Dedekind's Subtle Knife(mathenchant.wordpress.com)
mathenchant.wordpress.com
Dedekind's Subtle Knife
https://mathenchant.wordpress.com/2025/02/17/dedekinds-subtle-knife/
44 comments
That is a great paper. I came up with my own definition of real numbers [1] and using the simplified cut property of that paper was a pleasant pathway to establishing completeness.
[1]: https://github.com/jostylr/Reals-as-Oracles/blob/main/articl...
[1]: https://github.com/jostylr/Reals-as-Oracles/blob/main/articl...
How does this compare to the surreal numbers defined by Conway?
As far as I can tell, they are unrelated beyond the relation all definitions of real numbers would have with it. There is a superficial Left and Right boundary defining an interval that contains the number, but it does not seem like there are a family of intervals for a given real number in this approach.
I do not get the feeling that there is a narrowing down to a given number in the surreal numbers, but I know very little about them.
I also do not immediately see how my construction could be extended to infinitesimals or the infinite numbers.
I do not get the feeling that there is a narrowing down to a given number in the surreal numbers, but I know very little about them.
I also do not immediately see how my construction could be extended to infinitesimals or the infinite numbers.
would be nice to hear more about your self-directed learning, i'm in a similar boat right now.
how did you review fundamentals? how did you pick what topics to pursue? etc
how did you review fundamentals? how did you pick what topics to pursue? etc
Not the person you’re replying to, but I’ve been doing this and have documented my philosophy https://publish.obsidian.md/uncarved/3+Resources/Public/self...
And a syllabus https://publish.obsidian.md/uncarved/3+Resources/Public/self...
…and on that obsidian site you can see some of my notes. I curate my notes and then publish as part of my revision of a topic so I’m always a little bit ahead of what’s on the site but it’s directionally correct.
And a syllabus https://publish.obsidian.md/uncarved/3+Resources/Public/self...
…and on that obsidian site you can see some of my notes. I curate my notes and then publish as part of my revision of a topic so I’m always a little bit ahead of what’s on the site but it’s directionally correct.
I am a bit of a weirdo so my approach may not work, but for me, history, conceptual focus, and time are key.
I like reading original texts because they furnish the original logical arguments of the thinkers, which are still in many cases the best explanations. From there, viewing a modern axiomatic presentation further cements the ideas.
Time is also essential. I have to be comfortable with the fact that it might take me several weeks just to get comfortable with a single concept, so getting through a full work takes time.
Finally, I feel like mathematics and literature are equally hermeneutical—the parts inform the whole and vice versa, only, in the case of mathematics, these interactions and clarifications happen across the entire wide discipline, rather than within a single work. The more you wade out and explore, the more many other ideas become clearer as you can start to see them in a new light.
More practically, getting a firm grasp on set theory and predicate logic is essential imo. This is partly because I prefer axiomatic presentations—I simply do not do well dealing with a theory that doesn't begin at the ground floor (for example, many practitioners of calculus don't give a hoot about the logical soundness of its set-theoretic foundations and are comfortable working with it in a strictly operational sense, I however have a deep need for getting these foundations first, which is basically just a limitation on my part and probably why I was never good at math in school, where the presentations are strictly operational—I love the conceptual beauty of mathematics but I despise calculation!)
I like reading original texts because they furnish the original logical arguments of the thinkers, which are still in many cases the best explanations. From there, viewing a modern axiomatic presentation further cements the ideas.
Time is also essential. I have to be comfortable with the fact that it might take me several weeks just to get comfortable with a single concept, so getting through a full work takes time.
Finally, I feel like mathematics and literature are equally hermeneutical—the parts inform the whole and vice versa, only, in the case of mathematics, these interactions and clarifications happen across the entire wide discipline, rather than within a single work. The more you wade out and explore, the more many other ideas become clearer as you can start to see them in a new light.
More practically, getting a firm grasp on set theory and predicate logic is essential imo. This is partly because I prefer axiomatic presentations—I simply do not do well dealing with a theory that doesn't begin at the ground floor (for example, many practitioners of calculus don't give a hoot about the logical soundness of its set-theoretic foundations and are comfortable working with it in a strictly operational sense, I however have a deep need for getting these foundations first, which is basically just a limitation on my part and probably why I was never good at math in school, where the presentations are strictly operational—I love the conceptual beauty of mathematics but I despise calculation!)
