Not because I revel in destruction
A dessert-first policy. The answer was on page 21!
In this monthly News&Notes issue:
Upcoming events:
🕺In-person in Berkeley: Saturday, January 11th (next week)
࿇ Vajrayana Q&A Zoom
🙋 Live video AMA
Two nearly-posts about my creative process:
🧨 Not because I revel in destruction: dessert first! Getting to the good bits.
ℝ The answer was on page 21! Constructing the real numbers.
Stuff you may have missed:
🗒️ Best short videos 📺 and Substack Notes from the past month
🕺 Berkeley in-person events: yidam!
My spouse
and I will co-host an in-person discussion on Saturday, January 11th, from 6–9 p.m. (That’s a week from today!)We'll focus on yidam practice during our conversation, but we're happy to discuss any Vajrayana topic. Bring your questions and curiosities; or leave a comment here to let us know what else you'd like to hear about:
You do need to register to attend. It’s by donation (thank you!). The event will also be live-streamed, if you can’t make it in person.
Full-day retreat, January 12th
The next day, Sunday, January 12th, Charlie will host a four part day-long retreat: “A Day of Contemporary Vajrayana Practice.” (I won’t be there for this myself.) This would be a great follow-on for our joint event!
Charlie will introduce a contemporary Avalokiteshvara yidam practice. (That’s Avalokiteshvara in the photo above.) Plus! Methods from Charlie’s Liberating Shadow and Kaleidoscope of Interaction courses.
࿇ Vajrayana Q&A: also January 11th
Vajrayana is the unusual branch of Buddhism I discuss here on Substack, and on Vividness and Buddhism for Vampires. I offer live Zoom Q&A gatherings monthly. You can read more about the purpose and format of these sessions, and how to access them, here.
The next one is on Saturday, January 11th. It’s at 10:30 a.m. Eastern / 7:30 a.m. Pacific.
🙋 Substack Live video AMA: January 26th
I offer a monthly AMA (“Ask Me Anything”) via the Substack Live video feature. You can read more about these, and how to participate, here.
The next one will be Sunday, January 26th, at 9 a.m. Pacific Time; noon Eastern.
It helps me a lot if you suggest questions ahead of time, so I’ve opened a chat thread for that.
You can watch a twenty-minute excerpt from last month’s here:
🧨 Not because I revel in destruction
I have a recurring problem of an apparent negativity bias. Of the writing I have made public so far, too large a fraction is refutations of harmful wrong ideas. For example, of eternalism and nihilism, and rationalism, and the whole of philosophy.
That’s not because I revel in destruction! I mean, I do revel in destruction! But I revel even more gleefully in creating beautiful and useful new things!
The trouble is, the things I most want to create don’t look beautiful or useful at first glance. And they belong in places already occupied by things that come with attractive packaging and big advertising promises. Things like eternalism and rationalism and philosophy: their glossy exteriors conceal their harmful defects.
So if I just say “hey, here’s a better product!”, no one’s going to bother looking beyond its ugly box. The meta-rationality book is about how to use rationality accurately and effectively. But everyone with a STEM education thinks they already know that! What they know is rationalism. And rationalism promises, or at least stage-whisper hints, that you can achieve definitive understanding, emotional certainty of correctness, and control over your world.
My alternative can’t make any such promises. To use it, you have to accept nebulosity, which negates all those. However, meta-rationality can deliver better yet imperfect understanding; and a merely sensible degree of confidence; and significant influence, without absolute control.
So to make what I have to say available, I’ve believed that I first have to point out why the other leading brand is guilty of false advertising. And this may seem tedious and unpleasant! I wish there were a better way of going about this.
I have several major writing projects, and they each have this same problem, and they’re all incomplete. So mostly all you see is the negative preliminaries, because I haven’t gotten to the good bits yet!
I wish I could just jump to the good bits. Maybe some people would understand and appreciate them without the deconstruction. Maybe I will try that this year!
