Oh dear - this is about the best example of "smart people aren't necessarily great explainers"... IDK, maybe they didn't have time to prepare, that would explain it. But... come on, there are such great ways for intuition building towards category theory, none of them is explored for at least the first hour - though Tali Baynon does a pretty good job later on. Here's how I start when I explain category theory: 1. For the longest time, most mathematics we did was un-formalized in the sense that it had no canonical axiomatic foundation (euclidean geometry being the exception) 2. In the mid- to late 19th century, philosophical logic and mathematical thought came together and laid the foundation of everything for the next 100 years: predicate logic, propositional logic, boolean algebra, set theory, algebraic topology - and among them, set theory with propositional logic was thought to have the power to be able to serve as the language for grounding, formalizing - axiomatizing all of mathematics. 3. When we describe abstract mathematical structures in terms of sets, the foundational notions are of "inner structure" of the sets - and we represent everything as such inner structure of sets (including functions, relations, elements, unions, intersections, complements etc.) with the help of the "is element" diadic predicate. 4. This leads to loads of operations with powersets and inclusions and in general, there are many different encodings of structures in the language of sets (e.g. the natural number 2 might be represented by {∅,{∅}}, or by {1,1}, or by {0,0}, or by {0,{0}} ... and so on... the relational structure is not very intelligible when "flattened" into a set-theoretical representation. 5. What category fundamentally does differently is this: it talks only about objects and arrows/morphisms between them. The objects can certainly *have* inner structure (in that morphisms e.g. in the category of sets can be (non)injective and/or (non)surjective functions) - but category theory never looks "inside" the objects! They are black boxes - and all the internal structure is represented by conditions on the arrows/morphisms that go in and out! That's the fundamental idea - and the reason why category-theory (including topos-theory and homotopy type theory) is the language of structuralism: Things are defined in terms of their relations to themselves and to other things. It's the power of this concept that lies at the heart of everything. 6. An example: there is the notion of a categorical product. It is defined as an object AxB with two morphisms left:AxB -> A and right:AxB -> B such that any other object which has such morphisms to A and B factors *uniquely* through the categorical product, i.e. there is unique (up to *unique* isomorphism) morphism *g* from the other candidate Y with its candidate projections Yleft: Y->A and Yright: Y->B to the unique (up to unique isomorphism) actual categorical product AxB: g: Y->AxB such that left(g(Y)) = Yleft and right(g(Y) = Yright. This defined the "universal construction" of the categorical product: an object with morphisms to its left and right parts such that any other object with such morphisms factors through the actual product (in the above way). 7. Notice that at no point in the above description of the product have we needed to draw our objects as anything other than labelled black boxes - and being a product is certainly about inner structure! But it's the ways in which inner structure is revealed in or rather *defined through* how the object can relate to other things. 8. Now we can place constraints (are the morphisms epic, monic, both, none, what constraints are on the dual category etc.) and look for models. For example, in the category of Sets, the categorical product exactly describes the cartesian product - in this way we abstract and generalize notions of structure to make them universally discoverable and applicable. That is the magic of category theory.
@@sebastianmullerbalcazar6229 I came to a theoretical model of existence that I call Fractal Point Dynamic Theory that leans on the exact concepts you detailed above. Now I'm here listening to educated dudes dropping jargon left and right that reiterate the same exact concepts I discovered, but with the actual math terms. Oh, and they know wtf they're doing. I'm flailing in the dark trying to make the translation jump here
I think the problem is the person trying to explain this, Fabrizio,was constantly interrupted by Stephen. I wish Stephen, the person who contantly interrupted would park his ego and shut up so the presenter could have had the opportunity to run through it once and THEN deal with questions. I think any criticism of the Fabrizio's presentation/explanation skills are misplaced. This is what happens when dealing with an obnoxious person does not know how to shut up & listen to something that challenges his assumptions. As a mother, I see this in my children all the time. But one expects it from children, and in that case it is a sign of intelligence and intelectual curiosity. 🙄 The presenter, Fabrizio,was calm and polite. That's impressive. BRAVO! Hilariously Stephen did not "get it" when one other audience member he called on to comment said he had no comment and refused outright to comment. Clearly he was thinking what I was thinking: "Do shut up & listen, Stephen."
That's usually what I would be watching too and I can kind of understand it if I wound up here? This time I fell asleep watching Red Letter Media talking about The Snyder Cut and it immediately gave me 3 of these videos straight after. WTF algorithm
I loved this because I really struggle with Category Theory. I am always behind by months as I study to understand this breakthrough Wofram Theory! Exciting and I predict noble prizes in the future !!!! I got so much from this! Thank you to all the guests !!! Thank you Stephen Wolfram!!!!
By the way, being able to be a fly on the wall during conversations like this--its supremely awesome. I've struggled to understand sheaf cohomology since first reading Frankel. Hearing you guys discuss this and walk thru the concepts really opens up this stuff in my head while I'm listening to you.
I like the linguistic side the most from category theory, all those specific and absolutely exact terms for every possible abstract thing, like learning a new language with the maximum possible expressivness.
