We like to think about nouns rather thing things that relate these these nouns. In math, the functions which preserve the structures in sets are called morphisms, these are what they consider in category theory. For example, linear functions preserve linearity properties. Now apply this idea back onto itself. A lot of modern math now focuses on categorification, rather than trying to focus on the x level of abstraction and generalization, categorification takes that up a notch. Maybe an example most people are familiar with is the homeomorphism between a cup and a torus, we don’t look at the cup and the torus instead, we consider the homeomorphisms which all similar topological manifolds are under.
Lol. Category Theorists refer to their field as "Abstract Nonsense" Even published papers will gloss proof steps by saying "X is true by abstract nonsense"
@@nycholaus this is the meme amongst the math community but modern math is moving toward catigorification. Take any graduate class on algebra and it will be used or at least discussed in there.
She’s an amazing mathematician. She did her PhD at my school, the University of Chicago, under the legendary algebraic topologist J. Peter May, and has written several fantastic books. She’s currently a professor at Johns Hopkins.
Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
@@hyperduality2838 a lot of words to say humans are incapable of thinking past biological selfishness and that the mind is only capable of understanding two states: there is only "me" and "not me", this is where the notion of "duality" comes from. There is not even a "me" and "you", there is only "me" and "not me". You are not "you" to me or any other human, you are "not me" to me and every other human.
@@i-never-look-at-replies-lol The law of non contradiction -- Aristotle. Being is dual to non-being creates becoming -- Plato. Mind (internal mind/soul) is dual to matter (external mind/soul) -- Descartes. You can treat "the not me" as as external mind. Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic. The observed (external mind) is dual to the observer (internal mind) -- David Bohm. Photons or electro-magnetic waves connect the internal mind to the external mind. Electro is dual to magnetic -- Maxwell's equation. Photons or light are dual.
@Eugene Artin Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Always two there are" -- Yoda.
@@Chris-io2cs Glad you laughed, mate. Really it was a tongue in cheek observation of the tendency among the younger generation to affect a guttural or rasping vocalisation. The crazy spelling was, to me, what "trend" sounds like when vandalised thus.Not to be confused with uptalking, which also drives me nuts.
I'll just say that her "Category theory in context" is the best textbook on category theory there is so far, more focused than CTfWM and deeper than Awodey's text. It's superb.
One thing I've never understood: *If an operation is a function with closure, then aren't all functions also operations?* ( knowing that all operations are functions that map A^n -> A )... Since all functions belong to a category, the functions have closure within that category (relative closure) - a function with closure is an operation - therefor all functions are operations?
What worries me the most about Category Theory is that being so powerful at a language of relationships among objects, it is programming language isomorphism.
@lo hu I never saw it that way. My reading of that book left me with the feeling that she wanted to present that particular definition so it might feel familiar to folks coming from that background. On some level, if you're going to present on categories (at least while set theory is still the predominant foundation), you're going to have to elaborate hom-sets as a bridging mechanism. She's had no trouble also presenting formulations that are completely abstract and synthetic (in the Lawvere sense of the term). For example in her work with Mike Schulman and in her intro text on infinity categories.
More, more, more! Please more Emily and cosmoi and higher mathematics (higher categories, topoi, algebra, ....) Every little bit like this helps one, me anyway, understand and makes the literature easier to understand. Thank you Emily!
In a book Towards philosophy of the real mathematics I saw a statement that in XX century math was dealing with n-dim spaces and cathegories of smaller order, while the XXI century in math is about higher order cathegory theories and infinite cathegories.
Not gonna pretend like I understand it, but I’ve learned what I don’t know which expands my horizons in itself. Thanks for the clear explanation laying out the terms!
Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
@@Salmanul_ Categories are a result of reductionist thinking -- which is an entropic process of sub-division or differentiation into new states of information. Sub-division or exclusion comes from the Pauli exclusion principle in physics! Reductionism is dual to holism. Holism is a syntropic process of integrating or converging information into a single whole state. Teleological physics (syntropy) is dual to non-teleological physics (entropy). There is a 4th law of thermodynamics. "Science without religion is lame, religion without science is blind" -- Einstein. Science is dual to religion -- the mind duality of Albert Einstein. Einstein pinched his mind duality from Immanuel Kant. The conservation of duality (energy) will be known as the 5th law of thermodynamics, energy is duality, duality is energy. Potential energy is dual to kinetic energy, gravitational energy is dual. Apples fall to the ground because they are conserving duality. Electro is dual to magnetic, electro-magnetic energy is dual -- Maxwell's equations. Positive charge is dual to negative charge -- electric charge. North poles are dual to south poles -- magnetic fields. New laws of physics.
@@Salmanul_ Homotopic equivalence = duality. Antipodal points identify for the rotation group (spinors in physics). Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Non null homotopic = duality or topological holes cannot be shrunk down to zero.
@@kazedcat My line of reasoning is an axiom can be treated as a mathematical object and therefore it has its own category; and given mathematical objects can be understood by other members of their category, and categories themselves can be placed in categories/cosmoi to understand them -theoretically any axiom could be abstracted down to a finite category that accurately describes it. No? I’m sure there’s something here I’m missing or convoluting, but it’s an entertaining argument to toy with.
