone in thinking this way so starting with the Obama Administration a lot of money got pumped specifically into Neuroscience for finding better techniques finding more data and there's an equivalent of this kind of approach in Europe as well and yet here we are that just a couple weeks ago nature wrote that disruptive research is declining so if it's not a problem of getting more data then maybe we should rethink our approach and think if there's something we can do on the theoretical side and that will be my entire talk today so what I will argue is that other fields that had similar impasse they found that when they looked more into Theory and specifically into certain parts of mathematics things worked out for them so what I will highlight here is particle physics and just to familiarize real quick starting from you know last century where we realized that an atom is actually made out of smaller particles such as electrons and nucleus we found that the nucleus itself is made of smaller particles and you might have heard about this so the proton and the neutron each consists of three different smaller particles that have been turned quarks and people found out that there seems to be two types of these quarks the up quarks in a proton and one down quark and in the neutron we find two downparks and one up quark that doesn't blow anybody away but somebody realized that when it comes to mathematics and particular group Theory Things become very interesting so those scientists realize that if you take the neutron and the proton and you put them in a coordinate system specifically looking at the number of strange quarks the electric charge and the spin of these particles that there seems to be a symmetry they see they fall onto opposite sides and there is a large field of mathematics that's specifically interested in Symmetry and they they basically said that if we apply group Theory to these kind of findings there should be other particles as well as specifically there should be particles that have two up quarks and one strange Quark or let's say two strange rocks and one up quark and so on and then indeed when they went unlocked they found exactly that so in physics we're at the point where we go from Theory to new data it's almost the exact other way around of what we learn in Biology from diamond and more specifically they do this not just with the kind of high school or undergraduate level math that most of us are used to they do this with pure math and so I took this accent quote from Mark cullivan who wrote that sometimes the answers to scientific questions sit already on the Shelf after mathematics Department I just showed you one of these examples when it comes to physics but as I said I'm gonna make this point specifically about pure math not about applied math because applied math is what we already do so applied math is that if we look at certain phenomena in nature we measure we come up with certain numbers and then those numbers we can convert another numbers and eventually out of these numbers we can derive mathematical laws mathematical equations the mathematics that I will talk about today pure math is what some people would say higher level math and so when people hear higher level math they usually think about something like this well I'm here to tell you that this is wrong this is still applied math pure math doesn't usually have numbers for example so it's a completely different way of thinking about things and so this nice diagram that I modified a little bit shows you how we can go from regular theoretical thinking computer modeling all of these things that we readily do to Pure math and the difference as we deepen down in this hierarchy of theories is that pure math is a very high abstraction so of course numbers already are an abstraction there's no such thing as numbers it is something where we abstract from concrete things and allows us to do math math is Autobot abstraction but pure math math as you will see in a moment is abstraction from abstraction and in a way what we're doing here is we're going deeper into what math is itself we're trying to find out what have different fields that use use numbers for example in common and we're finding if you will the grammar or the rules of mathematics itself that is what a group Theory does and what has helped physics so maybe it's going to help us as well so we're looking at high abstraction because it comes with low complexity and new things can be discovered this again I find myself being not the only one thinking about it but maybe even being late to the party the last couple of years have seen an explosion of papers in various Neuroscience drones that do exactly that so the pure math that these groups are using is topology which in some ways is related geometry but for the longest time it was housed in mathematics department is pure it's non-applied something that you can't use in science or anything for good it was just math for the sake of mathematics and lo and behold there's an increasing number of papers that's coming out that are using pure math topology for data analysis well I would like to talk about a very different kind of pure math today called category Theory and I want to show you a success that category theory has had that I think is not very well known yet and then hopefully trigger a little bit of discussion or debate and what I will tell you about category theory is largely based of your blog taken from Titan a Bradley including actually some of her slides so I want to give credit where credit is to in fact the whole work the whole Insight the success that I will talk about today I know from the people that have done this work that they got inspired by titanist work but there's also a large number of other people that inspired me and allowed me to talk about this today people whose online work or books I read including people that have talked to me personally and educated me a little bit on category Theory now specifically what I would like to talk today about is what is category Theory and how does this apply to Neuroscience in particular to vision science or Consciousness which is my deep interest and to start out I would like to start on Common Ground so most of us are familiar with set the rich in some ways is a kind of pure math as well and a set is basically just a collection of things so here we have a collection of numbers here's a collection of letters those are two sets and set theory usually is using this as a starting ground to launch into deeper mathematics what category theory is not as interested in these objects what category theory is interested in is the relationships between them so in in set theory the relationships that we typically find are mappings from one set to another set and in the theory we call these functions so the definition of a function is basically a mapping a certain kind of mapping from one set to another set and it's