Conway takes this next level with On Numbers and Games. Very openly acknowledges the debt to Dedekind. Also Knuth did something similar with his Surreal Numbers book. I think he even wrote it as a kind of chaste romance novel, but I haven't read it so have no further comment.
An interesting essay on the history and precursors to Dedekind’s construction of the Reals.
I was disappointed that the author didn’t talk about some of the modern opponents of the “real numbers.” For example I heard this interesting podcast with a professor Norman Wildberger about the “Problem of Infinity in Math”. He seemed to say that a theory of real numbers should be rooted in their intrinsic computational properties. Constructing them using set theory thwarts this goal. Then again, I personally don’t see a viable alternative to the set theoretical construction.
https://youtu.be/l7LvgvunVCM?si=eGYdJGHmCLjc968b
I was disappointed that the author didn’t talk about some of the modern opponents of the “real numbers.” For example I heard this interesting podcast with a professor Norman Wildberger about the “Problem of Infinity in Math”. He seemed to say that a theory of real numbers should be rooted in their intrinsic computational properties. Constructing them using set theory thwarts this goal. Then again, I personally don’t see a viable alternative to the set theoretical construction.
https://youtu.be/l7LvgvunVCM?si=eGYdJGHmCLjc968b
I think Norm Wildberger's videos are very useful to think about and enjoy though certainly one should reflect carefully on what he says. His videos inspired my work on a new definition of real numbers. The basic idea is that it is the set of all intervals that contain the real number. Since this is circular, there are properties that describe when a set of rational intervals are defined. This approach is equivalent to Dedekind cuts and does not address Wildberger's concern.
But there is another definition which I call oracles which does a better job. It is much more constructive and is about a a procedure that one can ask whether a fuzzy version of a given interval contains the real number. It has various properties for a procedure to satisfy and, if so, then it will generate the set of intervals that contain the real number if taken out to infinite length.
So basically, it is a two part-definition. There is a theoretically perfect version and then there is another that yields to the practical problems of not being able to actually specify a real number entirely.
If interested, the papers are hosted on GitHub [1]. The most recent version going over what I just said is Real Numbers As Rational Betweenness Relations [2]
2: https://github.com/jostylr/Reals-as-Oracles/ 1: https://github.com/jostylr/Reals-as-Oracles/blob/a98472813e9...
But there is another definition which I call oracles which does a better job. It is much more constructive and is about a a procedure that one can ask whether a fuzzy version of a given interval contains the real number. It has various properties for a procedure to satisfy and, if so, then it will generate the set of intervals that contain the real number if taken out to infinite length.
So basically, it is a two part-definition. There is a theoretically perfect version and then there is another that yields to the practical problems of not being able to actually specify a real number entirely.
If interested, the papers are hosted on GitHub [1]. The most recent version going over what I just said is Real Numbers As Rational Betweenness Relations [2]
2: https://github.com/jostylr/Reals-as-Oracles/ 1: https://github.com/jostylr/Reals-as-Oracles/blob/a98472813e9...
Lucio Russo's The Forgotten Revolution points out somewhat indignantly that Dedekind was ("just") restating in modern number-theoretical language a definition of proportion or ratio that he found in Book 5 of Euclid.
The translation at https://mathcs.clarku.edu/~djoyce/elements/bookV/bookV.html says:
> Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
This is a bit hard to understand, but I think it does come out to the same thing: positing the existence of ratios of magnitudes that cannot be expressed as ratios of integers, but which divide the ratios of integers into those that are less than or greater than the ratio of magnitudes in question.
Further discussion at https://math.stackexchange.com/questions/499395/how-to-best-... seems to agree.
The translation at https://mathcs.clarku.edu/~djoyce/elements/bookV/bookV.html says:
> Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
This is a bit hard to understand, but I think it does come out to the same thing: positing the existence of ratios of magnitudes that cannot be expressed as ratios of integers, but which divide the ratios of integers into those that are less than or greater than the ratio of magnitudes in question.
Further discussion at https://math.stackexchange.com/questions/499395/how-to-best-... seems to agree.
