The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
I have been interested in and researching consciousness on and off since 9 years ago. I searched in computer science(AI), i searched in and did research in neuroscience and studied philosophy of mind on my own. With time i began to appreciate the mathematics in the sciences more and more. Now with this video, i have been convinced a mathematical fundamental in research is the way to go forward in my sciencetific career. Thank you so much for opening a partly closed door for me.
Thank you for the encouraging feedback. If you keep going in that direction, I highly recommend the late Mark Steiner's book "The Applicability of Mathematics as a Philosophical Problem". It's very dense, but worth it. Here's a free, thoughtful summary and review of the book: philpapers.org/archive/NUGMST-2.doc
This was playing in the background while watching porn, but from 7:04 onward my attention was fully fixed on category theory and I dropped what I was doing. The highest praise I can give you, or anyone, is listening to this managed to override my lizard brain behaviour. The way you translate math-ese to understandable language and images is amazing! Making arcane topics like the Yoneda lemma come alive not via rigorous mathematical chants but social webs and colour theory is pedagogically brilliant.
The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
Your brain knows what's important. Beautiful. Thanks for the share. Especially the part about your "porn brain" being overridden. Maybe, overshares are what people actually need.
The talk is nice but the final conclusion is hasty, to say the least. In particular, it seems to me that you showed (albeit informally) precisely why the inverted spectrum hypothesis does makes sense! Let's take the color wheel as an accurate representation of color perception. Every color is uniquely identified by the relative distances with all the other colors. Now rotate the circle, let's say 90° clockwise. The colors are still uniquely identified by their relative distances, but the absolute positions between the two circles differ: for example, what we call "red" is at the top in the first circle, but on the right in the second one (it took the place of the first circle's "blue" after the rotation). Now let's say we have two people, the first sees according to the first wheel and the second one according to the rotated version. Remember that for the two people here "seeing" (perceiving/experiencing colors if you prefer) is determined by the position on the color wheel, NOT by the hue we used in the actual diagrams (that is colors are NOT absolute); it may be useful to replace the actual colors with string of numbers (RGB encoding) to avoid confusion. So when i show a picture whose color is registered as being on the right side of the color wheel, both people will say that the color is "blue", because that's the only color compatible with the relative distances from all the other colors. In fact the second person lived all his life hearing that the thing (feeling?) he experienced when looking at objects similar to the one shown, is called "blue", even tough what he actually sees in his mind is the same as what the other person would describe as "red". It is crucial to notice that when passing from the first color wheel to the second one, we didn't just move a single color (for example putting "red" in the place of "blue"), but all the colors accordingly, in a way to preserve the relative distances. So for the second person is actually impossible to notice any inconsistencies with his mental representation of how colors works and the descriptions given by the first person of what colors are supposed to be. We can generalize these ideas to whatever mathematical object we think is an accurate representation of color perception (like the gamut in the final slides). The point is not that given a specific gamut, a certain color is uniquely determined by the relative distances with all the other colors (this is rather trivial). But what about all the possible gamuts? How are they related? Can we uniquely determine the gamut itself (that is the model which explains color perception) for all the people? To answer this question you would need to show that there can't possibly be a map between two different gamuts that preserves the relevant structure; in this case distances between colors (so it would be a continous isometry, if we think the gamut as a metric space). I hope that by now it's evident enough that "red" and "blue" are just labels and don't actually mean anything by themselves, but only when they are put in a broader context (when they are in relation with all the other colors). The act of fixing this context (which is a model of the phenomenon "colors") is analogous to putting labels to the vertices of a graph. There can be multiple labelling that are compatible, that is we can find a way to pass from the first labelling to the second one that preserves the structure of the graph (a sort of translation). An external viewer, that is not aware of the specific labelling used (let's say we cover the labels before showing the graph), cannot distinguish which one was used to arrive to a certain claim about the graph. Moreover, it doesn't even matter because the claims that can be reached by reasoning with the first labelling must be the same as the ones reachable by reasoning with the second one. In this sense we call these two labellings isomorphic (and hence the graphs), and this is what Yoneda lemma is about. For a structuralist/constructivist/empiricist/behaviorist/ecc. (i hope i'm not mislabelling) Yoneda lemma is a pretty trivial result (from a philosophical point of view): it's just an assertion of the way we think. Since we can't have access to the metaphysical/ontological level (the essence of the objects, what they "really are"), there is no use in try to reason about what we can't observe; therefore there is no practical difference between objects that we are not able to differentiate, regardless of their inner nature (that could be different as far as we know). If when we cover the labels (that is we make inaccessible the "true nature" of the object) we can't spot signs (discrepancies/inconsistencies/whatever) that could alert us of a possible difference, then the objects are equivalent/isomorphic; that is we treat them as they were the same. In the case of consciousness, the labels are always covered by default (unless we assume we can read people minds). There are others problem in the way category theory has been applied to this matter, that i'm not knowledgeable enough to discuss, but you can read this blog post (that i incidentally found as first result googling "color map yoneda lemma"): the title is "No, the Yoneda lemma doesn’t solve the problem of qualia" on Matteo Capucci website. [i don't put the link bcs youtube]
I think it's missing the point though: the inverted color perception thought experiment is one very specific way to bring attention to the main issue that no one has been able to specify how brain states determine the experience of color. More generally, no one has been able to specify how brain states determine any experience. In the best case, maybe the thought experiment is not well formed, which is very interesting in and of itself. However, what have we learned about the main issue of having no clue why the neural activity associated with red MUST BE that color? To be clear, I found the lecture to be enjoyable because I learned several new things.
Fair. Though, if you watch the other videos on the channel, you will see that there is a mathematical way to "specify how brain states determine any experience" which explains "why the neural activity associated with red MUST BE that color". The solution is inherent in Integrated Information Theory (IIT), but only gets fully apparent if you approach the result of IIT's mathematics from a categorical/structuralist perspective. In short, if you project the causal interactions that arise by brain activity into a (hyper)graph, you end up with a very specific structure. The claim is that this specific cause-effect structure is your qualia. In the strong form of the argument, there is an ontological equivalency. This might be hard to swallow, but similar notions exist in particle physics: In the end, there is nothing but structure (relations). That's our reality. Mass, energy, and maybe even time and space seem to be something different from what we naively assume. Mass is not made up of small particles, it arises from field interactions that can only be captured by abstract mathematics. In its weaker form, the argument is that mathematical description is that all there is to science. It is its endpoint. So, as psychology increasingly specifies qualia mathematically (think: color space), we end up with mathematical - in fact, structural/geometric - descriptions anyways. Taking the next step to assume that they are isomorphic to the cause-effect structure of the brain then does not seem such a big leap anymore. What's appealing about IIT is that it starts from a Cartesian stance of extreme skepticism, only needs five axioms, and enjoys increasing empirical support.
@@astonishinghypothesis I will argue structuralism does not solve the inverted spectrum sproblem. It more sweeps it into a corner. Let me explain. For a start, lets hypothesize that all humans experience the same color space in terms of psychophysical metrics. Now the yoneda lemma guarantees that the color spaces are the same up to a unique isomorphism. This in fact leaves the inverted spectrum problem entirely unresolved. After all, spectrum inversion was already a unique isomorphism. Replacing the spectrum inversion isomorphism by a yoneda isomorphism doesn't fix the problem: color qualia are still indiscernible except if translated /by some unknown and unknowable isomorphism/. *dramatic strings music* The thing is, we've pulled a bit of a con. For if we're happy with a structural perspective, then the spectrum inversion isn't a problem in the first place. A structuralist will already accept that we can only ever know another person's mind up to an isomorphism translating their qualia into our qualia. But then there is no reason to handwave Yonedishly and argue that categorical structualism solves the original problem. For a non-structuralist, a solution up to unique isomorphism is the original problem of the Inverted spectrum. They are freaked out by the indeterminacy and distance that puts between human minds, and the way that it seems to turn every mind into an unknowable "Other". Of course, structuralism shows us how this "Other" can be understood and even empathized with when viewed through the right morphism. But it is still an uncanny process to realize that our experienced realities are likely permuted in imperceivable mays with relation to other experienced realities.
I am also not sure about the colors. The distance map is a kind of average map derived from tests on many people. Meaning that if plotted for individuals there would be differences/distortions from personal map to personal map. This distortion between individuals could be that different colors would have the same distances.
For logicians, syntax and semantic are separate. The Yoneda Lemma shows that this separation is an optical illusion, because you can embed the syntax in Set ... The true category theory was found in 1965 by Max Kelly and Sammy Eilenberg: Enriched Category Theory, where "classical" category theory reduce to Set-enriched category theory ! I am myself category theorist (Enriched, but not rich 😂!). Thank you for your talk
@@rwtig I think you right, but you were speaking, I guess, about the completeness theorem. I mean this theorem put the semantic and the syntax in the same level. This deep (and beautiful !) result was necessary to see that. But the Yoneda is perhaps more direct for these purposes. But I don't want to debate the link about Yoneda and the completeness result of Godel, it is too difficult for me !
Holy cow this is exactly what I've been looking for! I'm in a seminar about scientific approaches to consciousness and recently I've been thinking a lot about functional explanations in science and how insights from pure math/philosophy can be used. Thank you!
If you can't appreciate the obvious usefulness of CT as an organizing language for several fields of mathematics it means you've only been working/studying in a field of math that is rather non-structural, like Analysis. There's nothing wrong with that. But, as soon as you need anything a bit structural (Idk, cohomology?), you'll immediately start appreciating CT. Ask an algebraic geometer. Ask an algebraic topologist. Ask an algebraist...
Physics is what we know, metaphysics is what we do not know. The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
@@bullpup1337 Enantiodromia is the unconscious opposite, opposame (duality) -- Carl Jung. Colours are differing aspects or frequencies of the same substance namely energy. Same is dual to different. Lacking is dual to non lacking. Black is the lack of colour and white is all colours (a spectrum) or non lacking. Electro is dual to magnetic -- electro-magnetic energy is dual, photons, light, colours. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought or the principle of equivalence, duality. Potential energy is dual to kinetic energy -- gravitational energy is dual. Energy is duality, duality is energy -- all energy is dual hence colours are dual. Your mind is using duality to create colours. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Mathematicians create new concepts all the time from their perceptions, observations or measurements. Conceptualization or creating new concepts is a syntropic process -- teleological. Thinking is a syntropic process. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! The word dual is the correct word to use here. Sine is dual to cosine or mutual sine -- the word co means mutual and implies duality. Mutual requires at least two perspectives. I would disagree with your comment.
@@bullpup1337 Left is dual to right, up is dual to down, in is dual to out -- space is dual. Space is dual to time -- Einstein. Time dilation is dual to length contraction -- Einstein, special relativity. Energy is dual to mass -- Einstein. Energy = force * distance (space) -- simple physics. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), push is dual to pull, stretch is dual to squeeze -- forces are dual. If forces are dual then energy by definition is dual. The conservation of duality (energy) will be known as the 5th law of thermodynamics! -- Generalized Duality. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. "The Force" is duality! -- Jedi teaching.
Great lecture, but I’m not sure the conclusion is correct. It shows that the color space of vision is isomorphic between people. That doesn’t mean the actual qualia is the same. It’s a basic confusion of map and territory. Personally, I’m largely skeptical of most attempts to apply really abstract math to most practical problems, partially because it just offloads the work into how you structure your category and partially because the conclusions are generally very nebulous. Some of my favorite books are by Robert Rosen, who pioneered relational biology (basically applying category theory to theoretical biology). Utterly fascinating work, and yet I have no idea if there is any substantive use for it. It’s entirely possible I just don’t see deeply enough into the problems.