The problem is, when the time comes to merge those thoughts with reality, we will fail. Our language/logic capabilities are just not suited for it, no matter how intelligent one is or familiar with the exact sciences, and even with the help of A.I. which is inevitably trained by our observations and later its own observations.
Hi all! With all respect, Don’t ask why im here randomly 3 yrs later of this being published but I do believe there should be a partnership dictionary/re-writing of these terms used, even tho people working on this for ages. Its extremely confusing. Hopefully not changing any of the discussed subject matters. Thanks and All the best to all.
When I was first told about category theory, my first thought was “oh, it is kind of like a combination between a group and a directed graph”, but that wasn’t quite right. In a group, all elements have an inverse, but the analogous thing doesn’t hold in category theory. Instead, a monoid is the thing, not a group. On the other hand, the “combined with a directed graph” idea, to make that actually work, would be the idea of a groupoid. A groupoid is like a group, except that instead of there being a composition of any 2 elements, only some elements can be composed, and as such there are multiple identity elements in order to fit with this. It can be thought of as each element having a domain and codomain and the composition of the groupoid elements works whenever those match up. Then, putting both sides of this together, the corrected version of what I thought is “oh, that’s like a combination of a groupoid and a monoid”. So, in the end, “a category is just a monoidoid”.
You end this on exactly what I've been wondering - which is how to express the symmetries of q.f.t. as a group, using category theory, and possibly Grothendieck equivalence to encode the group into the rule... i.e. what does a particle look like in rulial space? I think what it looks like is is an exceptional group as per Lisi, that emerges as you drive up the scale from hyperedge to electron size, where the exceptional group is showing you the stable vibrational modes in the spacelike graph.
I'm working on it. But here's something I don't really understand fully. q^-2Q/2=1 -2√2Q=q where q is a smaller quantity and Q is the larger, in a closed point pair system, and they must measure eachother in relation to nothing but themselves, and eachother. idk if it even plugs in or works though lmfao
Proof that transitions are also agent based via. Category Theory. Proof that SpaceTime is also agent based Via. Splicing (scientific observation). "Everything is everything else"- Everything is Agent-Based. "In this paper, we propose a particular style of semantic rules that make it visually clearer how changes at one level of a MAS require simultaneous changes in other levels of the system (where each component of each level is modelled as a separate transition system)."
Take a look at Tim Maudlin's _Theory of Linear Structures_ (book). He does exactly what was explained ~2:30 for sieves and open sets. He truncates conventional topology at the 0D point-set axioms, because they do not seem obvious or physical. He retains line elements as connectivity (for points that don't 'exist' :) then shows that the lines must be directed to derive the discrete equivalent of topology (e.g. open/closed). This obviously leads into one of your other sessions about rebuilding calculus over discrete structures. Tim goes on to discuss applications to physics. Perhaps arrange a live session with him! By all means start discussing the topology stuff, but he can also (perhaps mostly) contribute to the philosophical implications of your work. P.S. Echoes here of Rovelli's _Relational Quantum Mechanics_ which I like to call the _Zero Worlds Interpretation,_ because there are relations but no _relata_ (in Mermin's words), i.e. edges but no nodes : ) P.P.S. Also, not by coincidence, in Rovelli's LQG the lowest dimensional spatial operator is area! but the area appears on the incident edges. So neither the nodes nor the edges 'exist' spatially but there are area quanta with a spectrum, which presumably have normals in some limit. There are also volume quanta/spectrum. The outcome originates from one of Penrose's many amazing insights, that spin may be fundamental, not space, not time, not spacetime.
The ..."morphisms between morphisms between ... " property of infinity Cats looks very close to what we think of when we are looking at infinitely differentiable manifolds, aka the smoothness property: between any two points there is another point such that there is a way to go from a ball of one point to the ball of any other point.
Yes, let's please talk about categories of things other than graphs! Thank you! 31:40 The conversation went into this self-referential territory veeeery quickly.
You need to use category theory to figure out how exceptional groups, can help you organize space like graph behavior, to emulate the coupled quantum harmonic oscillators of quantum field theory, over time, where the vibrational states of the graph map to the behaviors listed in the standard model, and you need to do this in a way that the group symmetries show up clear at a range of 10^35 hyperedges. Adding a type submodule to Wolfram language could help provide a demonstration of how category theory provides mappings between types, where untyped wolfram language uses the isomorphic properties of category 'mappings' without the corresponding cateory types which have the equivalence relationship for the mapping property. Add a toy type implementation and watch the type confusion evaporate.
