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.
IIRC this is how we were taught in the first-term analysis course at Oxford: first the real numbers were introduced axiomatically, then we did a lot of material about convergence and divergence of sequences and series, then we were shown Cauchy sequences and proved that they obeyed the axioms. We used Mary Hart's textbook, which may not be an acknowledged classic but *is* admirably clear.
I think this is the pedagogical norm nowadays in reasonably good analysis courses. You can't do any analysis without knowing that the reals are a complete ordered field, so you might as well go ahead and prove that they are the *only* complete ordered field. But giving an actual construction is kind of stupid. It's quite obvious that the real numbers exist, or can be treated as existing--we can see the number line in our heads, after all. The constructions of Dedekind and Cantor are a symptom of an unhealthy mania in mathematics between about 1850 and 1950 for foundations, which was largely quite destructive in much the same way that any rationalism is.
I think we are talking about totally different approaches. "It's quite obvious that the real numbers exist" would defeat the real purpose of undergrad real analysis as I was taught it (25 years ago, gosh), which is to break students of the habit of reasoning loosely - at least until they've learned the formalisms necessary to turn their intuitive reasoning into rigorous proofs. If that's your real goal, then it's *essential* to give a construction of the reals at some point, as a proof that they're well-defined - it's all very well to reason axiomatically, but what's the point if your axioms don't admit a model? Student David was quite right to demand a construction; his professor was wrong to leave it out; Rudin was right from a paedagogical standpoint not to start with one (for most students, at least), but wrong to consign it to an appendix rather than folding it into the course at a later point, when students had enough experience with the necessary machinery.
I agree completely! The axioms of real numbers is what makes it useful and intuitive, and the construction of it often make it alien and weird.
I went back and looked up the textbook my professor based his lectures on: Analysis I by VA Zorich. It introduces the axioms in Chapter II. Given the preliminaries is often skipped, this would be introduced in the first lecture of a course. This approach is not without problem though: a good portion of my classmates got destroyed by this approach, even though I liked it a lot.
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”.
I'm a mathematician (no longer in academia, but many of my friends are) and I'll endorse most of this critique, at least when it comes to big or medium-sized research-focused universities like the places I did undergrad and grad school. The sad truth is that a lot of those professors see themselves as researchers first and educators second, and the incentives are set up to reinforce that. Many of them *personally* care about being good teachers, but they kind of have to figure out how to do that on their own; as it sounds like your sister is experiencing, there's not much institutional support, and that's generally not where you pick up career advancement points.
Also, I don't really have data to back this up, but my strong impression is that almost all of the energy that *is* being directed at improving undergrad math education is coming from ed-school types rather than math professors, and it's mostly directed at stuff like calculus rather than the proof-based math we're talking about here, which I think gets basically none of this sort of attention. And I'm not claiming that this has resulted in huge improvements to calculus classes!
Having said all that, David, I do think you might be extrapolating a bit too much from the experience you had in that analysis class at MIT. I'm certainly not defending your instructor, but I don't know that it's typical. For what it's worth, I took real analysis twice, once at the non-flagship-campus state school in my hometown, and once at the fancy-pants university where I did undergrad, and both times we definitely spent time early on constructing the reals. In the first class we did Dedekind cuts --- in fact using Rudin, which I agree is not a great textbook --- and in the second we did Cauchy sequences. Especially the second time around, I think the teacher did a very good job explaining that the point was to "fill in the gaps" in Q, which helped it not to feel like a weird symbol-pushing game from Mars.
I agree with every word of Plasma Messick's comment above. Your instructor definitely dropped the ball pretty badly, but I think if a student asked that question in a real analysis class today there's a decent chance they'd get an answer like Plasma Messick's. Depends on the instructor, of course, but it's basically exactly what I say to the students I work with in my tutoring practice.
I have to say that your description of mathematicians' personalities doesn't really sound like most mathematicians I know. Maybe I got lucky and all my math friends are unusually inquisitive people, but a lot of us care plenty about foundational questions! (And I'd be pretty shocked if someone I knew managed to get a pure math Ph.D. without ever learning that the real numbers can be constructed using Dedekind cuts or Cauchy sequences.) Depending on the field they might not need to think about foundational issues every day to do their research, but most mathematicians I know are aware of them.