Yes, but there is an important exception to this general rule of mappings and territories: The territory is also a 1:1 map of itself. This means that a 1:1 map is isomorphic with the territory, and not an impoverished model that fails to capture some aspects of the territory. So, the question is: if two perceptual color spaces are isomorphic, how can the associated qualias not be isomorphic? Some commenters on here seem to argue that there must be morphisms/relations that we are missing by limiting ourselves to perceptual color space. But in order to examine that argument we need to know what these missing puzzle pieces are. What relations other than degrees of perceived similarity are required to exhaustively characterize color qualia?
The quality can be 1:1 isomorphic, that doesn’t mean they are the same thing. Say my perceptual space with relations characterized by various morphisms, etc is a category C and your perceptual space is a category D. A more concrete example would be to say let’s assume that we arrive at some final version of physics, and say it’s something like the Wolfram model where we can fully simulate a segment of the universe in a truly isomorphic way to reality. Does that mean that the simulation is ontologically the same as the actual segment of the universe? Not impossible, but definitely not necessarily. Note that I’m not saying that you can I definitely don’t have the same perceptual space. I’m just pointing out that even if you show they are isomorphic, you still have more work ahead of you.
As for the inverted spectrum problem, one might also ask, "Did the dinosaurs build Stonehenge? And isn't it odd that someone has taken all the trouble to completely remove all the evidence of this?" (Monty Python, circa mid-20th century). Every time he makes the noise "green" he means red.
Great video! I'm getting into category theory but got lost when I got to natural transformations, so I obviously didn't understand Yoneda Lemma, but this video helped a lot. Thanks!
This whole presentation is just an exercise in sophistry. "Look at how well I can blend these arbitrary topics together." There's nothing innovative about it. The discussion about color and its perception is a philosophical one. The measurements of psychophysics bring nothing to the table; psychophisical measurements are philosophically on the same level as checking whether somebody sees a strawberry as red. You could have used the Yoneda lemma directly on those measurements just as well. "If we both call all red objects red then we must be talking about the same red." But the Yoneda lemma is grabbed out of context and not applied rigorously. If you believe people can percieve a color differently then that constitutes a different relation and so the Yoneda lemma wouldn't apply. BUt the whole discussion is foregone by the problem that saying I can perceive red as green presupposes a homunculus inside my head which again observes the color, which is ridiculous. Therefore the whole idea of perceiving colors as other colors didn;'t make sense in the first place.
As a scientist myself I see the main problem in the fact that you just can't afford to concentrate on high risk topics that might or might not produce groundbreaking results. You have to write several articles per year, fully aware that they're only mildly interesting. Because doing otherwise would be professional suicide. Another thing: you can't overestimate the value of discussions with top scientists. But this is not possible when 10% or more of a class go for a phd. Many of them don't even want to work in science. Some do a phd to further their career, others because they won't get a job without the title (biologists and chemists in some countries) and yet others because it's customary (medical "doctors")
i agree, "science" has become a buzzword to show society how competent one is and many in the scientific field are no longer interested in chasing the truth but in flexing
Stop being a chub and go against the publish or die mindset. True greatness wouldn't just go "Oh, I have to do this or my career!!!!" Einstein worked in a fcking patent office for god's sake. I did my undergrad at UC Berkeley, so I know how many midwits truly exist in academia
Yeah, I think is this why if you have a mapping from one distribution to another (F(X) => Y) you effectively have a mapping from x to y (f(x)=>y) (individual points in the distributions) when x is a high-dimensional object (which makes it very unique/specific). This is important because deep learning learns distributions, so now you have a certainty that individual data points can be mapped as well.
What does the asymptote on the right side represent? What do you think? x-axis = (Moral Egoism, 0) union [0, Moral Perfection) y-axis = (Hell, 0) union [0, Heaven) 0 = the Indifference point x is the independent variable, while y is the dependent variable x takes one parameter at a time from three possible: E, 0, P x's parameters are functions (Hell, 0) is the Injustice interval I [0, Heaven) is the Justice interval J f(E) = Moral Egoism = "Do that, which you want and can do." f(P) = Moral Perfection = "Do only that, which would be acceptable to all (mankind)." The Indifference point, 0, can also be expressed as, f(0) = "Do only that, for which all (mankind) would not want to punish you." It's also the threshold where morality has no net impact on circumstances. Thus, the Justice interval can also be expressed as, J = [f(0), f(P)) = ["Do only that, for which all (mankind) would not want to punish you," "Do only that, which would be acceptable to all (mankind).") Assuming justice is acceptable, while injustice is unacceptable because every reasonable human desires to continuously experience minimum injustice and maximum justice with respect to everything. We can plot this model too: arctanh(x) = ln[(1 + x)/(1 - x)]/2, |x| < 1 Above, I've presented the abstract functions for Moral Egoism and Moral Perfection. But here are general descriptions of these concepts: - Moral egoism is the normative position that moral agents ought to act in their own self-interest. It means that what is best for the individual is the highest morality. - Moral perfectionism is the conviction that human beings have a duty to aspire to moral perfection. It posits the pursuit of moral excellence as the ultimate goal of human life, as the path to physical perfection lies exclusively through moral perfection.
Physics is what we know, metaphysics is what we do not know. The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
This video does not make sense. Yoneda Lemma would say the color perception of the inverted spectrum guy be ISOMORPHIC to the color perception of the normal guy. Not literally the same. {Blue, Green, Red} is isomorphic to {Green, Orange, Pink} in Set. Doesn't mean they're literally the same. It misses the entire point of the hard problem of consciousness and the inverted spectrum idea. You could swap how two colors are subjectively experienced, and you'd get an isomorphic copy of color cognition. If they did not give rise to isomorphic ways of color cognition, there would be a way to test whether they are the same of not. So, Yoneda Lemma telling us they're isomorphic is not helpful, we already know they're isomorphic, thats the entire problem! At best its a way to reformulate the crux of the hypothetical.
I love this video and talk! The only nitpick I have is about the thought experiment with color qualia. The sets of relationships that define human color vision are part of a qualitative process- meaning, much like you explained in your video, consists of a set of qualitative relationships between each point in color space, which per the Yoneda Lemma, this means that each color is defined by its relation to other colors, thus meaning that no color could possibly be different than what it is, because it is defined by its relationship to other colors in color space. The thing about qualia is that it does not only involve qualitative processes- it involves the construction of qualitative processes, subjective processes, and distinction processes as well. I like to use the Duck-Rabbit illusion as an example of this. This is an example where a qualitative process (the empirical qualities of that image and how that pertains to how the photons enter the eye, optic nerve, etc.) takes on a new set of relationships because the subjective process (perception of the image) can impart distinction processes which create new sets of relationships where the subjective process and distinction process adds new sets of relationships to the diagram that involve the perception of the image one way or another from a subject. Since objects are defined by their relations, as per the Yoneda Lemma, this means that the changes between the relations between objects that can occur due to the perception of an image (not just seeing it) actually do change how that object (color) can be defined. Likewise, with color, things like different languages having different words for colors, or perceptual differences like synaesthesia, etc. may also contribute to additional subjective and distinction processes that can augment the sets of relationships between the qualitative process of the color space itself vs. any distinction processes resulting from its subjective perception. It's not to say that your statements about the rigorousness of the external qualitive process of human color space is incorrect- I 100% agree with you there. It's more so that I believe that these psychometrics can be augmented by other types of processes and relations besides just the qualitative ones, at least when it comes to qualia. I derive my reasoning from the compositional mathematics for subjective experience described in this paper. Most notably, the mathematical model proposed here contains a proof regarding the readability of the qualitative processes of others' experience, but the unreadability of others' subjective processes of experience- meaning that the inverse spectrum problem will never truly be answered, as per this mathematical model's postulates, at least. www.sciencedirect.com/science/article/pii/S1053810021000945#b0010
That's an interesting paper. Thanks for sharing! To your point (which is shared by many in this comment section). The counter point is to ask: what are these 'qualitative relations' that you postulate to exist in addition to the 'quantitative relations' when it comes to subjective color space? If you can make those concrete, we can just add them to the model (i.e., we add those 'qualitative' arrows to the 'quantitative' arrows), and the Yoneda lemma works exactly like it did before. So, even if that assumption were true, it does not change the logic of the argument. The concern, though, is that it seems difficult to point to what is 'missing' in a subjective color space that is described by 'quantitative' arrows. That is, the color space described by a singular metric - similarity judgments - seems to work just fine. Intuition might revolt at that, and demand that there is 'more', such as an 'intrinsic' quality to color or some kind of semantics. But Occam's razor asks for compelling justification for adding further assumptions. So, the ball is in the corner of the critique: if you believe that there is 'more' to color than metric color space, you will have to show what metric color space fails to explain. And, as a next step, you will have to be more concrete in explaining what exactly is missing: What is an 'intrinsic quality'? What are the missing relations? And the moment these are made concrete, we can just add them as arrows and apply the Yoneda lemma again. As a side note: It is arguable whether anything changes VISUALLY when Duck-Rabbit illusion changes interpretative meaning. That is, In terms of fundamental VISION, the illusion always looks the same. What changes is more like an inner voice saying "this is a rabbit". So, the change is more thought/cognition rather than phenomenal consciousness. Agnosias are a common example showing that these are dissociable mental functions: Certain brain lesions prevent people from making sense of what they see (i.e. they fail to identify a rabbit or a duck), but if you ask them to draw what they see, they create a perfect copy. Or imagine looking at a Jackson Pollock painting - you can see it, yet it will make no sense whatsoever. It is pure vision. This suggests that object recognition is secondary to phenomenal consciousness. You can have visual experiences without any semantics associated with them.
THAT IS A FREAKING AMAZING TEACHER !!! Math made FUN! THANK YOU!!!!!! It is genius to do what he has done - it is RARE to find someone that makes math this simple and fun !
I don't think this actually solves the inverted spectrum problem. Just looking at the very last figure in this talk - imagine red and green being swapped... each color would still have their own distinct set of relations, but the PERCEPTION of the color could be inverted and we'd never know.
I was thinking about the same thing, philosophically what you are saying is true. However, biologically this result makes it more unlikely. If we consider that humans have similarities in their neural structure and that color perception is something innate and not learnt, it is a good assumption that similar neural hardware firing should lead to similar qualia. With this assumption, the way that the inverted spectrum would manifest is if somehow the "hardware" for perceiving green was swapped with the ones for blue. I think this work is a good indication to show that this is not likely to happen because then we would see a different perception diagram. I think this work moves the problem up to a higher layer of processing, which is progress.
Right, but, without knowledge of the causal link between physical properties and subjective perception, even a consistent distinction between neurology underlying green versus red perception would not allow you to know what color a subject is actually perceiving (because they would report red as green, and vice versa). So I don't see this new theory advancing the inverted spectrum problem at all. It's just a more complicated way of saying "I assume there will be distinct neurologies underlying red and green color perception". It doesn't help us determine if, say, two people reporting the perception of red are actually having the same color experience or not - and *that's* the inverted spectrum problem.
The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
I get why we use colors as a way of talking about consciousness and how it could be different between people. Most of us have played around with changing colors on a photo editing program and have seen it for ourselves so it's easy to imagine as a thought experiment. But let's not forget color isn't the only aspect of visual conscious experience. The whole visual field that appears to us in first person experience is visual qualia. The spatial structure our brain produces of the world around us. Its not like a black and white version of the world that we see is the default neutral world. As if this is the underlying physical structure of the world that has always been around and will always be around for any conscious being to see for themselves, perhaps just filled in with different colors. As hard as it is to imagine this simply isn't the case. There's no inherent visual appearance of anything out there in the world. Without eyes to see nothing looks like anything in the same way that nothing smells like anything without a nose to sense that. Apparently dogs for instance can create a type of 3D volumetric map of their surroundings with the sense information they get from their nose. But does it "look" like a 3D map to them? Almost certainly not, it's an entirely different kind of experience. However to a dog it's the way the world inherently appears and cant be any ither way
My understanding is not that we necessarily see them the same, but that there IS a methodology (thanks to the work of Tsuchiya & Saigo) for determining HOW differently or similarly we see them.