The real breakthrough with Dedekind cuts (as far as I can see) is not that you can define irrational numbers this way, it’s that when you do, calculus works. In other words, they obey all the axioms you would want and in particular they have the Archimedean property and possess a least upper bound.
You could always say that Dedekind was “just” restating real numbers in modern analytic language, but the fact is that no-one had managed to do that up till then, and it wasn’t obvious how it could be done. This is what sparked the crisis in analysis with some people saying that calculus was more like a religion that you just had to believe in rather than being able to prove.
You could always say that Dedekind was “just” restating real numbers in modern analytic language, but the fact is that no-one had managed to do that up till then, and it wasn’t obvious how it could be done. This is what sparked the crisis in analysis with some people saying that calculus was more like a religion that you just had to believe in rather than being able to prove.
By the way, there is an accessible and (in my opinion) brilliantly clear explanation of the construction of numbers on the “Another Roof” youtube channel.
“What is a number?” https://youtu.be/dKtsjQtigag?si=EPzNVgk47gXzNLra Explains what natural numbers are and why we should care about this set of definitions
And “defining every number ever” goes from there up to the complex numbers https://youtu.be/dKtsjQtigag?si=DI39LQtGvBsOU00y
“What is a number?” https://youtu.be/dKtsjQtigag?si=EPzNVgk47gXzNLra Explains what natural numbers are and why we should care about this set of definitions
And “defining every number ever” goes from there up to the complex numbers https://youtu.be/dKtsjQtigag?si=DI39LQtGvBsOU00y
Oh and the reason I picked on the Archimedean property specifically is that it had been this that both Leibnitz’”infinitesimals” and Newton’s “fluxions” had handwaved around. They both relied no some sort of non-Archimedean numbers for limits and derivatives to work.
The Archimedean property is that given two numbers x and y there exists an integer n such that nx>y. Infinitesimals and fluxions relied on the idea that they were so small this wouldn’t work and therefore they didn’t fit the definition of real numbers.
The Archimedean property is that given two numbers x and y there exists an integer n such that nx>y. Infinitesimals and fluxions relied on the idea that they were so small this wouldn’t work and therefore they didn’t fit the definition of real numbers.
Is that not the same thing OP was talking about re: Eudoxos? Hard to tell because I'm not exactly sure what an "equimultiple" is, but comparing ratios is a constrained enough problem that I don't know what else it could be.
You're right, it is! I'm embarrassed to admit I stopped reading before that point.
I think the "Eudoxus reals" construction is worth thinking about too. It builds directly on integers instead of building on top of the rationals. https://ncatlab.org/nlab/show/Eudoxus+real+number
I wish history recorded how Eudoxus got to this. There was a paper speculating (iirc) that the Greeks started by thinking about continued fractions as a way to deal with incommensurables, and then realized you don't need that much machinery to define them. (You get continued fractions by running Euclid's algorithm on incommensurable input.)
I wish history recorded how Eudoxus got to this. There was a paper speculating (iirc) that the Greeks started by thinking about continued fractions as a way to deal with incommensurables, and then realized you don't need that much machinery to define them. (You get continued fractions by running Euclid's algorithm on incommensurable input.)
Richard Dedekind didn't just invent Dedekind cuts (and rings and ideals); he gave us a new way to think about (or avoid thinking about) the ultimate nature of mathematical reality.
All this assumes that numbers do exist in the physical world as some quantities. They don't. Two apples do not bring the number 2 into existence. Our perception of multiplicity at a macro-scale, is at the root of counting numbers, and it depends on defining the inside and outside of a thing. If you cut those 2 apples into 4 pieces, do the 2 apples still exist? If we grind the pieces into paste, do you still identify them as 2 apples?
Also, cuts are just another way of talking about infinity, which falls outside of the domain for logical foundations of mathematics or the physics of universe. So, cuts are just as alien as infinity.
The logical foundations require existence of multiplicity, comparison (smaller, greater), time-driven causalitiy, true/false and existence to be distinct from non-existence etc. All these would fail when you bring in things like cuts or infinity. So no point in putting them to the logic of our world and claiming that we understood something.
Also, cuts are just another way of talking about infinity, which falls outside of the domain for logical foundations of mathematics or the physics of universe. So, cuts are just as alien as infinity.