I tried to do that last year, with the meta-rationality book. I jumped ahead to the good bit! It’s Part Four, which is the Part of the book which is actually about meta-rationality. Part One just explains why the rationalist theory of how rationality works is wrong. Part Three explains how rationality actually does work, and Part Two is a preliminary for that. So let’s get to the best bit, and eat dessert first!
However, Part Four, which is supposed to be the dessert, or maybe I should say the main course, got completely out of control. There’s just way too much fascinating, beautiful, useful stuff to say! It may be unfinishable. At least in its current form, and in my lifetime. So I backburnered it several months ago.
I still think this book is probably the most important thing for me to work on, but I need to do a major re-think. Either I have to find a way to write a shorter version that’s still understandable, or I need to pace myself to spend several years on it. I hope to figure this out early in 2025.
In the mean time, “Undoing Philosophy” is much easier, and probably finite. I’ve been rethinking it somewhat, too! My current take on the main points:
Philosophy is bad. It has done pervasive, enduring damage to our ability to understand our lives and loves and work.
We already have pretty-good non-philosophical ways of making sense of our lives and loves and work. Our natural ways of understanding life work better than philosophy, although imperfectly. Similarly, the ways we already actually do use rationality work much better than the ways philosophical theories recommend.
We can think, feel, and act—within the natural way—more enjoyably and beneficially than we typically do. That’s the point of Meaningness. We can use rationality more accurately and effectively than we typically do. That’s the point of Meta-rationality.
If you read them as philosophy, as abstract “interesting ideas,” you will miss whatever value they may have. So the overall aim of “Undoing philosophy” is to encourage you to read Meaningness and Meta-rationality as practical pointers toward improving your ways of engaging with concrete circumstances.
This is not just a theoretical concern. From on-site comments, it seems many people do fall into this trap.
Nevertheless, maybe I will take a “dessert first” approach with “Undoing Philosophy.” I may just skip ahead now. Not bother to explain the ways philosophy is harmful, or what it actually is, or the history of how it got that way, or how it is that I don’t do it. Just talk about how we can improve our ways of thinking, feeling, and acting!
ℝ The answer was on page 21
In “When the proof comes as white light and angels,” I described a formative experience of asking a math professor a question he was unable, or unwilling, to answer: “What is a real number?”
(“Real” numbers are all the ones on the “number line.” They include “irrational” numbers like the square root of two, and “transcendental” numbers like π.)
I’ve had dreams about this in the week since writing it. In past I’ve tweeted the story, and told it in person, several times. It was emotionally significant, apparently, in ways I don’t fully understand. My best guess is that it provoked a sense of betrayal—that even here, at MIT, professors either don’t know the answers to basic questions, or refuse to answer them for some reason they also refuse to explain. The priests of the temple of learning have feet of clay.
And so, I must have concluded, I am still, once again, on my own. It is up to me to learn what I want to learn. And so, as my story goes, I spent a couple days in the library tracking down the answer: “Dedekind cuts, or Cauchy sequences.” Which meant: I can find things out for myself, even when the priests will not speak.
(Now, of course, you can get the answer instantly from a web search or chatbot. But that was decades in the future, and if you wanted to know something in 1980, you had to physically go to a library and hunt around in the card catalog and then the stacks. It’s astonishing how much we did know, considering the obstacles!)
I mentioned in “White light and angels” that the textbook we used in that class was Rudin’s Principles of Mathematical Analysis. I illustrated the post with a couple of snapshots of my copy—the only textbook I’ve kept from my student days, for its sentimental value. That class was the one in which I first got a glimpse of what it means to do mathematics, and a sense that this was possible for me.
I got it down from the shelf yesterday. I hadn’t looked in it in forty-four years. The first sentence of Chapter 1 is:
A satisfactory discussion of the main concepts of analysis must be based on an accurately defined number concept.
Yes, that was my point!
So, all right, why didn’t you do that? I flip through the rest of Chapter 1, which explains various properties of the real numbers without actually saying what one is. How annoying!
Blah, blah, blah, Appendix to Chapter 1, blah, blah, … WTF ⁉️
The members of ℝ will be certain subsets of ℚ called cuts.
