pgustafs

5 ms·

1.
by pgustafs 1y ago
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 different objects can be treated similarly. More generally, mathe
2.
by pgustafs 1y ago
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 fundamentals first and then pick a couple good books in areas yo
3.
by pgustafs 1y ago
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 roadmap things are meaningless. It’s like “how to join t
4.
Predict the Unpredictable(kdivergent.substack.com)
1 points by pgustafs 1y ago|0 comments
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by pgustafs 1y ago
it's called "angel" investing because it's only one step removed from charity
6.
by pgustafs 2y ago
There are three broad subareas of mathematics: geometry, algebra, and analysis. Geometry studies space, algebra studies time, and analysis studies infinity. They are not independent -- most professional mathematicians use some mixture o
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by pgustafs 3y ago
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 SOTA).
8.
by pgustafs 3y ago
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 hours. Also, there is a large difference between contest proble
9.
by pgustafs 3y ago
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 smart kid do a million multiplication/divis
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by pgustafs 3y ago
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 should be aiming for way above median. Aimin
11.
by pgustafs 5y ago
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 example, the wheat futures market has two natural participants
12.
by pgustafs 5y ago
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 explanation -- people don't want to walk around in the heat.
13.
by pgustafs 5y ago
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 same approach works for any graph, not just a
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by pgustafs 5y ago
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 measured against. The great thing about life is that it
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by pgustafs 5y ago
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
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by pgustafs 5y ago
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 the new edge (g.f). Not only that, we add h.(g.f) =
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by pgustafs 6y ago
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 anyone will do so for many years.” GH Hardy 1941
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by pgustafs 6y ago
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 attack Turing machines before the invention of the general pur
19.
by pgustafs 6y ago
“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 few implications for how to count things.“
20.
by pgustafs 6y ago
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 -- without real user-centric leadership, reddit will
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by pgustafs 6y ago
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 curve or insane, depending on how you look at it.
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by pgustafs 6y ago
Yep, that would be a better rewording. I missed it in my search for a <80 character title.
23.
Why is dependent type theory more suitable than set theory for proof assistants?(mathoverflow.net)
221 points by pgustafs 6y ago|59 comments
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by pgustafs 6y ago
If anyone is actually trying to learn about 3D rotations, I would recommend reading the Wikipedia entry [1] instead of this blog post. [1] https://en.wikipedia.org/wiki/3D_rotation_group
25.
by pgustafs 6y ago
1) A node _could_ run arbitrary code, but unless it trusts the code's provenance then it should use a white-list as you mention. > why not just settle on a fixed-but-extensible list of hashing algorithms that are known to work well
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by pgustafs 6y ago
Right, but this approach solves that by not defining any hashing algorithm at all. Every block includes its own hash algorithm source code. Every node decides for itself whether to accept the hash algorithm and how much to value it. If
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by pgustafs 6y ago
Thanks for commenting. First let me say that with current blockchains, we achieve "objective state" at the cost of centralized control of the algorithm by some core group of developers. This results in perverse incentives to ma
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Show HN: Toy Haskell blockchain with variable hash functions for ASIC resistance(github.com)
15 points by pgustafs 6y ago|7 comments
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by pgustafs 6y ago
Yep, and you could even prove "nat" is isomorphic to the type of binary numbers "Bin := List (0 | 1)" and transport proofs there too.
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by pgustafs 6y ago
We've hit the comment depth limit, so I'll reply here. > it's not clear to me why you necessarily want the language you use to rigorously investigate the foundations of mathematics to be the same one you use to do proof ve

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