I'm not sure there's no constituency for teaching arithmetic more effectively. People have worked a lot and very hard on solving that problem, and we do actually manage to teach most kids arithmetic (of integers, not fractions) these days although it's not clear whether that depends much on any pedagogical insights as opposed to just spending a lot of time on it. I agree there's no constituency for teaching proof-based mathematics well. It would be a kind of strange endeavor somehow since the only people who need to be able to write and understand proofs are mathematicians, and only a small proportion of even those students talented or lucky enough to make it through the standard poor curriculum will become mathematicians. Actually, my Russian program leader had exactly this kind of existential ambivalence about the success of his program--almost all of us ended up going to math grad school, and if teaching proof-based math well to a wider audience has that as a result, that's probably actually bad!
My equivalent was going to Russia for a semester, doing nothing but math, getting insulted a lot for being like a small Russian child as an American college junior, and coming back having jumped ahead about three years relative to my peers in the one semester.
I am very critical of general status of maths education, because the education experts are overly confident about general arguments devoid of context. A statistical analysis on whether homework is effective for learning is pointless, because how the homework is designed will make all the difference. Maths education research should be about the mathematical content and how the students are actually understanding it.
Another really annoying trend is that maths teachers are getting less education in mathematics and instead spending time studying education theories of questionable usefulness.
I for one am a fan of the negative portions where you explain whats wrong with certain concepts. A lot of that stuff is a set of mental tricks that are difficult to see through for the average person and pointing out what the trick is makes it much more obvious why it's wrong.
When it comes to philosophy I believe that many people turn to philosophy for self help as much as intellectual masturbation. They are confused about high level concepts that they can neither categorize nor articulate, and they are kinda aware that philosophy supposedly helps with that. So they read a bunch of it and find some answers that have the flaws you describe.
I think your work actually does what most people want from the "self help" mode of philosophy by clearly answering many of the questions that most people ask philosophy, like stuff about meaning, thinking good, religion and spiritual relations, etc. So that's what I mean when I say that I think you do "philosophy but good".
One of the "chapters" in the draft of "Undoing Philosophy" is "I don't do philosophy." It ends by addressing the objection "well, what DO you do, then?"
I offer several answers, but "it should get put on the self-help shelf in the bookstore" is probably the most convincing.
(The stuff that gets shelved there does not generally count as philosophy....)
Maybe it's the most convincing, but it's definitely insufficient. If we're to ditch "philosophy", there needs to be a positive short term for the good parts of what's generally associated with that bucket. With your other projects you propose positive replacements for flawed notions, like eternalism, nihilism -> complete stance, rationalism -> meta-rationality, so why doesn't philosophy get this treatment?
Well... I guess I think that once you remove the metaphysics, philosophy is no longer a cohesive thing. It was a bunch of unrelated bits stuck together with metaphysics. So individual bits worth rescuing, if any, would be better given separate names. Ethics, for example, is probably a fairly cohesive domain even after you rip the metaphysics out of it; and we can call that "ethics" unless someone comes up with a better word.
Fair enough. But is there a decent word for the domain about building world models and theories about how they relate to the world and to each other? I've seen some LessWrong people using "world modeling", but nobody else, including you.
Well... is that a coherent domain? If so, is it the same as "science"? (Also a vague term, but many definitions are pretty close to "building world models and theories about how they relate to the world"?)
I'm not sure myself if it's coherent, but I think that science mainly refers to work within established paradigms, while pre-paradigmatic stuff is relegated to philosophy. When you say things like "reality is both nebulous and patterned" or "there is a crisis of meaning", pretty much nobody would consider those scientific statements, but since you're so vehemently against calling them philosophical, it seems like an alternative positive term is called for.
+1 to going positive and going iterative. I would('ve) thought you'd thoroughly internalized "useful transmissions/writing need not be complete" but it sounds like Ch 4 needs some similar treatment to go iterative?
The content of Part Four will be mainly unfamiliar to most readers. This is an expositional challenge! Especially because there is so much of it.
My sense is that it needs a lot of structural overview first, so people can orient to where they are in the whole. However, different people learn differently. I need that map first, but many people do better with concrete examples first. So... not clear how to proceed.
Especially, again, as there is so much of it. Too much. So what can be left out while still leaving what remains comprehensible and useful?
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.
Yes. This would have been a good answer to my question :)
This MathOverflow post gives an IMHO great explanation of the relationship between the axiomatic treatment of e.g. the real numbers and the set-theoretic construction, and the true value of the latter: https://mathoverflow.net/questions/90820/set-theories-without-junk-theorems/90945#90945
IIRC this is how we were taught in the first-term analysis course at Oxford: first the real numbers were introduced axiomatically, then we did a lot of material about convergence and divergence of sequences and series, then we were shown Cauchy sequences and proved that they obeyed the axioms. We used Mary Hart's textbook, which may not be an acknowledged classic but *is* admirably clear.