@@mattwinward3168 You are missing that truths can be relationship between objects. Even if objects can be uniquely understood it does not mean you have full understanding of the objects relationship to other object. It just means you can identify the object base on how similar it is to other objects. If axioms are abstracted as objects then the relationship between them are the theorems. Incompleteness theorem tells you something about the theorems not the axioms. This theorem on the other hand tells you something about the objects (axioms).
@@mattwinward3168 Categories are defined by their morphisms, not their objects so it's insufficient to lump mathematical objects together to form a category. The morphisms must also follow certain axioms themselves which could lead to paradoxes of self-reference if you tried to put them into a category. I don't think it would work.
Really good communications! The lady had me drawing connections between the different definitions and rings and I haven't taken any classes on category theory! That's amazing
An object lives in a category. A category lives in a 2-category. A 2-category lives in a 3-category and so on. But an ∞ category is allowed to contain a copy of itself, so you don't need to go higher. They actually named them ∞-categories because ∞ = ∞ + 1.
"Emily Riehl hopes to make the powerful perspective more accessible to other mathematicians." from the intro. but I see your point. She has to suffer some anonymity for someone's editorial choice to use the women card.
What sorts of properties do Infinity-Cosmoi have? And what function do they serve in letting you analyze Infinity-Categories? I guess I’m imagining something sort of like the axiomatic basis for constructing arrows, or “containing” the contents of a category...?
Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
@ཧཱུཾ Philosophers should study more mathematics and mathematicians should study more philosophy. Divide and conquer, inclusion is dual to exclusion. Duality is the origin of the Pauli exclusion principle in physics which is used to model quantum stars. Immanuel Kant talks about creating new concepts in "The Critique of Pure Reason":- Comparison (contrast), reflection (same is dual to different), abstraction. Abstraction implies generalizing all of the "sameness" to form a new concept. "Philosophy is dead" -- Stephen Hawking.
Jack Crawford Mathematics ought to make the Natural Numbers its foundational entity and not all these unrelatable abstract concepts as the latter only lead to paradoxes.
@@cosmicwakes6443 That's just not true at all. Only inconsistent systems lead to paradoxes. Plus there's not exactly any way you *could* make the natural numbers "the foundation" - to say anything at all about them we'd need some higher-order objects. I think the foundation should be the calculus of inductive constructions.
@@cosmicwakes6443 actually if you look at the underlying math, the integers are really a nasty set to work with. I’m told they lead to all sorts of horrifying and bizarre implications when you look at large cardinals. Check out the work of Saharon Shelah
@@headlibrarian1996 if I remember correctly it’s actually a condition in model theory/stability theory for something called an unstable model, which as a mathematical object is particularly hard to work with. I will amend all my statements by saying that I myself am not a logician or model theorist, so I only know of such things from conversations with people who know much more than I do. If anyone reads this and has more background than I do (perhaps an interested grad student or something) Saharon Shelah, Maryanthe Malliaris, and Jerome Kiesler all do work somehow related to the topic I believe.
Anyone working at this level of mathematics should be provided with a real chalk board, and probably a larger one than the make-shift one she was able to obtain. Perhaps someone at John Hopkins could look into that.
whoa: (cosmoi, κόσμοι) => Proto-Indo-European: ḱens~ - To Greek => Kosmos (metaphysical world order) - To Latin => Censeo (a census administered by censors) - To English (toungue in cheek) => censor
Antinomies (Immanuel Kant) or paradoxes = dualities. Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
Someone recently notified me that the direction my Functional Programming (Haskell) course took was category theory. I had all this about catas and anas (combined as well) covered in 2 weeks and then had to do an exam on it... of course, I'm taking a resit for this now. Is this supposed to be a separate course/topic?
This is in effect an entirely different field (which is basics of category theory + concepts more relevant to Haskell then elsewhere) Higher category theory is, at this time, mostly useful for mathematics, and with particular dependent type theories, of which even dependent haskell is not a model (but Coq / Agda can be)
Yeah, you probably just click on thumbnails and keywords and then justify it to yourself, like we all do. It is similar enough to thumbnails on videos you have liked and new enough looking that your brain just clicks.
Your mind at a sub-conscious level understands the concept of duality! Attraction (convergence) is dual to repulsion (divergence). Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
Would defining an Infinity Cosmos with this logic not then allow further extensions in both directions, i.e. "our" infinity cosmos is its own infinity category in another "larger" infinity cosmos, and so on? This would mean "our" current defined categories we operate in would be both an insignificant and indisposable piece in an infinite blossom of infinity categories defining themselves both inwards and outwards to infinity.... A rigorous definition of an Infinity Cosmos would surely unlock a greater understanding, or even a definition of the concept of an infinity. anyone else feel a bit dizzy?
She is teaching you duality but not actually using the word "duality", here are some examples:- Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
Not being a mathematician myself; therefore I don't know what the hell she is saying...or, at least, not in any sort of depth...; but, I admit she sure sounds super-brainy enough(which I do greatly respect and admire)!
Can one define categories associated with each ordinal or cardinal number, even as high up the hierarchy into large cardinal numbers like Reinhardt cardinals?
Categories are defined by their morphisms, which must be associative and include the identity morphism for every object in the category. The category of sets with functions as morphisms is a category and would contain objects of every cardinality. Ordinal numbers can be well-ordered and so you can define morphism based on that ordering by providing a morphism from the smaller object to every object larger than it with respect to that relation. Reinhardt cardinals are incompatible with the axiom of choice which is a non-starter. Every diagram chase is going to use choice and Zorn's lemma is too powerful to do without.
