Leveling up technical work with context and purpose
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Likely you already know this if you’ve been following robotics, but I know of work on both compliant robots and emotional/social stuff.
Compliance has advantages for energy efficient walking as well as safety. One example is rethink robotics Baxter, http://www.rethinkrobotics.com/rethink-robotics-collaborative-robots/ (Rod Brooks founder) even in the late 90’s when I jumped to robotics, the MIT leg lab was already using compliant actuator assemblies.
I don’t know any details about emotion and social, but I’ve run across this at CMU. This is part of projects for nursing and tour guide robots. Attempts to read facial expressions, and to generate approximate expressions.
Completely agree with your point about representing ongoing behavior as a dynamic system, rather than (say) a symbolic structure. As someone who has done a lot of control work with high dynamics actuators, the main way that I interpret situated or embodied intelligence, is that vigorous behaviors such as sports or dancing involve inserting neural controller dynamic systems inside feedback loops embedded in the physical environment. For example, the “how to catch a fly ball” example.
The whole paragraph about kinesthetic representation etc has been on my mind for a couple weeks. I know Lakoff is a good source for that kinda stuff. Can you recommend anything more?
In exchange I’ll offer this paragraph from Ulam about von Neumann:
It seems curious to me that in the many mathematical conversations on topics belonging to set theory and allied fields, von Neumann even seemed to think formally. Most mathematicians, when discussing problems in these fields, seemingly have an intuitive framework based on geometrical or almost tactile pictures of abstract sets, transformations, etc. Von Neumann gave the impression of operating sequentially by purely formal deductions. What I mean to say is that the basis of his intuition, which could produce new theorems and proofs just as well as the "naive" intuition, seemed to be of a type that is much rarer. If one has to divide mathematicians, as Poincaré proposed, into two types—those with visual and those with auditory intuition—Johnny perhaps belonged to the latter. In him, the "auditory sense," however, probably was very abstract. It involved, rather, a complementarity between the formal appearance of a collection of symbols and the game played with them on the one hand, and an interpretation of their meanings on the other. The foregoing distinction is somewhat like that between a mental picture of the physical chess board and a mental picture of a sequence of moves on it, written down in algebraic notation.
The excerpt above really fascinates me but also seems to leave some detail out. How does he get from tactile representation to visual representation? What’s the relation between logical-symbolic representation and auditory representation? I haven’t been able to track down Poincare’s writings on this in particular to try to make sense of it.
A fun little tidbit: Jaron Lanier talks about experiments in VR where they’re trying to teach children via embodiment. E.g., becoming a triangle or molecule.
Thanks for these fascinating posts on your work in machine intelligence etc. It mind/blowing for sure, especially the way philosophical, cognitive and intelligent machine research overlap.
I am trying to imagine the complexity involved in two self-learning machines dancing with each other and, by means of internal representational feed-back loops and machine communication, creating a machine culture – an internal memory, a “personal”history”, a machine “subjectivity” and a machine inter-subjectivity or evolving culture.
What does it means to use the term human in contrast to intelligent machine —is there a difference other than a structural one? If there is a difference what is it? if there is not what are we in relation to a machine. I know — its master; or its slave . Anyway some sort of power relation is involved for sure, which says a lot about what it means to be a political animal. What will machine politic be like I wonder?
Introspectively, it seems to me that much of even abstract reasoning, when I’m proving a theorem for example, involves imagining performing bodily operations on imaginary spatial objects. (The vocabulary of mathematics supports this; we talk of “retractions” and “surgery” and “pumping” for example.)
That seems barely relatable to me.
I’ve proved plenty of things and program a lot. Neither activity involves anything that resembles sight, touch, moving, space-with-things-in-it (like a room with furniture) or any controlled bodily action at all, except maybe on the rare occasion when I draw a flow diagram.
The only direct sensations I have related to abstract reasoning are frustration, which is kind of like a mild (and sometimes, bad) headache, and untidiness, which is the painful feeling when things aren’t arranged neatly, when rituals aren’t followed, when something is off, when patterns are broken-and-it’s-not-a-joke, typically the thing you fight with activities that start with “obsessive”. (That is a bit vague, but it’s a very specific kind of pain all of these things invoke for me. Mostly a form of muscle tension.)