Jonathan speaks about "interpretation" informally, but in Computer Science interpretation is a formal (and formalized) notion. Interpretation is evaluation. LISP's eval() function. They are missing each other because the very notion of "time" doesn't exist in Mathematics, and so the distinctions between static and dynamic expressions doesn't exist. And so the Mathematicians speak about types and type-safety (which, happens at compile-time for computer scientists). while Stephen is talking about execution and evaluation (which happens at run-time for computer scientists). en.wikipedia.org/wiki/Meta-circular_evaluator en.wikipedia.org/wiki/Eval
This was awesome but what was annoying was this could be explain a lot more simpler with less fluff. you could see how Stephen Wolfram was trying bring things back to first principles but we kept getting this... Exaggerated Example: Stephen Wolfram: ok lets make it simple, does A = B? others: Well Its False but also not False because "A" is False but "B" is True so "A" must be True because "A" can be True and False but the equals-sign itself can be False or True but its mainly False but in this case its True but "A" and "B" can be mapped to each other so this makes "B" False but it depends on the position of "A". Stephen Wolfram: 🤦♂️ Stephen Wolfram: ok, does B = A? others: Yes, that's True Stephen Wolfram: but if B = A then A = B? does the order matter? others: No the order doesn't matter but they are still not the same. Stephen Wolfram: oh boy😅
The number of light bosons stems from the cyclotomic of 18 (divisors 1, 2, 3, 6, 9, 18 and new roots 1, 1, 2, 2, 6, 6) for 18 normal bosons (6 free ones as 18-12 [not fermion bound]) and if the equality of the mass independent free space view to zero is just an approximation to the reciprocal of a small oscillation then a differential equation for such is just scaled by units of Hz2 and having which would place the cyclotomy at 20 (divisors 1, 2, 4, 5, 10, 20 and new roots 1, 1, 2, 4, 4, 8) for 20 dark bosons perhaps? Or maybe it works inversely for reducing the cyclotomy to 16 (divisors 1, 2, 4, 8, 16 and new roots 1, 1, 2, 4, 8) or 16 dark bosons? Or “free dark bosons” at a tally of 2 (or -2)? I think I used η with a floating ~ (tilde) to indicate this secondary oscillation. Fermi exclusion unique factor domain expansion? Non-unique compaction “gravity”? What tickles my mind is the idea of 2 "ultra free dark bosons" as an idea. Put another way
The "events" comprising a spacetime manifold are possibly *not* atomic or irreducible primitives. Why not see them as fixed points of interactions between quantum systems - where the q-systems are not "in spacetime"?
Me too, woke up to this video when it had played about 2 hours. Must have been really tired. I will go into Grand Theft Auto Online (GTO) and see if I have subconsciously learned something 🤣 Maybe I will be able to categorize the try hards into all their sub segments from this sleep learning experience 😉
I am extremely unhappy that nobody could explain in this 4 hours how to 'talk' category theory. This only adds to the confusion and misunderstanding. Wolfram questions have mostly not been answered. One (worst) exapmle: Grothendieck topology is an unfortunate misnomer: open coverings are only one aspect of topology, this has little/nothing to do with 'continuity'.
I feel like the part about the Curry-Howard correspondence would have been clearer in a language with dependent types, instead of one without types. The proof functions thing, it feels to me like it doesn’t really capture the correspondence. In order to capture the correspondence, I feel that these things should be composable. A proof of “If A then B” should be composable with a proof of “If B then C” in order to produce a proof of “If A then C”. I don’t see a way to do that with these proof functions. The way of making pairs in untyped lambda calculus that I’m familiar with is lambda x . lambda y . lambda f . (f x y) Then you define fst as fst = lambda x . lambda y . x and snd as snd = lambda x . lambda y . y Then, if you give some pair the argument fst, you get the first thing in the pair, and if you give it the argument snd you get the second thing in the pair. If you want instead a function that you can apply to the pair instead of one you apply the pair to, just define a function which takes in the pair, and then applies the pair to fst (and another one that does the same thing but applies the pair to snd instead) Straightforwards enough. But, yeah, this doesn’t work so nicely when you want to make everything typed. Well, actually, I guess the type of the pair could be said to be, For any type C, accepts (a function which takes an input of type A and returns (a function which takes an input of type B and returns something of type C)) and returns something of type C . So, you can use types with this way of making tuples, but simply typed is not sufficient. You need, type arguments and dependent types? Although, I guess it only has the types depending on the type arguments, so I guess that isn’t really type arguments. You could do that in C++ if you wanted. And like, not using the newest stuff in C++ . As long as C++ has had function types I think. But, yeah, if you want to use the simply typed lambda calculus, you have to have a built-in way to make a product of 2 types (I.e. the type of pairs with the first entry being of the first type and the second entry being of the second type), and a built-in way to make pairs for those types. You can’t just use the definition of pairs from the untyped lambda calculus. And, it is better pedagogically, I think, to teach people the simply typed lambda calculus before you teach them any of the versions with dependent types. It’s just easier. Edit 2 : Ah, 1:21:50 really starts getting to the point for at least a little while. Especially at 1:24:30
A very impressive discussion leaving us with wanting to see this used to create sieves and something practical from this that we can use.... My question is how can we apply this to the Wolfram model... The hyperways ....?????????????????
I have done a little category theory but not much. (E.g. I don’t understand the Yoneda lemma, but have made the statement “a morphism in a concrete category is surjective iff it is right-orthogonal to a morphism from the initial object to the free object over a singleton set”). Are you speaking as a category theorist, and warning that this video is likely to make me confused and misunderstand things, or are you saying that someone who has almost no familiarity with category theory is likely to just be confused?
@@drdca8263 I am not sure how you want to draw the distinction. It is like a stormy sea that will drown a beginning swimmer for sure. And even if you know how to swim, you will appreciate the difference between a storm and a clear blue sky.
nice joke-I laughed @33:50 also at around @37:40 when they start talking about identity morphisms: at some point I think it is argued that eg on ints, the identity morphism is supposed to return the same int, which I think is wrong; aren't identities defined up to an isomorphism or am I mistaken here?
