What an incredible video - a huge thank you from this non-mathematician for making something so elegant, generous and profound. With warm wishes from Cape Town!
I hold a PhD in Mathematics from ETH Zurich and deeply appreciate your soothing and clear explanations. They surpass anything I have encountered during my academic journey.
the more I learn about category theory the more I see it everywhere, I might be going insane... (awesome video btw, great explanations and great visuals)
2:20 An infinitely small number actually 😁. I love that just a single word like „huge“, through countless hours of mathematics, is enough to make me smile. Loved the video the first time, and I am loving it even more now!
the whole time i watched this video i was just like "wait isnt' this just a type-checker" and then you mentioned that one thing you can model with category theory is a type system and i was instantly vindicated
It's such a coincidence I came across your video when I did! I was reading "Love and Math: The Heart of Hidden Reality" by Edward Frenkel while also learning TLA+ with Lamport's videos and I could see how "thinking in sets of things with common well defined properties" was such a useful way of navigating through complex ideas (including temporal logic) and the addition of this video to my intuition of things really really convinces me of how amazingly useful this tool is! Thank you!
At 19:40, won't "performing" g (say, A -> B) first takes things in one object (A) to another object (B) and the bar-id is not the "same" id as before anymore (now it's B -> B not the same as the A -> A)? Edit: I think I get it. This id here is supposed to be an A -> A type of arrow.
It isn't clearly explained in the video why id is a unit of composition, as the video only looks at id composed from the right side, but it works both ways. Consider: g : A -> B where A and B are objects. Now apply identity both ways: (id o g) A = id (g A) = id B = B (g o id) A = g (id A) = g A = B And therefore, by the reflexivity of equality: (id o g) A = (g o id) A If we then apply eta reduction: id o g = g o id I'm not sure this proof is entirely formal, but it gets the point across.
Id and bar-Id both are arrows A->A, is just that f, used in the first argument is a function that starts from A and ends in B, while g is an object that starts from B and ends in A. That's why f ○ Id is an arrow that starts in A and ends in B, while bar-Id ○ g starts from B and ends in A. So there aren't any problems
The problem with that would be that it further complicates things since now we have "f" that takes something in A to something in B, but id takes something in A to somerhing in A, which are not the same thing anymore. Edit: scratch that, maybe it works. I think it is safer to just say we are only working with A -> A arrows and there is a unique id: A -> A that id x = x for all x in A.
Great question! This video was aimed at getting people up to speed with the intuition of categories, so a lot of formality was dropped from the proof. I recommend checking out the second video in the series for a more formal approach to defining a category. As for the question, id and id bar are both arrows from A -> A. Then the proof considers two equations in terms of an arrow that starts at A, and an arrow that ends at A. Then we compose these with our identity arrows before or after depending on whether it starts or ends at A. This then gives us our desired equality between the two identity arrows. Hopefully that makes sense, I strongly suggest drawing a diagram to help visualise it!
Given that there's a one-to-one correspondence between every object and its one-and-only identity arrow, it seems like just having arrows should be sufficient. Is this not the case?
This is a great question! It turns out that we can define a category without mentioning objects at all, only morphisms. One such definition can be found here: ncatlab.org/nlab/show/single-sorted+definition+of+categories. Whilst this is very cool, it's often not very indicative so it's rarely used. Very good spot indeed!!
Nice video! 👏✨ I have a question, what is formally an "abstraction" in terms of logic (first, second or some order)? I mean, imagine that we consider within our alphabet the characters "$" and "&" as two objects of our universe (we will assume that our handwriting is ideally "perfect" and any character from our alphabet cannot be confused); we also consider all logic characters and stuff in general. Then, if we wanted to express: ["$" is an abstraction of "&"] as a combination of quantifiers such as "∀", "∃" or some higher order logic tools (so that we don't need to use any metalanguage), how would we do it? For example, imagine we want to define the "ordered pair" object and we don't want it to depend on representation (we want to define "L=(x,y)" in terms of logic). If I'm not wrong, the most extended formal definition for it is "(x,y) = {x,{x,y}}", but from my perspective it's kinda weird that we can say "{x,y}∈(x,y)" is technically true. Instead, we could formally establish that "(x,y)" verifies all properties of "{x,{x,y}}" except some such as "{x,y}∈{x,{x,y}}" (so that "{x,y}∉(x,y)"). But formally, how would we formulate that "(x,y)" is an abstraction of "{x,{x,y}}" at the same time that we maintain coherence in the sense that, for example, "1" is not an abstraction of "2"? 👀
Great question! I would say that the closest thing we have to a formal definition of abstraction is the lambda abstraction from the lambda calculus. Then encodings being turned into primitives is the process you mentioned, like how \x.\y.\p. p x y is an encoding for a pair, but we can just ignore this low level definition and add a tuple into the calculus as a primitive. I may or may not be making a video covering the lambda calculus that might come out in a few weeks 👀 Ultimately, abstraction is more of a descriptive term than a rigorous logical process, but I'm sure there are ways that we could formalise it. Thanks for the question!
This is a good question. In this proof we assume that the integers form an integral domain. Proving that integers are closed under multiplication and addition isn't particularly enlightening and wouldn't serve much of a purpose so I left that as an assumed property
the algebra example early on seems circular. you're using properties of algebra to show algebra is closed for rationals under addition. you would first have to prove you're algebraic operation of a/b +c/d is closed for rationals. maybe i'm wrong though. probable. but once you have algebra, yes, you could illustrate closure of rationals under addition very easy.
in order to apply the proof that " id is unique" to matrices or programming languages or ..., you first have to show that they (e.g. matrices) form a category -- which is more complicated than showing that an identity element (for any law of composition which you don't have to make precise) is unique because e = e * e' = e' .