There is colorblind people that doesn't realize about their condition until they have to drive and have confussion/trouble with the traffic light (happened to a friend of mine). That's almost 20 years without realizing it, thinking he saw the same colors as us.
This here is the calm before the storm. We are very quickly approaching the technological/cultural singularity where exponential growth becomes most significant compared the the rest of recorded history. It's just that now we're unsure if we can project our prior understandings of the world into the future to develop with them.
If the colour absorption spectrum was symmetrical the result would be the opposite. I do not see how one would need category theory to come to this conclusion and would be very happy if someone could shed some light on this. it is quite obvious even with simple logic following. Going further with asymmetrical colour absorption spectrum, the result still would be different if we do not "assume" individuals who might have inverted spectrum still have the same colour absorption/perception with the ones who do not have inverted. If inverted spectrum people also have different absorption, we still can not differentiate them. Another note is that this makes it possible for everyone to have different vision since we might have continuously different colour spaces with a corresponding absorption spectrum such that the distances between the colours are always the same for everyone.
The most common counter argument is that color perception is not the same as perception of electromagnetic wavelength. These two processes can be double dissociated, and hence are orthogonal: For example, there are different wavelengths of light that look the same (metamers), and there are colors that do not map onto any wavelengths such as pink, brown, or any neon colors (there are many more such "impossible colors"). Visual illusions that work on color demonstrate that we can even see color for gray or black and white stimuli.
@astonishinghypothesis Thank you for your response, this tells me that wawelenght and colour perception is a nonlinear mapping. So the question is still the same actually, is there only 1 possible nonlinear map for everyone to have? I think there might be infinitely many mappings that satisfy the same properties(distance of colours etc.. whatever is necessary). If there is only 1 possible way that means there is little to no room for an error, and the neurons resposible for this mapping does not change for a lifetime after those neurons are set possibly before we born. People live 80 -100 years and it is counterintiutive to think that those neurons are uneffected no matter what life throws at us :)
IDK, man. I appreciate the effort, but I feel like really what's the point? Explanations are a dime a dozen these days, to the point that people are no more enthused by it. For e.g., Integrated information in consciousness, or String theory in physics. Like you can fit an explanation to anything, & it can be beautiful. But, it often goes no where. Hence, the general disinterest.
The problem is that as short-term pro-business mentality infects every aspect of university nobody gets funding to research anything original, since markets only want incremental improvements on things that already exist.
After 100 years it really seems like the only REAL advancement in our understanding of the human brain is that we can say "we have no idea how the human brain works" in ever more complicated ways.
when it comes to applying the lemma to colors: different people demonstrably don't have the same relations between red and every other color. the obvious example are people with varying degrees of colorblindness. and in fact they don't necessarily have the same relations between colors and objects, because they may mistake red objects for green ones and vice versa (because the distance between those colors is much smaller for them than for the average person) and never mention it to be corrected. so i don't think this immediately rules out the idea of an inverted spectrum, it just makes it far less likely because most people's color relationships will be rather similar to each other's, and colorblindness has a known effect that does not represent an inverted spectrum.
Hi 1. so you say: we all experience the same colors more or less (as the video concludes), because we have very similar distances, except for some colorblind. Did I understand correct? 2. however isn't the colorblinds' difference of distances, caused by physio-anatomical differences/ difficiencies? Or is it the same but their experience differs? I don't know this.
@@SiEmG yes, colorblind color perception is caused by a particular difference in functioning of cone cells, but i'd imagine smaller differences occur throughout the population, presumably resulting in different distance relations between the same colored lights. the video's conclusion is likely "correct on average", but with some margin for error based on the extent of those differences, and excluding the colorblind unless they are to be considered part of the distribution rather than outliers (say, if their inclusion does not make the distribution bimodal).
@@rarebeeph1783 then its not the qualia that have different distance relations, but its caused anatomically. if the colorblind would have the "corect" cone cells, they would have the same distances. The quetion is: if 2 twins have the exact same anatomy, do they necesserily have the same color- differences experiencially?
@@SiEmG that's a good question that i suspect depends on to what extent visual processing itself is genetically determined vs. learned at a young age. given only the variables i'd laid out so far, identical twins should have the same color differences, so if it were to happen to be that they don't, that'd indicate a non-genetic component.
It seems like the category theory part doesn't pull any weight in this argument. It seems to me like there is no difference between the presented argument, and just stating that we all share the same brain structure in those specific parts, that maps bijectivity to our reported color experience... The same argument could have been made even if our receptors were 3 'bits' encoding 3 different colors... no real need to go into how colors corresponds to relations between those receptors...
Unfortunately the "solution" to the inverted color spectrum given really has nothing to do with category theory. He is merely shoehorning the concept of categories into the "solution", which is really just a statement about a certain metric space (space of colors) not having a symmetry (color inversion). One would actually need to add more structures, e.g. enrichment over real numbers, to make sense of distances in a category setting, and it really doesn't add anything to the discussion, and in fact obscures the main idea, which is, again, a lack of isometries. If the ideas of category theory inspired such a perspective, fine. But are there really no better examples to display the use of category theory? It sounds like a desperate attempt to ride a "applied category theory" wave that is creeping in academia. Perhaps the speaker wants to use buzzwords to get grants.
Interesting insights on the meaning of Cat th. But the application to cognitive neuroscience (psychophysics, etc) only begin at 24:00 (which is a bit disappointing for me but I cannot complain: it is very hard to present this topics for a reasonably large audience. There is an unavoidable tradeoff that depends on the targeted audience. )
Fascinating. I know one has to be careful to make connections betwenn science and religion, but this reminds me very much of how in Buddhism they say things have no existence in themselves but are only defined by ther relationships to all other things.
couldn't this imply inverted spectrum could still exist, just that there would be different relation of colors for those who would see "inversely" to me. That person could see and be able to better distinguish more of my "red" where I would see more of my "green"? We would still function the same in spite of seeing being able to see colors and distinguish distances in colors that the other wouldn't.
Still relations don´t define absolute colors, which do not even exist. We have no means of knowing in what subjective way another person experiences color. The relations could be the same though the actual experiences differ.
How was the lemma useful in a vision example exactly? I mean, I understand you could use it because it's such a general statement, but it also feels like using a cannon on a bunch of ants, the fact that the color space is not symmetrical should be enough on it's own.
Watch all the empiricists flood the comments with trash about how the thing that made their entire shallow careers possible (ie math) is completely pointless and just for people who "like puzzles"
Physics is what we know, metaphysics is what we do not know. The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
It is hilarious that you first taught the Category Theory and then explained the Inverted Qualia problem perfectly, AND THEN fell into the oldest mistake in the book! When the qualia of color changes, nothing stops the relationships between the qualias from changing as well. Hence they can be functionally (the "relationship between") swapped as well. Green can act like red! If we have no theories from voltages to qualities, that is. Because the quality and function is not equivalent without a solid theory for "red". The function of red is something we can measure, but the quality of the red seems completely random! Hence the Inverse Qualia problem: qualities seem arbitrary, so why not keep the functionality same and the quality different? I suggest you think about what quality of a sense mean.
This distorted sphere, does this come from measured data? Is that distorted sphere similar for all brains? Is there a noticeable uncertainty? Is it proven, that there is no way for this mathematically vaguely defined distorted sphere to be isomorphic to a flipped version of itself? It's not the same. It's isomorphic. And this color space maybe can be isomorphic to another ordering of the color spectrum.
You cannot have inverted spectrum, because qualia are meaning, they are not random, is like saying that you can have wings instead of legs and not noticing. Of course you notice. See my papers about consciousness, like "Meaning and Context: A Brief Introduction", author: Cosmin Visan.
what i am missing is a clearer explanation regarding the nature of this non-symmetrical color-map at the end - when we speak about colors, we might mean many different concepts, but in this talk, nearly all of these concepts have a relevance, so its unclear to me what concept of colors is the source of this map...
Roughly, I believe that distances in the map correspond to neurological signal differences (and hence, psychophysics claims, distinct experiences) So there are more shades of green that can be experienced as distinct in the brain, and for that reason green takes up a greater area in the map
@@tsawy6 i just dont see how this could ever tell us anything about qualia, this map can only ever tell us about the relationship between the stimulus (wavelengths of light) and our brains reaction to those, but not what qualia represent these inside anyones consciousness.
@@davejacob5208 yeah for sure, this criticism has come up a few times in the comments here. The author responded, almost seeming to go as far as to suggest that Qualia doesn't exist. It's very interesting, check them out
Science will (inevitably) stagnate once the universe is finite and everything in it is also, it will not solve all questions (and it isnt supposed to) and other types of knowledge not poluted by the senses like; philosophy, mathematics, logic etc will take protagonism
Lots of good points here, but it is ridiculous to divorce objective reality of matter from subjective reality of consciousness in this way! Indeed, i would say that the red qualia of human physical consciousness is inherently far more connected to the 600 nm wavelength than to the 530 nm wavelength or the 450 nm wavelength. Similar things hold true for green qualia, blue qualia, yellow qualia, turquise qualia, orange qualia, violet qualia, pink qualia, lime qualia etcetera. It MIGHT be true though that creatures who have a visual spectra ranging more than one "octave"/doubling of wavelength (like 400 nm - 800 nm), will have similar, but not quite isomorphic, qualia, inherently related to one electromagnetic light wavelength, and its double or its half! But "red" qualia is still fundamentally different from "green" qualia or "blue" qualia, and this fact of subjective consciousness is related to the objective fact of matter, that the corresponding wavelengths of electromagnetic light/radiation are related through very different ratios than powers of 2. The same is true for frequencies, that are evidently inversions of wavelengths. Possibly there is a similar "inversion" in consciousness of color qualia into something related, for example a physical mood. This is up to empiric psychological science to research however, not up to speculation to decide upon. Just as the objective world is governed by non-arbitrary Natural Laws, so are the subjective worlds also governed by non-arbitrary Laws of Life. Because all consciousness is grounded in matter (and all matter is more or less conscious), even non-physical consciousness (of various kinds) is grounded in non-physical matter (of corresponding kinds), just like physical consciousness is grounded in physical matter. This is a fact of reality, not a delusion of speculative arbitrary subjectivism or of ignorant physicalistic objectivism.
I was really happy to find out that this speculative subjectivistic argument of "perfect" color qualia inversion was indeed some kind of strawman! 😄💛 And that map of psycho-physiological distances of color qualia is indeed gorgeous, especially since it is related to the philosophy of the Yoneda Lemma❣️
I’m not sure that everything you said about the inverted spectrum was ultimately false. You made the example of saying your perception of red could be down in the blue corner, but aren’t you making a mistake there? If you inverted the colors of that image, the point you specified would still be in the same place and the relation to its context would remain the same, it would just be a different color.