it's again it's what category theory is interesting is this mapping rather than what is being met that's the big difference to set theory so you could compare in some ways set the to the semantics of mathematics whereas category theory is more syntax or rules it's about form rather than any substance and so this might become clearer more in a moment now the probably biggest finding in all of category theory is the so-called unit Lemma so the moment you hear about category Theory you will hear about the unit Lemma and it is exactly this finding that I would like to talk about today and want to show you how this has led to massive breakthrough so what is the unit dilemma the story of the Unita Lemma goes that to the 1954 where a young mathematician who just had just graduated went to Paris to work with other magician mathematicians there and got interested in this new field and this new buzz and this new Theory category theory that started out in mathematics and in fact one of the founders of category Theory Saunders McLane was interested in writing a book about category Theory and thought that a good way to do that would be to visit all of the various mathematicians that are already working on category Theory and he would do interviews with these mathematicians and so nobody and Sonos McLane they met the story goes in a tiny small Parisian Cafe in 1954 and it was at a train station because Nobu was waiting for a train and so there was limited time for sauna's nickname to chat in in this discussion they came towards this interesting finding and so the story goes that they had to rush towards the train and while they were walking to the train they were still talking about mathematics and in fact even though the train was only supposed to be taken by nubuyonida sanus McLane went on the train with nubu and barely made it out in time before the train left but at that point the Unita Lemma was established okay so what is DNA dilemma this is the unit dilemma and so this of course an equation by itself doesn't make any sense so let me write it out so if f is a functor from a category C to set then the natural Transformations from home C Dash 2f correspond by a bijection to F of C now if you're like me then this doesn't make any sense because of course without any help you will be wondering what do you mean what is this a factor what is a category what do you mean by set what is a natural transformation what's hum C Dash and what's a bijection so what I would like to do is to help to understand the unit Lemma basically take all of these words of jargon that might not make a whole lot of sense and then one by one just in slightly different order go over each of these terms and explain them to you so it's it's basically as you will see an introduction to category theory in a nutshell the first thing is not really category Theory it's standard in math what is a bijection and so if we go back to set theory where there's mappings between different elements of a set of course I said the mapping in between is a function and the definition of a function is that there is an assignment of each element of one set let's call it X exactly to one element of another set let's call it y so if we had set X here and set y here then what we should find in a function is that again there's an assignment where each element of X is assigned to exactly one element of Y so this is a legitimate function each element of X is assigned to exactly one element of Y even though of course two elements of X are assigned to exactly just one element and Y that's still a legitimate function this is what most people think immediately when they think of a function which is more straightforward you have an assignment of every element of x to exactly an element of Y and it's exactly only one of them and this is a bijection so another word another word for bijection is less fancy it's a one-to-one mapping so each time we have a one-to-one mapping between different elements of mathematics we call this a bijection okay so that's a bijection now I told you that category theory is interested in relations such as functions but it goes beyond that so what is a function I just told you it's a mapping but there's other forms as well we don't have to go to typically that I do it at the moment but for now all I want to tell you is that category Theory because it's abstracting from abstraction it's basically saying that anytime you have a mathematical structure and another mathematical structure such as one set and another set and there's some kind of mapping in between them we called it amorphism so it's a more general term for function if you will so another way that this is done in category theories is that we often don't talk about morphisms but just about an arrow so an arrow and category theory is just this mapping the the fact that there's a relation some kind of assignment between different elements in mathematics so from now on I might say morphism I might just say Arrow so as you can see category Theory at the Step at the get-go is very visual okay so here are different kinds of morphisms that some of you might be familiar with so when it comes to Vector spaces for example a linear transformation is just another mapping it's amorphism when we measure things measurable functions and Z functions are morphisms and there's other ways such as homomorphisms in group theory that are all basically structure preserving mappings they have one thing in common which that is that their relations their morphisms okay so we abstract away from these specifics and we say what do all of these have in common so we have mathematical objects and other other mathematical objects and arrows between them and that is already a definition of our next term that we have to define a category so a category are objects with relations between them each time you have something that relates to something else that's some kind of mapping between them some metamorphism we call it a category so it's a very broad starting point and as I said it's a very visual language so this beautiful drawing by Titan a it shows you that we can basically now use these dots as objects and then these arrows as relations and we can start drawing diagrams or if you will some kind of algebra how do these things relate to each other okay so if we have a category where you have an object and another object and an arrow in between and another category and an object an object amorphism in between what if there's amorphism between categories can we have a mapping from one category to another category the answer is yes of course there can be relations between them and we call that a functor so a functor is a mapping between categories it's an arrow between things that have errors between them and I told you this is just we can keep going this is what category Theory does so here's a beautiful diagram of a functor where you have one object and some kind of