The identity morphism for an object A is the unique morphism id : A -> A such that for any object B ( which does include A), and any morphism f from A to B, and any morphism g from B to A, the composition of id and f is equal to f, and the composition of g and id is g. So, if A is “the type of integers”, and we have the morphisms from A to A to be all the functions from A to A, with each of these functions being considered to be distinct, then the identity morphism will be the identity function. You can define a category where the morphisms are like, equivalence classes of functions, or something like that, but you don’t have to. You can have each morphism correspond to exactly one function. And because in the example they were talking about was the category of sets, in which “function” and “morphism” are the same thing, the identity morphism on the object “the set of integers” is exactly the identity function on the set of integers.
@@drdca8263 I really appreciate your comment; thank you very much for the clarity and level of detail! I'm pretty sure I probably misunderstood id(x)=x for id(x)=const eg 42, where xεint. I apologize for any confusion or frustration my comment might have caused. I'm really new to this and I'm struggling to get it right.
East Quack I’m glad my comment was clear! Also, thank you for explaining what was the likely cause for why you had been confused. It was a somewhat different reason than I had imagined, and both seem to me like an easy confusion to have. Please don’t worry about it causing any frustration; it didn’t cause any (at least for me, and I don’t see why it would frustrate anyone else). I’m not an expert in category theory, but to the extent that I can, I would be happy to answer other questions about it.
@@drdca8263 You are very kind, thank you. I might take you up on your offer. But I don't want to be a drag on anyone. I am not an expert in category theory, categorically! It's just that these days people on the internet are twitchy about almost everything. Although there are thoughtful and kind people willing and able to lend a helping hand when needed, this is by no means the rule. I stumbled upon some category theory videos unexpectedly and got hooked; I am trying to understand it ever since. I find myself going over and over those lectures to get acclimatized with the curriculum but I'm terrified to open a relevant textbook because I'm not a mathematician and I do not want to be scared away. Watching this video had the added bonus that I'm familiar with Mathematica (or Wolfram Language) so it felt like a safe space, in a sense. I don't expect to 'get' Stephen Wolfram or all the other extremely knowledgeable people on this video, but I'm hoping that eventually some of 'it' will rub off onto me as well and I'll eventually be ready to approach the material in a more standard way.
![Wolfram Physics Project: Working Session Aug 18, 2020 [Physicalization of Empirical Metamathematics]](/img/n.gif)
Find Stephen's notebook for this session here: www.wolframcloud.com/obj/wolframphysics/WorkingMaterial/2020/Categories-02.nb
I think that for your project you need to use programming Languages for a quantum computer this is the future
I have found it! It only took 3 years...
@@matrixgod8923🎉🎉🎉
I woke up to this video playing I have no idea how I came here or what this is, I’m truely confused
Bro. I had my phone in the pocket and this video started playing
every time i fall asleep to a youtube video i wake up with some guy doing physics equations and its becoming a weird pattern
I just woke up
Yea same
Same 😂
I love falling asleep to whatever UA-cam video I'm watching so I can wake up to smell the Category Theory.
I woke up to the sound of several men having a conversation about "A"s and "B"s.
Oh dear - this is about the best example of "smart people aren't necessarily great explainers"... IDK, maybe they didn't have time to prepare, that would explain it. But... come on, there are such great ways for intuition building towards category theory, none of them is explored for at least the first hour - though Tali Baynon does a pretty good job later on.
Here's how I start when I explain category theory:
1. For the longest time, most mathematics we did was un-formalized in the sense that it had no canonical axiomatic foundation (euclidean geometry being the exception)
2. In the mid- to late 19th century, philosophical logic and mathematical thought came together and laid the foundation of everything for the next 100 years: predicate logic, propositional logic, boolean algebra, set theory, algebraic topology - and among them, set theory with propositional logic was thought to have the power to be able to serve as the language for grounding, formalizing - axiomatizing all of mathematics.
3. When we describe abstract mathematical structures in terms of sets, the foundational notions are of "inner structure" of the sets - and we represent everything as such inner structure of sets (including functions, relations, elements, unions, intersections, complements etc.) with the help of the "is element" diadic predicate.
4. This leads to loads of operations with powersets and inclusions and in general, there are many different encodings of structures in the language of sets (e.g. the natural number 2 might be represented by {∅,{∅}}, or by {1,1}, or by {0,0}, or by {0,{0}} ... and so on... the relational structure is not very intelligible when "flattened" into a set-theoretical representation.
5. What category fundamentally does differently is this: it talks only about objects and arrows/morphisms between them. The objects can certainly *have* inner structure (in that morphisms e.g. in the category of sets can be (non)injective and/or (non)surjective functions) - but category theory never looks "inside" the objects! They are black boxes - and all the internal structure is represented by conditions on the arrows/morphisms that go in and out! That's the fundamental idea - and the reason why category-theory (including topos-theory and homotopy type theory) is the language of structuralism: Things are defined in terms of their relations to themselves and to other things. It's the power of this concept that lies at the heart of everything.