The logical foundations require existence of multiplicity, comparison (smaller, greater), time-driven causalitiy, true/false and existence to be distinct from non-existence etc. All these would fail when you bring in things like cuts or infinity. So no point in putting them to the logic of our world and claiming that we understood something.
You're criticizing a strawman of Platonism. The people who believe that numbers such as 2 do exist as some quantities are Platonists, and they do not believe that numbers exist in the physical world. Rather, Platonists believe that the way the physical world exists is just a pale shadow of the way numbers like 2 exist.
Separately, you are advocating finitism, though apparently without much understanding of the logical issues involved, and you seem to be conflating formalism with finitism.
Nobody has come up with a convincing proof of either mathematical Platonism or finitism or their inverses. Consequently, plenty of mathematicians and logicians working at the frontiers of their fields subscribe to any of the four possible combinations of these doctrines.
Separately, you are advocating finitism, though apparently without much understanding of the logical issues involved, and you seem to be conflating formalism with finitism.
Nobody has come up with a convincing proof of either mathematical Platonism or finitism or their inverses. Consequently, plenty of mathematicians and logicians working at the frontiers of their fields subscribe to any of the four possible combinations of these doctrines.
Nobody came up with a proof of finitism because the semantics of a mathematical proof can forbid an expression of reality of the physical world. Physical world abhors infinite and mathematics was supposed to be a tool to describe the physical world. Instead, math went into hallucinations about infinity etc. Physics works perfectly well even when you replace infinite with a relatively large number, giving you as much precision as you need. Physics never needed PI to be calculated to millions of digits, or multiple infinities to be defined. Computers are working fine without this poetry of infinite. I'm not sure why infinity can't be redefined as a relatively large finite number.
> Physics works perfectly well even when you replace infinite with a relatively large number
Has this actually been studied? I reckon you'd run into problems pretty quickly if you tried to impose a "maximum" allowed number (call it N). For example, you might set N equal to the number of quarks in the universe. But now you have no way to represent the number of different configurations of quarks, taking into account their possible quantum states (i.e. taking the factorial). But you also have a time dimension (which I assume you're splitting into discrete chunks to avoid infinitesimals). To decsribe the quantum state over time requires your previous estimate multiplied by the number of time points. And so on... ad infinitum!
Has this actually been studied? I reckon you'd run into problems pretty quickly if you tried to impose a "maximum" allowed number (call it N). For example, you might set N equal to the number of quarks in the universe. But now you have no way to represent the number of different configurations of quarks, taking into account their possible quantum states (i.e. taking the factorial). But you also have a time dimension (which I assume you're splitting into discrete chunks to avoid infinitesimals). To decsribe the quantum state over time requires your previous estimate multiplied by the number of time points. And so on... ad infinitum!
I can't make any sense of your comment; it seems like it has missing words, circular logic, and fundamental misconceptions, but it's not really coherent enough to tell.
ℝ is mind boggling. Here's a puzzler at a layman+ (high school or freshman) level: Between two reals there's at least one rational number [1], and between two rationals there's at least one real number. So there must be an equal number of them. (You can replace "at least one" with "infinitely many" and it's still true.)
[1]: https://math.stackexchange.com/questions/2602418/proof-there...
[1]: https://math.stackexchange.com/questions/2602418/proof-there...
This one always breaks my brain, or at least my intuition. I understand that the reals infinitely outnumber the rationals, and the diagonal proof makes perfect sense to me. But the "between two reals there is always a rational" just beats my intuition like there's candy inside it.
Between any two reals, there’s a whole copy of R.
The original comment assumes its conclusion: it reduces to this smaller copy — then just jumps to the conclusion. It never actually tells us how to measure the ratio.
The original comment assumes its conclusion: it reduces to this smaller copy — then just jumps to the conclusion. It never actually tells us how to measure the ratio.
There's a simple constructive proof using high-school level thinking. ... Two different reals have different decimal expansions. Go out far enough that they differ. Since this is about intuition, let's just assume the larger one is positive and irrational, and thus has an infinitely long expansion. Since the truncation has a finitely long decimal expansion, it's rational. And it's between the two original reals. Q.E.D. ... A full proof for all cases can be built similarly.
It doesn't break my brain that there's a rational between any two reals; it breaks my brain that this doesn't imply equivalent sizes between the reals and the rationals.
