The classic methods of defining numbers are “wrong” in the sense that it doesn’t match how people actually think about numbers

Peano Arithmetic and ZFC pretty much do define addition and multiplication recursively in terms of successor and addition, respectively.

My guess would be that we actually want to view there as being multiple basic/intuitive cognitive starting points, and they'd correspond to different formal models. As an example, consider steps / walking. It's pretty intuitive that if you're on a straight path, facing in one fixed direction, there's two types of actions--walk forward a step, walk backward a step--and that these cancel out. This corresponds to addition and subtraction, or addition of positive numbers and addition of negative numbers. In this case, I would say that it's a bit closer to the intuitive picture if we say that "take 3 steps backward" is an action, and doing actions one after the other is addition, and so that action would be the object "-3"; and then you get the integers. I think there just are multiple overlapping ways to think of this, including multiple basic intuitive ones. This is a strange phenomenon, one which Sam has pointed out. I would say it's kinda similar to how sometimes you can refactor a codebase infinitely, or rather, there's several different systemic ways to factor it, and they are each individually coherent and useful for some niche, but there's not necessarily a clear way to just get one system that has all the goodnesses of all of them and is also a single coherent system. (Or maybe there is, IDK. Or maybe there's some elegant way to have it all.)

Another example might be "addition as combining two continuous quantities" (e.g. adding some liquid to some other liquid, or concatenating two lengths). In this case, the unit is NOT basic, and the basic intuition is of pure quantity; so we really start with $R$.

The usual definition of numbers in lambda calculus is closer to what you want; numbers are iterators, which given a zero z and a function f, iterate f some number of times. I played around with defining numbers in lambda calculus plus an iota operator. (iota [p]) returns a term t such that (p t) = true, if such a term exists. ("true" is just $\lambda xy.x$) This allows us to define negative numbers as the things that inverse-iterate, define imaginary numbers, etc, all in one simple formalism.

iota plus lambda allows for higher-order logic already ($\exists x.p\left(x\right)$ is just $p\left(\iota p\right)$; $\exists p.px$ is just $\left[\lambda p\right(px\left)\right](\iota \lambda p\left(px\right))$) so there are definitely questions of consistency. However, it feels like some suitable version of this could be a really pleasing foundation close to what you had in mind.

The distributivity property is closely related to multiplication being repeated addition. If you break one of the numbers apart into a sum of 1s and then distribute over the sum, you get repeated addition.

It sounds like you might be looking for Peano's axioms for arithmetic (which essentially formalize addition as being repeated "add 1" and multiplication as being repeated addition) or perhaps explicit constructions of various number systems (like those described here).

The drawback of these definitions is that they don't properly situate these numbers systems as "core" examples of rings. For example, one way to define the integers is to first define a ring and then define the integers to be the "smallest" or "simplest" ring (formally: the initial object in the category of rings). From this, you can deduce that all integers can be formed by repeatedly summing $1$s or $-1$s (else you could make a smaller ring by getting rid of the elements that aren't sums of $1$s and $-1$s) and that multiplication is repeated addition (because $a\cdot b=a\cdot (1+\cdots +1)=a+\cdots +a$ where there are $b$ terms in these sums).

(It's worth noting that it's not the case in all rings that multiplication, addition, and "plus 1" are related in these ways. E.g. it would be rough to argue that if $A$ and $B$ are matrices then the product $AB$ corresponds to summing $A$ with itself $B$ times. So I think it's a reasonable perspective that multiplication and addition are independent "in general" but the simplicity of the integers forces them to be intertwined).