What, literally, do you mean with that description?
That you see things that somehow correspond to things about your proof, and you can make them move, transform them, etc?
That you have the sensations of moving a real object around, like a small wooden cube say, and that somehow corresponds to proof properties? If so, how?
Could you please be as literal and explicit as possible? I find all talk about phenomenology frustrating because half the people are literal and half the people are metaphorical, and they typically think only their version is real at all, and thus logically the only way things can be. (And even when they are literal, they use super-vague terminology like “spatial”.) (ie Galton’s visualization survey and all that)
Good example how to do it right: Feynman talking about how he counts by talking in his mind vs another student seeing a labeled tape that moves forward; or how he visually imagines objects as part of a proof, like starting with an empty ball and adding hair as a stand-in for property A, and making it green for B, etc
(PS: Obviously there can be variation in how people do things, but if some people are competent in X, and they don’t use Y as far as anyone can tell, then Y is not a fundamental requirement of X, and unlikely to be very important.)
Rob—Thanks, yes, compliant motion was just coming in as the hot new thing when I left MIT in 1990.
I was going to put some stuff about Baxter in a footnote, but didn’t because the Rethink Robotics web site has absolutely no content; it’s all advertising hype, so I couldn’t tell anything about how it works. I half-remember some details from things I read when they first announced, but didn’t want to rely on half-memories.
Re “emotions and social,” I came across this a few days ago:
which frames it as “look how sexist everything is”, but has some interesting info on work in this area, being done by women (most of whom I know vaguely from 25 years ago; it was interesting seeing what they are up to now!).
Duckland (and relevant also to the comment from Anonymous): Thanks for that quote from Ulam; I had not seen it before!
From talking with other math students, I concur that there are people who do math purely formally, by symbol manipulation, and others who do it visually and/or kinaesthetically, by active imagination. I would describe the former as “linguistic” rather than “auditory,” although I suppose I do hear the symbol names in my head as I read them. “Dee ecks dee tee equals two pi theta squared.”
I started out being able to do math only formally, which made calculus and differential equations a major pain for me. My roommate could just imagine the shape of a surface and see the integral as a simple solid, whereas I had to fight my way through a thicket of squiggly symbols. After my first year of required courses in continuous math, I switched to discrete math, and particularly logic, where it seemed that the formal approach was more the way to go. Ironically, the ability to do math visually/kinaesthetically kicked in sometime late in my sophomore year, so I did mathematical logic kinaesthetically and differential equations formally, which is probably exactly wrong in both cases. (In some spare lifetime, I’d like to re-do differential equations, with feeling!)
I think, though, that the most power comes from being able to work back and forth between the two. That is, facility both at turning a string of symbols into a visual/kinaesthetic model, and facility in turning your experience manipulating the model back in formal terms.
This seems closely related to the skill of reasoning back and forth between a program text and a mental model of the state of the runtime environment during program execution. This seems to be critical to learn for skilled debugging.
My gigantic outline says that the “meta-systematic cognition” series will include a post about this. The tentative title is “What they don’t teach you in STEM school”—a play on the (excellent) book “What they don’t teach you at Harvard Business School.”
Also in your comment:
I know Lakoff is a good source for that kinda stuff. Can you recommend anything more?
Uh… the only thing that is coming immediately to mind is Heinz Werner‘s work, which is quite relevant, but I don’t think I’d actually recommend it. (Very dated conceptual framework.) Maybe I’ll think of something else later.
Can anyone else recommend something?
What does it means to use the term human in contrast to intelligent machine ---is there a difference other than a structural one
Well, so far we don’t have intelligent machines (depending on what you mean by “intelligent,” and on the whole I’m quite skeptical about the rate of progress in AI, so I don’t expect them soon. Since there aren’t any, it’s hard to think usefully about what they might be like, or how we might relate to them.
Anonymous—Unfortunately, it would take a long essay to answer your questions in a way that would do justice to them!
It sounds like you may be one of those people who, like von Neumann in the Ulam quote, works entirely in formal domain. Since von Neumann was quite a decent mathematician :-) , as you say, imagery may be “unlikely to be very important.” On the other hand, I found math much easier once I learned it. (I might not be quite as smart as von Neumann, so I need all the help I can get!)
