Viewing profile — pgustafs

The definition of bijection is much more interesting than comparing cardinals. Many everyday use cases where (structure-preserving) bijections make it clear that two apriori differ…

The books are good, but way too many and wildly varying in difficulty. No one can read all that in 2 years starting without knowledge of linear algebra. just worry about the fundam…

Nah, just study linear algebra (Shilov or Hoffman & Kunze) and baby Rudin. Then read the most famous books in geometry, analysis, and algebra (do proofs + get a mentor). All these …

it's called "angel" investing because it's only one step removed from charity

There are three broad subareas of mathematics: geometry, algebra, and analysis. Geometry studies space, algebra studies time, and analysis studies infinity. They are not independen…

Follow your innate curiosity, and respect the edifice of knowledge constructed by those who came before us (i.e., don't get sucked into quackery without a deep understanding of the…

I have nothing against contest math (I was a USAMO qualifier in high school), but contest math isn’t enough if a kid has to sit through years of tedium during regular classroom hou…

I think problem solving math is definitely fun and can be a huge source of confidence, but I don't see why "racing" through the standard curriculum is a negative. Why should a smar…

Agree, but I don't like the framing of "accelerating." Math in school is for the median student. If you want a quantitative career or just want to have quantitative skills, you sho…

Adding to your point, most successful markets are positive-sum -- hedgers gain value from mitigating their structural risks, and speculators get paid to assume price risk. For exam…

One point that seems to be missing from this discussion is climate. Walkable cities: NYC, Chicago, SF, Boston, Seattle Urban hellscapes: LA, Houston, Atlanta, Phoenix Simple explan…

Great post, one nitpick -- I wouldn't say that a matrix is a "sparsely defined" function, but rather a function defined on a finite grid. It might also be worth pointing out that s…

I vehemently disagree. Not with your explicit reasoning, but with the implicit assumption that there is some 1-dimensional metric of specialness or greatness that we're all being m…

By 2030: 90% more advanced type systems 50% IDEs, github, and slack will merge into one monolithic social developer experience 10% end user programming using NLP

The main thing categories add to graphs is identifications of paths. Every pair of edges f: A -> B and g : B -> C generates a third edge g.f, and we identify the path (g)(f) with t…

Here’s another great quote along the same lines: “No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyo…

I was referring to the fact that computers themselves are the descendants of logicians — Church, Turing, von Neumann. Your argument above could have just as easily been used to att…

“It seems unlikely that there will be any profound insights from logic on how to compute, similarly to how thousands of years of advances in the theory of numbers have had notably …

Centralized opaque moderation ruins every large subreddit. It's too bad because reddit could be 10x more interesting if it were more open. Network effects can only take you so far …

The problem is that the sites don't have feedback loops to the real world, just imaginary internet points. People who optimize for imaginary internet points are either ahead of the…

Yep, that would be a better rewording. I missed it in my search for a <80 character title.