The Mathematician's Weapon | Category Theory and Why We Care 1.0 

Many thanks!

Very true! Density of Q in R is pretty brain bending, but a really cool result!

Infinitely small and large!

Thank you!

Glad you found it useful! :D

Thank you :D Not an intentional reference but a happy accident haha

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!!

Thank you!

I use Manim for the animated maths, pivot for the stick figures and Da Vinci resolve to splice them all together :)

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.

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. 👍✨️

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.

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.

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!

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.

As is the case with most algebraic structures, attempting to read into the name too much is rarely beneficial. A group or a ring both have names used in common language but have very tenuous links to their commonly used meaning

@Eyesomorphic yeah.. forgot math tends to do that.... thank you for answer

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

A joint honours in mathematics and computer science

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.

@@empathogen75 you forgot to say that a monad is just a monoid in the category of endofunctors

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 RU-vid.

Lo

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.

Thanks :D