That you see things that somehow correspond to things about your proof, and you can make them move, transform them, etc? That you have the sensations of moving a real object around, like a small wooden cube say, and that somehow corresponds to proof properties?
Yes to all those. I’m sorry that going into detailed examples would take more time than I can spare at the moment. (Partly because it’s been decades since I proved anything!) Feynman’s examples are good and similar to my experience also.
Gian-Carlo Rota (with whom I studied universal algebra and category theory, and who was a close friend of Ulam’s) said “there’s two kinds of proof: ones where you can draw a picture, and then you actually understand what’s going on; and the kind where you grind through all the steps and verify they are correct, but you have no idea what it means.” He apparently only approved of math done kinaesthetically/visually, although he was certainly capable of the other mode as well.
David, interesting that you started off formal and picked up the visual/kinesthetic way of working. I’m very much the opposite. I’d like to become better at the game-with-symbols stuff so that I can, as you point out, switch between them when I want.
I’m fascinated with the idea of different cognitive styles for doing math (in particular). It would seem the primary senses involved are sight, touch, and to a lesser extent hearing. What about taste and smell? Are they of no value at all for math? Why? Also, how does the symbolic fit in here? I think the visual sense is involved in the symbolic way of working. For example, when I remember a formula I see an image of the symbols, and when a formula is slightly wrong we say it “looks off”. Or, when one is looking to simplify an algebraic expression one recalls “seeing” a similar pattern before.
It seems plausible to me that there may be rare and valuable styles of doing math that could be taught deliberately if we paid enough attention to different styles.
Anonymous, I can give some specific examples. When I remember what the Intermediate Value Theorem says I tend to visualize a well-behaved continuous function defined between two points on the x-axis, and then imagine a vertical line in between the two points intersecting the graph. Once I have that image getting the symbolic statement of the theorem isn’t much trouble.
In R^n with the usual metric, the closest distance between a line and a point off the line is just the length of the segment starting from the point that runs orthogonal to the line and then meets the line. Draw a picture in R^2 and this is totally obvious.
I can give more examples but those seem sufficient. I find that combinatorics problems are most amenable to visualization and the tactile sense. As part of my obsession with these issues I like to watch my teachers’ bodies as they teach. Some of them never really use their body, while others do almost constantly. For example, when my topology teacher recalls the general definition of continuity in topology as inverse images of open sets being open he would gesture with his finger as if moving the open set from the co-domain to the domain. I find this amusing since topology is arguably one of the most spatially counterintuitive subjects.
Another interesting tidbit, iirc Korzybski was the one who pioneered air quotes with your fingers. He believed that using the body in this way would reinforce the intention of the air quotes – this term isn’t well-defined, etc.
Would be interesting to think about how this ties into mudras etc. I’m going off the rails at this point. Would love to hear any thoughts.
David, I was reading through something about von Neumann and I recall someone saying that von Neumann was actually envious of mathematicians who worked in a way different from him! Gotta track it down now…
Very interesting.. So the passage I pasted above was from a von Neumann eulogy type article written in 1958. Here’s another excerpt from that same article:
In spite of his great powers and his full consciousness of them, he lacked a certain self-confidence, admiring greatly a few mathematicians and physicists who possessed qualities which he did not believe he himself had in the highest possible degree. The qualities which evoked this feeling on his part were, I felt, relatively simple-minded powers of intuition of new truths, or the gift for a seemingly irrational perception of proofs or formulation of new theorems.
Then, in a conversation with Rota I found,
For a man of his stature he was curiously insecure, but his understanding, intelligence, mathematical breadth, and appreciation of what mathematics is for, historically and in the future, was unsurpassed.
It is curious to me that in our many mathematical conversations on topics belonging to set theory and allied fields, he always seemed to think formally. Most mathematicians, when discussing problems in these fields, seem to have an intuitive framework based on geometrical or almost tactile pictures of abstract sets, transformations, and such. Johnny gave the impression of operating sequentially by formal deductions. His intuitions seemed very abstract; they involved a complementarily between the formal appearance of a collection of symbols, the games played with them, and the interpretation of their meanings. Something like the distinction between a mental picture of the physical chess board and a mental picture of a sequence of moves on it written down in algebraic notation!