I believe you are correct. Under one premise - that is that there is more to color qualia than mathematics can describe. This reasoning is best explored by taking consciousness out of the equation for a moment: Take the example of a human being defined by the totality of their social relations. One could argue that there is more that makes us individuals than our interactions with other humans (some sociologists dispute that, but let's put that aside for a moment). You could argue that humans are also affected by their relations to other living and non-living things, such as pets, food, geography etc., for example. Very well. But this only creates temporary embarrassment for the argument put forward here. Why? Because you can simply add all of these relations to the categorical model - and the argument holds again. So, the argument only fails if you can name some kind of influence on what makes an individual that could not be expressed in a web of relations. Now, when it comes to color, you could similarly argue that the psychophysical color space under-defines color perception. That is, it seems logically possible that one could see an inverted color while establishing the same color space behaviorally. As you point out, a proponent of the original inverse spectrum argument would probably insist on exactly that. But then the question arises what is missing to fully define color. If whatever is missing can be measured and/or expressed mathematically, it would be easy to add to the categorical model and thus resolve the issue again. It is only when we assume that there is something to color that goes beyond mathematical/structural description that the argument falls flat. Now, some philosophers believe exactly that. But, the problem is that most arguments that aim to show that this is the case (e.g., the knowledge argument/Mary's room) are not logically compelling. These arguments tend to rest on intuition, which we all know can be unreliable. Plus, similar views were long held about the nature of life (vitalism) that did not stand the test of time. In fact psychophysics was a response to the notion that subjective experience is something that can neither be measured, nor assigned numbers to. These notions had to be dropped as well. Thus, there is a certain similarity between the "God of the gaps" argument and the clinging onto the notion that consciousness falls outside the rest of reality in that it escapes naturalism. That is certainly a valid view, but as science progresses, it seems to cease more and more ground. tl;dr: The argument rests on the view that structure is all there is to color qualia. This in turn makes most sense on the ontology that relational structure is all there is to everything (hence, Yoneda "worldview"). This should have probably been made more clear. Here is an interesting snippet with Carlo Rovelli discussing a very similar view: ruclips.net/video/r_fUPbBNmBw/видео.html
@@astonishinghypothesis It seems if structure is all there is to color qualia we should long ago have evolved language to convey the nature of our color experiences sufficiently to put inverse spectra to rest, along with being able to explain to the color vision deficient what they are missing seeing and having what *we* are missing explained to us by tetrachromatic individuals. If a tetrachromatic individual (or a category theorist studying tetrachromatic psychophysics) were to completely describe to us the structural relations of the tetrachromatic gamut, how far would this enlighten us as to the nature of the tetrachromatic experience of color? If we explain the structural differences between red and green to a red-green color blind individual, how much insight as to the nature of the standard experience of red would they gain? It's hard for me to conceive that after the full structural description is given there would be no surprises in store for such a person suddenly gifted the ability to experience directly the color gamut whose description they had received, but had previously not experienced!
@@astonishinghypothesis ["which we all know can be unreliable"] Even so, it's extremely essential for the development of mathematics itself, isn't it? It's fun to see how you try to throw intuition out the window when mathematics without it is just a set of letters, or rather, colors, or better yet, particles (which we wouldn't even know are particles without our intuitions allowing us to grasp the world). Mathematics simply does not exist without intuition (even when math leads us to have seemingly anti-intuitive beliefs). The inability to answer things like Mary's room argument or to define consciousness in precise terms should not be an excuse to spit on its strength, or to try to force it into a box in which you feel more comfortable, but rather should be a reason for you to wonder if it is even possible for a conscious being to understand its own consciousness, or if we are going in the right direction at all when we talk about consciousness and end up reducing a phenomenon that is fundamentally part of our lives, our whole lives, to a handful of numbers.
@@astonishinghypothesis: The assumption that one can "discretely" separate different types of qualia (e.g. we experience color as a 'packet' of info completely orthogonal to other mental experiences such as smell or 'inner' emotions) is an extremely strong assumption and, and very possibly false. When I see red, I don't just _see_ red: I go through a space of experiences all bundled together; I e.g. feel the emotions conjured by the memory of the red dress of my ex girlfriend; I'm reminded of the picture of the red mushroom in the book my mum used to read me when I was 3, which brings me a memory of smells and/or of tastes that I newly discovered at that age; etc etc. So, my "red" is all that structure together. Even if the non "purely visual" mental relations are subconscious, it may be the case that some aspect of them resurfaces as an ingredient of my "red" qualia. It may even be the case that my "red" qualia of today aren't the same as my "red" qualia of yesterday, and they are just related to each other by a sense of continuity as complex as the sense of continuity of my own identity. _In theory,_ you may apply a structural/mathematical framework to try to model all that - but in practice it's impossible because of the complexity of the thing. tl;dr: the space of qualia _types_ may not be discrete.
I would bet a large sum of money that with a sensitive enough measurement you would not find two humans in the world whose color-to-color mapping would be exactly the same. So I don't find this argument very compelling. Nevertheless, since qualia (experiences) are epiphenomenal, we can pretty much ignore them, so the inverted spectrum problem is a non-problem. Join the camp of qualia denyers, everything is easier on this side.
Physics is what we know, metaphysics is what we do not know. The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
Are you familiar with the philosophies of Martin Heidegger and Ludwig Wittgenstein? Heidegger was interested in the meaning of being and Wittgenstein was interested in the philosophy of language. They operated very independently of one another but interestingly converged on ideas very similar to what you mentioned here with the Yoneda Lemma. Wittgenstein argued that a private language would be impossible to learn. And your argument on color perception is a more precisely stated version Wittgenstein's private language argument. I want to go even further than you here. The universe is inherently meaningful because of its networked nature. Meaning is the relationship between a node and the network it is embedded in. And this holds true for all levels of nature. Being is the internal structure of a node (its existence) plus the relationship it has with the network it's embedded in. For this reason, meaning and being are intertwined with one another. I refer to this geometry as 'Situatedness.' Information is the signaling of Situatedness. Learning and knowledge are the storage and remembering of Situatedness. Meaning makes information possible, which in turn makes life possible, which in turn makes intelligence possible, which in turn makes consciousness possible. In other words, meaning makes information possible, not the other way around.
I disagree with the inverted spectrum theory. You would be able to tell a difference with comparison colors. Such as someone saying this dress goes with my eyes being on the opposite side of the color wheel. The spectator, though, will disagree because his eyes would not match the color the spectator sees. It would be a totally different color.
Physics is what we know, metaphysics is what we do not know. The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
You know that if some quantity grows "fast" it doesn't mean that it grows exponentially? There is an actual mathematical meaning to this term. These graphs look more linear or at most quadratic. This is only two minutes in, which is quite disappointing for a talk about "Category Theory for Neuroscience". I will give it a couple more minutes hoping that it improves.
Physics is what we know, metaphysics is what we do not know. The Yoneda lemma = Duality. Domains are dual to co domains. Injective is dual to surjective synthesizes bijective or isomorphism (duality). Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. The rule of two -- Darth Bane, Sith lord. "Always two there are" -- Yoda. Synchronic lines/points are dual to enchronic lines/points. Points are dual to lines -- the principle of duality in geometry. The colour black is dual to the colour white -- colours are dual.
From what we know in neuroscience. Consciousness isn’t material in the sense that it’s a physical object. Rather, it is an emergent property of sufficiently advanced and complex neurological systems.
Define "metaphysical". In my dictionary, Metaphysics describes a method to gain knowledge about our world from a priori assumptions by purely logical means (which has been proven by Kant to be unreliable). Whatever consciousness is, it is not a method to gain knowledge.
The inverted spectrum thought experiment (as it's normally presented) is pure idiocy. Essentially it says that laws of physics are arbitrary. For color perception to be different there must be an underlying cause, a physical alteration either in the eye or in the brain, the way it is with color blindness (daltonism). An abnormal perception, if it doesn't interfere with any aspect of life, would be undetectable. However, it would then also be of no consequence, as if it didn't exist at all. More over, it's quite possible that everyone perceives color SLIGHTLY differently (rather than inverted), and it's also of no consequence whatsoever. However, I fail to see why mapping NORMAL color perception in any way contradicts any ABERRANT perception and the colloquial use of color (which is a very rough division of the visible into 6-8 colors at the most). If we had 10s or 100s of colors, a biologically determined mapping would clash with the naming conventions and perceptions in the aberrant individual, but with a rough division into red-green-blue-yellow-purple? Unlikely.
Seems to me like you could get to this conclusion through a simple application of Leibniz's law & second order logic. Ie, Identical(x,y) iff for any relation R, function f (R(x) iff R(y) /\ f(x) = f(y)). Given the manifold defined, which represents the colors (and I'm assuming is measurable), we define a metric d(x,y) which is not unlike the ones in a 3D Hilbert or even Euclidean space. Then it's just a geometric proof that for any three colors, a,b,c, d(a, b) = d(b,c) iff Identical(a, b) or Identical(b,c) or Identical(a,b) Different formalism but same idea. His introduction juxtaposing set theory as concerned with objects whereas category theory concerns itself with relations just simply is not correct (see Benacerraf's paper "What Numbers Could Not Be"). Category theory is possibly a better syntax for describing relations between these larger fields in mathematics, but there are serious differences in terms of semantics. I mean category theorists have posited the existence of some very large objects which, according to work done on large cardinals in set theory, are of dubious existence, in that they are independent of our axioms in ZFC. Furthermore, the very existence of so called abstract categories, which cannot map into a set (as opposed to concrete categories) is something which though set theory allows, eg, in the case of proper classes, such as V or the ordinals; are much more restricted.
Decades of physicists and mathematicians wasted their time on totally unprovable String Theory, and have gotten nowhere. Yet still they persist on the dead end, instead of doing something useful. String Theory is pure bunk.
This lemma cannot prove that we experience the same red. You can reverse the role of green and red on the gamut as well. I mean, you can construct a device, an AR glasses which makes this trick. Whith no math you have access to personal experience.
Perfect balance between high level and technical, in this presentation.
Well, it's the only way the subject should be taught, so there's not really a need to praise him
@@Flaystrayit’s true that praise isn’t always necessary, but it’s still nice to give it to people
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
you'll be okay little guy @@Flaystray
@@hyperduality2838 you cannot say that because two colours are dual, all colours are.....
I have been interested in and researching consciousness on and off since 9 years ago. I searched in computer science(AI), i searched in and did research in neuroscience and studied philosophy of mind on my own. With time i began to appreciate the mathematics in the sciences more and more. Now with this video, i have been convinced a mathematical fundamental in research is the way to go forward in my sciencetific career. Thank you so much for opening a partly closed door for me.
Thank you for the encouraging feedback. If you keep going in that direction, I highly recommend the late Mark Steiner's book "The Applicability of Mathematics as a Philosophical Problem". It's very dense, but worth it.
Here's a free, thoughtful summary and review of the book:
philpapers.org/archive/NUGMST-2.doc
@@astonishinghypothesis thank you for the recommendation! I'll look into it.
@@astonishinghypothesis Again thanks for the recommendation. I have just started reading it.
This was playing in the background while watching porn, but from 7:04 onward my attention was fully fixed on category theory and I dropped what I was doing. The highest praise I can give you, or anyone, is listening to this managed to override my lizard brain behaviour.
The way you translate math-ese to understandable language and images is amazing! Making arcane topics like the Yoneda lemma come alive not via rigorous mathematical chants but social webs and colour theory is pedagogically brilliant.
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
Sir I cannot stop laughing at this comment 😂
I’m interested to know which porn had this playing in the background
Your brain knows what's important. Beautiful. Thanks for the share. Especially the part about your "porn brain" being overridden. Maybe, overshares are what people actually need.
this is the best comment I ever seen in 10 years of youtube addiction
The talk is nice but the final conclusion is hasty, to say the least.
In particular, it seems to me that you showed (albeit informally) precisely why the inverted spectrum hypothesis does makes sense!
Let's take the color wheel as an accurate representation of color perception. Every color is uniquely identified by the relative distances with all the other colors.
Now rotate the circle, let's say 90° clockwise. The colors are still uniquely identified by their relative distances, but the absolute positions between the two circles differ: for example, what we call "red" is at the top in the first circle, but on the right in the second one (it took the place of the first circle's "blue" after the rotation).