Category theory is the study of duality. Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
I just got out of a Semantics class for Linguistics and whoa… the applications of understanding infinity cosmoi could be incredibly useful for understanding universal constants cross-linguistically in seemingly unrelated languages.
Number theory Set theory Group theory Ring theory Graph theory Lattice theory Category theory Model theory Proof theory Topology .. Logic . Universal logic😍😍😍😍 SO GREAT 😇😇😇
Yeah, I am applying categorical logic / topos theory to AGI (artificial general intelligence) in my research. Category theory is really a key component in that direction of AI research. I am using a form of universal logic combining topos, modal, (homotopy) type theory, and fuzzy-probabilistic logic.
Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
@@osamakhalouf35 Gravitation is equivalent or dual to acceleration --- Einstein's happiest thought, the principle of equivalence (duality). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Space is dual to time -- Einstein. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is therefore dual, gravitational energy is dual. Apples fall to the ground because they are conserving duality. Potential energy is dual to kinetic energy. Action is dual to reaction -- Sir Isaac Newton. Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. Notice the pattern in physics! This means that there must be a dual process to that of increasing entropy namely syntropy! -- the 4th law of thermodynamics. "Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy. Spinoza is accurately describing syntropy which means that there is a 4th law -- duality. Falling apples means that duality is real -- gravity.
What a beautiful person! - I understand there are only two categories of people. The people that believe there are only two categories, and the people who don't. Hmmm??
Being is dual to non-being creates becoming -- Plato. Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic. Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
Injective is dual to surjective creates bijective (isomorphism). Thesis is dual to anti-thesis creates synthesis -- the time independent Hegelian dialectic. Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
@@telecorpse1957 No, it's quite literally called "abstract nonsense". It's an endearing name for the techniques used in the field. It's sort of like when you work with the tensor product and cite a universal property rather than construct it explicitly- that's you doing abstract nonsense. It's just a friendly name/joke.
@@telecorpse1957 It's a joke among mathematicians to call category theory abstract nonsense. There is a whole historical context that involves foundational questions in mathematics and logic, Gödel's incompleteness theorems and the Curry-Howard correspondence but the joke is that all of mathematics is abstract nonsense so calling something that just means it's actually math.
"I'm using notation that will be familiar to mathematicians". I got lost 20 seconds in. I am not a mathematician but I have a degree in mathematics. What is the notation in the expression "Z/2"? Normally when X is a ring. X/Y is denotes a quotient ring formed with an ideal Y in ring X. But how is 2 an ideal in the ring Z? I don't understand the notation, could someone please clarify?
Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
@@Bratjuuc No physics. The principle of equivalence (duality):- Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. The big bang is a Janus hole/point -- two faces = duality! Topological holes cannot be shrunk down to zero. Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. Mathematics, physics & philosophy are all built upon a pattern of duality.
So cool. But I feel a recursion paradox looming. Applying category theory to itself just sounds a little bit "immoral," so if they can escape a Russell or Gödelian type paradox I'd be very excited, for it might suggest categories (or the general framework of "arrows") really are deep, deeper than most n-Cat theorists even assume. Foundational bedrock deep. I'd want to know what's to stop one from giving different inequivalent definitions of infinity cosmoi. Infinity multi-cosmoi? How about aleph-n hyper-cosmoi, and so on, and so on...
Metacategories = dualities! Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
@@StevenNess Both Aristotle & Kant developed a theory of categories. Fear is dual to to anger, anger is dual to hate, hate is dual to suffering -- the Yoda dualities. Ignorance is dual to strength, freedom is dual to slavery, war is dual to peace -- the Orwellian dualities. "Sith lords come in pairs (duals)" -- Obi Wan Kenobi. "Entropy is a measure of randomness" -- Sir Roger Penrose. Syntropy is a measure of order and predictability. The conservation of duality (energy) will be known as the 5th law of thermodynamics, energy is duality, duality is energy. Apples fall to the ground because they are conserving duality. Potential energy is dual to kinetic energy. Duality is "the force".
Lawvere's book: Conceptual Mathematics even high-school kids can understand it and written by a master (with the other co-author, I forgot his name...)
Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Mind (intangible) is dual to matter (tangible) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. "Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality. Active matter (life) is dual to passive matter (atoms, forces) -- matter duality. Mind duality is dual to matter duality. Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Union is dual intersection. Integration is dual to differentiation. The dot product is dual to the cross product. Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant. Absolute truth is dual to relative truth -- Hume's fork. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics. "Always two there are" -- Yoda. Duality creates reality.
where was this channel when I was 11 I would have liked math if I knew this stuff lol. I wouldn't have understood it but I would have been baffled enough to be in awe
Was Stephen Hawking know as the 'disabled' Physicist who rewrote our understanding about Black Holes? It would be massively condescending to refer to him as a great 'disabled' researcher. Why do people want their gender to be the first thing that is know about them? Guess you gotta break them stereotypes... but at what cost.
@@Bratjuuc Yeh, you think in context of a channel with videos like 'Carolina Araujo on Supporting Women in Mathematics'... it's not putting that word on a pedestal?
@@StuartDesign I just stumbled upon this video, never checked the channel. I still find the video title neutral, tbh. Besides, most video titles on this channel are neutral too. I get that feminism and such liberal shit is cancer, but you've got a false alarm here.
