# Comments on “Robots That Dance”

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### Styles of thinking

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.

### robotics

Hi David,

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

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.)

### *air quotes* insight *air quotes*

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

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…

### Found some stuff

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.

and

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

### Becoming the official recommendator

@Duckland: Russell T. Hurlburt is who you want for the most precise, most encompassing description of thinking phenomenology(ies)

### Huh, that's really

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”.

### Styles and intelligence and stuff

David,

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

### Personality Cults and Foundations

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.

### Ah. Eternally F5ing that page

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.

### Chaitin

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!

### David's Intervention

“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.

### Wow! Chaitin: CS theory as post-rational performance art

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.

### Selfish reasons

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.

### Clarity

Thanks for the links! I was expecting something much more arcane, but it looks like I already have the necessary background via programming.

### 13 Ways Godel Proved God is Real

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

### Mathematical Platonism

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!)

### Zalamea on post-analytic philosophy

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 atheorematic manner: the register ofuniversals capable of unmooring themselves from any 'primordial' absolute, relative universals regulating the flow of knowledge."

### Qualifications

Zalamea is a professor of mathematics, so that much is definitely solid. There’s also a detailed review which I haven’t finished reading.

### Zalamea review review

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:

- "contemporary" (1950-2000) mathematics got over the foundational crisis and started studying questions like "what different kinds of foundations are possible? what different things are they useful for? how can we translate ideas from one branch of mathematics to another?"
- analytic philosophers of mathematics are focused on ZF set theory and predicate logic, and mostly have no idea about any of this new stuff; some continental philosophers have done better
- it might be fun to invent some philosophies based on other parts of math!

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!

### Two types of mathematicians

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.

### Nerd matchmaking based on

Nerd matchmaking based on autistic niche obsession might be one of my favorite parts of the internet.

Bookmarking your tumblr.

### thanks!

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.

## related robotics work

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.

Rob