Some other notes:

1. Defining $-a$ to be the additive inverse of $a$ is the same as defining it as the solution to $a+x=0$. No matter which approach you take, you need to prove the same theorems to show that the notion makes sense (e.g. you need to prove that $-a+-b=-(a+b)$).
2. Similarly, taking $Q$ to be the field of fractions of $Z$ is equivalent to adding insisting that all equations $ax=b$ have a solution, and the set of theorems you need to prove to make sure this is reasonable are the same.
3. In general, note that giving a definition doesn't mean that there's actually any object that actually satisfies that definition. E.g. I can perfectly well define $\alpha$ to be an integer such that $\alpha \cdot 0=3$ but I would still need to prove that there exists such an integer $\alpha$. No matter how you define the integers, rational numbers, etc., you need to prove that there exists a set and some operations that satisfy that definition. Proving this typically requires giving a construction along the lines of what you seem to be looking for. So these definitions aren't really meant to be a substitute for constructing models of the number systems.

You may be interested in this article and its successors which looks at a specific type of commutative hyperoperator.

This makes me think of eurisko/automated mathematicion, and wonder what minimal set of heuristic and concepts you can start with to get to higher math. 

A hot math take

I was expecting a take on a particular tune by Andrew Bird.

I disagree maths “should be” done differently. I have a strong feeling the way stuff is defined usually nowadays has a property of being maximally easy to use. We don’t really need the definitions to look exactly like the intuition we had to invent them as long as the resulting objects behave exactly the same, and the less intuitive definitions are easier to use in proofs. For example, defining all powers directly as the Taylor series of e^x makes defining complex and matrix exponentials much easier / possible at all, and ad hoc proof this coincides with the naive version is simple. Also simplifies checking well-definedness a lot. Many more such examples. 

There’s a piece of folklore whose source I forget, which says. “If someone in the hallway asks you a question about probability, then with probability 1/e the answer will be 1/e. The rest of the time it will be ‘you should switch.’”

Oddly enough, you're the second person to post about this.

Of course, at least in the context of startups, the success of the startups will be correlated, for multiple reasons, partly selection effects (selected by the same funders), partly network effects (if they are together in a batch, they will benefit (or harm) each other).

It's the exponential map that's more fundamental than either e or 1/e. Alon Amit's essay is a nice pedagogical piece on this.

Books are quite long. I think we should go with the shortest possible description of his life.

Hard to say, there is no good evidence either way, but I lean toward speed-reading not being a real thing. Based on a quick search, it looks like the empirical research suggests that speed-reading doesn't work.

The best source I found was a review by Rayner et al. (2016), So Much to Read, So Little Time: How Do We Read, and Can Speed Reading Help? It looks like there's not really direct evidence, but there's research on how reading works, which suggests that speed-reading shouldn't be possible. Caveat: I only spent about two minutes reading this paper, and given my lack of ability to speed-read, I probably missed a lot.

If anyone claims to be able to speed-read, the test I would propose is: take an SAT practice test (or similar), skip the math section and do the verbal section only. You must complete the test in 1/4 of the standard time limit. Then take another practice test but with the full standard time limit. If you can indeed speed-read, then the two scores should be about the same.

(To make it a proper test, you'd want to have two separate groups, and you'd want to blind them to the purpose of the study.)

As far as I know, this sort of test has never been conducted. There are studies that have taken non-speed-readers and tried to train them to speed read, but speed-reading proponents might claim that most people are untrainable (or that the studies' training wasn't good enough), so I'd rather test people who claim to already be good at speed-reading. And I'd want to test them against themselves or other speed-readers, because performance may be confounded by general reading comprehension ability. That is, I think that I personally could perform above 50th percentile on an SAT verbal test when given only 1/4 time, but that's not because I can speed-read, it's just because my baseline reading comprehension is way above average. And I expect the same is true of most LW readers.

Edit: I should add that I was already skeptical before I looked at the psych research just now. My basic reasoning was

1. Speed reading is "big if true"
2. It wouldn't be that hard to empirically demonstrate that it's real under controlled conditions
3. If such a demonstration had been done, it would probably be brought up by speed-reading advocates and I would've heard of it
4. But I've never heard of any such demonstration
5. Therefore it probably doesn't exist
6. Therefore speed reading probably doesn't work

Another similar topic is polyphasic sleep—the claim that it's possible to sleep 3+ times per day for dramatically less time without increasing fatigue. I used to believe it was possible, but I saw someone making the argument above, which convinced me that polyphasic sleep is unlikely to be real.