Now let's say we have two people, the first sees according to the first wheel and the second one according to the rotated version. Remember that for the two people here "seeing" (perceiving/experiencing colors if you prefer) is determined by the position on the color wheel, NOT by the hue we used in the actual diagrams (that is colors are NOT absolute); it may be useful to replace the actual colors with string of numbers (RGB encoding) to avoid confusion.
So when i show a picture whose color is registered as being on the right side of the color wheel, both people will say that the color is "blue", because that's the only color compatible with the relative distances from all the other colors. In fact the second person lived all his life hearing that the thing (feeling?) he experienced when looking at objects similar to the one shown, is called "blue", even tough what he actually sees in his mind is the same as what the other person would describe as "red".
It is crucial to notice that when passing from the first color wheel to the second one, we didn't just move a single color (for example putting "red" in the place of "blue"), but all the colors accordingly, in a way to preserve the relative distances. So for the second person is actually impossible to notice any inconsistencies with his mental representation of how colors works and the descriptions given by the first person of what colors are supposed to be.
We can generalize these ideas to whatever mathematical object we think is an accurate representation of color perception (like the gamut in the final slides). The point is not that given a specific gamut, a certain color is uniquely determined by the relative distances with all the other colors (this is rather trivial). But what about all the possible gamuts? How are they related? Can we uniquely determine the gamut itself (that is the model which explains color perception) for all the people? To answer this question you would need to show that there can't possibly be a map between two different gamuts that preserves the relevant structure; in this case distances between colors (so it would be a continous isometry, if we think the gamut as a metric space).
I hope that by now it's evident enough that "red" and "blue" are just labels and don't actually mean anything by themselves, but only when they are put in a broader context (when they are in relation with all the other colors). The act of fixing this context (which is a model of the phenomenon "colors") is analogous to putting labels to the vertices of a graph. There can be multiple labelling that are compatible, that is we can find a way to pass from the first labelling to the second one that preserves the structure of the graph (a sort of translation).
An external viewer, that is not aware of the specific labelling used (let's say we cover the labels before showing the graph), cannot distinguish which one was used to arrive to a certain claim about the graph. Moreover, it doesn't even matter because the claims that can be reached by reasoning with the first labelling must be the same as the ones reachable by reasoning with the second one. In this sense we call these two labellings isomorphic (and hence the graphs), and this is what Yoneda lemma is about.
For a structuralist/constructivist/empiricist/behaviorist/ecc. (i hope i'm not mislabelling) Yoneda lemma is a pretty trivial result (from a philosophical point of view): it's just an assertion of the way we think. Since we can't have access to the metaphysical/ontological level (the essence of the objects, what they "really are"), there is no use in try to reason about what we can't observe; therefore there is no practical difference between objects that we are not able to differentiate, regardless of their inner nature (that could be different as far as we know). If when we cover the labels (that is we make inaccessible the "true nature" of the object) we can't spot signs (discrepancies/inconsistencies/whatever) that could alert us of a possible difference, then the objects are equivalent/isomorphic; that is we treat them as they were the same. In the case of consciousness, the labels are always covered by default (unless we assume we can read people minds).
There are others problem in the way category theory has been applied to this matter, that i'm not knowledgeable enough to discuss, but you can read this blog post (that i incidentally found as first result googling "color map yoneda lemma"): the title is "No, the Yoneda lemma doesn’t solve the problem of qualia" on Matteo Capucci website.
[i don't put the link bcs youtube]
This was an amazing introduction to both category theory and the neuroscience of visual perception.
I think it's missing the point though: the inverted color perception thought experiment is one very specific way to bring attention to the main issue that no one has been able to specify how brain states determine the experience of color. More generally, no one has been able to specify how brain states determine any experience.
In the best case, maybe the thought experiment is not well formed, which is very interesting in and of itself. However, what have we learned about the main issue of having no clue why the neural activity associated with red MUST BE that color?
To be clear, I found the lecture to be enjoyable because I learned several new things.
Fair. Though, if you watch the other videos on the channel, you will see that there is a mathematical way to "specify how brain states determine any experience" which explains "why the neural activity associated with red MUST BE that color". The solution is inherent in Integrated Information Theory (IIT), but only gets fully apparent if you approach the result of IIT's mathematics from a categorical/structuralist perspective.
In short, if you project the causal interactions that arise by brain activity into a (hyper)graph, you end up with a very specific structure. The claim is that this specific cause-effect structure is your qualia.
In the strong form of the argument, there is an ontological equivalency. This might be hard to swallow, but similar notions exist in particle physics: In the end, there is nothing but structure (relations). That's our reality. Mass, energy, and maybe even time and space seem to be something different from what we naively assume. Mass is not made up of small particles, it arises from field interactions that can only be captured by abstract mathematics.
In its weaker form, the argument is that mathematical description is that all there is to science. It is its endpoint. So, as psychology increasingly specifies qualia mathematically (think: color space), we end up with mathematical - in fact, structural/geometric - descriptions anyways. Taking the next step to assume that they are isomorphic to the cause-effect structure of the brain then does not seem such a big leap anymore.
What's appealing about IIT is that it starts from a Cartesian stance of extreme skepticism, only needs five axioms, and enjoys increasing empirical support.
@@astonishinghypothesis I will argue structuralism does not solve the inverted spectrum sproblem. It more sweeps it into a corner.
Let me explain.
For a start, lets hypothesize that all humans experience the same color space in terms of psychophysical metrics. Now the yoneda lemma guarantees that the color spaces are the same up to a unique isomorphism. This in fact leaves the inverted spectrum problem entirely unresolved.
After all, spectrum inversion was already a unique isomorphism. Replacing the spectrum inversion isomorphism by a yoneda isomorphism doesn't fix the problem: color qualia are still indiscernible except if translated /by some unknown and unknowable isomorphism/. *dramatic strings music*
The thing is, we've pulled a bit of a con. For if we're happy with a structural perspective, then the spectrum inversion isn't a problem in the first place. A structuralist will already accept that we can only ever know another person's mind up to an isomorphism translating their qualia into our qualia. But then there is no reason to handwave Yonedishly and argue that categorical structualism solves the original problem.
For a non-structuralist, a solution up to unique isomorphism is the original problem of the Inverted spectrum. They are freaked out by the indeterminacy and distance that puts between human minds, and the way that it seems to turn every mind into an unknowable "Other".
Of course, structuralism shows us how this "Other" can be understood and even empathized with when viewed through the right morphism. But it is still an uncanny process to realize that our experienced realities are likely permuted in imperceivable mays with relation to other experienced realities.
I am also not sure about the colors. The distance map is a kind of average map derived from tests on many people. Meaning that if plotted for individuals there would be differences/distortions from personal map to personal map. This distortion between individuals could be that different colors would have the same distances.
Exactly right, they've smuggled the conclusion into the premises.@@maxsuica6144
i think he will argue that navigating thru that color space (which is not determined by physics, but is psychophysical), is having an experience.
For logicians, syntax and semantic are separate. The Yoneda Lemma shows that this separation is an optical illusion, because you can embed the syntax in Set ... The true category theory was found in 1965 by Max Kelly and Sammy Eilenberg: Enriched Category Theory, where "classical" category theory reduce to Set-enriched category theory ! I am myself category theorist (Enriched, but not rich 😂!). Thank you for your talk
In terms of logic, isn't this what Godel showed in order to prove his incompleteness theorem (syntax and semantic separation as an illusion)?
@@rwtig I think you right, but you were speaking, I guess, about the completeness theorem. I mean this theorem put the semantic and the syntax in the same level. This deep (and beautiful !) result was necessary to see that. But the Yoneda is perhaps more direct for these purposes. But I don't want to debate the link about Yoneda and the completeness result of Godel, it is too difficult for me !
Holy cow this is exactly what I've been looking for! I'm in a seminar about scientific approaches to consciousness and recently I've been thinking a lot about functional explanations in science and how insights from pure math/philosophy can be used. Thank you!
If you can't appreciate the obvious usefulness of CT as an organizing language for several fields of mathematics it means you've only been working/studying in a field of math that is rather non-structural, like Analysis.
There's nothing wrong with that. But, as soon as you need anything a bit structural (Idk, cohomology?), you'll immediately start appreciating CT. Ask an algebraic geometer. Ask an algebraic topologist. Ask an algebraist...
Great talk. From pure math to applied math. I don't have math background but I love type theory and category theory.
Physics is what we know, metaphysics is what we do not know.
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
neither black nor white is a color, and they are opposite, not dual.
@@bullpup1337 Enantiodromia is the unconscious opposite, opposame (duality) -- Carl Jung.
Colours are differing aspects or frequencies of the same substance namely energy.
Same is dual to different.
Lacking is dual to non lacking.
Black is the lack of colour and white is all colours (a spectrum) or non lacking.
Electro is dual to magnetic -- electro-magnetic energy is dual, photons, light, colours.
Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought or the principle of equivalence, duality.
Potential energy is dual to kinetic energy -- gravitational energy is dual.
Energy is duality, duality is energy -- all energy is dual hence colours are dual.
Your mind is using duality to create colours.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Mathematicians create new concepts all the time from their perceptions, observations or measurements.
Conceptualization or creating new concepts is a syntropic process -- teleological.
Thinking is a syntropic process.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The word dual is the correct word to use here.
Sine is dual to cosine or mutual sine -- the word co means mutual and implies duality.
Mutual requires at least two perspectives.
I would disagree with your comment.
@@bullpup1337 Left is dual to right, up is dual to down, in is dual to out -- space is dual.
Space is dual to time -- Einstein.
Time dilation is dual to length contraction -- Einstein, special relativity.
Energy is dual to mass -- Einstein.
Energy = force * distance (space) -- simple physics.
Action is dual to reaction -- Sir Isaac Newton or the duality of force.
Attraction (sympathy) is dual to repulsion (antipathy), push is dual to pull, stretch is dual to squeeze -- forces are dual.
If forces are dual then energy by definition is dual.
The conservation of duality (energy) will be known as the 5th law of thermodynamics! -- Generalized Duality.
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
"The Force" is duality! -- Jedi teaching.
@@hyperduality2838 I get it, everything has a dual. I fail to see the big insight in that.
Great lecture, but I’m not sure the conclusion is correct. It shows that the color space of vision is isomorphic between people. That doesn’t mean the actual qualia is the same. It’s a basic confusion of map and territory.
Personally, I’m largely skeptical of most attempts to apply really abstract math to most practical problems, partially because it just offloads the work into how you structure your category and partially because the conclusions are generally very nebulous.
Some of my favorite books are by Robert Rosen, who pioneered relational biology (basically applying category theory to theoretical biology). Utterly fascinating work, and yet I have no idea if there is any substantive use for it.
It’s entirely possible I just don’t see deeply enough into the problems.
Yes, but there is an important exception to this general rule of mappings and territories: The territory is also a 1:1 map of itself. This means that a 1:1 map is isomorphic with the territory, and not an impoverished model that fails to capture some aspects of the territory.
So, the question is: if two perceptual color spaces are isomorphic, how can the associated qualias not be isomorphic?
Some commenters on here seem to argue that there must be morphisms/relations that we are missing by limiting ourselves to perceptual color space. But in order to examine that argument we need to know what these missing puzzle pieces are. What relations other than degrees of perceived similarity are required to exhaustively characterize color qualia?
The quality can be 1:1 isomorphic, that doesn’t mean they are the same thing. Say my perceptual space with relations characterized by various morphisms, etc is a category C and your perceptual space is a category D.
A more concrete example would be to say let’s assume that we arrive at some final version of physics, and say it’s something like the Wolfram model where we can fully simulate a segment of the universe in a truly isomorphic way to reality.
Does that mean that the simulation is ontologically the same as the actual segment of the universe? Not impossible, but definitely not necessarily.