The latter passage is very similar to the one I pasted above but differs in some ways. Fascinating that styles differ enough that even von Neumann was envious. Sorry if this was too nerdy and lengthy! Hope it’s valuable for someone else
@Duckland: Russell T. Hurlburt is who you want for the most precise, most encompassing description of thinking phenomenology(ies)
Huh, that’s really interesting info about von Neumann!
Unfortunately, I’m also super low on time, so I’m not gonna engage with the rest of the post. I’m trying to stick to the principle of “first cause the Singularity, then correct people who are wrong on the internet”.
Though I did sketch out a few pieces, and just wanna post this fragment, useless as it may be, because I like some of these words:
There’s two parts: the physical operation you perform on something that acts as a scratchpad, which may be an image in your head, or an ink shape on paper, or noises you subvocalize. That’s the transformation to get from A to B, which can vary drastically. While mechanically necessary for thinking, it isn’t really the interesting part of thinking.
And then there’s the part that makes goals happen, that tells you what shape to turn something into, which is pure instant magic. Every time I think about it, I feel like a reinforcement-based, demon-possessed p-zombie.
The phenomenology of thinking should really be titled “Confessions of a Lookup Table”.
(Combined reply to the last several comments.)
Silver V.—Thanks for the Hurlburt recommendation; I had never heard of him. Looks very interesting from the brief googling I’ve done so far!
Duckland—I find cognitive style fascinating in general! There’s not much written about it, that I know of.
I would be happy if anything were taught about how to do math! No one was willing to explain that when I was a math student (with the partial exception of Rota). Perhaps this was only a local phenomenon. Have you had any useful advice from math professors about how?
For example, when my topology teacher recalls the general definition of continuity in topology as inverse images of open sets being open he would gesture with his finger as if moving the open set from the co-domain to the domain.
That’s a wonderful example, yes!
how this ties into mudras
Yow! That’s a connection I would never have made, but yes I think it might be productive…
Anonymous—I’m not sure I agree that only one part is interesting, but your two-part phenomenology seems accurate.
In obsessing about this stuff I have found some books of interest. Handbook of Intellectual Styles edited by Zhang et al. Visualization, Explanation and Reasoning Styles in Mathematics edited by Mancosu et al. Psychology of Invention in the Mathematical Field by Hadamard. Dunno how links are working atm but there’s also a “Mathematical Styles” article on SEP.
The former two sounded promising, but unfortunately they seem to be dry, academic, and mostly not interesting to me. If you look at the tables of content you’ll see.
The Hadamard book seems the most promising. Hadamard was actually an accomplished mathematician who went through sincere efforts to survey many professional mathematicians. I’ve yet to read it carefully. I feel that it would have been good to cite in your How to Think Real Good article. It illustrates well that even math, the most rational subject, proceeds by non-mechnical processes.
No, none of my teachers have talked about learning math. At most, one will hear banalities about “study habits”. It seems to me that accommodation or even consideration of learning style is rare to nonexistent. It occurred to me after one of my previous posts that it may be that mathematicians inclined to write textbooks overrepresent a certain mathematical style. Then, other styles may be going through unnecessary strain in learning from them.
On the other hand, I understand reasons why people mock this styles point of view. At its extreme people claim that “everyone is intelligent in their own way” etc. The strong, dogmatically egalitarian form of this argument seems silly to me. But, certainly there are different dimensions and styles of intelligence. To me that’s exciting. I’d like to understand and nurture my own style while trying to pick up others (as you say in Think Real Good).
I wonder, have you read Art of Learning by Josh Waitzkin? Some of the themes he explores there are relevant to this conversation, and even your whole project (imo).
Btw, it’s sort of strange to think that you knew Rota. I have Rota’s autobiography ready to read. Got any inside scoops? :P
Thanks—I knew about the Hadamard book, but not the others. I haven’t read Hadamard; if you do and it’s good, please let us know!
I’ve thought about writing a page about my memories of Rota, but I didn’t know him well, so it’s probably not useful. He was an inspiring teacher in many ways. For me, the most important thing was that he did explain a little about how math is done. Much less than I wished for, but more than any of my other teachers.