relationship let's say a function onto another object this whole thing is a category and we have a factor that maps that onto a second category and that's how we get the f of x and the F of F at the F of Y because there's a mapping this Factor tells us how these relate and it's structure preserving if we have an object here another object here with an arrow in between that's exactly what we come up with over here as well now if there can be a functor as an arrow between categories could there be an arrow between arrows so could we have an arrow from one factor to another could there be a mapping from one function to another and again why not so this arrow in category theory is called it's a very important one a natural transformation so the natural transformation is just another arrow it's an arrow from one Arrow to another arrow you can see it's very very abstract so everything that I told you so far basically is that we can have mathematical objects could be anything could be a set could just be a number and morphisms between them and it's this structure this relationship this web of relations between these objects that category theory is interested in so it finds maps for different other categories that have the same kind of structure that would be a functor and if there's multiple factors we can find the natural transformation between them so the relation between the relations between the relations okay so now we already made most of these terms and you might be wondering why why would you bother what is this why is this interesting I got some data I want to do some analysis I'm writing computer code thing that I told you so far which seems very abstract is just diagrammatic you can Implement in computer code you can do calculations just as you used to it's just a different way of thinking about things not only this there's also emerging programming languages in Python for example that really go to the heart of it where you can just put in these diagrams and you can let the computer do the math follow it on this diagram so you don't even have to do the translation from the diagrams two simple symbols and then back again so there is practical application to everything that I'm telling you okay so we talked about what a bijection is what a category is what a functor is and what natural Transformations are what I still owe you is what set is and what this is home well set is the category of set that's it so if there's a category where you have different sets and the functions between them which is called a set set is the category of sets that's very simple and if there are two objects that say A and B in a category then the the harm factor a b assigns them to a new object that consists of all the morphisms from A to B so there's a set of all morphisms between these objects that's why this home factor is so important and it's a very important part of the unit Lemma okay so this is the crucial point so if we look at the network of relationships that say this object C sheds with all objects in capital c in the category then what we see is there's different kinds of relationships if you will different kinds of arrows some pointing in and some pointing out and that corresponds to these two different harm factors where we're leaving the stash for all of the other objects that might exist in the category and you can see that we're going from those objects towards C or from C towards these objects so you can think of these as almost like two sets of morphisms of relationships and the important thing that seems immediately obvious is that if there are two objects that say c and a and they have the exact same factors this functor and this function is the same then they also must be the same that's in a nutshell the principle inside of the unit Lemma and I find if with with this abstraction it might actually be quite difficult to fully appreciate that I find that if you break this down in a practical example it immediately makes sense oh the one thing I want to say first is that I am going to use for now the term isomorphic and being the same is almost synonymous okay so isomorphic just means that it has the exact same structure and so when it comes to for example you and all of the people that you have relationships with you can think of yourself in a way as having in or outgoing arrows these are all the relationships that you do that you have and so everything that I just told you is that if there would be another object another you and it would have the exact same hum fun fact as the exact same relationships with other people that you have that person would have to be isomorphic with you it would have to be the same with you it has to be you so you were born to certain people you might give birth or be apparent to other people you will have certain romantic Partners friends colleagues and if you take all of these relationships there cannot be another person that has exactly those relationships to be born to the exact same people give birth to the exact same people have exactly the same colleagues and exact same friends so the totality of your relationships defines you uniquely there cannot be another person that has the exact same relationships but is not you that's the fundamental Insight of the unit Lemma that just by the relations anything is uniquely determined and so if I explain it this way in concrete terms it can seem awfully trivial but it is mathematic so I I haven't just shown you something that in a way is Trivial I have shown it to you finally that this works not just for humans in relationships but for anything that can be described mathematically so this what Titanic caused the Unita world view the generalization of this that objects are fully defined by their relations I just showed you not just in something that's trivially trivial but also I showed you and this is not a nice quote that the purpose of category theory is to show that which is formal is formally formal so I showed you formal definitions for this very simple statement I did not show you the formal proof I let a lot of rigor go in the end but I hope that you still caught the gist of it okay so now I'm going to take a big break and I'm going to tell you a very big problem and then I'm gonna show you how the Unita solves this problem this problem is in the Neuroscience of vision in fact it generalizes to all of Neuroscience in particular it bugs the Neuroscience of Consciousness and this problem is called the inverted Spectrum problem so the idea is the following and it started with Berkeley thinking about these things you know more than 400 years ago all of us see a spectrum of colors more or less some of us have some color weakness but we can order colors from in this case as we think about it long wavelength light to short weight wave inside and the idea is that if there would be some people who for whatever reason would see that spectrum that I showed you left to right