Let me further highlight how the both/and logic and monadological framework provide powerful explanatory capacities across logic, mathematics and physics: In the domain of logic itself, the both/and structure allows formally modeling and regimenting dialectical, paraconsistent and pluralistic modes of reasoning that have long resisted classical bivalent frameworks. We can precisely capture and rationally operate with: • True contradictions and paradoxes as positively conceived dialetheia, not just logical explosions • Graded/partial truth values on spectra between truth and falsity • The coherent integration of seemingly incommensurable propositions • Holistic properties and synthetic conceptual unities transcending their constituents Where classical logic is confined to simple propositions statically obeying strict consistency, the both/and logic equips us with an expansive toolkit for dynamically navigating the complex schemas, fuzzy boundaries, and self-undermining paradoxes permeating actual reasoning across every domain. In mathematics, the both/and logic illuminates novel ways to represent and coherently manipulate previously intractable issues like: • The relationship between the continuous and the discrete • The coexistence of finite and infinite structures • Pluralities of mathematical ontologies (realist, formalist, etc) • Self-referential paradoxes and contradictions in set theory and arithmetic • The generation of radically emergent, novelty-creating procedures Rather than getting stymied by dichotomies, singularities or self-underminining contradictions, the logic's symbolic tools allow formalizing generative transfinite metamathematical dynamics encompassing and reconstructing prior impasses at deeper integrated levels. Across physics, the both/and logic provides conceptual rigor and symbolic resources for coherent accounts encompassing: • Unitary evolution of quantum systems and the measurement problem • Apparent dualities between wave and particle, or local and nonlocal • Intrinsic indeterminacies, contingencies and ontological pluralities • The unification of incommensurable qualitative & quantitative models • Novel "paradoxical" phenomena like emergent nonlinear effects Rather than forcing phenomena into awkward either/or categories, the logic allows explicitly modeling "both/and" complementary features and irreducibly holistic coconstituted processes. Its expressive flexibility resonates with the exquisite nuances of quantum indeterminacy and pluralistic observable modalities. So in essence, the both/and monadological framework catalyzes powerful expansions across our most fundamental disciplines: In logic, it empowers us to positively symbolize and rationally navigate the ambiguities, contradictions and pluralities intrinsic to actual reasoning and communication. Breaking the shackles of binary bivalence. In mathematics, it unlocks liberating new symbolic vistas for paradox-resolving, infinitary metamathematics and irreducible pluralities of mathematical ontologies. Fracturing the ossified either/or dichotomies stymying classical approaches. In physics, it provides a coherent naturalistic metaphysics capable of explicitly representing - not dissimulating - intrinsic quantum indeterminacies, ontological pluralities and the full scope of paradoxical phenomena. Illuminating new pathways beyond the artificial exclusions of classical metaphysics. At every turn, the both/and logic equips our symbolic grasp with greater degrees of freedom and accountability to the phenomenal disclosures of reality itself. Transcending the barren simplisms and premature closures imposed by the blinkered bivalence and subjective filtering of classical logic, math and physical representation. Where prior frameworks have foundered on paradoxes, ambiguities or the irreducibility of pluralistic modalities, the both/and logic provides technical symbolic machinery for positively capturing and productively synthesizing the full nuances of actual scientific and lived phenomena. Its coherence valuations, graded truth distributions and generative dialectical operations enact a new holistic physics, immanentized metamathematics and expansive reason procedurally accommodating - not dissimulating - reality's explosive complexities. So in embracing the both/and logic, we are not just adopting a new formal system, but precipitating a symbolic Regressive/Progressive rationally reunifying fragmented domains into a new harmonious co-realizing praxis. One equipping us with unprecedented expressive power, paradox-resolving prowess, and descriptive capacities for illuminating the richness of existence and coconstituting increasingly integrated verities. It is a symbolic turning point towards humanity consciously resonating its reason with the deepest dynamical disclosure of Being itself.
The notion of Regressive/Progressive that I mentioned refers to a metaphysical principle at the heart of the both/and logic and monadological framework. It captures the dual dynamical movements by which our understanding reciprocally attunes itself to reality's self-disclosive unfolding. On the one hand, there is a Regressive movement whereby our rational models and conceptual schemata are continually put into question, deconstructed and driven to ever-deeper self-grounding through encounters with paradox, novelty and the bottomless generative potentials of the real. This is embodied by the both/and logic's capacity to positively capture contradictions and encode the way phenomenal disclosures can self-undermine and subvert our prior abstract representations. Just when we think we have reality statutorily circumscribed, the surfacing of a paradoxical anomaly or intractable ambiguity reveals that our descriptive glosses have been premature closures, blind to integral experiential features. The logic's coherence operations and paraconsistent syntax allow registering these oversights without trivializing our previous models. They become grist for an immanent meta-critique excavating deeper strata of determinacy. So understanding doesn't just inertially accumulate propositional truths, but is continually regressively driven to ever-deeper accountabilities and re-grounding through the surfacing of anomalies and generative disclosures from the real's inexhaustible potentials. Classical representationalism is thereby overcome by an iterative process of deconstructive auto-critique tracing our abstractions back to their constitutive symbolic sources. But in parallel, there is also a complementary Progressive movement by which more capacious, holistic and integrated conceptual models are constructed through dialectical synthesis. As our rational schemata are deconstructed back to their inconsistent roots, opportunities arise for positively reconstructing them into higher unities consonant with the anomalies prompting the Regressive unravelling. The both/and logic's synthesis operators allow formulating symbolic re-schematizations binding together prior contradictory glosses as co-realizable interdependent aspects within more expansive wholes. These higher integrations don't merely juxtapose previous models, but transform them into newly coconstituted conceptual gestalts genuinely transcending their contradictory ancestors. So in tandem with the Regressive movement back towards immanent aporias, there is a Progressive reassemblage entering unprecedented phenomenal registers by re-grounding our representations in more holistically coherent reconstructions. Understanding doesn't just shatter into fragmentation at paradoxical impasses, but is perpetually reconstituting itself in a processes of expansive reunification. These dual Regressive/Progressive