chobytes

Sounds amazing. Wish we could get them that low here. Biking on the road next to multiple ton, blocks of metal going 30mph (48km/h) is never pleasant.

A fun wrinkle is that there are (infinitely many) models of set theory which are pointwise definable. Thus in the right model of zfc, every mathematical object is uniquely addressable by a certain finite formula. (see https://arxiv.org/pdf/1105.4597.pdf )

I believe it. Lockdowns and covid suck, but living in the city is still better than the alternative.

That use-case seems not strictly useful, but things become trickier with more elaborate expressions; like when we want to abstract over what f looks like.

Eg One might have f(x,y,z) = (...). If f is in some class of functions with some properties (homogeneous, linear, smooth, etc) we can operate on it abstractly. We could even derive properties of surfaces f(x,y,z) = C.

No tbh I think that's fantastic. Engineering and physics are a great way to get the right intuition. Having a firm grasp of the basics and having lots of possible examples in mind is very useful! Basically I would try to round out the basic pure math stuff for sure, but I think youre better equipped than most.

Another intro book I thought of was Peter Eccle's introduction to mathematical reasoning. Might be worth looking at.

If you want a nice leisurely introduction to groups Nathan Carter's Visual group theory is nice.

I got a lot of use out of the princeton encyclopedias of math. Dont expect for them to really teach you anything, but the articles are nice for seeing whats out there.

Definitely try to learn basic analysis and algebra. I dont think any book I know is amazing, but basically any will do the job.

Importantly, dont be afraid to try to learn something out of your depth. In fact I think its important to try! If something really grabs you, try to read more and backfill what you don't know.

Yeah I think that's basically the idea. Just solve whatever problems you find interesting via whatever means you can.

Eg Galois theory kind of looks like this: Applying an abstract model of symmetry to a model of polynomials to answer a question about solvability.

I dont know what level youre at, but if you don't already know how to write proofs then something on that. Its an extremely important foundation for everything else.

I learned in a class and we didnt use a book so I cant recommend one. I dont think it should matter too much though.

Aside from that, what kind of things are you curious about?

Projects in pure math are basically just research. I would say just follow your curiosity and try to figure things out. You might not be scratching novel work for awhile, but its still enjoyable.

I don't know exactly what you like to work on, but perhaps theres some related mathematical area youre curious to know more about?

Ive never read this book in particular, but had a similar experience with another book (Serge Lang, Basic Mathematics) that changed my life trajectory. If you find this stuff remotely interesting please give it a go! Math is really wonderful!

Learning math in some prescribed way (book or sequence of books) is the mathematics equivalent of "what programming language should I learn first". The most important thing is simply doing anything at all! Don't get planning paralysis!

If you think you'll actually do or read anything, then give it a try. Definitely push yourself some, but if it becomes a slog don't be afraid to move on to something that looks more interesting. Youll find your way back to anything that was actually important anyway. :)

Seconding this a little. The best way to learn math is to spend a long time with it and see it in as many ways as you can. You have to build an intuition for it so you can move past definitions and theorems and just "get it".

Although I also dont want to discourage anyone from trying this either. Anything that'll get people to learn is better than apathy. :)

I think the trouble with LA education is that LA is just so ubiquitous and can look very different dependent on context. No style is going to suit everyones needs.

Yup, notebooks are what I had in mind. I make heavy use of mathematica when I want to explore a new topic.

I think I prefer physical textbooks for many reasons I wont get in to... but I do really like the interactive and visualization elements computers offer.

I feel like teaching kids some amount of coding to be able to play with math could be very helpful.

Yeah thats very true and I didnt think to mention it. I very much appreciate how concise latex is, and that I don't need any special editor for it.

Should the toolmakers priority not be to make a tool its users will use?

This sort of dictating opinions via tech seems common in the browser world, and it honestly makes me thankful we primarily target pdfs and not the web.

Anecdata, but thats not been my experience. Ive found latex to be a much smoother experience to use than languages like js or python.

I get the impression this is a tool made by technologists for technologists, rather than for people who spend a lot of time typesetting math. Academics (mathematicians, physicists, etc) don't have the same concerns as technologists, and the specifics of browser tech in particular are't relevant.

Latex is basically the mathematical (computer) lingua franca right now. Theres an entire cultural and technological ecosystem around latex thats been built up for many years. We even message each in latex notation when rendering isn't available! Theres decades of documentation online. Most important of all, its dead easy to use for most use cases!

Perhaps MathML is technically superior in some ways, but none that the people typesetting math are concerned with. That this is browser oriented tech in particular is already a disadvantage, as the browser is not a priority to us. In this way mathjax feels like the superior tool for users. It attempts to work with users rather than dictating to them.

Not sure who this is for. The notation looks extremely verbose and unusable, and latex won a long time ago already.

Similar. Ive been looking into "dumb phone" options, and moving many tasks to old school solutions. Eg desk calendars, notepads, physical books, etc.

For tech stuff I do need Id like to pick dedicated "appliance like" tools, but each of those requires some research.

Presumptuous. I certainly dont want this pseudo-righteous power grab done for me.

Ive already been itching to de-cloud, and de-tech my life. If were already getting to this stage of surveillance I guess thats just another sign I should be getting on top of it.

Today its csam. Tomorrow "misleading information". etc.

That's definitely a good place to start. Having the right intuition for groups from the beginning makes everything much easier.

As for other materials, what kind of math background do you have?

Physics is probably one of the best ways to "get" calculus. Its the application calculus was made for so I think it conveys the intuition well.

Honestly philosophers getting ahold of set theory is probably one of the worst things to happen to it. I feel like it can be hard for people to take set theory seriously when the conversation about it is dominated by this kind of drivel.

Basically this kind of dichotomy, and this image of set theory as the source of all truth in math or whatever... is essentially a myth made up by people peddling their own (new and improved!) foundations project as the source of all truth.

Edtech is a blight. Hopefully the US follows suit.

Were currently doing lattice calculations for a class im in, using "big cloud company"s quantum computers. That eats qubits quick, so 5000 qubits sounds luxurious!

The organizers dont seem to know specifics about D-wave when Ive asked, but do you think these kinds of simulations will ever be able to be run on D-wave hardware?

Me and my classmates were running code on real quantum computer literally earlier today.

10 qubits is enough to simulate a pair of particles interacting already.

To be honest if you're sufficiently motivated and you find it interesting then go with it. Better to be motivated about an idiosyncratic method than ambivalent about the whole thing.

A functor is a certain kind of function (Theres a pedantic point to be made but it's mostly irrelevant). However, most functions are not functors.

To give a simpler example, we say a function, f, is monotone when for x<y, f(x)<f(y); that is, it preserves order. Every monotone function is obviously a function, but functions like x^2 are not monotone.

Basically a functor is a function which preserves some other properties. I wont go into detail as many have before me, but thats the gist.

Math is at its heart about intuition; formalism is a just tool. One uses the formalism but one has to see past it. Roughly its the difference between knowing that a group is a set with a certain binary function, and knowing that a group is a particular formalization of symmetry.

While a shortcut to understanding sounds nice, I don't think you'll find it by focusing on abstract formalism.