Pronouns have worked this way for hundreds of years. Whatever nonsense you've conjured in your mind to explain the this is a phantom of your own imagination.
@@abebuckingham8198 Look for a video on this channel that stats with 'The Man who...' Gender is clearly being emphasised because... 'she' is a 'woman'' Also 'Woman' is not a pronoun 'she/her' is.
You are a rather interesting person... and the deeper you look the more interesting you are (I hope you don't mind my off-topic comments). I have been working (avocationally) in the areas of foundations for a while now and am quite interested in your work. I have reached out to a few experts in the areas of category and set theory, as well as topology and manifold theories (to include Saunders Mac Lane and Michael Freedman, et al.)... with regard to a number of questions that I have had having to do with abstract extend- ability/exhaustiveness, incomplete- ness, independence... and so on. In addition I am quite interested in the areas of hierarchy theories, ontolog- ic commitments/entailments, infi- nity-categories, class and conglom- erate hierarchies... and Infinite-re- gress, in general. Recently, I have been interested in how the vast array of (category, class, conglom- erate?) and set theoretic systems (at all orders) will effect the infinitistic, or large cardinal hierarchy...and also, whether or not they will all be affect- ed somewhat similarly, by independ- ence phenomena and the limitative results of Kunen. If you find any of this of interest I would be honored to hear from you. I feel that the work you (and your colleagues) do is among the most important (and difficult) that we as a species, can try to understand. Thank you for your time and understanding! M.L. Viar
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I have no idea what is going-on, but I’m excited that you’re excited.
Is your comment part of the three dimensional comment cosmos? (or just a member of an infinite comment category)
@@tomrock6134 Yes? But also no?
I like toroids.
We like to think about nouns rather thing things that relate these these nouns. In math, the functions which preserve the structures in sets are called morphisms, these are what they consider in category theory. For example, linear functions preserve linearity properties. Now apply this idea back onto itself. A lot of modern math now focuses on categorification, rather than trying to focus on the x level of abstraction and generalization, categorification takes that up a notch.
Maybe an example most people are familiar with is the homeomorphism between a cup and a torus, we don’t look at the cup and the torus instead, we consider the homeomorphisms which all similar topological manifolds are under.
Lol. Category Theorists refer to their field as "Abstract Nonsense"
Even published papers will gloss proof steps by saying "X is true by abstract nonsense"
@@nycholaus this is the meme amongst the math community but modern math is moving toward catigorification. Take any graduate class on algebra and it will be used or at least discussed in there.
Carry on!
She’s an amazing mathematician. She did her PhD at my school, the University of Chicago, under the legendary algebraic topologist J. Peter May, and has written several fantastic books. She’s currently a professor at Johns Hopkins.
Wow!
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
@@hyperduality2838 a lot of words to say humans are incapable of thinking past biological selfishness and that the mind is only capable of understanding two states: there is only "me" and "not me", this is where the notion of "duality" comes from. There is not even a "me" and "you", there is only "me" and "not me". You are not "you" to me or any other human, you are "not me" to me and every other human.
@@i-never-look-at-replies-lol The law of non contradiction -- Aristotle.
Being is dual to non-being creates becoming -- Plato.
Mind (internal mind/soul) is dual to matter (external mind/soul) -- Descartes.
You can treat "the not me" as as external mind.
Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic.
The observed (external mind) is dual to the observer (internal mind) -- David Bohm.
Photons or electro-magnetic waves connect the internal mind to the external mind.
Electro is dual to magnetic -- Maxwell's equation.
Photons or light are dual.
@Eugene Artin Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
"Always two there are" -- Yoda.
She knows what she's talking about.
Her language sounds so clear. She knows how to speak.
Yes, but croakingly, as is the modern trrrgggrend.
@@stephenl7048 what the fuck does that mean lol
@@Chris-io2cs Glad you laughed, mate. Really it was a tongue in cheek observation of the tendency among the younger generation to affect a guttural or rasping vocalisation. The crazy spelling was, to me, what "trend" sounds like when vandalised thus.Not to be confused with uptalking, which also drives me nuts.
She has zero future as a mumble rapper fo sho
@@stephenl7048 Oh.
What?
I'll just say that her "Category theory in context" is the best textbook on category theory there is so far, more focused than CTfWM and deeper than Awodey's text. It's superb.
Thank you, I try to find and read some of this books.
sweet -- I haven't heard slate like that is over 50 years..
I like the magazine's logo. What is it? Why does it resonate with me?
Maybe I like that you left it open, evolving? We need to open all the logos.
One thing I've never understood: *If an operation is a function with closure, then aren't all functions also operations?* ( knowing that all operations are functions that map A^n -> A )... Since all functions belong to a category, the functions have closure within that category (relative closure) - a function with closure is an operation - therefor all functions are operations?
Small trashcan: *is full*
Mathematicians: what we're trying to do is build a bigger trashcan around it
if your trash can is full, just toss it into the trash can trash can.
We are building a universe in which the image of this trashcan is a bigger trashcan by construction. That is a much more elegant solution!
What worries me the most about Category Theory is that being so powerful at a language of relationships among objects, it is programming language isomorphism.
Love you Emily
she wrote the clearest Intro to Category Theory textbook im aware of
badass mathematician
@lo hu any chance you could elaborate? im not objecting but what are the alternatives to this plague as you call it?