Note that I’m not saying that you can I definitely don’t have the same perceptual space. I’m just pointing out that even if you show they are isomorphic, you still have more work ahead of you.
That's the longest "I don't know" answer to consciousness.
As for the inverted spectrum problem, one might also ask, "Did the dinosaurs build Stonehenge? And isn't it odd that someone has taken all the trouble to completely remove all the evidence of this?" (Monty Python, circa mid-20th century). Every time he makes the noise "green" he means red.
Great video! I'm getting into category theory but got lost when I got to natural transformations, so I obviously didn't understand Yoneda Lemma, but this video helped a lot. Thanks!
This whole presentation is just an exercise in sophistry. "Look at how well I can blend these arbitrary topics together." There's nothing innovative about it. The discussion about color and its perception is a philosophical one. The measurements of psychophysics bring nothing to the table; psychophisical measurements are philosophically on the same level as checking whether somebody sees a strawberry as red. You could have used the Yoneda lemma directly on those measurements just as well. "If we both call all red objects red then we must be talking about the same red." But the Yoneda lemma is grabbed out of context and not applied rigorously. If you believe people can percieve a color differently then that constitutes a different relation and so the Yoneda lemma wouldn't apply.
BUt the whole discussion is foregone by the problem that saying I can perceive red as green presupposes a homunculus inside my head which again observes the color, which is ridiculous. Therefore the whole idea of perceiving colors as other colors didn;'t make sense in the first place.
As a scientist myself I see the main problem in the fact that you just can't afford to concentrate on high risk topics that might or might not produce groundbreaking results. You have to write several articles per year, fully aware that they're only mildly interesting. Because doing otherwise would be professional suicide.
Another thing: you can't overestimate the value of discussions with top scientists. But this is not possible when 10% or more of a class go for a phd. Many of them don't even want to work in science. Some do a phd to further their career, others because they won't get a job without the title (biologists and chemists in some countries) and yet others because it's customary (medical "doctors")
i agree, "science" has become a buzzword to show society how competent one is and many in the scientific field are no longer interested in chasing the truth but in flexing
Stop being a chub and go against the publish or die mindset. True greatness wouldn't just go "Oh, I have to do this or my career!!!!"
Einstein worked in a fcking patent office for god's sake. I did my undergrad at UC Berkeley, so I know how many midwits truly exist in academia
Excellent presentation and communication skills, was very easy to follow.
Yeah, I think is this why if you have a mapping from one distribution to another (F(X) => Y) you effectively have a mapping from x to y (f(x)=>y) (individual points in the distributions) when x is a high-dimensional object (which makes it very unique/specific). This is important because deep learning learns distributions, so now you have a certainty that individual data points can be mapped as well.
What does the asymptote on the right side represent? What do you think?
x-axis = (Moral Egoism, 0) union [0, Moral Perfection)
y-axis = (Hell, 0) union [0, Heaven)
0 = the Indifference point
x is the independent variable, while y is the dependent variable
x takes one parameter at a time from three possible: E, 0, P
x's parameters are functions
(Hell, 0) is the Injustice interval I
[0, Heaven) is the Justice interval J
f(E) = Moral Egoism = "Do that, which you want and can do."
f(P) = Moral Perfection = "Do only that, which would be acceptable to all (mankind)."
The Indifference point, 0, can also be expressed as, f(0) = "Do only that, for which all (mankind) would not want to punish you." It's also the threshold where morality has no net impact on circumstances.
Thus, the Justice interval can also be expressed as, J = [f(0), f(P)) = ["Do only that, for which all (mankind) would not want to punish you," "Do only that, which would be acceptable to all (mankind).")
Assuming justice is acceptable, while injustice is unacceptable because every reasonable human desires to continuously experience minimum injustice and maximum justice with respect to everything.
We can plot this model too:
arctanh(x) = ln[(1 + x)/(1 - x)]/2, |x| < 1
Above, I've presented the abstract functions for Moral Egoism and Moral Perfection. But here are general descriptions of these concepts:
- Moral egoism is the normative position that moral agents ought to act in their own self-interest. It means that what is best for the individual is the highest morality.
- Moral perfectionism is the conviction that human beings have a duty to aspire to moral perfection. It posits the pursuit of moral excellence as the ultimate goal of human life, as the path to physical perfection lies exclusively through moral perfection.
Word 🥗
In 10:07 it went from what is a set to Yoneda's lemma, first day of math degree to last day of math degree ;-)
Great talk given here. Very interesting and insightful
Physics is what we know, metaphysics is what we do not know.
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
This video does not make sense. Yoneda Lemma would say the color perception of the inverted spectrum guy be ISOMORPHIC to the color perception of the normal guy. Not literally the same. {Blue, Green, Red} is isomorphic to {Green, Orange, Pink} in Set. Doesn't mean they're literally the same.
It misses the entire point of the hard problem of consciousness and the inverted spectrum idea. You could swap how two colors are subjectively experienced, and you'd get an isomorphic copy of color cognition. If they did not give rise to isomorphic ways of color cognition, there would be a way to test whether they are the same of not. So, Yoneda Lemma telling us they're isomorphic is not helpful, we already know they're isomorphic, thats the entire problem!
At best its a way to reformulate the crux of the hypothetical.
I love this video and talk! The only nitpick I have is about the thought experiment with color qualia.
The sets of relationships that define human color vision are part of a qualitative process- meaning, much like you explained in your video, consists of a set of qualitative relationships between each point in color space, which per the Yoneda Lemma, this means that each color is defined by its relation to other colors, thus meaning that no color could possibly be different than what it is, because it is defined by its relationship to other colors in color space.
The thing about qualia is that it does not only involve qualitative processes- it involves the construction of qualitative processes, subjective processes, and distinction processes as well.
I like to use the Duck-Rabbit illusion as an example of this. This is an example where a qualitative process (the empirical qualities of that image and how that pertains to how the photons enter the eye, optic nerve, etc.) takes on a new set of relationships because the subjective process (perception of the image) can impart distinction processes which create new sets of relationships where the subjective process and distinction process adds new sets of relationships to the diagram that involve the perception of the image one way or another from a subject. Since objects are defined by their relations, as per the Yoneda Lemma, this means that the changes between the relations between objects that can occur due to the perception of an image (not just seeing it) actually do change how that object (color) can be defined.
Likewise, with color, things like different languages having different words for colors, or perceptual differences like synaesthesia, etc. may also contribute to additional subjective and distinction processes that can augment the sets of relationships between the qualitative process of the color space itself vs. any distinction processes resulting from its subjective perception.
It's not to say that your statements about the rigorousness of the external qualitive process of human color space is incorrect- I 100% agree with you there. It's more so that I believe that these psychometrics can be augmented by other types of processes and relations besides just the qualitative ones, at least when it comes to qualia.
I derive my reasoning from the compositional mathematics for subjective experience described in this paper. Most notably, the mathematical model proposed here contains a proof regarding the readability of the qualitative processes of others' experience, but the unreadability of others' subjective processes of experience- meaning that the inverse spectrum problem will never truly be answered, as per this mathematical model's postulates, at least.
www.sciencedirect.com/science/article/pii/S1053810021000945#b0010
That's an interesting paper. Thanks for sharing!
To your point (which is shared by many in this comment section). The counter point is to ask: what are these 'qualitative relations' that you postulate to exist in addition to the 'quantitative relations' when it comes to subjective color space? If you can make those concrete, we can just add them to the model (i.e., we add those 'qualitative' arrows to the 'quantitative' arrows), and the Yoneda lemma works exactly like it did before. So, even if that assumption were true, it does not change the logic of the argument.
The concern, though, is that it seems difficult to point to what is 'missing' in a subjective color space that is described by 'quantitative' arrows. That is, the color space described by a singular metric - similarity judgments - seems to work just fine. Intuition might revolt at that, and demand that there is 'more', such as an 'intrinsic' quality to color or some kind of semantics. But Occam's razor asks for compelling justification for adding further assumptions. So, the ball is in the corner of the critique: if you believe that there is 'more' to color than metric color space, you will have to show what metric color space fails to explain. And, as a next step, you will have to be more concrete in explaining what exactly is missing: What is an 'intrinsic quality'? What are the missing relations? And the moment these are made concrete, we can just add them as arrows and apply the Yoneda lemma again.
As a side note: It is arguable whether anything changes VISUALLY when Duck-Rabbit illusion changes interpretative meaning. That is, In terms of fundamental VISION, the illusion always looks the same. What changes is more like an inner voice saying "this is a rabbit". So, the change is more thought/cognition rather than phenomenal consciousness. Agnosias are a common example showing that these are dissociable mental functions: Certain brain lesions prevent people from making sense of what they see (i.e. they fail to identify a rabbit or a duck), but if you ask them to draw what they see, they create a perfect copy. Or imagine looking at a Jackson Pollock painting - you can see it, yet it will make no sense whatsoever. It is pure vision. This suggests that object recognition is secondary to phenomenal consciousness. You can have visual experiences without any semantics associated with them.
THAT IS A FREAKING AMAZING TEACHER !!! Math made FUN! THANK YOU!!!!!! It is genius to do what he has done - it is RARE to find someone that makes math this simple and fun !
average american brain, it is not rare to find good mathematics teachers.
I didn't even understand
I don't think this actually solves the inverted spectrum problem. Just looking at the very last figure in this talk - imagine red and green being swapped... each color would still have their own distinct set of relations, but the PERCEPTION of the color could be inverted and we'd never know.
I was thinking about the same thing, philosophically what you are saying is true. However, biologically this result makes it more unlikely.
If we consider that humans have similarities in their neural structure and that color perception is something innate and not learnt, it is a good assumption that similar neural hardware firing should lead to similar qualia.
With this assumption, the way that the inverted spectrum would manifest is if somehow the "hardware" for perceiving green was swapped with the ones for blue.
I think this work is a good indication to show that this is not likely to happen because then we would see a different perception diagram.
I think this work moves the problem up to a higher layer of processing, which is progress.
Right, but, without knowledge of the causal link between physical properties and subjective perception, even a consistent distinction between neurology underlying green versus red perception would not allow you to know what color a subject is actually perceiving (because they would report red as green, and vice versa). So I don't see this new theory advancing the inverted spectrum problem at all. It's just a more complicated way of saying "I assume there will be distinct neurologies underlying red and green color perception". It doesn't help us determine if, say, two people reporting the perception of red are actually having the same color experience or not - and *that's* the inverted spectrum problem.
does anyone here know the name of the presenter, and other details (venue/occasion/date) of this talk?
Absolutely brilliant video!
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
I get why we use colors as a way of talking about consciousness and how it could be different between people. Most of us have played around with changing colors on a photo editing program and have seen it for ourselves so it's easy to imagine as a thought experiment. But let's not forget color isn't the only aspect of visual conscious experience.
The whole visual field that appears to us in first person experience is visual qualia. The spatial structure our brain produces of the world around us.
Its not like a black and white version of the world that we see is the default neutral world. As if this is the underlying physical structure of the world that has always been around and will always be around for any conscious being to see for themselves, perhaps just filled in with different colors.
As hard as it is to imagine this simply isn't the case. There's no inherent visual appearance of anything out there in the world. Without eyes to see nothing looks like anything in the same way that nothing smells like anything without a nose to sense that. Apparently dogs for instance can create a type of 3D volumetric map of their surroundings with the sense information they get from their nose. But does it "look" like a 3D map to them? Almost certainly not, it's an entirely different kind of experience. However to a dog it's the way the world inherently appears and cant be any ither way
So, just to be clear, we all see the same colors basically, right? Finally I can give my kids an answer.
My understanding is not that we necessarily see them the same, but that there IS a methodology (thanks to the work of Tsuchiya & Saigo) for determining HOW differently or similarly we see them.
