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 different objects can be treated similarly.

More generally, mathematics is experimental not just in the sense that it can be used to make physical predictions, but also (probably more importantly) in that definitions are "experiments" whose outcome is judged by their usefulness.

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 you’re interested in. The main thing is deep understanding, not superficial breadth.

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 the NBA.” Lift weight, condition, and practice fundamentals. Nothing else matters.

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 independent -- most professional mathematicians use some mixture of the three, and virtually every mathematician understands the basics of all three.

The most important object in modern geometry is the manifold. This is a space that looks locally like n-dimensional Euclidean space -- 1-dimensional manifolds are curves, 2-dimensional manifolds are surfaces, and higher dimensional manifolds are simply called n-manifolds. All of physics takes place on manifolds. Differential equations correspond to vector fields on manifolds. The manifold hypothesis says that much of the high-dimensional data we see actually lives on much lower-dimensional manifolds (partially explaining the unreasonable effectiveness of deep learning on very high-dimensional datasets).

The most important object in algebra is the group. The collection of symmetries of any object (e.g. a Rubick's cube, a piece of paper, or three-dimensional space) forms a group under composition. A group that is also a manifold is called a Lie group. These are everywhere -- n-dimensional rotations form groups, fundamental particles correspond to representations of Lie groups, invertible matrices form a group. Spherical harmonics and Fourier series are both naturally viewed in terms of representations of Lie groups.

The most important object in analysis is the limit. Limits first appear in the construction of the real line by adjoining limits of Cauchy sequences to the rational numbers. Using the real line, one can measure volumes, probabilities, and distances in geometric spaces such as manifolds, but also in spaces of functions, sequences, and more abstract objects. The proof of the fundamental theorem of calculus (that derivatives and integrals are roughly inverse operations) requires rigorous analysis of the definitions of derivative and integral as limits.

To learn math, you should begin by understanding what a proof is. All of mathematics is based on proving theorems. A mathematical proof is a sequence of statements that explains the logical steps required to use the assumptions of the theorem to verify the result. Just as a computer program cannot "almost output" the correct answer, there is no such thing as an "almost correct" proof. A proof either describes a correct chain of logic to reach the conclusion, or it does not. The reason math is based on proofs is because more advanced math and science builds upon more basic math. An error in a mathematical theorem or an imprecise definition will lead to bigger problems down the line, so every step must be carefully validated. For an individual student as well, only through proving theorems can one deeply understand a mathematical subject, and a solid understanding of basic subjects is required to understand more advanced topics.

Fortunately, you can learn to prove theorems at the same time as learning the foundations of math. The first books you should work through are "Principles of Mathematical Analysis" by Walter Rudin, and "Linear Algebra" by Georgi Shilov. This will be hard, not for an arbitrary reason, but because assimilating new math into your brain is intrinsically difficult, especially at the beginning. If possible, try to find a teacher.

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).

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 problems which focus on cleverness and real-world problems which focus on conceptual understanding. Many kids prefer one or the other, and I think it’s a mistake to assume that contest math works for all kids who might be mathematically inclined.

Re: goals —- the goal is to let the kid learn as fast as they want assuming they have solid foundations. If they like proofs let them do proofs, if they like applications let them do that. Just don’t force them to sit in a classroom doing busywork for the most formative years of their lives.

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/division problems for 5 years when they would have a ton more fun and get a lot more long-term utility from learning some stat/algebra/geometry? If a kid demonstrates mastery of a concept, it's a lot more bizarre and potentially damaging to force them to relearn the same material over and over.

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. Aiming for median outcomes makes zero sense in the current world. Find your niche and hit it hard.

Kids intuitively understand this -- they like doing what they're good at. Unfortunately, most schools are not good at serving this need. A very important part of being a parent is to encourage kids to start compounding positive habits/learning early, and to prevent the schools from dragging them back to the median.

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 -- farmers and bakers. Farmers can sell future produce to buy seeds right now. Bakers can hedge their wheat price exposure to reduce their chance of getting ruined by a bad harvest. Speculators get paid to hold onto wheat futures contracts if a farmer wants to sell a future when the bakers aren't around to buy (presumably baking), selling to a baker later for a higher price reflecting the price risk assumed. All of these participants derive value from the market.