@lo hu I never saw it that way. My reading of that book left me with the feeling that she wanted to present that particular definition so it might feel familiar to folks coming from that background. On some level, if you're going to present on categories (at least while set theory is still the predominant foundation), you're going to have to elaborate hom-sets as a bridging mechanism. She's had no trouble also presenting formulations that are completely abstract and synthetic (in the Lawvere sense of the term). For example in her work with Mike Schulman and in her intro text on infinity categories.
she needs to be on numberphile!
she is
@@pmcate2 what video?
@@mayabartolabac just search numberphile emily riehl
@@pmcate2 ah the stable marriage girl
kinda disappointed that she didn't talk about category theory
Does the cosmoi run into well-foundation problem?
More, more, more! Please more Emily and cosmoi and higher mathematics (higher categories, topoi, algebra, ....) Every little bit like this helps one, me anyway, understand and makes the literature easier to understand. Thank you Emily!
The book of Marquis introduces category theory without too much tears.
Don't forget your friendly neighborhood cosmoticians, too! If you have any questions about these topics hit me up.
@@chemistryguy9679 depends on what's your current mathematical background
@Chemistry Guy I just found this very nice and informative video of Riehl: ruclips.net/video/A6hXn6QCu0k/видео.html
@@chemistryguy9679 u need to be specific...what u want..."heigher level pure maths" it's very relative nd broad term.
"...this gets a bit hard to draw on paper." Beautiful understatement.
I didnt know that paper chalkboard existed and its the most mathematical invention I’ve seen in my life. I want one so bad now.
Isn't it just black paper lol
In a book Towards philosophy of the real mathematics
I saw a statement that in XX century math was dealing with n-dim spaces and cathegories of smaller order, while the XXI century in math is about higher order cathegory theories and infinite cathegories.
No longer will the asterisk be enough for footnotes.
Spatial hyperlink nodes for all info!😀
One of the greatest mind in category theory.just respect
I really hope she is mentally healthy, I studied some category theory a while ago and I became an addict.
She seems a wonderful person,
What
@@eustab.anas-mann9510 Category theory and math obsession does that to people
I was once an addict too. As I grew older, I got dumber, which cured my addiction.
Is it only me who's seeing a new kind of self reference paradox ahead?
We essentially deal with Russell's paradox by saying "just don't do that" and this is no different.
i couldnt continue listening because of the hissing noise in the background
Not gonna pretend like I understand it, but I’ve learned what I don’t know which expands my horizons in itself. Thanks for the clear explanation laying out the terms!
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
@@Salmanul_ Categories are a result of reductionist thinking -- which is an entropic process of sub-division or differentiation into new states of information.
Sub-division or exclusion comes from the Pauli exclusion principle in physics!
Reductionism is dual to holism.
Holism is a syntropic process of integrating or converging information into a single whole state.
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
There is a 4th law of thermodynamics.
"Science without religion is lame, religion without science is blind" -- Einstein.
Science is dual to religion -- the mind duality of Albert Einstein.
Einstein pinched his mind duality from Immanuel Kant.
The conservation of duality (energy) will be known as the 5th law of thermodynamics, energy is duality, duality is energy.
Potential energy is dual to kinetic energy, gravitational energy is dual.
Apples fall to the ground because they are conserving duality.
Electro is dual to magnetic, electro-magnetic energy is dual -- Maxwell's equations.
Positive charge is dual to negative charge -- electric charge.
North poles are dual to south poles -- magnetic fields.
New laws of physics.
@@Salmanul_ Homotopic equivalence = duality.
Antipodal points identify for the rotation group (spinors in physics).
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Non null homotopic = duality or topological holes cannot be shrunk down to zero.
come over
I understand. Go on.
What a fantstic person!
I don't even know what an integer is but she is such an excellent speaker that it sounds exciting.
Nonsense, a category including all possibilities (infinity) is no category.
A bit too meta for me
0:40 Doesn’t that theorem fly in the face of Gödel's incompleteness theorems?
Why? Incompleteness theorem just states that there are truths that cannot be proven given a finite set of axioms.
@@kazedcat My line of reasoning is an axiom can be treated as a mathematical object and therefore it has its own category; and given mathematical objects can be understood by other members of their category, and categories themselves can be placed in categories/cosmoi to understand them -theoretically any axiom could be abstracted down to a finite category that accurately describes it. No?
I’m sure there’s something here I’m missing or convoluting, but it’s an entertaining argument to toy with.
@@mattwinward3168 You are missing that truths can be relationship between objects. Even if objects can be uniquely understood it does not mean you have full understanding of the objects relationship to other object. It just means you can identify the object base on how similar it is to other objects. If axioms are abstracted as objects then the relationship between them are the theorems. Incompleteness theorem tells you something about the theorems not the axioms. This theorem on the other hand tells you something about the objects (axioms).
@@mattwinward3168 Categories are defined by their morphisms, not their objects so it's insufficient to lump mathematical objects together to form a category. The morphisms must also follow certain axioms themselves which could lead to paradoxes of self-reference if you tried to put them into a category. I don't think it would work.
Really good communications! The lady had me drawing connections between the different definitions and rings and I haven't taken any classes on category theory! That's amazing
what about the category of infinity cosmoi? And the category of categories of infinity cosmoi?
"whataboutism"
An object lives in a category. A category lives in a 2-category. A 2-category lives in a 3-category and so on.