He was highly charismatic and quirky, and created—or allowed to develop—a sort of personality cult around himself. I was wary of that, so I didn’t join the “inner circle.”
Will certainly report back if I find something interesting. I’m really not just saying that. It’s rare to find people up for discussing this, it seems to me anyway.
Interesting about the personality cult. When I read the bio I’ll keep it in mind.
This is almost entirely off topic at this point. But, do you have any opinions about the foundations of math? Historically or today? It seems to me that if we can’t even find a strong foundation for math how will we have a foundation for any systematic thought? This seems to fit in perfectly with your writing.
Yes, supposedly I’m going to write about the problem of mathematical foundations on this page, which currently only contains a brief summary.
We had a conversation about this also a couple weeks ago in this comment thread: http://meaningness.com/stem-fluidity-bridge/comments
Ah. Eternally F5ing that page.
Laughed at the “Godel woo”. True, and unfortunate since I think it obscures that Godel’s theorems were actually important. Are you going to write of the theorems of Church, Turing, Tarski, and Chaitin? I just Googled all those names together and didn’t find a good source that includes them all among commentary on the limits of formal systems. Seems to me to be a glaring omission.
Chaitin is an interesting case since he’s still alive and writes/talks openly about what he believes are the philosophical consequences. He’s said in a few places that he thinks mathematicians are largely just ignoring the results of Godel, in a sense.
Are you going to write of the theorems of Church, Turing, Tarski, and Chaitin?
Probably not in any detail, because this is well-understood stuff, and had no significant cultural impact outside geekdom, as far as I know.
Chaitin is perhaps an exception to both of those. I think his recent-ish rants are excellent and highly relevant to Meaningness. Explaining them to anyone who doesn’t have the mathematical background would be extremely difficult, though.
Could either of you suggest some especially relevant work from Chaitin? I probably don’t have the background to understand him just yet, but I’m working on it!
Yes, http://arxiv.org/html/nlin/0004007 is a great, lively, clear overview. It’s a talk transcript. It’s a good introduction to the whole conundrum of mathematical foundations, as well. (Presented from a somewhat computer-science-centric point of view; there would be a different historical emphasis if you were doing an overview from the point of view of pure logic.)
The rules of the game should be absolutely explicit, it should be an artificial language and then mathematics will give you absolute truth. “Absolute truth” should be underlined in a very beautiful font and you should hear the angels singing when you say these words! This was the thought that we mathematicians have absolute truth. It's ours—no one else has it, only us! That was the idea. So it turns out this doesn't quite work.
That’s stage 4 rationalist eternalism—and then in the end it doesn’t work.
You read essays by Hermann Weyl or John von Neumann saying things like this: I became a mathematician because this was my religion, I believed in absolute truth, here was beauty, the real world was awful, but I took refuge in number theory. And all of a sudden Gödel comes and ruins everything, and I want to kill myself!
There’s stage 4.5 nihilism!
I must say that philosophers have not picked up the ball. I think logicians hate my work, they detest it! And I'm like pornography, I'm sort of an unmentionable subject in the world of logic, because my results are so disgusting!
This is the “messiness” of stage 5.
Chaitin is great because his math is absolutely solid, there is no woo anywhere, and it just totally implodes mathematical rationalism.
“Chaitin is great because his math is absolutely solid, there is no woo anywhere, and it just totally implodes mathematical rationalism.”
Word. I feel the same.
Chaitin has his own sense of what the resolution should be – something like treating math as more empirical. Do you agree? Will your sense of the way forward be laid out in that placeholder article you linked?
“Probably not in any detail, because this is well-understood stuff, and had no significant cultural impact outside geekdom, as far as I know.”
Yes, but aside from a few wiki pages I don’t really see any one place where all of this addressed in a way such that an educated laymen can understand it. True, there are plenty of Godel books, but I’ve yet to find one that mentions the whole history of Discoveries in the Limits of Formal Systems: Foundational Crises in Math (something like what the book should be called).
I’m waiting on several of your articles. You’re such a tease. Everyone who knows David irl send him numerous copies of productivity self-help books. The sheer vapidity of it all should kick him into high gear.