> So there must be an equal number of them.
(Pretending to dig out my imaginary high school/freshman hat.)
If you said between every two reals there's one rational (and between every two rational there's one real), I'd assume you meant exactly one (i.e. in the conversational sense), and then there's an equal number of them. But you said at least one, so it could be two or three, so now I'm not sure. And if there are infinitely many of them... but infinity is not even a number, so who knows?!
(Pretending to dig out my imaginary high school/freshman hat.)
If you said between every two reals there's one rational (and between every two rational there's one real), I'd assume you meant exactly one (i.e. in the conversational sense), and then there's an equal number of them. But you said at least one, so it could be two or three, so now I'm not sure. And if there are infinitely many of them... but infinity is not even a number, so who knows?!
“So there must be an equal number of them” is not correct.
Q is indeed dense in R, but firstly it’s very clear that there isn’t an equal number of them because rational numbers are a subset of the real numbers and there exists at least one irrational number (I pick “e”) that is in R but not in Q. So R must be at least bigger than Q.
Additionally you can’t say that between any two rationals there must be a real number because all rational numbers are also real numbers. You can say that Q is dense in R, but if you try to say R is dense in Q what you’re trying to say is “the bits of R that are not in Q are dense in the rest of R” which ends up with a bit of set logic to just be the same statement as the first one.
Q is indeed dense in R, but firstly it’s very clear that there isn’t an equal number of them because rational numbers are a subset of the real numbers and there exists at least one irrational number (I pick “e”) that is in R but not in Q. So R must be at least bigger than Q.
Additionally you can’t say that between any two rationals there must be a real number because all rational numbers are also real numbers. You can say that Q is dense in R, but if you try to say R is dense in Q what you’re trying to say is “the bits of R that are not in Q are dense in the rest of R” which ends up with a bit of set logic to just be the same statement as the first one.
> Q is indeed dense in R, but firstly it’s very clear that there isn’t an equal number of them because rational numbers are a subset of the real numbers and there exists at least one irrational number (I pick “e”) that is in R but not in Q. So R must be at least bigger than Q.
This isn't a correct explanation, because I can use the same explanation to show that there are more integers than that there are even integers.
"it’s very clear that there isn’t an equal number of them because even numbers (let's call it E) are a subset of the natural numbers (let's call that N) and there exists at least one odd number (I pick 1) that is in N but not in E. So N must be at least bigger than E."
This isn't a correct explanation, because I can use the same explanation to show that there are more integers than that there are even integers.
"it’s very clear that there isn’t an equal number of them because even numbers (let's call it E) are a subset of the natural numbers (let's call that N) and there exists at least one odd number (I pick 1) that is in N but not in E. So N must be at least bigger than E."
But as I explain in another thread, that doesn’t apply because E and N are in 1-to-1 correspondence which is not the case with Q and R
That’s what makes this statement incorrect:
> firstly it’s very clear that there isn’t an equal number of them because rational numbers are a subset of the real numbers and there exists at least one irrational number (I pick “e”) that is in R but not in Q
There are N not in E, but E and N have the same cardinality.
You have a second technical mistake as well:
> Additionally you can’t say that between any two rationals there must be a real number because all rational numbers are also real numbers.
They’re obviously referring to Q as a subset of R, and for any two elements of subset Q there is indeed a member of R not in Q.
> firstly it’s very clear that there isn’t an equal number of them because rational numbers are a subset of the real numbers and there exists at least one irrational number (I pick “e”) that is in R but not in Q
There are N not in E, but E and N have the same cardinality.
You have a second technical mistake as well:
> Additionally you can’t say that between any two rationals there must be a real number because all rational numbers are also real numbers.
They’re obviously referring to Q as a subset of R, and for any two elements of subset Q there is indeed a member of R not in Q.
Why does this work? Just because there is 1 number which is in R but not in Q doesn’t prove anything, does it?
If it did, why are Even N and (Even + Odd N) the same size?
If it did, why are Even N and (Even + Odd N) the same size?
Even N and (even + odd) N have the same size because there is a 1 to 1 correspondence between the sets. For every even number x in N I can pair it with x/2 which is also in N and that gives me all of N so N must be at least as large as even N. Likewise for x in N I can pair x with 2x and that gives me all the even N, so even N must be at least as large as N. So N and even N must have the same size.