There is colorblind people that doesn't realize about their condition until they have to drive and have confussion/trouble with the traffic light (happened to a friend of mine). That's almost 20 years without realizing it, thinking he saw the same colors as us.
so interesting! You just motivated me to find my thesis subject ^.^
This here is the calm before the storm. We are very quickly approaching the technological/cultural singularity where exponential growth becomes most significant compared the the rest of recorded history. It's just that now we're unsure if we can project our prior understandings of the world into the future to develop with them.
There was a clue at 27:14 when the berry color was not inverted, this was a foreshadow of the un-invert ability of the color spectrum.
Wow. This really motivates me to understand CT further.
Category theory: a solution in search of a problem.
If the colour absorption spectrum was symmetrical the result would be the opposite. I do not see how one would need category theory to come to this conclusion and would be very happy if someone could shed some light on this. it is quite obvious even with simple logic following. Going further with asymmetrical colour absorption spectrum, the result still would be different if we do not "assume" individuals who might have inverted spectrum still have the same colour absorption/perception with the ones who do not have inverted. If inverted spectrum people also have different absorption, we still can not differentiate them. Another note is that this makes it possible for everyone to have different vision since we might have continuously different colour spaces with a corresponding absorption spectrum such that the distances between the colours are always the same for everyone.
The most common counter argument is that color perception is not the same as perception of electromagnetic wavelength. These two processes can be double dissociated, and hence are orthogonal: For example, there are different wavelengths of light that look the same (metamers), and there are colors that do not map onto any wavelengths such as pink, brown, or any neon colors (there are many more such "impossible colors"). Visual illusions that work on color demonstrate that we can even see color for gray or black and white stimuli.
@astonishinghypothesis Thank you for your response, this tells me that wawelenght and colour perception is a nonlinear mapping. So the question is still the same actually, is there only 1 possible nonlinear map for everyone to have? I think there might be infinitely many mappings that satisfy the same properties(distance of colours etc.. whatever is necessary). If there is only 1 possible way that means there is little to no room for an error, and the neurons resposible for this mapping does not change for a lifetime after those neurons are set possibly before we born. People live 80 -100 years and it is counterintiutive to think that those neurons are uneffected no matter what life throws at us :)
Very well explained! Thank you for sharing.
IDK, man. I appreciate the effort, but I feel like really what's the point? Explanations are a dime a dozen these days, to the point that people are no more enthused by it. For e.g., Integrated information in consciousness, or String theory in physics. Like you can fit an explanation to anything, & it can be beautiful. But, it often goes no where. Hence, the general disinterest.
The problem is that as short-term pro-business mentality infects every aspect of university nobody gets funding to research anything original, since markets only want incremental improvements on things that already exist.
beautiful presentation!
This is by far the best intro to category theory in the youtube
Thank you. Amazing presentation.
After 100 years it really seems like the only REAL advancement in our understanding of the human brain is that we can say "we have no idea how the human brain works" in ever more complicated ways.
when it comes to applying the lemma to colors: different people demonstrably don't have the same relations between red and every other color. the obvious example are people with varying degrees of colorblindness. and in fact they don't necessarily have the same relations between colors and objects, because they may mistake red objects for green ones and vice versa (because the distance between those colors is much smaller for them than for the average person) and never mention it to be corrected. so i don't think this immediately rules out the idea of an inverted spectrum, it just makes it far less likely because most people's color relationships will be rather similar to each other's, and colorblindness has a known effect that does not represent an inverted spectrum.
Hi
1. so you say: we all experience the same colors more or less (as the video concludes), because we have very similar distances, except for some colorblind. Did I understand correct?
2. however isn't the colorblinds' difference of distances, caused by physio-anatomical differences/ difficiencies? Or is it the same but their experience differs? I don't know this.
@@SiEmG yes, colorblind color perception is caused by a particular difference in functioning of cone cells, but i'd imagine smaller differences occur throughout the population, presumably resulting in different distance relations between the same colored lights. the video's conclusion is likely "correct on average", but with some margin for error based on the extent of those differences, and excluding the colorblind unless they are to be considered part of the distribution rather than outliers (say, if their inclusion does not make the distribution bimodal).
@@rarebeeph1783 then its not the qualia that have different distance relations, but its caused anatomically. if the colorblind would have the "corect" cone cells, they would have the same distances. The quetion is: if 2 twins have the exact same anatomy, do they necesserily have the same color- differences experiencially?
@@SiEmG that's a good question that i suspect depends on to what extent visual processing itself is genetically determined vs. learned at a young age. given only the variables i'd laid out so far, identical twins should have the same color differences, so if it were to happen to be that they don't, that'd indicate a non-genetic component.
It seems like the category theory part doesn't pull any weight in this argument. It seems to me like there is no difference between the presented argument, and just stating that we all share the same brain structure in those specific parts, that maps bijectivity to our reported color experience...
The same argument could have been made even if our receptors were 3 'bits' encoding 3 different colors... no real need to go into how colors corresponds to relations between those receptors...
I miss specifics... It started great but finished with extra abstractions without enough time to pause and ponder
Super interesting, thank you! 🙏🏻
Psycho Matics?
What Kind Category?
How Connected A Point In A Black Box?
Conductual Reductionist And Massive Tendency Like A Behaviour?
Unfortunately the "solution" to the inverted color spectrum given really has nothing to do with category theory. He is merely shoehorning the concept of categories into the "solution", which is really just a statement about a certain metric space (space of colors) not having a symmetry (color inversion). One would actually need to add more structures, e.g. enrichment over real numbers, to make sense of distances in a category setting, and it really doesn't add anything to the discussion, and in fact obscures the main idea, which is, again, a lack of isometries.
If the ideas of category theory inspired such a perspective, fine. But are there really no better examples to display the use of category theory? It sounds like a desperate attempt to ride a "applied category theory" wave that is creeping in academia. Perhaps the speaker wants to use buzzwords to get grants.
Thanks for the video presentation.
Interesting insights on the meaning of Cat th. But the application to cognitive neuroscience (psychophysics, etc) only begin at 24:00 (which is a bit disappointing for me but I cannot complain: it is very hard to present this topics for a reasonably large audience. There is an unavoidable tradeoff that depends on the targeted audience. )
Yes, but what we call red is red only when we see it. Otherwise it is red and green at the same time :)
Fascinating. I know one has to be careful to make connections betwenn science and religion, but this reminds me very much of how in Buddhism they say things have no existence in themselves but are only defined by ther relationships to all other things.
couldn't this imply inverted spectrum could still exist, just that there would be different relation of colors for those who would see "inversely" to me. That person could see and be able to better distinguish more of my "red" where I would see more of my "green"? We would still function the same in spite of seeing being able to see colors and distinguish distances in colors that the other wouldn't.
ah neverminded, because the color spectrum wouldn't be inverted properly
I don't think it'd prove my perception of color aligns with someone else's though.
Still relations don´t define absolute colors, which do not even exist. We have no means of knowing in what subjective way another person experiences color. The relations could be the same though the actual experiences differ.
Can I have the link to the paper of this presentation?
Thank you so much for sharing!
How was the lemma useful in a vision example exactly? I mean, I understand you could use it because it's such a general statement, but it also feels like using a cannon on a bunch of ants, the fact that the color space is not symmetrical should be enough on it's own.
It’s giving Haskell 💅🏽
*ahem for those from outer-space or cyber-space @ 16:57 he meant "functional"
What if the person with inverted color spectrum starts painting, wouldn't we see his choice of colors rather odd ?
Watch all the empiricists flood the comments with trash about how the thing that made their entire shallow careers possible (ie math) is completely pointless and just for people who "like puzzles"
Physics is what we know, metaphysics is what we do not know.
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
Bro was like, if A=B and B=C, then A=C, but with functors LOL
Stimulating video, outstanding.
Great lecture.
It is hilarious that you first taught the Category Theory and then explained the Inverted Qualia problem perfectly, AND THEN fell into the oldest mistake in the book!
When the qualia of color changes, nothing stops the relationships between the qualias from changing as well. Hence they can be functionally (the "relationship between") swapped as well. Green can act like red! If we have no theories from voltages to qualities, that is. Because the quality and function is not equivalent without a solid theory for "red". The function of red is something we can measure, but the quality of the red seems completely random! Hence the Inverse Qualia problem: qualities seem arbitrary, so why not keep the functionality same and the quality different?
I suggest you think about what quality of a sense mean.
What is that last equation at the end?
This distorted sphere, does this come from measured data? Is that distorted sphere similar for all brains? Is there a noticeable uncertainty? Is it proven, that there is no way for this mathematically vaguely defined distorted sphere to be isomorphic to a flipped version of itself? It's not the same. It's isomorphic. And this color space maybe can be isomorphic to another ordering of the color spectrum.
Interesting 🤔
You cannot have inverted spectrum, because qualia are meaning, they are not random, is like saying that you can have wings instead of legs and not noticing. Of course you notice. See my papers about consciousness, like "Meaning and Context: A Brief Introduction", author: Cosmin Visan.
amazing
what i am missing is a clearer explanation regarding the nature of this non-symmetrical color-map at the end - when we speak about colors, we might mean many different concepts, but in this talk, nearly all of these concepts have a relevance, so its unclear to me what concept of colors is the source of this map...
Roughly, I believe that distances in the map correspond to neurological signal differences (and hence, psychophysics claims, distinct experiences) So there are more shades of green that can be experienced as distinct in the brain, and for that reason green takes up a greater area in the map
@@tsawy6 i just dont see how this could ever tell us anything about qualia,
this map can only ever tell us about the relationship between the stimulus (wavelengths of light) and our brains reaction to those, but not what qualia represent these inside anyones consciousness.
@@davejacob5208 yeah for sure, this criticism has come up a few times in the comments here. The author responded, almost seeming to go as far as to suggest that Qualia doesn't exist. It's very interesting, check them out
amazing video!
Science will (inevitably) stagnate once the universe is finite and everything in it is also, it will not solve all questions (and it isnt supposed to) and other types of knowledge not poluted by the senses like; philosophy, mathematics, logic etc will take protagonism
Lots of good points here, but it is ridiculous to divorce objective reality of matter from subjective reality of consciousness in this way!
Indeed, i would say that the red qualia of human physical consciousness is inherently far more connected to the 600 nm wavelength than to the 530 nm wavelength or the 450 nm wavelength. Similar things hold true for green qualia, blue qualia, yellow qualia, turquise qualia, orange qualia, violet qualia, pink qualia, lime qualia etcetera.
It MIGHT be true though that creatures who have a visual spectra ranging more than one "octave"/doubling of wavelength (like 400 nm - 800 nm), will have similar, but not quite isomorphic, qualia, inherently related to one electromagnetic light wavelength, and its double or its half!
But "red" qualia is still fundamentally different from "green" qualia or "blue" qualia, and this fact of subjective consciousness is related to the objective fact of matter, that the corresponding wavelengths of electromagnetic light/radiation are related through very different ratios than powers of 2. The same is true for frequencies, that are evidently inversions of wavelengths. Possibly there is a similar "inversion" in consciousness of color qualia into something related, for example a physical mood. This is up to empiric psychological science to research however, not up to speculation to decide upon.
Just as the objective world is governed by non-arbitrary Natural Laws, so are the subjective worlds also governed by non-arbitrary Laws of Life.
Because all consciousness is grounded in matter (and all matter is more or less conscious), even non-physical consciousness (of various kinds) is grounded in non-physical matter (of corresponding kinds), just like physical consciousness is grounded in physical matter.
This is a fact of reality, not a delusion of speculative arbitrary subjectivism or of ignorant physicalistic objectivism.