Yeah, this is all stuff I really want to write about. It’s important, and it’s bizarre that there’s no book Discoveries in the Limits of Formal Systems: Foundational Crises in Math. There should be one. Done well and properly marketed, it might even be highly commercially successful.
However, there are way too many other things I want to write about, and I try to focus on ones where I have something unusual to contribute. There’s lots of people who could write well about the foundational crisis—and better than me—even if no one is.
From my notes, two relatively short overviews worth reading:
I’d heard of computational information measures, but didn’t understand that the whole area was based on proofs that its object of study was intractable. In hindsight it seems kind of inevitable that, just as almost all numbers are irrational, the universe of possible specific formal statements (programs, etc.) is almost entirely filled with objects with deny any sort of generalization. (A meta-generalization, I guess.)
I also like the framework of the “number of bits of axioms needed”. I’ve always been pretty skeptical of the idea that Godel incompleteness says anything good about human mental capacities. (the common spin that “we can see it’s true, but a formal system can’t so…”) I’ve been pretty intensively following neuro and genetics stuff for a while, and what we’re learning is that, if regarded as formal systems, they require lots and lots of bits of axioms. Evolved systems resist being compressed down to humanly comprehensible narratives because comprehensibility wasn’t a design requirement.
Neat. Thanks for the notes.
I respect that your time is limited. But, perhaps if it adds a bit of motivation for doing that page you linked with more detail and completeness [teehee] (when you get around to it), I’ll explain why I personally think it’s important.
Many educated people probably think, “Yeah, sure, no system gives us all the answers. But, we still have science to get us incrementally closer to truth, and math (its foundation) to give us absolute truth”. There are variants of this like we’ll-compute-our-way-to-truth, or compute-our-way-to-heaven or whatever.
This quoted statement is plausible and I think I agree with a qualified and weaker form of it. The problems I see are at least:
1) When you actually look at the most prominent philosophers of math and science you don’t find much agreement. Similarly when you look at the philosophies of prominent mathematicians and scientists.
2) The limits of knowing is a big theme in 20th century math and physics (despite Godelian, quantum, and relativity appropriation, ‘woo’, etc)
In my limited observation, scientists/mathematicians look down upon 1) as “just philosophy, I’ve got real work to do”, etc. Seen Neil deGrasse Tyson’s comments like this? Then, for 2) they sort of ignore it or say “well math still works”, “QM doesn’t really say there’s limits to our knowing”, etc.
Which leads me to wonder, “Precisely how much should we put our faith in math and science?” I’ve yet to come across a clear exposition by a qualified person that really takes on the question honestly. It troubles me.
Thanks for the links! I was expecting something much more arcane, but it looks like I already have the necessary background via programming.
the idea that Godel incompleteness says anything good about human mental capacities. (the common spin that "we can see it's true, but a formal system can't so...")
Yeah, that’s “Gödel woo.” It’s closely similar to “quantum consciousness” woo. It’s not even wrong, because it’s not stated clearly enough, but it’s clear that any similar statement that’s actually coherent would be factually false.
Both are emotionally attractive because they suggest that we humans have some special ontological significance that allows us to transcend mere mechanism, and therefore all our spiritual fantasies of omnipotence and immortality might somehow be true.
Yes, the foundational problems in math seem to be ignorable in practice. That wasn’t clear in the early 20th century; it genuinely seemed like math might completely collapse. When I learned math in the early ’80s, there was still hidden worry about this. I’m not sure if that’s true now.
“This system does seem to work reliably, but it has some internal problems and limits, so it’s not quite clear why; and it’s not altogether clear how it relates to things outside the system; so we ought to try and understand those issues better, starting empirically” is a characteristically “stage 5” approach. It’s what I’d advocate for math.
As far as science goes, we suddenly have the “replication crisis,” which to me seems like a huge step forward. The underlying problem has always been there; the step forward is that many people are admitting it, and trying to figure out how to do better.
This seems like a major cultural shift away from rationalist eternalism. That may have extensive repercussions even outside science, comparable to the early-20th-century foundational crises with relativity and quantum.