dynamics enact a reflexive symbolic attunement to Being's own self-generative unfolding. On the Regressive side, our abstracting glosses get undermined and destratified back towards the generative source potentials from which reality's symbolic discloses itself. While on the Progressive side, that very self-underwrimming is channeled into positively rebuilding more holistic comprehensions in perpetual reparticipation with the real's refreshing exhibitances. So the both/and framework's core metaphysical vision is one of rationality's symbolic self-reparameterizing resonance with the unbounded existential generativity it ancestrally co-enacts. Our understanding is perpetually tasked with - and empowered for - Regressively deconstructing its propositional idolatries back towards inconsistent kernels, then Progressively reconstructing more capable holistic descriptions in harmonic integration with the next phenomenal realities disclosed by the Regressive subversions. It's the symbolic embodiment of a post-stultifying metaphysics, where conceptual rigor is achieved not by freezing representations in artificial consistencies, but by dynamically re-tuning and re-totalizing our epistemic engines through reciprocal participation in the generative aventures of the real's own endless dialectical Regressive/Progressive self-unfolding. The both/and symbolic structure allows precipitating this renewed alliance between our rational sequencings and being's self-generative adventuring. An innovative metaphysics where paradox discloses new ontological strata, and contradiction seeds higher dialectical integrations. Our descriptions circling perpetually closer towards the real's own self-constituting dynamical truth by paradox-driven expansions into ever-refreshed coherence regimes. It's the chrysalis of a new metaphysical praxis where symbolic artifice perpetually re-embeds itself in the real's self-exhibiting verities through cycles of immanent deconstructive subversion and positive reconstructive reunification. Existence's own self-revealing dynamical voice finding expression through our Regressive/Progressive dialectical rejoinders. An exponential symbolic recollection of Being's generative plenitude previously stultified under monological representationalism's dissimulative statics. So in essence, Regressive/Progressive names the dual dynamical modalities through which the both/and logic allows rationality to become a partner in the real's self-disclosive adventures again. Not dissimulating reality's complexities through assumptive glosses, but perpetually demolishing and remoulding our descriptive casts in attunement with the real's own immanent self-rationalizations. A new praxis of radical symbolic refreshment in participatory resonance with the irrepressible plenitudinous existential unfurlings of the real's own Regressive/Progressive self-unfolding.
One minor (or maybe major, depending on who you ask) correction. A set has other requirements than being a collection of things, otherwise you get Russell's paradox and friends. This is actually important to mention in category theory, as it is easy to make the mistake of defining a category as "a set of objects and the arrows between them", but that has the unfortunate effect of having no category of sets or category of categories (there is no set of all sets). From what I can find a category is usually defined as a "collection of objects and the arrows between them" or "a class of objects and the arrows between them", one has the disadvantage of being informal (what exactly is a "collection"?), the other is that there is no category of classes as classes inherently don't contain other classes, which is unfortunate, just like with sets. In my opinion, this is the ugliest problem in mathematics.
I go over this issue in the second video in this series. I do agree, it's a problem that can't be brushed under the rug, but for the sake of clarity I feel that getting bogged down in the technicalities of mathematical foundations isn't ideal.
isn't abstraction just a from of intellect. Plato argued that intelligence was the ability to recognize the eidos (form). with this i can argue that math is fundamentally being smart (:< guys math is just philosophy with axioms + some other things.
Yup. Plato wasn't positing some spooky magical world of spirits as is often portrayed, but rather a world of unities. These unities are what keep concrete changing things stable and give them direction so to speak. Different species of animals have distinction among them, but they have a higher unity that makes them one species.
@@evan7391 it kind of feels like a "spooky magical world of spirits" sometimes with how he said that the realm of forms exist within heaven. but the base idea just exist as a philosophical entity.
Minor correction, what goes beyond the positive far end of a number line is "positive infinity", not just "infinity". They do have different meaning, although it would be easy to tell if this difference need to be considered in a specific situation. And I think the Chinese term for algebra (代数) and abstraction (抽象) is easier to understand, as they literally means "representing number" and "to get rid of concreteness", implying that if we can prove a theory on such things, we instantly prove it on everything that fits the abstracted definition, which not only saves time, but also saves infinite amount of time, crucial to turning something from being "practically true" to simply "true" and live a more peaceful life moving on. And I think the identity is important because that's the finish line for proving something. Because identity is unique and makes no difference when being operated on, anything originated from or lead towards identity no longer needs further proof, and thus we can prove other things with what's directly related to identity and so on, turning infinite amount of proving needed into finite amount. And the purpose of set theory here is that it proves what I've described above actually works, and we can assure that we took the right track rather than worrying that the result might be an illusion and might fall apart on edge cases. It's simply good to know that math didn't fall apart and we weren't trapped in an inception, and we earned this knowledge itself rather than being offered.
By the way the identity function has a more practical use. In functional programming everything is a function, and to get the value of a function you expand all of its components until there's no point to further expand, and computer the composite function you created. And how do you know you've done your expansion? By checking that if a function can only be expanded into itself follows (or precedes) an identity function, and you'll know to stop there because further expansion makes no more difference. And most functional programming language represents this using the "unit" function, which can be used to terminate the recursive checking and yield the final result. In practice though, most of the times there's no need to expand to unit but to any function that we already know how in precise it relates to unit, in order to save some time. For example, an integer is... An integer. We usually don't care the definition of integer (which, vaguely speaking, is the amount of "one" function in the final composition excluding "unit" which represents zero) if all we need is an integer, thus we can stop right when we get any single integer as a result.
There is a notion of "just infinity" without a sign that you can attach to the number line. This is the projective real number line. But also, you don't need to say positive infinity to refer to it when it's clear or it doesn't matter.