But an ∞ category is allowed to contain a copy of itself, so you don't need to go higher. They actually named them ∞-categories because ∞ = ∞ + 1.
cosmoi
Mr. Wolfram might also have some ideas on that ;-)
Does this woman has a name?
I'm sure she does. Why address her as This Woman?
"Emily Riehl hopes to make the powerful perspective more accessible to other mathematicians." from the intro. but I see your point. She has to suffer some anonymity for someone's editorial choice to use the women card.
@@elmersbalm5219 It's not the women card it's just a standard curiosity trap. Clickbait doesn't need gender.
Isn't the fundamental theorem of category theory the Yoneda lemma?
That's basically what was said in the video at 0:45, just in an accessible way.
9 or 5 in pi
How should one start learning higher infinity categories? Any idea? I am not a topologist, but rather representation theorist.
What sorts of properties do Infinity-Cosmoi have? And what function do they serve in letting you analyze Infinity-Categories? I guess I’m imagining something sort of like the axiomatic basis for constructing arrows, or “containing” the contents of a category...?
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
@ཧཱུཾ Philosophers should study more mathematics and mathematicians should study more philosophy. Divide and conquer, inclusion is dual to exclusion.
Duality is the origin of the Pauli exclusion principle in physics which is used to model quantum stars.
Immanuel Kant talks about creating new concepts in "The Critique of Pure Reason":-
Comparison (contrast), reflection (same is dual to different), abstraction.
Abstraction implies generalizing all of the "sameness" to form a new concept.
"Philosophy is dead" -- Stephen Hawking.
Absolutely need more category theory content!
Jack Crawford
Mathematics ought to make the Natural Numbers its foundational entity and not all these unrelatable abstract concepts as the latter only lead to paradoxes.
@@cosmicwakes6443 That's just not true at all. Only inconsistent systems lead to paradoxes. Plus there's not exactly any way you *could* make the natural numbers "the foundation" - to say anything at all about them we'd need some higher-order objects. I think the foundation should be the calculus of inductive constructions.
@@cosmicwakes6443 actually if you look at the underlying math, the integers are really a nasty set to work with. I’m told they lead to all sorts of horrifying and bizarre implications when you look at large cardinals. Check out the work of Saharon Shelah
@@dmr11235 Bizarre I can see, but horrifying? How so?
@@headlibrarian1996 if I remember correctly it’s actually a condition in model theory/stability theory for something called an unstable model, which as a mathematical object is particularly hard to work with. I will amend all my statements by saying that I myself am not a logician or model theorist, so I only know of such things from conversations with people who know much more than I do. If anyone reads this and has more background than I do (perhaps an interested grad student or something) Saharon Shelah, Maryanthe Malliaris, and Jerome Kiesler all do work somehow related to the topic I believe.
Anyone working at this level of mathematics should be provided with a real chalk board, and probably a larger one than the make-shift one she was able to obtain. Perhaps someone at John Hopkins could look into that.
Yes. I understand. I indeed understand. This makes sense. I'm happy. Categories. Yes.
whoa: (cosmoi, κόσμοι) =>
Proto-Indo-European: ḱens~
- To Greek => Kosmos (metaphysical world order)
- To Latin => Censeo (a census administered by censors)
- To English (toungue in cheek) => censor
I get the feeling she needs to stand in front of a real MIT slides up chalk board and expand on this for about an hour.
She’s incredibly intelligent.
Bravo!
Genius
Interesting! Are categories::cosmoi like sets::classes with a corresponding Russell-like paradox that categories themselves might otherwise create?
yes
Antinomies (Immanuel Kant) or paradoxes = dualities.
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
@@hyperduality2838 u spammmed every comment??
Someone recently notified me that the direction my Functional Programming (Haskell) course took was category theory. I had all this about catas and anas (combined as well) covered in 2 weeks and then had to do an exam on it... of course, I'm taking a resit for this now. Is this supposed to be a separate course/topic?
This is in effect an entirely different field
(which is basics of category theory + concepts more relevant to Haskell then elsewhere)
Higher category theory is, at this time, mostly useful for mathematics, and with particular dependent type theories, of which even dependent haskell is not a model (but Coq / Agda can be)
anyone know, why in the world I am watching this video at 11:37 PM ET? I ain't know sh**t about mathematics..
Yeah, you probably just click on thumbnails and keywords and then justify it to yourself, like we all do. It is similar enough to thumbnails on videos you have liked and new enough looking that your brain just clicks.
Your mind at a sub-conscious level understands the concept of duality!
Attraction (convergence) is dual to repulsion (divergence).
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
The category of oo categories and so on? But whats the category universe u start with?
Why did youtube recommend this to me? She has neat handwriting for a mathematician is about the most intelligent comment on the subject I can make!
Would defining an Infinity Cosmos with this logic not then allow further extensions in both directions, i.e. "our" infinity cosmos is its own infinity category in another "larger" infinity cosmos, and so on? This would mean "our" current defined categories we operate in would be both an insignificant and indisposable piece in an infinite blossom of infinity categories defining themselves both inwards and outwards to infinity.... A rigorous definition of an Infinity Cosmos would surely unlock a greater understanding, or even a definition of the concept of an infinity.
anyone else feel a bit dizzy?
I have no clue what she just said. I wish I did, but alas.
She is teaching you duality but not actually using the word "duality", here are some examples:-
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
Wonderful explanations! I find myself at turning higher category theory.
sounds like she's trying to quantify the unquantifiable
I dig the pants-chalk matching combo. That's my contribution to this video
But can I use it to balance my checking account?