However, while I can pair every rational number up with a real number, if I try to go the other way I find there is no rational number to pair with some (actually infinitely many) real numbers. I picked one, e, to show this. So the real numbers must be strictly larger than the rationals. Because the rationals are a strict subset of the reals it’s simpler than the Even/odd natural numbers because you don’t need the correspondence- every x in Q is in R but there exists at least one x in R not in Q, and no amount of correspondence shenanigans can get around that so |R|>|Q|.
However, while I can pair every rational number up with a real number, if I try to go the other way I find there is no rational number to pair with some (actually infinitely many) real numbers. I picked one, e, to show this. So the real numbers must be strictly larger than the rationals. Because the rationals are a strict subset of the reals it’s simpler than the Even/odd natural numbers because you don’t need the correspondence- every x in Q is in R but there exists at least one x in R not in Q, and no amount of correspondence shenanigans can get around that so |R|>|Q|.
> every x in Q is in R but there exists at least one x in R not in Q, and no amount of correspondence shenanigans can get around that so |R|>|Q|.
This also applies to evens and naturals: evens are a strict subset of naturals and there exists naturals which are not evens.
A counter example in the reals is (0,1) and R have the same cardinality, despite there being real numbers not in the interval (0,1).
You have the fact incorrect: for infinite sets, all B being a strict subset of A shows is that |A| >= |B|.
This also applies to evens and naturals: evens are a strict subset of naturals and there exists naturals which are not evens.
A counter example in the reals is (0,1) and R have the same cardinality, despite there being real numbers not in the interval (0,1).
You have the fact incorrect: for infinite sets, all B being a strict subset of A shows is that |A| >= |B|.
> So there must be an equal number of them
There definitely are not. Rationals are countable, reals are not.
It's tricky to prove things about the sizes of infinite sets, and "there's infinitely many of them interleaved with each other" doesn't do it, at least to my knowledge. I don't know of a way to prove equipotence except by creating a 1-1 relationship between the sets in question.
There definitely are not. Rationals are countable, reals are not.
It's tricky to prove things about the sizes of infinite sets, and "there's infinitely many of them interleaved with each other" doesn't do it, at least to my knowledge. I don't know of a way to prove equipotence except by creating a 1-1 relationship between the sets in question.
> I don't know of a way to prove equipotence except by creating a 1-1 relationship between the sets in question.
The usual definition of two sets having the same size (cardinality) is that they are in bijection. So ultimately, any proof that |A| = |B| will reduce to a "creating a 1-1 relationship between the sets in question", or at least showing that one exists.
The usual definition of two sets having the same size (cardinality) is that they are in bijection. So ultimately, any proof that |A| = |B| will reduce to a "creating a 1-1 relationship between the sets in question", or at least showing that one exists.
Ok cool, I didn't remember that was the actual definition.
> I don't know of a way to prove equipotence except by creating a 1-1 relationship between the sets in question.
That relies on first defining it as "there is 1-to-1 mapping". But for sequences of numbers you could instead compare growth rates. Then the natural numbers and the even natural numbers wouldn't be the same size. Which is nice, because the latter sequence is a proper subset of the former. The former grow twice as fast, so you could say they are twice as large. But it only works for ordered sequences. Anyway, it all depends on definitions.
That relies on first defining it as "there is 1-to-1 mapping". But for sequences of numbers you could instead compare growth rates. Then the natural numbers and the even natural numbers wouldn't be the same size. Which is nice, because the latter sequence is a proper subset of the former. The former grow twice as fast, so you could say they are twice as large. But it only works for ordered sequences. Anyway, it all depends on definitions.
You can get infinitely many of such perplexing puzzles when you bring in infinity into our logic framework. The existence of these nonsensical puzzles prove that infinite doesn't fit into our logical framework, and one can get any number of astounding and amusing results they want, when you mix common mathematical logic and infinite.
You'd think so, but there aren't equal numbers of them.
I had a subpar mathematics education growing up which prevented my seriously learning it for a long time. Now, everything I know about math stems from my own, self-directed reading, and papers like yours (if you are the same person) are an absolute godsend. Thank you for all your work. I'm looking forward to digging into this post, in either case, as it seems to fit within the same category!