6. An example: there is the notion of a categorical product. It is defined as an object AxB with two morphisms left:AxB -> A and right:AxB -> B such that any other object which has such morphisms to A and B factors *uniquely* through the categorical product, i.e. there is unique (up to *unique* isomorphism) morphism *g* from the other candidate Y with its candidate projections Yleft: Y->A and Yright: Y->B to the unique (up to unique isomorphism) actual categorical product AxB: g: Y->AxB such that left(g(Y)) = Yleft and right(g(Y) = Yright. This defined the "universal construction" of the categorical product: an object with morphisms to its left and right parts such that any other object with such morphisms factors through the actual product (in the above way).
7. Notice that at no point in the above description of the product have we needed to draw our objects as anything other than labelled black boxes - and being a product is certainly about inner structure! But it's the ways in which inner structure is revealed in or rather *defined through* how the object can relate to other things.
8. Now we can place constraints (are the morphisms epic, monic, both, none, what constraints are on the dual category etc.) and look for models. For example, in the category of Sets, the categorical product exactly describes the cartesian product - in this way we abstract and generalize notions of structure to make them universally discoverable and applicable. That is the magic of category theory.
Great!!! loved and totally agree!
@@sebastianmullerbalcazar6229 I came to a theoretical model of existence that I call Fractal Point Dynamic Theory that leans on the exact concepts you detailed above. Now I'm here listening to educated dudes dropping jargon left and right that reiterate the same exact concepts I discovered, but with the actual math terms. Oh, and they know wtf they're doing. I'm flailing in the dark trying to make the translation jump here
As a graduate student, I always found the abstraction of Category Theory to be too much.
I think the problem is the person trying to explain this, Fabrizio,was constantly interrupted by Stephen. I wish Stephen, the person who contantly interrupted would park his ego and shut up so the presenter could have had the opportunity to run through it once and THEN deal with questions. I think any criticism of the Fabrizio's presentation/explanation skills are misplaced. This is what happens when dealing with an obnoxious person does not know how to shut up & listen to something that challenges his assumptions. As a mother, I see this in my children all the time. But one expects it from children, and in that case it is a sign of intelligence and intelectual curiosity. 🙄 The presenter, Fabrizio,was calm and polite. That's impressive. BRAVO! Hilariously Stephen did not "get it" when one other audience member he called on to comment said he had no comment and refused outright to comment. Clearly he was thinking what I was thinking: "Do shut up & listen, Stephen."
Fell asleep watching real civil engineer play poly bridge 3 and woke to a physics lesson.
That's usually what I would be watching too and I can kind of understand it if I wound up here? This time I fell asleep watching Red Letter Media talking about The Snyder Cut and it immediately gave me 3 of these videos straight after. WTF algorithm
woke up to this and i just can't sleep through it. where did you bring me youtube
out of category theory comes the principle of irreducible confusability
this is the deepest rabbit hole youtube's taken me to and i am genuinely afraid
just wait until you understand the shit and your mind goes with you down that rabbithole.
Auto play went crazy with this one
@@jaymethodus3421were there a ee😅😅😅😅😅www ewrwwwwkpdo😂oh❤️❤️❤️❤️❤️😙😙🤣😒😙😙🤣😌😂😂😌😮
Bro same lmao
I loved this because I really struggle with Category Theory. I am always behind by months as I study to understand this breakthrough Wofram Theory! Exciting and I predict noble prizes in the future !!!!
I got so much from this!
Thank you to all the guests !!! Thank you Stephen Wolfram!!!!
By the way, being able to be a fly on the wall during conversations like this--its supremely awesome. I've struggled to understand sheaf cohomology since first reading Frankel. Hearing you guys discuss this and walk thru the concepts really opens up this stuff in my head while I'm listening to you.
Spoiler alert- you *are* the fly on the wall
I like the linguistic side the most from category theory, all those specific and absolutely exact terms for every possible abstract thing, like learning a new language with the maximum possible expressivness.
The problem is, when the time comes to merge those thoughts with reality, we will fail. Our language/logic capabilities are just not suited for it, no matter how intelligent one is or familiar with the exact sciences, and even with the help of A.I. which is inevitably trained by our observations and later its own observations.
Hi all! With all respect, Don’t ask why im here randomly 3 yrs later of this being published but I do believe there should be a partnership dictionary/re-writing of these terms used, even tho people working on this for ages. Its extremely confusing. Hopefully not changing any of the discussed subject matters. Thanks and All the best to all.
I fell asleep on this tab and woke up to this stream. Apparently I've been watching Sam O'Nella reactions for the past 5 hours.
starts at 9:48
When I was first told about category theory, my first thought was “oh, it is kind of like a combination between a group and a directed graph”, but that wasn’t quite right. In a group, all elements have an inverse, but the analogous thing doesn’t hold in category theory. Instead, a monoid is the thing, not a group.
On the other hand, the “combined with a directed graph” idea, to make that actually work, would be the idea of a groupoid. A groupoid is like a group, except that instead of there being a composition of any 2 elements, only some elements can be composed, and as such there are multiple identity elements in order to fit with this. It can be thought of as each element having a domain and codomain and the composition of the groupoid elements works whenever those match up.