A fun talk. I'd love to follow a course taught by her for beginning grad students, with this level of informality . No need to wear a suit!
Not being a mathematician myself; therefore I don't know what the hell she is saying...or, at least, not in any sort of depth...; but, I admit she sure sounds super-brainy enough(which I do greatly respect and admire)!
Can one define categories associated with each ordinal or cardinal number, even as high up the hierarchy into large cardinal numbers like Reinhardt cardinals?
Categories are defined by their morphisms, which must be associative and include the identity morphism for every object in the category. The category of sets with functions as morphisms is a category and would contain objects of every cardinality. Ordinal numbers can be well-ordered and so you can define morphism based on that ordering by providing a morphism from the smaller object to every object larger than it with respect to that relation. Reinhardt cardinals are incompatible with the axiom of choice which is a non-starter. Every diagram chase is going to use choice and Zorn's lemma is too powerful to do without.
I don't get the joke.
Okay, I will pretend that I understood all..😂😂
Category theory is the study of duality.
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
Then how do you prove the infinity category is valid?!
I just got out of a Semantics class for Linguistics and whoa… the applications of understanding infinity cosmoi could be incredibly useful for understanding universal constants cross-linguistically in seemingly unrelated languages.
Is the integer to ring correspondence at 1:04 a consequence of Yoneda's lemma?
Admit it...
U understood nothing
Number theory
Set theory
Group theory
Ring theory
Graph theory
Lattice theory
Category theory
Model theory
Proof theory
Topology ..
Logic .
Universal logic😍😍😍😍
SO GREAT 😇😇😇
Yeah, I am applying categorical logic / topos theory to AGI (artificial general intelligence) in my research.
Category theory is really a key component in that direction of AI research.
I am using a form of universal logic combining topos, modal, (homotopy) type theory, and fuzzy-probabilistic logic.
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
@@hyperduality2838 thank you ..so great ..more more please ......
@@osamakhalouf35 Gravitation is equivalent or dual to acceleration --- Einstein's happiest thought, the principle of equivalence (duality).
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Space is dual to time -- Einstein.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is therefore dual, gravitational energy is dual.
Apples fall to the ground because they are conserving duality.
Potential energy is dual to kinetic energy.
Action is dual to reaction -- Sir Isaac Newton.
Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle.
Notice the pattern in physics! This means that there must be a dual process to that of increasing entropy namely syntropy! -- the 4th law of thermodynamics.
"Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy.
Spinoza is accurately describing syntropy which means that there is a 4th law -- duality.
Falling apples means that duality is real -- gravity.
@@hyperduality2838 you are the legend of world .....all love from me to you ..best wishes in 2021...
What a beautiful person! - I understand there are only two categories of people. The people that believe there are only two categories, and the people who don't. Hmmm??
Being is dual to non-being creates becoming -- Plato.
Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic.
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
Emily goes in so you don't have to. But how about you choose to anyway?
Most of the time I feel like I can follow and gain some sort of understanding... but not this. Lol I’m so lost.
I guess one needs more fundamental understanding. I'm totally clueless, too.
@@jollyjokress3852
You need more mathematical maturity. If you want to study rings first, study abstract algebra.
So it all boils down to find/prove isomorphisms between infinity categories?
Injective is dual to surjective creates bijective (isomorphism).
Thesis is dual to anti-thesis creates synthesis -- the time independent Hegelian dialectic.
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
@@hyperduality2838 what the fq did u wrote!!!?? 🤯🤯🤯🤯🤯
In another lecture, she talks about "equivalence", a weaker condition than isomorphism. But I'm not sure she applies that to infinity-categories.
We live in an elegant recursive function that mirrors itself at its integers.
Mirrors itself as a complex numbers..
We live in a whole that transcends analysis
@@Actualhumanlive We live in the product of the working of balance.
THIS IS AWESOME AND SHE IS AWESOME
Abstract nonsense.
Why do you think it's nonsense?
-Norman Steenrod
@@telecorpse1957 No, it's quite literally called "abstract nonsense". It's an endearing name for the techniques used in the field. It's sort of like when you work with the tensor product and cite a universal property rather than construct it explicitly- that's you doing abstract nonsense.
It's just a friendly name/joke.
@@telecorpse1957 It's a joke among mathematicians to call category theory abstract nonsense. There is a whole historical context that involves foundational questions in mathematics and logic, Gödel's incompleteness theorems and the Curry-Howard correspondence but the joke is that all of mathematics is abstract nonsense so calling something that just means it's actually math.
"I'm using notation that will be familiar to mathematicians". I got lost 20 seconds in. I am not a mathematician but I have a degree in mathematics. What is the notation in the expression "Z/2"? Normally when X is a ring. X/Y is denotes a quotient ring formed with an ideal Y in ring X. But how is 2 an ideal in the ring Z? I don't understand the notation, could someone please clarify?
late comment but it's just shorthand for Z/2Z
A master piece
I am astounded by her eloquence and manner. Truly a great communicator.
Dear Emily,
If you read this, I want you to know that I love you!
Thankyou.
No content, only classification into three most broadest categories
its a 4 minute video bro
Thank you for this!
She's so cool!!
Category theory is a social construct
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
ohh here u have it huh
I think you've overdosed on dialectics. Heavily.