What’s really important (in terms of the “civilizational collapse” risk) is to make sure the public understands that science can work, and can’t be replaced with woo. This may be difficult to get across.
Yes, the math is not really difficult.
I think a foundations course ought to be taught to every math undergraduate, along with a “how to do math” course. Both of those are unthinkable in the current cultural climate.
I believe that the foundational problems are deliberately censored or suppressed by math departments, because the senior faculty are almost all committed to some version of mathematical eternalism. (Mathematical Platonism, mostly, in practice.)
I believe that “how to do math” is deliberately suppressed in order to make the field more difficult, thereby inflating mathematician’s feelings of being special and superior people.
Godel was himself a Platonist, and hoped to prove Platonism using rigorous methods. So, I think Godel thought that his theorems provided evidence that there’s something special going on with humans. I’m personally not willing to rule out the possibility that there’s something special going on with humans. I only believe it on Wednesdays though.
Godel published a short list of 14 of his philosophical views. For anyone interested, Google “Godel fourteen philosophical views”. List should be in all of the top results.
“Yes, the foundational problems in math seem to be ignorable in practice. “
I’m not so confident about this. Maybe if we add on the qualifier “for now”. Also, what a person considers ignorable would seem to differ.
The “replication crisis” thing is amusing, but it doesn’t seem to include the hard sciences so it isn’t all that interesting to me. I have the least confidence of all in soft sciences already.
“What’s really important (in terms of the “civilizational collapse” risk) is to make sure the public understands that science can work, and can’t be replaced with woo. This may be difficult to get across.”
I’m really not sure about this either. What does it mean for science to work? Certainly there are tangible results like technology, medicine, etc. But, people have the urge to then conclude “yeah, so science is real and true“. I’m not so sure that that follows or what it even means. It seems to me that if there’s not a clear alternative view for our relationship to science then rationalist eternalism is going to continue to be compelling.
“I believe that “how to do math” is deliberately suppressed in order to make the field more difficult, thereby inflating mathematician’s feelings of being special and superior people.”
^This x 1000.
I probably shouldn’t be trying to work out my philosophical angst in this comment thread. Hope someone else is getting something out of this
Ironically, from a really thoroughly (Neo-)Platonist perspective all of this seems deeply satisfying and exciting.
Like, guys, you’re from the physical plane and its laws freak you out; did you expect the World of Forms to somehow be less freaky than quantum mechanics? Besides, don’t you know that Unknowability is even closer to God than Truth? It’s like the Academy isn’t even teaching basic emanation theory these days.
(Chaitin’s QM connection is really fun to think about from this angle!)
Duckland— Thanks, I hadn’t see that list of Gödel’s! It’s a really nice statement of rationalist eternalism. I know little about him; his aim, then, was disproving logicism/formalism in favor of Platonism, I guess?
I'm personally not willing to rule out the possibility that there's something special going on with humans.
I agree. On the other hand, if there is, it probably has nothing to do with the mathematical foundations issues. And, it’s unlikely to underwrite our spiritual fantasies. (But these are just guesses!)
what a person considers ignorable would seem to differ
Definitely; I spent a lot of time working with the foundational questions, and didn’t consider them ignorable at all! But for a mathematician working in, say, the outer reaches of analytic geometry, I think ignoring is fine (unless/until there’s some dramatic unexpected new foundational crisis).
people have the urge to then conclude "yeah, so science is real and true". I'm not so sure that that follows or what it even means.
if there's not a clear alternative view for our relationship to science then rationalist eternalism is going to continue to be compelling
I hope so! My point in that post is that rationalist eternalism is not quite right, but it’s less bad than anti-rationalism, so we’d better keep it going for a while. And we’d better get that better alternative worked out and communicated to the public soon, because there’s increasing public knowledge that the underpinnings of science are somewhat shaky.
Dan— This is really funny! Thank you.
A friend just recommended to me Fernando Zalamea’s Synthetic Philosophy of Contemporary Mathematics, which seems highly relevant (though possibly beyond my reading level!). This passage in particular caught my eye:
The entire work of contemporary mathematics, carefully recounted by Zalamea, aimed at the production of "remarkable invariants … without any need of being anchored in an absolute ground. We will therefore take up a revolutionary conception which has surfaced in contemporary mathematics in a theorematic manner: the register of universals capable of unmooring themselves from any 'primordial' absolute, relative universals regulating the flow of knowledge."