(“ℝ” means “the real numbers” and ℚ is the rational numbers.)
Several pages of dense proof-stuff… flip, flip, flip, page 21:
The cuts in ℚ which we used here were invented by Dedekind. The construction of ℝ from ℚ by means of Cauchy sequences is due to Cantor.
So… wait…
The answer to my question was in Chapter 1 of our textbook!
Why didn’t the professor just say that?!
Why did Rudin bury what should have been the first thing in the book in an Appendix?
I flip back to the start of Chapter 1, and see there’s a Preface before it:
Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, Dedekind’s construction is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time seems ripe.
Folded into the book are my hand-written summary notes for the course. They cover Chapters 2–9. Apparently, we didn’t do Chapter 1 at all. The professor thought that our intuitive understanding of real numbers, from high school math, was adequate to tackle his course’s agenda. The aim was to teach the two main topics of introductory calculus—differentiation and integration—with fewer lies. He never read Chapter 1 himself, because he thought he too knew well enough what a real number was, and didn’t care about logical foundations.
As I observed in “Philosophy isn’t…”,
I love logic, but most mathematicians don’t like it. They use it as little as possible, and wish it would go away.
So… now, forty-four years later, I finally understand what happened in that emotionally critical interaction in math class!
Out of curiosity, I did a little web investigation of Rudin’s book. It is still commonly used, in the same edition, without revision. It is considered a classic, and one of the most famous mathematics textbooks ever written. Here’s 28 seconds of someone enthusing about it in 2023:
🗒️ Best videos and Notes from the past month
I use Substack Notes for short videos, and for bits of writing that aren’t worked-out enough for full Posts. You can see all my Notes here. I also include some of the more popular, interesting, or substantial ones in my Monthly News&Notes posts. Here they are!
If the question is “what is a real number?”, then Dedekind cuts or Cauchy sequences are relevant, but still not quite the answer. Those are methods of “constructing” the reals.
The real numbers are best defined axiomatically — as an ordered field with the least-upper-bound property — with the aforementioned construction methods being treated as existence proofs. The real numbers are a self-consistent mathematical pattern with a particular bundle of properties that relate to other things we care about in especially useful ways.
Side note: the reason we can speak of “the” real numbers is that any two models of those axioms are canonically isomorphic. This fact and its proof should also be regarded as part of the existence proof, just as important as showing that Dedekind cuts (or whatever) satisfy all the necessary algebraic properties and so on.
Sometimes mathematicians will say things like “we define the real numbers to be sets of rational numbers such that...”. This is especially likely in a formal text. However, this is a lie. If you take it at face value you will end up believing things which are mathematically meaningless by convention, such as “the rational number 2 is a member of the real number π”, which is utter nonsense.
ok this is tragic!
i learned about Dedekind cuts (and Cauchy sequences, I think?) in a math summer camp in high school. it took weeks to get there and was presented as a grand finale, and of course we got hyped up for it.
tbh i don’t know what i’d have done without math camp. high school classes don’t have proofs apart from geometry, college classes don’t spend enough time & attention teaching you how to prove (and there’s too much time & grade pressure getting in the way) so unless you’re smart & motivated enough to be totally self-taught (I wasn’t) or have unusual early educational opportunities like me, you’re just screwed! you will not learn (higher) math! mathematics departments do not prioritize teaching you!
nobody cares because few people learn higher math at all, and those who do are strongly pressured to identify student confusion with student stupidity/laziness rather than bad pedagogy.
if we talk about the problem at all it’s in a ~woke way that too often conflates making math accessible with watering it down.
There’s no constituency for teaching a given curriculum more effectively -- ie if you’re teaching arithmetic, ensuring most students learn arithmetic; if you’re teaching calculus, ensuring most students learn calculus; if you’re teaching a first proof-based class, ensuring most students learn to write proofs.
My sister teaches college math and she is basically alone on team “AcTuAlLy TeAcH sKiLls” as opposed to “sift for geniuses who don’t need teaching” or “make the class pathetically easy so nobody gets bad grades”.