@@purewaterruler I'd say if negative infinity is mentioned then the opposite of that should be positive infinite to make it more accurate. Even if it's "clear" I guess it would be a good thing to stop some wild thoughts that keeps extending the discussion. There is no more handrails in math after highschool education, after all.
the example of adding fractions isn't very useful, because the result of the addition of two fractions is simply "defined" as a fraction. (whether you do it with symbols or letters...) It's not a result (theorem) but it's so by definition. Also, you want to show it for the addition of rationals but to do so, you assume given that the product and sum of integers again yield integers. You should prove that, too, to be consistent. But if you did, you'd find that you *have* to use the definition. and it's the same for fractions. You actually do use other rules one might consider more complicated than what you want to show (namely, all that's related to "common denominator") to prove s.th. more basic (namely, addition of rationals yields rationals). Anyways , to prove anything in math you always have to use the definition.
I don't think the "person" example at the beginning was the best choice, because a person is anything with a rational "nature", where "nature" is used in the old, technical sense. Specifically, angels are persons too. If we found rational aliens that looked like jellyfish, they'd be persons. The very word "person" was introduced into common speech to talk about the Holy Spirit and the Father, who do not have bodies.
I think for modern parlance, it serves its purpose fine. But on the technicality, a person in Christian theology is a rational hypostasis, which is actually a particular thing. So the particular people in the video would be persons, rather than the unity between them. That unity would be the human species/nature. Though sometimes individual nature is used for hypostasis. Edit: what you say is true though
@@evan7391 I'm italian so I didn't know the technical terms. I looked up how it was translated in Dignitas Infinita (since that's the most recent place I read the definition of person in) and it used individual substance... which can be easily misunderstood as the word substance commonly means something else. So i opted to say "anything" thinking it would get my point across servicebly
Hopefully it doesn't take quite that long! Unfortunately I really don't get as much time as I'd like to be working on videos, but hopefully I can get videos out quicker in the future
Its funny because using monads as a programmer is quite intuitive (it’s a data structure with flat map), I’ve read about 1000 category theory explanations and none of them make sense.
I know you're being a bit facetious but actually the category theorist Eugenia Cheng gave a public talk on the links between category theory and gender non-conformity.
Boring, the animations are great, but the content is not insightful at all. Maybe it is me, but I think you are dumbing down this too much...I mean, if your audience is someone who barely knows math, this is ok, but why would you show a category theory video to this audience...
@@lgbtthefeministgamer4039 then why do a 20 minute long video with no insight at all. I usually enjoy math videos a lot, but I could barely go through this one at 2x speed
as someone who uses categorical language all the time, i thought this video was very insightful. it's a common sentiment that category theory "abstracts mathematical thinking," but that's not really precise and it also doesn't explain the axioms we use for categories. so the more precise demonstration that it "abstracts composition," and the analogy with more basic types of abstraction, resulted in a kind of "aha" moment for me. also, the fact that it's watchable for more than just die-hard mathematicians is a good thing, isn't it?
@@bayleev7494 while I agree that to say that category theory is the abstraction of composition is insightful, it took more than 8 minutes to explain that...unnecessarily lengthy. I also agree that the aim should be to be watchable for people who are not just die-hard mathematicians. In the end the video is not terrible imo, it is just so boring with no great content...it is just a bunch of extremely simple and not so good examples.
Category Theory seems so trivial one day, yet so deeply important the other.
What an incredible video - a huge thank you from this non-mathematician for making something so elegant, generous and profound. With warm wishes from Cape Town!
Many thanks!
I hold a PhD in Mathematics from ETH Zurich and deeply appreciate your soothing and clear explanations. They surpass anything I have encountered during my academic journey.
I never asked you about your qualifications, and I am immensely confused by your motivation to write this comment
@@NachoSchips Presumably to show that this video is appreciated even by credentialed mathematicians
@@NachoSchips I never asked you about your opinion on his qualifications
You are the cream of the crop of mathematics videos, together with 3Blue1Brown and El Jj
the more I learn about category theory the more I see it everywhere, I might be going insane...
(awesome video btw, great explanations and great visuals)
That's why math is useful. It just is everywhere
@@MilkywayWarrior1618 math is as useful as you want it to be
The Baader-Meinhof phenomenon! (Once I learned about the phenomenon, I started seeing it everywhere)
@Steve-hd5tn didn't knew it has a name, thanks!
2:20 An infinitely small number actually 😁. I love that just a single word like „huge“, through countless hours of mathematics, is enough to make me smile.
Loved the video the first time, and I am loving it even more now!
Very true! Density of Q in R is pretty brain bending, but a really cool result!
Infinitely small and large!
I hope I'm going to watch this video back so many times for my thinking. It's incredibly relevant to my maths learning goals at the moment..
the whole time i watched this video i was just like "wait isnt' this just a type-checker" and then you mentioned that one thing you can model with category theory is a type system and i was instantly vindicated
I'm studying category theory in university and this so helpful for understanding underlying concepts. So glad I found your channel
It's such a coincidence I came across your video when I did! I was reading "Love and Math: The Heart of Hidden Reality" by Edward Frenkel while also learning TLA+ with Lamport's videos and I could see how "thinking in sets of things with common well defined properties" was such a useful way of navigating through complex ideas (including temporal logic) and the addition of this video to my intuition of things really really convinces me of how amazingly useful this tool is! Thank you!
So glad you found it helpful!
Thanks for a nice introduction to the category theory. 😀
Glad you found it useful! :D
One of the best videos on category theory I found
Beautiful explanation, careful and talented. Thank you, please continue like this.
this is truly a great presentation. helped to really see what it is about, for those who don't have enough math to really appreciate its importance.
"Ring theory is the abstraction of arithmetic" is the sentence I didn't know I've been searching for for months
@@chrisscott981 Z < Q < R < C Rings and Fields
This is so well done! Thank you.