A positive example is caffeine. If caffeine worked as well as people say, then it wouldn't be hard to demonstrate under controlled conditions. And indeed, there are dozens of controlled experiments on caffeine, and it does work.

I find that the type of thing greatly affects how I want to engage with it. I'll just illustrate with a few extremal points:

• Philosophy: I'm almost entirely here to think, not to hear their thoughts. I'll skip whole paragraphs or pages if they're doing throat clearing. Or I'll reread 1 paragraph several times, slowly, with 10 minute pace+think in between each time.
• History: Unless I'm especially trusting of the analysis, or the analysis is exceptionally conceptually rich, I'm mainly here for the facts + narrative that makes the facts fit into a story I can imagine. Best is audiobook + high focus, maybe 1.3x -- 2.something x, depending on how dense / how familiar I already am. I find that IF I'm going linearly, there's a small gain to having the words turned into spoken language for me, and to keep going without effort. This benefit is swamped by the cost of not being able to pause, skip back, skip around, if that's what I want to do.
• Math / science: Similar to philosophy, though with much more variation in how much I'm trying to think vs get info.
• Investigating a topic, reading papers: I skip around very aggressively--usually there's just a couple sentences that I need to see, somewhere in the paper, in order to decide whether the paper is relevant at all, or to decide which citation to follow. Here I have to consciously firmly hold the intention to investigate the thing I'm investigating, or else I'll get distracted trying to read the paper (incorrect!), and probably then get bored.

Afair the usual culprit is subvocalizing as you read. Try https://www.spreeder.com/app.php?

So, I'm not really asking "how can I read a page in 30 seconds?". I'm more looking for something like, are there systematic things I could be doing wrong that would make me way faster?

A thing I've noticed as I read more is a much greater ability to figure out ahead of time what a given chapter or paragraph is about based on a somewhat random sampling of paragraphs & sentences.

Its perhaps worthwhile to explicitly train this ability if it doesn't come naturally to you, eg randomly sample a few paragraphs, read them, then predict what the shape of the entire chapter or essay is & the arguments & their strength, then do an in-depth reading & grade yourself.

Or is it all just trading-off with comprehension?

Probably depends on the book. Some books are dense with information. Some books are a 500-page equivalent of a bullet list with 7 items.

It definitely is trading off with comprehension, if only because time spent thinking about and processing ideas roughly correlates with how well they cement themselves in your brain and worldview (note: this is just intuition). I can speedread for pure information very quickly, but I often force myself to slow down and read every word when reading content that I actually want to think about and process, which is an extra pain and chore because I have ADHD. But if I don't do this, I can end up in a state where I technically "know" what I just read, but haven't let it actually change anything in my brain—it's as if I just shoved it into storage. This is fine for reading instruction manuals or skimming end-user agreements. This is not fine for reading LessWrong posts or particularly information-dense books.

If you are interested in reading quicker, one thing that might slow your reading pace is subvocalizing or audiating the words you are reading (I unfortunately don't have a proper word for this). This is when you "sound out" what you're reading as if someone is speaking to you inside your head. If you can learn to disengage this habit at will, you can start skimming over words in sentences like "the" or "and" that don't really enhance semantic meaning, and eventually be able to only focus in on the words or meaning you care about. This still comes with the comprehension tradeoff and somewhat increases your risk for misreading, which will paradoxically decrease your reading speed (similar to taking typing speed tests: if you make a typo somewhere you're gonna have to go back and redo the whole thing and at that point you may as well have just read slower in the first place.)

Hope this helps!

Reminds me of the classic "Rakkaus matemaattisena relaationa" (Love as a mathematical relation).

Partial English translation by GPT-5

1. The Love Relation

Definition 1. Let $I$ be the set of all people and $a,b\in I.$ Define the relation “$♡$” by

$a♡b⟺a$  loves $b$. 