I think this is the pedagogical norm nowadays in reasonably good analysis courses. You can't do any analysis without knowing that the reals are a complete ordered field, so you might as well go ahead and prove that they are the *only* complete ordered field. But giving an actual construction is kind of stupid. It's quite obvious that the real numbers exist, or can be treated as existing--we can see the number line in our heads, after all. The constructions of Dedekind and Cantor are a symptom of an unhealthy mania in mathematics between about 1850 and 1950 for foundations, which was largely quite destructive in much the same way that any rationalism is.
I think we are talking about totally different approaches. "It's quite obvious that the real numbers exist" would defeat the real purpose of undergrad real analysis as I was taught it (25 years ago, gosh), which is to break students of the habit of reasoning loosely - at least until they've learned the formalisms necessary to turn their intuitive reasoning into rigorous proofs. If that's your real goal, then it's *essential* to give a construction of the reals at some point, as a proof that they're well-defined - it's all very well to reason axiomatically, but what's the point if your axioms don't admit a model? Student David was quite right to demand a construction; his professor was wrong to leave it out; Rudin was right from a paedagogical standpoint not to start with one (for most students, at least), but wrong to consign it to an appendix rather than folding it into the course at a later point, when students had enough experience with the necessary machinery.
I agree completely! The axioms of real numbers is what makes it useful and intuitive, and the construction of it often make it alien and weird.
I went back and looked up the textbook my professor based his lectures on: Analysis I by VA Zorich. It introduces the axioms in Chapter II. Given the preliminaries is often skipped, this would be introduced in the first lecture of a course. This approach is not without problem though: a good portion of my classmates got destroyed by this approach, even though I liked it a lot.
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”.
I'm a mathematician (no longer in academia, but many of my friends are) and I'll endorse most of this critique, at least when it comes to big or medium-sized research-focused universities like the places I did undergrad and grad school. The sad truth is that a lot of those professors see themselves as researchers first and educators second, and the incentives are set up to reinforce that. Many of them *personally* care about being good teachers, but they kind of have to figure out how to do that on their own; as it sounds like your sister is experiencing, there's not much institutional support, and that's generally not where you pick up career advancement points.
Also, I don't really have data to back this up, but my strong impression is that almost all of the energy that *is* being directed at improving undergrad math education is coming from ed-school types rather than math professors, and it's mostly directed at stuff like calculus rather than the proof-based math we're talking about here, which I think gets basically none of this sort of attention. And I'm not claiming that this has resulted in huge improvements to calculus classes!
Having said all that, David, I do think you might be extrapolating a bit too much from the experience you had in that analysis class at MIT. I'm certainly not defending your instructor, but I don't know that it's typical. For what it's worth, I took real analysis twice, once at the non-flagship-campus state school in my hometown, and once at the fancy-pants university where I did undergrad, and both times we definitely spent time early on constructing the reals. In the first class we did Dedekind cuts --- in fact using Rudin, which I agree is not a great textbook --- and in the second we did Cauchy sequences. Especially the second time around, I think the teacher did a very good job explaining that the point was to "fill in the gaps" in Q, which helped it not to feel like a weird symbol-pushing game from Mars.
I agree with every word of Plasma Messick's comment above. Your instructor definitely dropped the ball pretty badly, but I think if a student asked that question in a real analysis class today there's a decent chance they'd get an answer like Plasma Messick's. Depends on the instructor, of course, but it's basically exactly what I say to the students I work with in my tutoring practice.
I have to say that your description of mathematicians' personalities doesn't really sound like most mathematicians I know. Maybe I got lucky and all my math friends are unusually inquisitive people, but a lot of us care plenty about foundational questions! (And I'd be pretty shocked if someone I knew managed to get a pure math Ph.D. without ever learning that the real numbers can be constructed using Dedekind cuts or Cauchy sequences.) Depending on the field they might not need to think about foundational issues every day to do their research, but most mathematicians I know are aware of them.
I'm not sure there's no constituency for teaching arithmetic more effectively. People have worked a lot and very hard on solving that problem, and we do actually manage to teach most kids arithmetic (of integers, not fractions) these days although it's not clear whether that depends much on any pedagogical insights as opposed to just spending a lot of time on it. I agree there's no constituency for teaching proof-based mathematics well. It would be a kind of strange endeavor somehow since the only people who need to be able to write and understand proofs are mathematicians, and only a small proportion of even those students talented or lucky enough to make it through the standard poor curriculum will become mathematicians. Actually, my Russian program leader had exactly this kind of existential ambivalence about the success of his program--almost all of us ended up going to math grad school, and if teaching proof-based math well to a wider audience has that as a result, that's probably actually bad!
