chobytes

The concrete instances are demonstrably useful, since theyre... actually used! However in Haskell theyre basically one kind of very concrete object. Invoking the abstract machinery of CT to explain list concatenation just has the hierarchy backward. Its like telling someone that inner-product induced norms on hilbert spaces are extremely useful when youre talking about taking the absolute value.

Haha, I feel like thats how I started to get CT concepts too. Topology and geometry really feel like the "natural" context for those ideas.

For programming... I feel like they're basically trying to do logic in a roundabout way. I suspect that (finite) model theory might be more useful for such applications if they really want theory.

Is there any benefit to using these? I imagine that maybe it could save power by keeping light out and uaing better insulation... but I have no idea.

I got a bit obsessed with this stuff (CT, HoTT, haskell, etc) and ended up studying math in college because of it. My advice is not to bother with CT. Its out of context, very abstract math, and you wont find any compelling examples or applications at any accessible level.

I dont want to discourage you from learning math, but I think this isnt a good place spend too much thought at first.

To be honest I feel like the functor concept is too abstract and subtle to the point where it's probably not very useful in the context of programming.

Worse... their usage in programming languages is very unlike how homomorphisms are typically used in their usual context, so appealing to the math seems like its just confusing learners without payoff.

To give an analogy... it feels like trying to explain the chain rule by explaining pullbacks between tangent spaces. Not technically wrong, but now they're confused and asking questions whose answers wont help them.

Im skeptical of this kind of educational material. It seems like they want to provide the feeling and the aesthetic of learning, but aren't willing to dig in to the hard work of it.

Personally I don't think that researchers have any more business doing this kind of surveillance than Google and company do.

The idea that this will benefit society seems naive to me. I feel like it will only serve to legitimize the practice by putting ostensibly trustworthy faces on the packaging.

This is an awfully generous way to present what looks like yet another in-browser advertising platform.

When I was younger I got a bit notable online by doing just this. You can pick up thousands of followers on twitter in a couple weeks by just picking hot topics and big targets, and publicly dunking on them.

Ive grown out of the behavior but its definitely left me very cynical about politics.

This helped me a lot when I was taking calculus. We were assigned Stewart, but I basically learned the from this.

Analysis from Rudin, and later some manifold theory from Lee, have converted me to more mainstream views on the subject... However I think this is probably how calculus should be introduced at first, particularly to say highschoolers.

I think it really depends what part of math and what part of computer science you're talking about. Certainly foundations of computation use finite objects from countable sets... But there is more to CS than turing machines and encodings of numbers.

For example: Computer vision, machine learning, data science, cryptography, etc are all rife with infinite objects! Proof assistant software and SMT solvers can also prove things about infinite mathematical structures from number theory, ZFC, topology, etc.

Pure mathematics also cares about finite algorithms. Every proof is a finite sequence of deductions on finite objects from a countable set... eg a computer program! Other examples include: computing bounds, integrals, roots of polynomials, divisors, bases, fundamental groups, etc. Pure math is full of computation!

tl;dr the line between math and cs is extremely fuzzy.

Agreed. With normal math notation I can just see how something works. In programming language math notation I cant even always parse the hierarchy.

In my experience you learn math by doing. Its takes personal time and effort and it cannot be spoon fed.

A teacher can provide a little guidance, but mostly its up to the student.

Works well in Chicago with the CTA. Havent owned a car in years, and just rent when I need one.

Im not opposed to the use of proof assistant software for difficult theorems. And if one foundation or another is more applicable to that software, then thats fine.

Its more the narrative of the grand unification of mathematics thats spun around 'foundational theories' that irks me. In reality its just various emulation schemes.

Rephrasing other mathematicians work and claiming fundamental status, while not producing many novel theorems... just strikes me as very distasteful.

Bothers me to no end how many once simple devices have gone this way. I feel like I would be willing to pay more for simpler appliances at this point.

I wish we had started with Spivak! Reading Rudin after 3 semesters of Stewart was a lot.

...This could be a brilliant technique for collecting funding.

This kind of foundational (ZFC,CT,HTT) stuff is pretty niche in mathematics. Set theory, though touted as the foundation of math, isn't really used to its full extremes (HOD, etc) by anyone but set theorists. Its more like... A common datatype? And while CT could also be a foundation, as far as im aware its only practical uses as a tool are found in commutative algebra.

I think the truth is that most math is pretty simple to describe and doesn't require a grand framework to formalize.

I went into my math undergrad having gotten hyped online about CT and HoTT. ...And basically the only time i've seen CT come up is in algebra (they use aluffi). And the only time Ive heard anything about HoTT was when I tried bringing it up with the logic graduate students and got some light teasing.

I dont think so. Topology is ubiquitous in mathematics. From Zariski topologies and Stone spaces, to concrete objects like manifolds and topological vector spaces.

Differential forms aren't exactly comparable to GA... I would instead look at the relation between exterior algebra and GA.

Differentials are a concept that the comes from doing calculus on manifolds, and exterior products of differentials are just used for tracking information about oriented volumes.

To answer your question (switching differentials forms for for exterior algebras), you wont miss anything, as the wedge product is part of a GA.

An important example from CS would be the semigroup action on the set of states of a deterministic finite automaton.

The action takes a state from Q, and a symbol from E, and returns a new state. ie f:QxE->Q. In the case when you have some "no op" action the semigroup action is actually a monoid action.

This all corresponds to the DFA moving through states as it "eats" the string.

Yeah, I was thinking this was going to be an article about making colors easier to see for colorblind people... but it actually appears to be the opposite.

Having some degree color blindness is extremely common. Its depressing how designers just refuse to take this into account.

Yeah, I have to agree. It's frustrating because CT and these kinds of diagrams can actually be kind of interesting... They are just not nearly as grand or as useful as their proponents would have us believe.

(My spicier opinion is that the machinery of CT is often completely unnecessary for many applications, and actually only serves to obfuscate simple concepts.)

This is a recipe for a very poor education in my opinion. This sort of mass produced educational content, taught by professional educators rather than researchers, produces students who merely know facts and can perform basic algorithms.

Real understanding of a field comes from long hours engaging with the subject and playing with ideas, and from interacting with others in the field. Working through educational material (where it exist) is obviously helpful, but it's not enough.

tl;dr: Nobody becomes in expert in anything by working through a textbook.

there’s lots of visualizations for spherical harmonics; those are what you’d want to look for. eg https://m.youtube.com/watch?v=Ziz7t1HHwBw

the models already account for this. eg in compartmental models both SIS or SIRS have this assumption.

Yeah, this sort of stuff is fine supplementary material... but nobody really learns anything from a blog post or some videos.

Learn proofs well and you get pretty good and knowing when you’re right. Enough for almost any problem you’ll be likely to encounter in a math textbook anyway.