It's not clear to me that prediction markets usually have natural hedging participants (maybe political operatives, but the tx costs are probably too high relative to the value at stake).

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.

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 grid.

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's so multidimensional. If you want to be the richest person in the world, of course you're setting yourself up for failure. But if you want to be the best version of yourself, you can easily be the best father-husband-son-coder-blogger-walker-painter to your children+wife+colleagues in your city in July 2021.

More than that, you can do things no one else has done. If you like research, the frontier is endless and extremely high dimensional. Find some niche that you enjoy and crush it. If you like helping people, there will never be an end of people you can help. You don't have to be average -- you can be in the 1% of what you're passionate about, easily, because there are so many possible choices of passion.

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 the new edge (g.f). Not only that, we add h.(g.f) = (h.g).f for any h: C -> D, which basically says that "." acts like concatenation of paths (it doesn't matter how you group the edges in a path, you get the same thing).

Those are the basic axioms to say you're working with a nice operation that turns paths into edges in a reasonable way. It gets interesting when you add more identifications (commutative diagrams) and operations (functors). Prototypical example: vertices are sets, edges are functions, you get identification of paths (sequential compositions of functions) whenever the underlying composite functions are pointwise equal.

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

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 purpose computer.

“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.“

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 go the way of tumblr/digg in the next five years.

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.

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

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

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 today?"

Sure, that's what a whitelist will be, but with one major difference: each node gets to pick its own whitelist, there's no built-in centralization via the "one true source code."

2) Yes, miners will choose hashing algorithms that are easy for themselves and hard for others, but the nodes get to value each block from their own perspective. If they think a miner is using a weird hash function, they can penalize it or just blacklist the function.

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 people suspect an algorithm is getting ASIC'd, they can devalue it (if they want -- maybe the majority of nodes want ASICs for some reason, and that's fine too). The point is to let the users to choose the algorithm, not a core group of developers.

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 maintain the status quo (c.f. bitcoin transaction cost problems).

Getting back to your question, I have no idea if the scaling functions will stabilize on a singular chain or not. However, I think it would be a Pareto improvement over the status quo. Indeed, you could embed BTC and ETH into DCC (with the proviso that all users must use the core-developer-chosen hash functions and scaling functions), and both chains could live side-by-side.

Yep, and you could even prove "nat" is isomorphic to the type of binary numbers "Bin := List (0 | 1)" and transport proofs there too.

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 verification.

Among other things, I want theorems about foundations to be useful to non-foundational mathematicians. Working on type theory can improve, at the very least, formal verification tools. In my view, traditional ZFC-based mathematical logic has a much smaller likelihood of being useful for regular mathematicians. For example, the vast majority of mathematicians will never explicitly assume the existence of a large cardinal in a paper. It seems to me that the difference in usefulness is probably due to their relative distance from common discourse.

Thanks for the interesting discussion. My main point of disagreement re:foundations is that in my view a central criterion for a foundational system is easy translation of common mathematical discourse into formal proofs. Otherwise, it seems that the coherence of ordinary mathematical discourse must be an article of faith.

Mathematicians use implicit DSLs approximately embedded in natural language to write proofs. Formalizing these DSLs in ZFC is a nightmare because its axioms are arcane and far from common mathematical discourse (much like assembly). On the other hand, the type formers and rules of (say) MLTT are natural generalizations of the rules of ordinary logic.

Mainstream mathematicians are not being prevented from proving theorems because they lack "better ways of reasoning about equality of types."

How do you know? I've made the mistake of assuming that the definition of a certain morphism is natural with respect to a certain object when it wasn't (coevaluation in a rigid monoidal category). Better language for equality and naturality are precisely what HoTT provides. More importantly, the hope is that type theory will not only provide better language, but relieve us of the tedium of checking proofs by hand.

The importance of ZFC is that it ensures our common language reasoning isn't incoherent.

I don't think this is true. AFAIK, no one has written down the definition of (say) a scheme from first principles in ZFC. In fact, if you care about coherence/verification, you should care even more about type theory. IIRC this is what drew Voevodsky to type theory in the first place -- https://mathoverflow.net/questions/234492/what-is-the-mistak...

Seconded, but it might be worth learning a language with algebraic data types first (such as Haskell or OCaml).