Then, putting both sides of this together, the corrected version of what I thought is “oh, that’s like a combination of a groupoid and a monoid”.
So, in the end, “a category is just a monoidoid”.
You end this on exactly what I've been wondering - which is how to express the symmetries of q.f.t. as a group, using category theory, and possibly Grothendieck equivalence to encode the group into the rule... i.e. what does a particle look like in rulial space? I think what it looks like is is an exceptional group as per Lisi, that emerges as you drive up the scale from hyperedge to electron size, where the exceptional group is showing you the stable vibrational modes in the spacelike graph.
I'm working on it. But here's something I don't really understand fully.
q^-2Q/2=1
-2√2Q=q
where q is a smaller quantity and Q is the larger, in a closed point pair system, and they must measure eachother in relation to nothing but themselves, and eachother. idk if it even plugs in or works though lmfao
@@jaymethodus3421 if 0^0=1 then q=Q=0
I fell asleep learning about ice ages and methane. Woke up learning about proofs to infinity and morphisms 😭
Genius video!!! Physics will never be the same!!!
#OpenPhysics
#WolframPhysicsProject
I fell asleep watching Joe Bartilozi and I wake up to this
Proof that transitions are also agent based via. Category Theory. Proof that SpaceTime is also agent based Via. Splicing (scientific observation). "Everything is everything else"- Everything is Agent-Based.
"In this paper, we propose a particular style of semantic rules that make it visually clearer how changes at one level of a MAS require simultaneous changes in other levels of the system (where each component of each level is modelled as a separate transition system)."
Thank you Prof. Wolfram to clarify the cat. theory - decompose the abtractions into concret explanation. Save lots of time to decode them.
1:21:33 - Category Theory strikes again!
35:09 - To the Monad we go.
Take a look at Tim Maudlin's _Theory of Linear Structures_ (book).
He does exactly what was explained ~2:30 for sieves and open sets.
He truncates conventional topology at the 0D point-set axioms,
because they do not seem obvious or physical.
He retains line elements as connectivity (for points that don't 'exist' :)
then shows that the lines must be directed
to derive the discrete equivalent of topology (e.g. open/closed).
This obviously leads into one of your other sessions
about rebuilding calculus over discrete structures.
Tim goes on to discuss applications to physics.
Perhaps arrange a live session with him!
By all means start discussing the topology stuff,
but he can also (perhaps mostly) contribute to
the philosophical implications of your work.
P.S. Echoes here of Rovelli's _Relational Quantum Mechanics_
which I like to call the _Zero Worlds Interpretation,_
because there are relations but no _relata_ (in Mermin's words),
i.e. edges but no nodes : )
P.P.S. Also, not by coincidence, in Rovelli's LQG
the lowest dimensional spatial operator is area!
but the area appears on the incident edges.
So neither the nodes nor the edges 'exist' spatially
but there are area quanta with a spectrum,
which presumably have normals in some limit.
There are also volume quanta/spectrum.
The outcome originates from one of Penrose's
many amazing insights, that spin may be fundamental,
not space, not time, not spacetime.
Left my laptop on after watching some legal debates and woke up to this. And it seems I'm not the only one
The ..."morphisms between morphisms between ... " property of infinity Cats looks very close to what we think of when we are looking at infinitely differentiable manifolds, aka the smoothness property: between any two points there is another point such that there is a way to go from a ball of one point to the ball of any other point.
I fell asleep watching Red Letter Media talking about The Snyder Cut and it immediately gave me 3 of these videos straight after. WTF algorithm
I fell asleep watching gaming videos and woke up here 3 hours in
lol same
Yes, let's please talk about categories of things other than graphs! Thank you! 31:40
The conversation went into this self-referential territory veeeery quickly.
Well, that didn't help a lot. :/
so glad i found this through autoplay
You need to use category theory to figure out how exceptional groups, can help you organize space like graph behavior, to emulate the coupled quantum harmonic oscillators of quantum field theory, over time, where the vibrational states of the graph map to the behaviors listed in the standard model, and you need to do this in a way that the group symmetries show up clear at a range of 10^35 hyperedges.
Adding a type submodule to Wolfram language could help provide a demonstration of how category theory provides mappings between types, where untyped wolfram language uses the isomorphic properties of category 'mappings' without the corresponding cateory types which have the equivalence relationship for the mapping property. Add a toy type implementation and watch the type confusion evaporate.
Jonathan speaks about "interpretation" informally, but in Computer Science interpretation is a formal (and formalized) notion.
Interpretation is evaluation. LISP's eval() function.
They are missing each other because the very notion of "time" doesn't exist in Mathematics, and so the distinctions between static and dynamic expressions doesn't exist.
And so the Mathematicians speak about types and type-safety (which, happens at compile-time for computer scientists).
while Stephen is talking about execution and evaluation (which happens at run-time for computer scientists).
en.wikipedia.org/wiki/Meta-circular_evaluator
en.wikipedia.org/wiki/Eval
A powerful set of ideas about Category Theory... I think this is a important video!!!!
i fell asleep and woke up to this, how the fu-
bro same 😭😭
Thank you Prof.
i’ve finished the entire video... while sleeping. atleast i dreamed of science
You need a 'super' category with:
AAAAA (over:)
BBBBB
Where you can have true or false for each combination.