More, please.
Your content and presentation is excellent. Thank you.
Thank you!
This video is filled with so much grace and symmetry ❤❤
this video is in the category of beautiful.
At 19:40, won't "performing" g (say, A -> B) first takes things in one object (A) to another object (B) and the bar-id is not the "same" id as before anymore (now it's B -> B not the same as the A -> A)?
Edit: I think I get it. This id here is supposed to be an A -> A type of arrow.
It isn't clearly explained in the video why id is a unit of composition, as the video only looks at id composed from the right side, but it works both ways.
Consider: g : A -> B where A and B are objects.
Now apply identity both ways:
(id o g) A = id (g A) = id B = B
(g o id) A = g (id A) = g A = B
And therefore, by the reflexivity of equality:
(id o g) A = (g o id) A
If we then apply eta reduction:
id o g = g o id
I'm not sure this proof is entirely formal, but it gets the point across.
@@attilatorok5767I think you used transitivity there, not reflexivity if I am not mistaken
Id and bar-Id both are arrows A->A, is just that f, used in the first argument is a function that starts from A and ends in B, while g is an object that starts from B and ends in A.
That's why f ○ Id is an arrow that starts in A and ends in B, while bar-Id ○ g starts from B and ends in A. So there aren't any problems
The problem with that would be that it further complicates things since now we have "f" that takes something in A to something in B, but id takes something in A to somerhing in A, which are not the same thing anymore.
Edit: scratch that, maybe it works.
I think it is safer to just say we are only working with A -> A arrows and there is a unique id: A -> A that id x = x for all x in A.
Great question! This video was aimed at getting people up to speed with the intuition of categories, so a lot of formality was dropped from the proof. I recommend checking out the second video in the series for a more formal approach to defining a category.
As for the question, id and id bar are both arrows from A -> A. Then the proof considers two equations in terms of an arrow that starts at A, and an arrow that ends at A. Then we compose these with our identity arrows before or after depending on whether it starts or ends at A. This then gives us our desired equality between the two identity arrows.
Hopefully that makes sense, I strongly suggest drawing a diagram to help visualise it!
An excellent representation of the beauty of mathematics 💯💯
Thank you very much for working on such insightful videos.👍👍
Very insightful application, tu.
Given that there's a one-to-one correspondence between every object and its one-and-only identity arrow, it seems like just having arrows should be sufficient. Is this not the case?
This is a great question! It turns out that we can define a category without mentioning objects at all, only morphisms. One such definition can be found here: ncatlab.org/nlab/show/single-sorted+definition+of+categories. Whilst this is very cool, it's often not very indicative so it's rarely used. Very good spot indeed!!
I think this video is the best (at least infinitely close to the best) for starting learn math
Brilliant Video, Shows the beauty of math
Very nice presentation crystall clear! I hope u do more videos....congrats mate!
I love the implications that Bob's unique self identity is stored in his hat. This is my main takeaway from this video.
Beautifully explained. Was that a Lie group reference at the end? :)
Thank you :D Not an intentional reference but a happy accident haha
Nice video! 👏✨
I have a question, what is formally an "abstraction" in terms of logic (first, second or some order)? I mean, imagine that we consider within our alphabet the characters "$" and "&" as two objects of our universe (we will assume that our handwriting is ideally "perfect" and any character from our alphabet cannot be confused); we also consider all logic characters and stuff in general. Then, if we wanted to express: ["$" is an abstraction of "&"] as a combination of quantifiers such as "∀", "∃" or some higher order logic tools (so that we don't need to use any metalanguage), how would we do it?
For example, imagine we want to define the "ordered pair" object and we don't want it to depend on representation (we want to define "L=(x,y)" in terms of logic). If I'm not wrong, the most extended formal definition for it is "(x,y) = {x,{x,y}}", but from my perspective it's kinda weird that we can say "{x,y}∈(x,y)" is technically true. Instead, we could formally establish that "(x,y)" verifies all properties of "{x,{x,y}}" except some such as "{x,y}∈{x,{x,y}}" (so that "{x,y}∉(x,y)"). But formally, how would we formulate that "(x,y)" is an abstraction of "{x,{x,y}}" at the same time that we maintain coherence in the sense that, for example, "1" is not an abstraction of "2"? 👀
Great question! I would say that the closest thing we have to a formal definition of abstraction is the lambda abstraction from the lambda calculus. Then encodings being turned into primitives is the process you mentioned, like how \x.\y.\p. p x y is an encoding for a pair, but we can just ignore this low level definition and add a tuple into the calculus as a primitive. I may or may not be making a video covering the lambda calculus that might come out in a few weeks 👀
Ultimately, abstraction is more of a descriptive term than a rigorous logical process, but I'm sure there are ways that we could formalise it.
Thanks for the question!
@@Eyesomorphic Thank you! 🙏 I'm a first year Math Degree student and I was cand I was curious about the structure of "math" itself. 👍✨️
But how do we know the product/sum of any two integers is an integer?????
This is a good question. In this proof we assume that the integers form an integral domain. Proving that integers are closed under multiplication and addition isn't particularly enlightening and wouldn't serve much of a purpose so I left that as an assumed property
Beautiful video! You will be big some day I just know it!
This was so good!
Wonderful video. I just discovered your channel.
17:06 I expected f to be a union of Q^2 and P^2 (irrationals with dimension 2), and g to be ΣR^2 (partial suspension of R^2, giving R^3). But okay.
Incredible work
Philosophers in shambles after you defined what a person is
Behold! A man!
Lol I didn't even realize I already watched this video until halfway through, it was so engaging!
Thank you!