Remark 2. There are many mutually equivalent definitions for the notion “$a$ loves $b$”. Invent and prove equivalent definitions.^[1]

Remark 3. “$♡$” is not an equivalence relation.

Proof. It would suffice to observe the lack of reflexivity, symmetry, or transitivity, but we’ll do them all to highlight how difficult a relation we’re dealing with.

• Not reflexive: Clearly “$a\text{/}♡/a$” for some self-destructive $a\in I$.
• Not symmetric: From $a♡b$  it does not necessarily follow that $b♡a$. Think about it.
• Not transitive: From $a♡b$ and $b♡c$ does not follow that $a♡c$, since if it did, nearly all of us would be bisexual. $\square$

Remark 4. The lack of transitivity also implies that the relation “$♡$” is not an order.

For it to work well in practice, the relation “$♡$” would need at least symmetry and transitivity. Reflexivity is not necessary, but a rather desirable property. However, as is common in mathematics, we can forget practice and study partitions of sets via the relation. We soon notice that the lack of symmetry allows very different kinds of sets.

2. Inward-Warming Sets

We begin our investigation by looking at inward-warming sets. Then we can restrict ourselves to sets whose size is greater than two. To simplify definitions we assume that no one loves themselves. In other words, we silently assume that everyone loves themselves, so it can be left unmarked separately.

Convention 5. From now on, assume $A\subset I$ and $\#A\ge 3$, unless otherwise stated. Also assume that for all $a\in A$ we have $a\text{/}♡a$.

If the notation in this section feels complicated, read Section 3 and then take another look. Drawing pictures often helps. In mathematics it’s really just about representing complicated things with simple notations—whose understanding nonetheless requires several years of study.

Definition 6. We say that a set $A$ is

1. inward-warming if, for

$B=b\in I:\exists a\in A\text{with}a\ne b\text{and}a♡b$,

we have $A\cap B\ne \varnothing$.

2. strongly inward-warming if the set

$B=b\in I:\exists a\in A\text{with}a\ne b\text{and}a♡b$

is a nonempty subset of $A$.

3. together if for every $a\in A$ there exists $b\in A$ with $a♡b$ and $a\ne b$, and in addition the set

$B=b\in I:\exists a\in A\text{with}a\ne b\text{and}a♡b$

is a subset of $A$.

4. a hippie commune if for all $a,b\in A$ we have $a♡b$.

Remark 7.
i) A set that is together is strongly inward-warming. Conversely, a strongly inward-warming set is inward-warming.
ii) In a hippie commune, the relation “$♡$” is an equivalence relation.

3. Basic Notions

The definitions in the previous section admittedly look a bit complicated. For that we develop a bit of terminology:

Convention 8. From now on, also assume $a,b\in A$ with $a\ne b$, unless otherwise stated.

Definition 9. We say that $a$ is, in the set $A$,

1. cold if there is no $b\in A$ with $a♡b$.
2. lonely if there is no $b\in A$ with $b♡a$.
3. popular if there exists $b\in A$ with $b♡a$.
4. warm if there exists $b\in A$ with $a♡b$.
5. a recluse if it is cold and lonely in $A$.
6. a player if it is popular and warm in $A$.
7. a slut if there exist $b,c\in A$ with $b\ne c$ and $a♡b$ as well as $a♡c$.
8. a hippie if $a♡b$ for all $b\in A$.
9. loyal in $A$ if there exists exactly one $b\in A$ with $a♡b$. Then we say that $a$ is loyal to $b$.
10. $b$'s partner in $A$ if $a♡b$ and $b♡a$. Then $(a,b)$ is called a couple, and $A$ contains a couple. The couple $(a,b)$ is a strong couple in $A$ if $a$ is loyal to $b$ in $A$ and $b$ is loyal to $a$ in $A$.

Remark 10. Every hippie is a slut.