That does sound highly relevant! If anyone has read this, I’d love to hear more.
FWIW, the first thing I’d do is see whether he actually understands the math he writes about, or if he’s relying on natural-language explanations of it. If the latter, is likely to be nonsense (but not necessarily). If he does know the math, it still might be nonsense. I’ve come across some pomo stuff written by serious French mathematicians that, as far as I could tell, was Sokal-style gibberish. I don’t know what to make of that!
Zalamea is a professor of mathematics, so that much is definitely solid. There’s also a detailed review which I haven’t finished reading.
Thanks for that, Dan! I’ve started reading it, and it seems promising, but I’m short for time so I haven’t gotten very far yet. I hope to finish this coming weekend perhaps.
In case you never finished that: I’ve now read every review I could find of SPCM, and some excerpts, so maybe I can spare you the trouble!
The consensus seems to be that it’s solid; it’s been reviewed by several mathematicians and philosophers, and I can’t find anyone accusing it of being nonsense. At the same time, it is very math-heavy, and while professional mathematicians seem to like it a lot, non-mathematicians find it mostly incomprehensible. (They figure it’s over their head but probably not nonsense.)
Since I am not a mathematician my understanding is vague. However, the gist seems to be that:
Part Two of the book illustrates the new Stage-5-style (?) math with a long series of case studies from formal logic, reverse mathematics, topology, algebraic geometry, model theory, etc. etc. etc. If you can follow the explanations, this part’s probably interesting in its own right.
Zalamea draws some philosophical conclusions in Part Three, making connections between C.S. Peirce and (mostly) category theory, but you apparently have to have understood the case studies to make much sense of it. Otherwise, as my friend puts it, it is “a somewhat eye-glazing combination of the incomprehensible and the vague”.
Still, again, it seems like whatever he’s saying does mean something to qualified readers. My favourite bit of jargon was “the sheafification of epistemology”.
If you find yourself with a copy, a quick “is it worth reading this” test might be to try skimming the first few pages of Chapter 7 (dense math) and the first few pages of Chapter 9 (dense philosophy).
Hope that helps!
Thank you—that is very interesting and helpful!
I really want to dive into this—I was holding off replying until I had taken another look—but realistically I’m not going to get a chance for some considerable while. I hope other readers are motivated to investigate further, however!
Ah, the discussion on this article is really interesting! The subject of learning styles in mathematics really fascinates me, but I’ll try not to ramble on too much here.
Poincaré’s distinction between two types of mathematician that Duckland mentioned in their first post is in ‘Intuition and logic in mathematics’ - there’s a translated version here. It’s part of a small genre that fascinates me: mathematicians dividing mathematicians into two groups, which normally roughly correspond to symbolic-algebraic/intuitive-geometric. I made a start at assembling a list on my disorganised tumblr blog here. (I’m very much in the second category and found the abstract approach hard going in my degree, which is how I got interested in the subject in the first place.)
Not sure I’m convinced that advice on ‘how to do maths’ is deliberately suppressed - I always thought noone bothered for more-or-less the same reasons as Rota describes in his differential equations rant:
Why is it that no one has undertaken the task of cleaning the Augean stables of elementary differential equations? I will hazard an answer: for the same reason why we see so little change anywhere today, whether in society, in politics, or in science. Vested interests dominate every nook and cranny of our society, even the society of mathematicians. A revamped elementary differential equations course would require Professor Neanderthal at Oshkosh College to learn the subject anew.
Personally, I’d love to hear your memories of Rota :) I found his books inspiring when I was drowning in a sea of unexplained formal derivations that I had no intuition for.
Thanks for all that!
Have been browsing your tumblr; it’s great fun!
Nerd matchmaking based on autistic niche obsession might be one of my favorite parts of the internet.
Bookmarking your tumblr.
Duckland: haha, thanks! Yes, it’s always nice to know you’re not the only one obsessed with something, particularly once your real-life friends are all sick of the subject! The blog will be quiet for a little bit, but I’m sure to write more of this stuff soon.
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