Great video
More of this!
the algebra example early on seems circular. you're using properties of algebra to show algebra is closed for rationals under addition. you would first have to prove you're algebraic operation of a/b +c/d is closed for rationals. maybe i'm wrong though. probable. but once you have algebra, yes, you could illustrate closure of rationals under addition very easy.
in order to apply the proof that " id is unique" to matrices or programming languages or ..., you first have to show that they (e.g. matrices) form a category -- which is more complicated than showing that an identity element (for any law of composition which you don't have to make precise) is unique because e = e * e' = e' .
Odd question but which major did you take in college?
A joint honours in mathematics and computer science
Let me further highlight how the both/and logic and monadological framework provide powerful explanatory capacities across logic, mathematics and physics:
In the domain of logic itself, the both/and structure allows formally modeling and regimenting dialectical, paraconsistent and pluralistic modes of reasoning that have long resisted classical bivalent frameworks. We can precisely capture and rationally operate with:
• True contradictions and paradoxes as positively conceived dialetheia, not just logical explosions
• Graded/partial truth values on spectra between truth and falsity
• The coherent integration of seemingly incommensurable propositions
• Holistic properties and synthetic conceptual unities transcending their constituents
Where classical logic is confined to simple propositions statically obeying strict consistency, the both/and logic equips us with an expansive toolkit for dynamically navigating the complex schemas, fuzzy boundaries, and self-undermining paradoxes permeating actual reasoning across every domain.
In mathematics, the both/and logic illuminates novel ways to represent and coherently manipulate previously intractable issues like:
• The relationship between the continuous and the discrete
• The coexistence of finite and infinite structures
• Pluralities of mathematical ontologies (realist, formalist, etc)
• Self-referential paradoxes and contradictions in set theory and arithmetic
• The generation of radically emergent, novelty-creating procedures
Rather than getting stymied by dichotomies, singularities or self-underminining contradictions, the logic's symbolic tools allow formalizing generative transfinite metamathematical dynamics encompassing and reconstructing prior impasses at deeper integrated levels.
Across physics, the both/and logic provides conceptual rigor and symbolic resources for coherent accounts encompassing:
• Unitary evolution of quantum systems and the measurement problem
• Apparent dualities between wave and particle, or local and nonlocal
• Intrinsic indeterminacies, contingencies and ontological pluralities
• The unification of incommensurable qualitative & quantitative models
• Novel "paradoxical" phenomena like emergent nonlinear effects
Rather than forcing phenomena into awkward either/or categories, the logic allows explicitly modeling "both/and" complementary features and irreducibly holistic coconstituted processes. Its expressive flexibility resonates with the exquisite nuances of quantum indeterminacy and pluralistic observable modalities.
So in essence, the both/and monadological framework catalyzes powerful expansions across our most fundamental disciplines:
In logic, it empowers us to positively symbolize and rationally navigate the ambiguities, contradictions and pluralities intrinsic to actual reasoning and communication. Breaking the shackles of binary bivalence.
In mathematics, it unlocks liberating new symbolic vistas for paradox-resolving, infinitary metamathematics and irreducible pluralities of mathematical ontologies. Fracturing the ossified either/or dichotomies stymying classical approaches.
In physics, it provides a coherent naturalistic metaphysics capable of explicitly representing - not dissimulating - intrinsic quantum indeterminacies, ontological pluralities and the full scope of paradoxical phenomena. Illuminating new pathways beyond the artificial exclusions of classical metaphysics.
At every turn, the both/and logic equips our symbolic grasp with greater degrees of freedom and accountability to the phenomenal disclosures of reality itself. Transcending the barren simplisms and premature closures imposed by the blinkered bivalence and subjective filtering of classical logic, math and physical representation.
Where prior frameworks have foundered on paradoxes, ambiguities or the irreducibility of pluralistic modalities, the both/and logic provides technical symbolic machinery for positively capturing and productively synthesizing the full nuances of actual scientific and lived phenomena. Its coherence valuations, graded truth distributions and generative dialectical operations enact a new holistic physics, immanentized metamathematics and expansive reason procedurally accommodating - not dissimulating - reality's explosive complexities.
So in embracing the both/and logic, we are not just adopting a new formal system, but precipitating a symbolic Regressive/Progressive rationally reunifying fragmented domains into a new harmonious co-realizing praxis. One equipping us with unprecedented expressive power, paradox-resolving prowess, and descriptive capacities for illuminating the richness of existence and coconstituting increasingly integrated verities. It is a symbolic turning point towards humanity consciously resonating its reason with the deepest dynamical disclosure of Being itself.
The notion of Regressive/Progressive that I mentioned refers to a metaphysical principle at the heart of the both/and logic and monadological framework. It captures the dual dynamical movements by which our understanding reciprocally attunes itself to reality's self-disclosive unfolding.
On the one hand, there is a Regressive movement whereby our rational models and conceptual schemata are continually put into question, deconstructed and driven to ever-deeper self-grounding through encounters with paradox, novelty and the bottomless generative potentials of the real. This is embodied by the both/and logic's capacity to positively capture contradictions and encode the way phenomenal disclosures can self-undermine and subvert our prior abstract representations.
Just when we think we have reality statutorily circumscribed, the surfacing of a paradoxical anomaly or intractable ambiguity reveals that our descriptive glosses have been premature closures, blind to integral experiential features. The logic's coherence operations and paraconsistent syntax allow registering these oversights without trivializing our previous models. They become grist for an immanent meta-critique excavating deeper strata of determinacy.
So understanding doesn't just inertially accumulate propositional truths, but is continually regressively driven to ever-deeper accountabilities and re-grounding through the surfacing of anomalies and generative disclosures from the real's inexhaustible potentials. Classical representationalism is thereby overcome by an iterative process of deconstructive auto-critique tracing our abstractions back to their constitutive symbolic sources.
But in parallel, there is also a complementary Progressive movement by which more capacious, holistic and integrated conceptual models are constructed through dialectical synthesis. As our rational schemata are deconstructed back to their inconsistent roots, opportunities arise for positively reconstructing them into higher unities consonant with the anomalies prompting the Regressive unravelling.