@@Bratjuuc No physics.
The principle of equivalence (duality):- Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought.
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
The big bang is a Janus hole/point -- two faces = duality!
Topological holes cannot be shrunk down to zero.
Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle.
Mathematics, physics & philosophy are all built upon a pattern of duality.
@@hyperduality2838 ignoring me?
edit: ohh, u have replied. didnt saw that.
♡
What? I would be great if you explain those concepts slower and one by one :)
So cool. But I feel a recursion paradox looming. Applying category theory to itself just sounds a little bit "immoral," so if they can escape a Russell or Gödelian type paradox I'd be very excited, for it might suggest categories (or the general framework of "arrows") really are deep, deeper than most n-Cat theorists even assume. Foundational bedrock deep. I'd want to know what's to stop one from giving different inequivalent definitions of infinity cosmoi. Infinity multi-cosmoi? How about aleph-n hyper-cosmoi, and so on, and so on...
Metacategory theory!
Metacategories = dualities!
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
@@hyperduality2838 yoda quote is 🔥
@@StevenNess Both Aristotle & Kant developed a theory of categories.
Fear is dual to to anger, anger is dual to hate, hate is dual to suffering -- the Yoda dualities.
Ignorance is dual to strength, freedom is dual to slavery, war is dual to peace -- the Orwellian dualities.
"Sith lords come in pairs (duals)" -- Obi Wan Kenobi.
"Entropy is a measure of randomness" -- Sir Roger Penrose.
Syntropy is a measure of order and predictability.
The conservation of duality (energy) will be known as the 5th law of thermodynamics, energy is duality, duality is energy.
Apples fall to the ground because they are conserving duality.
Potential energy is dual to kinetic energy.
Duality is "the force".
@@hyperduality2838 what arrrre u?
@@hyperduality2838 is there a compilation of all these things you are saying so that i can read it later rather than in youtube comments!!??
How to learn category theory easily ? Any trick
Lawvere's book: Conceptual Mathematics
even high-school kids can understand it
and written by a master (with the other co-author, I forgot his name...)
There's no trick. You study it carefully and meticulously over long periods of time.
@@geometerfpv2804 , it depends. Always there is some strategy and trick to understand a topic. I found it helpful in my career. 👍
@@kingyinyan254 , thanks 👍
Points (objects) are dual to lines (representations) -- the principle of duality in geometry.
Mind (intangible) is dual to matter (tangible) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
"Concepts without percepts are empty, percepts without concepts are blind" -- Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions) or waves are dual to particles -- quantum duality or matter duality.
Active matter (life) is dual to passive matter (atoms, forces) -- matter duality.
Mind duality is dual to matter duality.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual intersection.
Integration is dual to differentiation.
The dot product is dual to the cross product.
Noumenal (rational, analytic) is dual to phenomenal (empirical, synthetic) -- Immanuel Kant.
Absolute truth is dual to relative truth -- Hume's fork.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics.
"Always two there are" -- Yoda.
Duality creates reality.
where was this channel when I was 11 I would have liked math if I knew this stuff lol. I wouldn't have understood it but I would have been baffled enough to be in awe
That was great
🤔.
Was Stephen Hawking know as the 'disabled' Physicist who rewrote our understanding about Black Holes? It would be massively condescending to refer to him as a great 'disabled' researcher. Why do people want their gender to be the first thing that is know about them? Guess you gotta break them stereotypes... but at what cost.
The "woman" word isn't on some pedestal. It's the same as saying "The Man Who Did X".
@@Bratjuuc Yeh, you think in context of a channel with videos like 'Carolina Araujo on Supporting Women in Mathematics'... it's not putting that word on a pedestal?
@@StuartDesign I just stumbled upon this video, never checked the channel. I still find the video title neutral, tbh. Besides, most video titles on this channel are neutral too.
I get that feminism and such liberal shit is cancer, but you've got a false alarm here.
Pronouns have worked this way for hundreds of years. Whatever nonsense you've conjured in your mind to explain the this is a phantom of your own imagination.
@@abebuckingham8198 Look for a video on this channel that stats with 'The Man who...' Gender is clearly being emphasised because... 'she' is a 'woman''
Also 'Woman' is not a pronoun 'she/her' is.
Yes.
You are a rather interesting person... and the deeper you look the more interesting you are (I hope you don't mind my off-topic comments).
I have been working (avocationally) in the areas of foundations for a while now and am quite interested in your work. I have reached out to a few experts in the areas of category and set theory, as well as topology and manifold theories (to include Saunders Mac Lane and Michael Freedman, et al.)... with regard to a number of questions that I have had having to do with abstract extend- ability/exhaustiveness, incomplete- ness, independence... and so on. In addition I am quite interested in the areas of hierarchy theories, ontolog- ic commitments/entailments, infi- nity-categories, class and conglom- erate hierarchies... and Infinite-re- gress, in general. Recently, I have been interested in how the vast array of (category, class, conglom- erate?) and set theoretic systems (at all orders) will effect the infinitistic, or large cardinal hierarchy...and also, whether or not they will all be affect- ed somewhat similarly, by independ- ence phenomena and the limitative results of Kunen. If you find any of this of interest I would be honored to hear from you. I feel that the work you (and your colleagues) do is among the most important (and difficult) that we as a species, can try to understand. Thank you for your time and understanding!
M.L. Viar
Meaningless video.
Is this about OO programming?
No