This was awesome but what was annoying was this could be explain a lot more simpler with less fluff.
you could see how Stephen Wolfram was trying bring things back to first principles
but we kept getting this...
Exaggerated Example:
Stephen Wolfram: ok lets make it simple, does A = B?
others: Well Its False but also not False because "A" is False but "B" is True so "A" must be True because "A" can be True and False but the equals-sign itself can be False or True
but its mainly False but in this case its True but "A" and "B" can be mapped to each other so this makes "B" False but it depends on the position of "A".
Stephen Wolfram: 🤦♂️
Stephen Wolfram: ok, does B = A?
others: Yes, that's True
Stephen Wolfram: but if B = A then A = B? does the order matter?
others: No the order doesn't matter but they are still not the same.
Stephen Wolfram: oh boy😅
I fell asleep watching Vsauce and now I am here
man i fell asleep to a different video and woke up to this what on earth
Woke up again to this 😍
The number of light bosons stems from the cyclotomic of 18 (divisors 1, 2, 3, 6, 9, 18 and new roots 1, 1, 2, 2, 6, 6) for 18 normal bosons (6 free ones as 18-12 [not fermion bound]) and if the equality of the mass independent free space view to zero is just an approximation to the reciprocal of a small oscillation then a differential equation for such is just scaled by units of Hz2 and having which would place the cyclotomy at 20 (divisors 1, 2, 4, 5, 10, 20 and new roots 1, 1, 2, 4, 4, 8) for 20 dark bosons perhaps? Or maybe it works inversely for reducing the cyclotomy to 16 (divisors 1, 2, 4, 8, 16 and new roots 1, 1, 2, 4, 8) or 16 dark bosons?
Or “free dark bosons” at a tally of 2 (or -2)? I think I used η with a floating ~ (tilde) to indicate this secondary oscillation. Fermi exclusion unique factor domain expansion? Non-unique compaction “gravity”?
What tickles my mind is the idea of 2 "ultra free dark bosons" as an idea. Put another way
A proof is a proof of course of course! As long as the proof has proof of course! ^.^
That's a very destructive meme.
The "events" comprising a spacetime manifold are possibly *not* atomic or irreducible primitives. Why not see them as fixed points of interactions between quantum systems - where the q-systems are not "in spacetime"?
The talk: www.appliedcategorytheory.org/wp-content/uploads/2018/03/Michael-Robinson-Sheaf-Methods-for-Inference.pdf
Me too, woke up to this video when it had played about 2 hours. Must have been really tired. I will go into Grand Theft Auto Online (GTO) and see if I have subconsciously learned something 🤣
Maybe I will be able to categorize the try hards into all their sub segments from this sleep learning experience 😉
I read a new kind of science in 2003 fascinating stuff. Also check out the quantum enigma and Seth speaks, two equally mind blowing books.
Dark mode should be standard practice everywhere. This is the day that I finally disable autoplay for better sleep.
Edit: AGI when?
I am extremely unhappy that nobody could explain in this 4 hours how to 'talk' category theory. This only adds to the confusion and misunderstanding. Wolfram questions have mostly not been answered. One (worst) exapmle: Grothendieck topology is an unfortunate misnomer: open coverings are only one aspect of topology, this has little/nothing to do with 'continuity'.
WOw very interesting. I think that could help us.
Thanks
3:29:30 The Aharonov-Bohm effect???!
I feel like the part about the Curry-Howard correspondence would have been clearer in a language with dependent types, instead of one without types.
The proof functions thing, it feels to me like it doesn’t really capture the correspondence. In order to capture the correspondence, I feel that these things should be composable. A proof of “If A then B” should be composable with a proof of “If B then C” in order to produce a proof of “If A then C”. I don’t see a way to do that with these proof functions.
The way of making pairs in untyped lambda calculus that I’m familiar with is lambda x . lambda y . lambda f . (f x y)
Then you define fst as fst = lambda x . lambda y . x
and snd as snd = lambda x . lambda y . y
Then, if you give some pair the argument fst, you get the first thing in the pair, and if you give it the argument snd you get the second thing in the pair.
If you want instead a function that you can apply to the pair instead of one you apply the pair to, just define a function which takes in the pair, and then applies the pair to fst
(and another one that does the same thing but applies the pair to snd instead)
Straightforwards enough.
But, yeah, this doesn’t work so nicely when you want to make everything typed.
Well, actually, I guess the type of the pair could be said to be,
For any type C, accepts (a function which takes an input of type A and returns (a function which takes an input of type B and returns something of type C)) and returns something of type C .
So, you can use types with this way of making tuples, but simply typed is not sufficient. You need,
type arguments and dependent types? Although, I guess it only has the types depending on the type arguments, so I guess that isn’t really type arguments. You could do that in C++ if you wanted. And like, not using the newest stuff in C++ . As long as C++ has had function types I think.