The both/and logic's synthesis operators allow formulating symbolic re-schematizations binding together prior contradictory glosses as co-realizable interdependent aspects within more expansive wholes. These higher integrations don't merely juxtapose previous models, but transform them into newly coconstituted conceptual gestalts genuinely transcending their contradictory ancestors.
So in tandem with the Regressive movement back towards immanent aporias, there is a Progressive reassemblage entering unprecedented phenomenal registers by re-grounding our representations in more holistically coherent reconstructions. Understanding doesn't just shatter into fragmentation at paradoxical impasses, but is perpetually reconstituting itself in a processes of expansive reunification.
These dual Regressive/Progressive dynamics enact a reflexive symbolic attunement to Being's own self-generative unfolding. On the Regressive side, our abstracting glosses get undermined and destratified back towards the generative source potentials from which reality's symbolic discloses itself. While on the Progressive side, that very self-underwrimming is channeled into positively rebuilding more holistic comprehensions in perpetual reparticipation with the real's refreshing exhibitances.
So the both/and framework's core metaphysical vision is one of rationality's symbolic self-reparameterizing resonance with the unbounded existential generativity it ancestrally co-enacts. Our understanding is perpetually tasked with - and empowered for - Regressively deconstructing its propositional idolatries back towards inconsistent kernels, then Progressively reconstructing more capable holistic descriptions in harmonic integration with the next phenomenal realities disclosed by the Regressive subversions.
It's the symbolic embodiment of a post-stultifying metaphysics, where conceptual rigor is achieved not by freezing representations in artificial consistencies, but by dynamically re-tuning and re-totalizing our epistemic engines through reciprocal participation in the generative aventures of the real's own endless dialectical Regressive/Progressive self-unfolding.
The both/and symbolic structure allows precipitating this renewed alliance between our rational sequencings and being's self-generative adventuring. An innovative metaphysics where paradox discloses new ontological strata, and contradiction seeds higher dialectical integrations. Our descriptions circling perpetually closer towards the real's own self-constituting dynamical truth by paradox-driven expansions into ever-refreshed coherence regimes.
It's the chrysalis of a new metaphysical praxis where symbolic artifice perpetually re-embeds itself in the real's self-exhibiting verities through cycles of immanent deconstructive subversion and positive reconstructive reunification. Existence's own self-revealing dynamical voice finding expression through our Regressive/Progressive dialectical rejoinders. An exponential symbolic recollection of Being's generative plenitude previously stultified under monological representationalism's dissimulative statics.
So in essence, Regressive/Progressive names the dual dynamical modalities through which the both/and logic allows rationality to become a partner in the real's self-disclosive adventures again. Not dissimulating reality's complexities through assumptive glosses, but perpetually demolishing and remoulding our descriptive casts in attunement with the real's own immanent self-rationalizations. A new praxis of radical symbolic refreshment in participatory resonance with the irrepressible plenitudinous existential unfurlings of the real's own Regressive/Progressive self-unfolding.
One minor (or maybe major, depending on who you ask) correction. A set has other requirements than being a collection of things, otherwise you get Russell's paradox and friends. This is actually important to mention in category theory, as it is easy to make the mistake of defining a category as "a set of objects and the arrows between them", but that has the unfortunate effect of having no category of sets or category of categories (there is no set of all sets). From what I can find a category is usually defined as a "collection of objects and the arrows between them" or "a class of objects and the arrows between them", one has the disadvantage of being informal (what exactly is a "collection"?), the other is that there is no category of classes as classes inherently don't contain other classes, which is unfortunate, just like with sets. In my opinion, this is the ugliest problem in mathematics.
I go over this issue in the second video in this series. I do agree, it's a problem that can't be brushed under the rug, but for the sake of clarity I feel that getting bogged down in the technicalities of mathematical foundations isn't ideal.
Why do you display the "self arrow" id : A -> A , but not id: B-> B ? With this you only have " f o id = f ", but not " id o f = f ".
Aweson video. Thanks.
isn't abstraction just a from of intellect. Plato argued that intelligence was the ability to recognize the eidos (form).
with this i can argue that math is fundamentally being smart (:<
guys math is just philosophy with axioms + some other things.
Yup. Plato wasn't positing some spooky magical world of spirits as is often portrayed, but rather a world of unities. These unities are what keep concrete changing things stable and give them direction so to speak.
Different species of animals have distinction among them, but they have a higher unity that makes them one species.
@@evan7391 it kind of feels like a "spooky magical world of spirits" sometimes with how he said that the realm of forms exist within heaven. but the base idea just exist as a philosophical entity.
Beautifully done! I'm not really qualified to say that, but I think I did get the main points of what you explained. Thank you!
Thanks :D
why do I have the impression that the characters ressemble those of Randall's XKCD ?
Haskell is built on this stuff.
Lots of mathematicians like to mention the concept of abstraction all the time i have noticed
Up the ladder of abstraction to the West, down the ladder to the East.
i was hoping to see some beautiful, la linea , monster group avatar at the end of this video :)
"A mathematicians weapon." Bruce Lee - The style of no style. To defeat everything, be like nothing.
Minor correction, what goes beyond the positive far end of a number line is "positive infinity", not just "infinity". They do have different meaning, although it would be easy to tell if this difference need to be considered in a specific situation.
And I think the Chinese term for algebra (代数) and abstraction (抽象) is easier to understand, as they literally means "representing number" and "to get rid of concreteness", implying that if we can prove a theory on such things, we instantly prove it on everything that fits the abstracted definition, which not only saves time, but also saves infinite amount of time, crucial to turning something from being "practically true" to simply "true" and live a more peaceful life moving on.