I was really happy to find out that this speculative subjectivistic argument of "perfect" color qualia inversion was indeed some kind of strawman! 😄💛
And that map of psycho-physiological distances of color qualia is indeed gorgeous, especially since it is related to the philosophy of the Yoneda Lemma❣️
Astonishing
I’m not sure that everything you said about the inverted spectrum was ultimately false.
You made the example of saying your perception of red could be down in the blue corner, but aren’t you making a mistake there? If you inverted the colors of that image, the point you specified would still be in the same place and the relation to its context would remain the same, it would just be a different color.
I believe you are correct. Under one premise - that is that there is more to color qualia than mathematics can describe.
This reasoning is best explored by taking consciousness out of the equation for a moment: Take the example of a human being defined by the totality of their social relations. One could argue that there is more that makes us individuals than our interactions with other humans (some sociologists dispute that, but let's put that aside for a moment). You could argue that humans are also affected by their relations to other living and non-living things, such as pets, food, geography etc., for example. Very well. But this only creates temporary embarrassment for the argument put forward here. Why? Because you can simply add all of these relations to the categorical model - and the argument holds again. So, the argument only fails if you can name some kind of influence on what makes an individual that could not be expressed in a web of relations.
Now, when it comes to color, you could similarly argue that the psychophysical color space under-defines color perception. That is, it seems logically possible that one could see an inverted color while establishing the same color space behaviorally. As you point out, a proponent of the original inverse spectrum argument would probably insist on exactly that. But then the question arises what is missing to fully define color. If whatever is missing can be measured and/or expressed mathematically, it would be easy to add to the categorical model and thus resolve the issue again. It is only when we assume that there is something to color that goes beyond mathematical/structural description that the argument falls flat.
Now, some philosophers believe exactly that. But, the problem is that most arguments that aim to show that this is the case (e.g., the knowledge argument/Mary's room) are not logically compelling. These arguments tend to rest on intuition, which we all know can be unreliable. Plus, similar views were long held about the nature of life (vitalism) that did not stand the test of time. In fact psychophysics was a response to the notion that subjective experience is something that can neither be measured, nor assigned numbers to. These notions had to be dropped as well. Thus, there is a certain similarity between the "God of the gaps" argument and the clinging onto the notion that consciousness falls outside the rest of reality in that it escapes naturalism. That is certainly a valid view, but as science progresses, it seems to cease more and more ground.
tl;dr: The argument rests on the view that structure is all there is to color qualia. This in turn makes most sense on the ontology that relational structure is all there is to everything (hence, Yoneda "worldview"). This should have probably been made more clear.
Here is an interesting snippet with Carlo Rovelli discussing a very similar view: ruclips.net/video/r_fUPbBNmBw/видео.html
@@astonishinghypothesis It seems if structure is all there is to color qualia we should long ago have evolved language to convey the nature of our color experiences sufficiently to put inverse spectra to rest, along with being able to explain to the color vision deficient what they are missing seeing and having what *we* are missing explained to us by tetrachromatic individuals.
If a tetrachromatic individual (or a category theorist studying tetrachromatic psychophysics) were to completely describe to us the structural relations of the tetrachromatic gamut, how far would this enlighten us as to the nature of the tetrachromatic experience of color?
If we explain the structural differences between red and green to a red-green color blind individual, how much insight as to the nature of the standard experience of red would they gain?
It's hard for me to conceive that after the full structural description is given there would be no surprises in store for such a person suddenly gifted the ability to experience directly the color gamut whose description they had received, but had previously not experienced!
@@astonishinghypothesis
["which we all know can be unreliable"]
Even so, it's extremely essential for the development of mathematics itself, isn't it? It's fun to see how you try to throw intuition out the window when mathematics without it is just a set of letters, or rather, colors, or better yet, particles (which we wouldn't even know are particles without our intuitions allowing us to grasp the world). Mathematics simply does not exist without intuition (even when math leads us to have seemingly anti-intuitive beliefs). The inability to answer things like Mary's room argument or to define consciousness in precise terms should not be an excuse to spit on its strength, or to try to force it into a box in which you feel more comfortable, but rather should be a reason for you to wonder if it is even possible for a conscious being to understand its own consciousness, or if we are going in the right direction at all when we talk about consciousness and end up reducing a phenomenon that is fundamentally part of our lives, our whole lives, to a handful of numbers.
@@astonishinghypothesis: Taking the hard problem of consciousness seriously doesn't mean to think that consciousness somehow "escapes naturalism"!
@@astonishinghypothesis: The assumption that one can "discretely" separate different types of qualia (e.g. we experience color as a 'packet' of info completely orthogonal to other mental experiences such as smell or 'inner' emotions) is an extremely strong assumption and, and very possibly false.
When I see red, I don't just _see_ red: I go through a space of experiences all bundled together; I e.g. feel the emotions conjured by the memory of the red dress of my ex girlfriend; I'm reminded of the picture of the red mushroom in the book my mum used to read me when I was 3, which brings me a memory of smells and/or of tastes that I newly discovered at that age; etc etc. So, my "red" is all that structure together.
Even if the non "purely visual" mental relations are subconscious, it may be the case that some aspect of them resurfaces as an ingredient of my "red" qualia.
It may even be the case that my "red" qualia of today aren't the same as my "red" qualia of yesterday, and they are just related to each other by a sense of continuity as complex as the sense of continuity of my own identity.
_In theory,_ you may apply a structural/mathematical framework to try to model all that - but in practice it's impossible because of the complexity of the thing.
tl;dr: the space of qualia _types_ may not be discrete.
Oh no, category theory went the way of Lisp and AI in the 80's
Interesting presentation, but actually yonedas lemmas still doesn't solve the problem with qualia
I am surprised that neuroscientists found an application of Category theory before mathematicians did😂
I would bet a large sum of money that with a sensitive enough measurement you would not find two humans in the world whose color-to-color mapping would be exactly the same. So I don't find this argument very compelling.
Nevertheless, since qualia (experiences) are epiphenomenal, we can pretty much ignore them, so the inverted spectrum problem is a non-problem. Join the camp of qualia denyers, everything is easier on this side.
Check out this work by the Tsuchiya lab that speaks directly to your concern:
scholar.google.com/scholar?cluster=2143039580810330465&hl=en&as_sdt=0,43
Physics is what we know, metaphysics is what we do not know.
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
Are you familiar with the philosophies of Martin Heidegger and Ludwig Wittgenstein? Heidegger was interested in the meaning of being and Wittgenstein was interested in the philosophy of language. They operated very independently of one another but interestingly converged on ideas very similar to what you mentioned here with the Yoneda Lemma. Wittgenstein argued that a private language would be impossible to learn. And your argument on color perception is a more precisely stated version Wittgenstein's private language argument.
I want to go even further than you here. The universe is inherently meaningful because of its networked nature. Meaning is the relationship between a node and the network it is embedded in. And this holds true for all levels of nature. Being is the internal structure of a node (its existence) plus the relationship it has with the network it's embedded in. For this reason, meaning and being are intertwined with one another.
I refer to this geometry as 'Situatedness.' Information is the signaling of Situatedness. Learning and knowledge are the storage and remembering of Situatedness. Meaning makes information possible, which in turn makes life possible, which in turn makes intelligence possible, which in turn makes consciousness possible.
In other words, meaning makes information possible, not the other way around.
27:48 hahaha, it's like he wrote it, then realized that one of the lines was too long.
I disagree with the inverted spectrum theory. You would be able to tell a difference with comparison colors. Such as someone saying this dress goes with my eyes being on the opposite side of the color wheel. The spectator, though, will disagree because his eyes would not match the color the spectator sees. It would be a totally different color.
Brilliant!
Thanks , is an interesting approach to this problem, i would like to help develop it theoretically
Physics is what we know, metaphysics is what we do not know.
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
We have quantum + AI making an entry into reality within my lifetime and we have science LLMs.
28:29 something about complex numbers in quantum mechanics
You know that if some quantity grows "fast" it doesn't mean that it grows exponentially? There is an actual mathematical meaning to this term. These graphs look more linear or at most quadratic.
This is only two minutes in, which is quite disappointing for a talk about "Category Theory for Neuroscience". I will give it a couple more minutes hoping that it improves.
Yoneda Lema: If it walks like a duck and it quacks like a duck then it is a duck. :)
Revolutionaries 😊
Is consciousness material? Is consciousness not metaphysical?
Physics is what we know, metaphysics is what we do not know.
The Yoneda lemma = Duality.
Domains are dual to co domains.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
The rule of two -- Darth Bane, Sith lord.
"Always two there are" -- Yoda.
Synchronic lines/points are dual to enchronic lines/points.
Points are dual to lines -- the principle of duality in geometry.
The colour black is dual to the colour white -- colours are dual.
From what we know in neuroscience. Consciousness isn’t material in the sense that it’s a physical object. Rather, it is an emergent property of sufficiently advanced and complex neurological systems.
Define "metaphysical".
In my dictionary, Metaphysics describes a method to gain knowledge about our world from a priori assumptions by purely logical means (which has been proven by Kant to be unreliable). Whatever consciousness is, it is not a method to gain knowledge.
@Prodigious147 haha congratulations on your masters
The inverted spectrum thought experiment (as it's normally presented) is pure idiocy. Essentially it says that laws of physics are arbitrary. For color perception to be different there must be an underlying cause, a physical alteration either in the eye or in the brain, the way it is with color blindness (daltonism). An abnormal perception, if it doesn't interfere with any aspect of life, would be undetectable. However, it would then also be of no consequence, as if it didn't exist at all. More over, it's quite possible that everyone perceives color SLIGHTLY differently (rather than inverted), and it's also of no consequence whatsoever.
However, I fail to see why mapping NORMAL color perception in any way contradicts any ABERRANT perception and the colloquial use of color (which is a very rough division of the visible into 6-8 colors at the most). If we had 10s or 100s of colors, a biologically determined mapping would clash with the naming conventions and perceptions in the aberrant individual, but with a rough division into red-green-blue-yellow-purple? Unlikely.
Pure Maths Could Be A Psychosis Delirium...
Seems to me like you could get to this conclusion through a simple application of Leibniz's law & second order logic. Ie, Identical(x,y) iff for any relation R, function f (R(x) iff R(y) /\ f(x) = f(y)). Given the manifold defined, which represents the colors (and I'm assuming is measurable), we define a metric d(x,y) which is not unlike the ones in a 3D Hilbert or even Euclidean space. Then it's just a geometric proof that for any three colors, a,b,c, d(a, b) = d(b,c) iff Identical(a, b) or Identical(b,c) or Identical(a,b) Different formalism but same idea. His introduction juxtaposing set theory as concerned with objects whereas category theory concerns itself with relations just simply is not correct (see Benacerraf's paper "What Numbers Could Not Be"). Category theory is possibly a better syntax for describing relations between these larger fields in mathematics, but there are serious differences in terms of semantics. I mean category theorists have posited the existence of some very large objects which, according to work done on large cardinals in set theory, are of dubious existence, in that they are independent of our axioms in ZFC. Furthermore, the very existence of so called abstract categories, which cannot map into a set (as opposed to concrete categories) is something which though set theory allows, eg, in the case of proper classes, such as V or the ordinals; are much more restricted.
Decades of physicists and mathematicians wasted their time on totally unprovable String Theory, and have gotten nowhere. Yet still they persist on the dead end, instead of doing something useful. String Theory is pure bunk.
This lemma cannot prove that we experience the same red. You can reverse the role of green and red on the gamut as well. I mean, you can construct a device, an AR glasses which makes this trick. Whith no math you have access to personal experience.
Ah The 'epicycle' problem... when theory becomes dogma.
Ugh. 20 minutes I'll never get back.
Mathematicians pave the way for pretty much every other science.
But what's it gonna cost me?