But, yeah, if you want to use the simply typed lambda calculus, you have to have a built-in way to make a product of 2 types (I.e. the type of pairs with the first entry being of the first type and the second entry being of the second type), and a built-in way to make pairs for those types.
You can’t just use the definition of pairs from the untyped lambda calculus.
And, it is better pedagogically, I think, to teach people the simply typed lambda calculus before you teach them any of the versions with dependent types.
It’s just easier.
Edit 2 :
Ah, 1:21:50 really starts getting to the point for at least a little while. Especially at 1:24:30
A very impressive discussion leaving us with wanting to see this used to create sieves and something practical from this that we can use.... My question is how can we apply this to the Wolfram model... The hyperways ....?????????????????
UA-cam... why why have you brought my brain here in my sleep... just to suffer...
1:54 Stephen needs to comprehend that those "empty' or meaningless functions have stubs for that action and fibers can use.
3:10 you discovered curvature!
3:05 Does this work if the base is not compact?
56:16 - I thought that was Meatloaf?
I need some videos about that
I also woke up to this. Why did it happen to so many people?
Computer, what's the best factoring of this codebase?
Categorization is computationally irreducible.
🤔 a substitution system that counts
F(Result[]) -> Result 🤔 locality if sibling?
Easy to reverse engineer a topology (topography?): functions bumping around. 😜
Repeating structure classification...AI problem?
„So you’re saying“
as I understand this guy is big fan of ABBA
There are pretty many mistakes in this conversation, philosophical and technical, which will confuse anyone who is not already a category theorist.
I have done a little category theory but not much. (E.g. I don’t understand the Yoneda lemma, but have made the statement “a morphism in a concrete category is surjective iff it is right-orthogonal to a morphism from the initial object to the free object over a singleton set”). Are you speaking as a category theorist, and warning that this video is likely to make me confused and misunderstand things, or are you saying that someone who has almost no familiarity with category theory is likely to just be confused?
@@drdca8263 I am not sure how you want to draw the distinction. It is like a stormy sea that will drown a beginning swimmer for sure. And even if you know how to swim, you will appreciate the difference between a storm and a clear blue sky.
Joel Sjögren Thank you, that answers my question quite well. I appreciate it.
LOL! When are y'all going to use dualities, aka Adjunctions?
I woke up like this
where's the "mind blown" meme when you need it
We always end here
nice joke-I laughed @33:50 also at around @37:40 when they start talking about identity morphisms: at some point I think it is argued that eg on ints, the identity morphism is supposed to return the same int, which I think is wrong; aren't identities defined up to an isomorphism or am I mistaken here?
The identity morphism for an object A is the unique morphism id : A -> A such that for any object B ( which does include A), and any morphism f from A to B, and any morphism g from B to A, the composition of id and f is equal to f, and the composition of g and id is g.
So, if A is “the type of integers”, and we have the morphisms from A to A to be all the functions from A to A, with each of these functions being considered to be distinct, then the identity morphism will be the identity function.
You can define a category where the morphisms are like, equivalence classes of functions, or something like that, but you don’t have to. You can have each morphism correspond to exactly one function. And because in the example they were talking about was the category of sets, in which “function” and “morphism” are the same thing, the identity morphism on the object “the set of integers” is exactly the identity function on the set of integers.
@@drdca8263 I really appreciate your comment; thank you very much for the clarity and level of detail! I'm pretty sure I probably misunderstood id(x)=x for id(x)=const eg 42, where xεint. I apologize for any confusion or frustration my comment might have caused. I'm really new to this and I'm struggling to get it right.
East Quack I’m glad my comment was clear! Also, thank you for explaining what was the likely cause for why you had been confused. It was a somewhat different reason than I had imagined, and both seem to me like an easy confusion to have. Please don’t worry about it causing any frustration; it didn’t cause any (at least for me, and I don’t see why it would frustrate anyone else).
I’m not an expert in category theory, but to the extent that I can, I would be happy to answer other questions about it.
@@drdca8263 You are very kind, thank you. I might take you up on your offer. But I don't want to be a drag on anyone. I am not an expert in category theory, categorically! It's just that these days people on the internet are twitchy about almost everything. Although there are thoughtful and kind people willing and able to lend a helping hand when needed, this is by no means the rule. I stumbled upon some category theory videos unexpectedly and got hooked; I am trying to understand it ever since. I find myself going over and over those lectures to get acclimatized with the curriculum but I'm terrified to open a relevant textbook because I'm not a mathematician and I do not want to be scared away. Watching this video had the added bonus that I'm familiar with Mathematica (or Wolfram Language) so it felt like a safe space, in a sense. I don't expect to 'get' Stephen Wolfram or all the other extremely knowledgeable people on this video, but I'm hoping that eventually some of 'it' will rub off onto me as well and I'll eventually be ready to approach the material in a more standard way.
I understood all the words, but none of the sentences 😢
Spoiler: Category Theory is really, really hard.
Blah blah blah it's all JavaScript big deal I do this at work every day. These guys aren't smarter than me.
wonderwall
wtf did i wake up to??????
Mmmm..
YOUR MICROPHONES ARE HORRIBLE TO LISTEN TO !!!!!!!! BE MORE CONCIOUS !!!!!!!!!!!!!!!