Using these definitions, we can rewrite the earlier concepts:

Theorem 11. A set $A$ is

1. inward-warming if and only if there exists at least one $a\in A$ that is warm in $A$.
2. strongly inward-warming if every $a\in A$ is cold in $I\setminus A$ and there exists at least one $a\in A$ that is warm in $A$.
3. together if every $a\in A$ is warm in $A$ and cold in $I\setminus A$.
4. a hippie commune if $(a,b)$ is a couple in $A$ for all $a,b\in A$.

Proof. Exercise. □

[...]

7. Connection to Graph Theory

It becomes easier to visualize different sets when the relationships between elements are represented as graphs. We call the elements of the set the vertices of the graph and say that there is an edge between vertices $a$ and $b$b if $a♡b$. Since “$♡$” is not symmetric, the graphs become directed. Thus we draw the set’s elements as points and edges as arrows. In the situation $a♡b$, we draw an arrow from vertex $a$ to vertex $b$. Figure 1 shows a simple situation where $a$ loves $b$, and Figure 2 shows the definitions from Section 3 and a few more complicated situations.

Convention 28. We will henceforth call the graphs formed via $♡$ simply graphs.

[...]

8. Graph Colorings, Genders, and Hetero-ness

In many cases it is helpful for visualization to color the vertices of graphs so that “neighbors” always differ in color. Since blue is the boys’ color and red the girls’ color, we want to color graphs with only two colors.

Definition 33. A vertex $b$ is a neighbor of a vertex $a$ if $a♡b$.

Definition 34. A vertex $a\in G$ is hetero if all its neighbors are of a different gender than $a$. A pair $(a,b)$ is a hetero couple if $a$ and $b$ are of different genders.

Definition 35. A graph $G$ is a rainbow if it contains a vertex that is not hetero.

Remark 36. The name “rainbow” comes from the fact that a rainbow graph cannot be colored with two colors so that neighbors always differ in color.

Definition 37. A path (from $a_1$​ to $a_n$​) in a graph $G$ is an ordered set {$a_1,\dots ,a_n$} whose elements satisfy $a_i♡a_{i+1}$​ or $a_{i+1}♡a_i$ for all $i=1,2,\dots ,n-1$. The path is a cycle if $a1=a_n$​. The length of a path/cycle is $n$.

Theorem 38. If a graph contains a cycle of odd length, it contains a vertex that is not hetero.

Proof. Exercise.

1. ^{^}

   Hint: Help can be sought from rock lyrics or lager beer.

Yes, pretty much it. Concern with path-connectedness is equivalent to considering the transitive closure of the relation.

Graphs studied in graph theory are usually more sparsely connected than those studied in binary relations.

Also I have a not-too-relevant nerd snipe for you: You have a trillion bitstrings of length 1536 bits each, which you must classify them into a million buckets of approximately a million vectors each. Given a query bitstring, you must be able to identify a small number of buckets (let's say 10 buckets) that contain most of its neighbours. Distance between two bitstrings is their Hamming distance.

At a certain point, graph theory starts to include more material - planar graph properties, then graph embeddings in general. I haven't ever heard someone talking about a 'planar binary relation'.

you’ll probably have more luck spelling the problems out explicitly

Taleb has made available a technical Monograph that parallels that book, and all of his books. You can find it here: https://arxiv.org/abs/2001.10488

The pdf linked by @CstineSublime is definitely towards the textbook. I’ve started reading it and it has been an excellent read so far. Will probably write a review later.

Might look at Wolfram's work. One of the major themes of his CA classification project is that chaotic (in some sense, possibly not the rigorous ergodic dynamics definition) rulesets are not Turing-complete; only CAs which are in an intermediate region of complexity/simplicity have ever been shown to be TC.

I like that idea! I definitely welcome people to do that as practice in distillation/research, and to make their own polished posts of the content. (Although I'm not sure how interested I would be in having said person be mostly helping me get the posts "over the finish line".)