exactly flipped we would not be able to find out so the idea is that there are two people and they talk about color and the behavior throughout color exactly identical but one c-screen is red and the other person sees red as red and if you think this through you will not find any logical contradiction so why why does this make sense well let's imagine that I would be the odd one here and all of you see apples or let's say the strawberry all of you see that the same way as red red is red for you but this would be my strawberry but my whole life ever since I was a baby whenever I see this everybody calls this red so how would I know that this isn't red if each time anybody sees Red Ice and shows me something that is red that's the color that I get so any attempt now of us talking about what makes a bread I would always agree if you would say well red is the color of fire it's warm it's what you see if something heats up and I would say yeah because that's what I see So based on just talking about it in fact even psychophysics if we just discriminate it seems that we cannot tell apart two people where one would have an inverted Spectrum from the other person and what's worse is that there's no logical reason that this couldn't happen even with the exactly identical brains so you could say well if we look in Alex's brain we would find things that different in a weird and that's why Alex Alex's strawberry appears different to him but if you really think this through there is no metaphysical reason that if my brain is exactly like your brain and we're seeing the same thing and it's the same retinal processes the same neuronifying in higher areas that this would be my mental experience so that is this why is a very troubling finding so there are some arguments that maybe there are even people that have inverted Spectra we would never find out but the really troubling finding is that identical brains could have more than one experience because it doesn't seem to be a hardening there's no mathematical mapping if you will between certain brain States and a mental state at least not one we found yet or that we're aware of and that allows for this scenario okay so does this mean that even if we know everything including the connectivity the past the current states everything about all neurons I give you all glia cells this the blood vessels everything about the brain we could never find out what exactly somebody is conscious of because it could be greenish it could be reddish we can't prefer from the brain well that's why the unit dilemma comes in so the first thing I have to basically refine a little bit is that when we think about the color spectrum of course we put it on a spectrum because color perception in its most simple way is dependent on the wavelength of light and we go from very long wavelength to short wavelength and we have three receptors most of us cone receptors in the retina that sample the Spectrum and it's the activation the relative activation between the three of them that gives us color perception but if you think about this and if you if that is your knowledge about how we perceive color you would not be able to explain this which is that we can order color continuously so somehow this wraps around so how do we take a linear phenomena and we make it circular well the answer to that is in the early visual system because of color opponency so we have a cone circuit diagram that leads us to oppose yellow from blue and red versus green they're literally inhibiting each other and that gives us two opposing poles a yellow blue pole and a green red pole that gives rise to a two-dimensional structure so this is how we wrap that one-dimensional linear Spectrum around in a two-dimensional spectrum and in fact this right here would only be a slice of something more complicated because we can go all the way from Black from complete darkness to White complete brightness so this is not just a two-dimensional Circle each time I'm showing you a color circle like that we're actually taking a slice through a three-dimensional structure such as a sphere and we can do psychophysics on that so one thing we can do for example is we can take the color red and we can psychophysically measure precisely with numbers the distances that exist between red and another color and then find out if that is a certain difference that people assign to that is the difference between this red and this this yellow or greenish color is that twice as large or three three times as large as this distance so that is very simple to do with standard psychophysics so we can actually for each of these colors we can come up with these distances and psychophysicists do that and it turns out it's not a perfect sphere where we see color it actually is this distorted space this is stored in color gamut that we perceive that really means which is interesting that if you map the distance of any point in here it's Unique so you would not be able to take the green over here and then measure all distance with other all other colors and come up with a similar Matrix of differences so the difference is that we find for the color of red are unique for the color of red so in other words if we just look at these arrows what emerges is the Oneida Lemma and this was first realized by now to Kia and his co-worker yato saigo so remember the unit Lemma is that objects are fully defined by their relations I told you that relations are just the structure of arrows what that means this this structure of arrows this relation between red and all the other colors uniquely defines the color red this means everything I told you about the inverted Spectrum doesn't make any sense because if my perception of red would be over here and we would map all of these distances I would actually yield a different map it's mathematically impossible that you would have the exact same relation of red in all other colors but the spectrum is inverted that is what the unit shows and so if you're more interested more deeply into that if you want to read the full formalism there's two there's a white paper and a published paper on that I'm just gonna point this out really quickly so that actually is about it another one of the arguably one of the greatest mathematicians and certainly one of the most interesting mathematicians of the last century was Alexander and so he had this beautiful quote the development of General abstract Theory so pure math by abstracting away from everything eventually brings with it effortless solutions to concrete problems that if you take a knot you can't quite crack it but if you put it in salt water again and again and again eventually it softens it up to the point that you can loosen it and with that thank you foreign [Applause]

Channel: Astonishing Hypothesis

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Id: 4GJ4UQZvCNM

Length: 32min 15sec (1935 seconds)

Published: Mon Jun 26 2023