And I think the identity is important because that's the finish line for proving something. Because identity is unique and makes no difference when being operated on, anything originated from or lead towards identity no longer needs further proof, and thus we can prove other things with what's directly related to identity and so on, turning infinite amount of proving needed into finite amount.
And the purpose of set theory here is that it proves what I've described above actually works, and we can assure that we took the right track rather than worrying that the result might be an illusion and might fall apart on edge cases. It's simply good to know that math didn't fall apart and we weren't trapped in an inception, and we earned this knowledge itself rather than being offered.
By the way the identity function has a more practical use. In functional programming everything is a function, and to get the value of a function you expand all of its components until there's no point to further expand, and computer the composite function you created. And how do you know you've done your expansion? By checking that if a function can only be expanded into itself follows (or precedes) an identity function, and you'll know to stop there because further expansion makes no more difference.
And most functional programming language represents this using the "unit" function, which can be used to terminate the recursive checking and yield the final result. In practice though, most of the times there's no need to expand to unit but to any function that we already know how in precise it relates to unit, in order to save some time. For example, an integer is... An integer. We usually don't care the definition of integer (which, vaguely speaking, is the amount of "one" function in the final composition excluding "unit" which represents zero) if all we need is an integer, thus we can stop right when we get any single integer as a result.
There is a notion of "just infinity" without a sign that you can attach to the number line.
This is the projective real number line.
But also, you don't need to say positive infinity to refer to it when it's clear or it doesn't matter.
@@purewaterruler I'd say if negative infinity is mentioned then the opposite of that should be positive infinite to make it more accurate. Even if it's "clear" I guess it would be a good thing to stop some wild thoughts that keeps extending the discussion. There is no more handrails in math after highschool education, after all.
its not a structure but associating function.
the example of adding fractions isn't very useful, because the result of the addition of two fractions is simply "defined" as a fraction. (whether you do it with symbols or letters...) It's not a result (theorem) but it's so by definition. Also, you want to show it for the addition of rationals but to do so, you assume given that the product and sum of integers again yield integers. You should prove that, too, to be consistent. But if you did, you'd find that you *have* to use the definition. and it's the same for fractions. You actually do use other rules one might consider more complicated than what you want to show (namely, all that's related to "common denominator") to prove s.th. more basic (namely, addition of rationals yields rationals). Anyways , to prove anything in math you always have to use the definition.
I don't think the "person" example at the beginning was the best choice, because a person is anything with a rational "nature", where "nature" is used in the old, technical sense. Specifically, angels are persons too. If we found rational aliens that looked like jellyfish, they'd be persons.
The very word "person" was introduced into common speech to talk about the Holy Spirit and the Father, who do not have bodies.
I think for modern parlance, it serves its purpose fine.
But on the technicality, a person in Christian theology is a rational hypostasis, which is actually a particular thing. So the particular people in the video would be persons, rather than the unity between them. That unity would be the human species/nature. Though sometimes individual nature is used for hypostasis.
Edit: what you say is true though
@@evan7391 I'm italian so I didn't know the technical terms. I looked up how it was translated in Dignitas Infinita (since that's the most recent place I read the definition of person in) and it used individual substance... which can be easily misunderstood as the word substance commonly means something else. So i opted to say "anything" thinking it would get my point across servicebly
Category theory? I've watched few recently, good coz that implies more people are getting more interested in math.
Thanks
Can't wait for a monad explanation sometime in 2034
Hopefully it doesn't take quite that long! Unfortunately I really don't get as much time as I'd like to be working on videos, but hopefully I can get videos out quicker in the future
Its funny because using monads as a programmer is quite intuitive (it’s a data structure with flat map), I’ve read about 1000 category theory explanations and none of them make sense.
@@empathogen75 you forgot to say that a monad is just a monoid in the category of endofunctors
They made OOP in math🗿
but its functional...
Category theory then OOP
2:18 ah yes, my favorite huge amount, 0.0%
he said a huge number not a huge proportion (;
Category Theory confuses me 😂😢😮😅😊🌙😅😢
category theory says gender abolition?!
Yes if you have no (abstract away) qualities. Or in day-to-day terms: boring.
And there is no such thing as a boring person! But no harm in trying
I know you're being a bit facetious but actually the category theorist Eugenia Cheng gave a public talk on the links between category theory and gender non-conformity.
@@cosmicwakes6443 no actually i'm a trans woman and that's the firs thing i thought of. I'm gonna have to see this talk
@@jenreiss3107Sorry for misunderstanding your comment, I meant no offence. As for the videos on Eugenia Cheng they're easily found here on TH-cam.
Lo
wow....
Boring, the animations are great, but the content is not insightful at all. Maybe it is me, but I think you are dumbing down this too much...I mean, if your audience is someone who barely knows math, this is ok, but why would you show a category theory video to this audience...
because it's not actually that difficult and they can understand it given the correct explanation (i.e. this one)
@@lgbtthefeministgamer4039 then why do a 20 minute long video with no insight at all. I usually enjoy math videos a lot, but I could barely go through this one at 2x speed
as someone who uses categorical language all the time, i thought this video was very insightful. it's a common sentiment that category theory "abstracts mathematical thinking," but that's not really precise and it also doesn't explain the axioms we use for categories. so the more precise demonstration that it "abstracts composition," and the analogy with more basic types of abstraction, resulted in a kind of "aha" moment for me. also, the fact that it's watchable for more than just die-hard mathematicians is a good thing, isn't it?
@@bayleev7494 while I agree that to say that category theory is the abstraction of composition is insightful, it took more than 8 minutes to explain that...unnecessarily lengthy. I also agree that the aim should be to be watchable for people who are not just die-hard mathematicians. In the end the video is not terrible imo, it is just so boring with no great content...it is just a bunch of extremely simple and not so good examples.
That's awesome, you introduced the term of math function, but what cathegory theory actually is?
They should have called it abstralgebra.