My equivalent was going to Russia for a semester, doing nothing but math, getting insulted a lot for being like a small Russian child as an American college junior, and coming back having jumped ahead about three years relative to my peers in the one semester.
I am very critical of general status of maths education, because the education experts are overly confident about general arguments devoid of context. A statistical analysis on whether homework is effective for learning is pointless, because how the homework is designed will make all the difference. Maths education research should be about the mathematical content and how the students are actually understanding it.
Another really annoying trend is that maths teachers are getting less education in mathematics and instead spending time studying education theories of questionable usefulness.
Yes, strong agreement with all that! If I had a few spare lifetimes, improving mathematics education is the way I would spend one.
I for one am a fan of the negative portions where you explain whats wrong with certain concepts. A lot of that stuff is a set of mental tricks that are difficult to see through for the average person and pointing out what the trick is makes it much more obvious why it's wrong.
When it comes to philosophy I believe that many people turn to philosophy for self help as much as intellectual masturbation. They are confused about high level concepts that they can neither categorize nor articulate, and they are kinda aware that philosophy supposedly helps with that. So they read a bunch of it and find some answers that have the flaws you describe.
I think your work actually does what most people want from the "self help" mode of philosophy by clearly answering many of the questions that most people ask philosophy, like stuff about meaning, thinking good, religion and spiritual relations, etc. So that's what I mean when I say that I think you do "philosophy but good".
Thank you! I feel understood :)
One of the "chapters" in the draft of "Undoing Philosophy" is "I don't do philosophy." It ends by addressing the objection "well, what DO you do, then?"
I offer several answers, but "it should get put on the self-help shelf in the bookstore" is probably the most convincing.
(The stuff that gets shelved there does not generally count as philosophy....)
Maybe it's the most convincing, but it's definitely insufficient. If we're to ditch "philosophy", there needs to be a positive short term for the good parts of what's generally associated with that bucket. With your other projects you propose positive replacements for flawed notions, like eternalism, nihilism -> complete stance, rationalism -> meta-rationality, so why doesn't philosophy get this treatment?
> so why doesn't philosophy get this treatment?
Well... I guess I think that once you remove the metaphysics, philosophy is no longer a cohesive thing. It was a bunch of unrelated bits stuck together with metaphysics. So individual bits worth rescuing, if any, would be better given separate names. Ethics, for example, is probably a fairly cohesive domain even after you rip the metaphysics out of it; and we can call that "ethics" unless someone comes up with a better word.
Fair enough. But is there a decent word for the domain about building world models and theories about how they relate to the world and to each other? I've seen some LessWrong people using "world modeling", but nobody else, including you.
Well... is that a coherent domain? If so, is it the same as "science"? (Also a vague term, but many definitions are pretty close to "building world models and theories about how they relate to the world"?)
I'm not sure myself if it's coherent, but I think that science mainly refers to work within established paradigms, while pre-paradigmatic stuff is relegated to philosophy. When you say things like "reality is both nebulous and patterned" or "there is a crisis of meaning", pretty much nobody would consider those scientific statements, but since you're so vehemently against calling them philosophical, it seems like an alternative positive term is called for.
+1 to going positive and going iterative. I would('ve) thought you'd thoroughly internalized "useful transmissions/writing need not be complete" but it sounds like Ch 4 needs some similar treatment to go iterative?
Thanks for the vote!
The content of Part Four will be mainly unfamiliar to most readers. This is an expositional challenge! Especially because there is so much of it.
My sense is that it needs a lot of structural overview first, so people can orient to where they are in the whole. However, different people learn differently. I need that map first, but many people do better with concrete examples first. So... not clear how to proceed.
Especially, again, as there is so much of it. Too much. So what can be left out while still leaving what remains comprehensible and useful?
Any chance you are recording the yidam talk?
The venue (Berkeley Alembic) apparently will. My understanding is that to get access, you need to register for the event (even if you won't be there in person), and then the Alembic will send you information about how to access the live stream, or watch the recording later. (That's all I know! You could contact the Alembic directly if you want more explanation.) https://www.eventbrite.com/e/an-evening-of-vajrayana-conversation-with-david-chapman-and-charlie-awbery-tickets-1105055032349
Amazing. Thank you David!