DRAFT: Three Intellectual Temperaments: Birds, Frogs and Beavers

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32 comments, sorted by Click to highlight new comments since: Today at 4:19 PM

I skimmed most of this because I can't handle quote-dumps. I wanted to comment similarly on some of your earlier maths quote-dumps but since you said there that you were just trying to organise your thoughts, I assumed that you would convert more of the quotes into prose.

On the post itself: The 'beaver' part kind of appears out of nowhere, I suggest putting more foreshadowing/summarising at the beginning. I'm also not sure I understand what a beaver does that's different to the other groups. Frogs and birds seem to straightforwardly correspond to bottom-up and top-down thinking, or Sensing versus Intuitive in Myers-Briggs jargon. Beavering seems quite top-down to me.

Frogs and birds seem to straightforwardly correspond to bottom-up and top-down thinking

This seems like an intuitive summary to me. Let me elaborate on why.

One way the birds might build theories is by noticing the possibility of large patterns, and then asking themselves questions about such patterns. Perhaps they might be thought of as solving blurry *conceptual* problems like "How are algebraic geometry and algebraic number theory connected?". To build theory they would have to ask progressively more specific questions, until their answers could be written as mathematical definitions.

The frogs could instead start by asking a very concrete question like "Do the prime numbers contain arbitrarily long arithmetic progressions?". If the question couldn't be solved with elementary means they would have to develop new concepts, and understand the relationships between these concepts. To do this they could ask progressively more abstract questions, until they had built enough theory to solve their original problem.

I skimmed most of this because I can't handle quote-dumps. I wanted to comment similarly on some of your earlier maths quote-dumps but since you said there that you were just trying to organise your thoughts, I assumed that you would convert more of the quotes into prose.

I've wondered what to do about this. I like the idea of quoting really good people verbatim at length because (a) they have more credibility than I do and (b) I feel a little squeamish about paraphrasing them for fear of skewing the truth.

Would appreciate more detailed suggestions here if you have any to offer.

On the post itself: The 'beaver' part kind of appears out of nowhere, I suggest putting more foreshadowing/summarising at the beginning.

Okay, right.

I'm also not sure I understand what a beaver does that's different to the other groups. Frogs and birds seem to straightforwardly correspond to bottom-up and top-down thinking, or Sensing versus Intuitive in Myers-Briggs jargon. Beavering seems quite top-down to me.

As I said to ThomasR, my subjective impression is that there are examples both of bird/beaver hybrids and frog/beaver hybrids. Maybe there's a bird vs. frog axis and an independent axis measuring beaver-likeness. There is *something* real that I'm trying to get at here, but I'll have to think more about what it is.

Felix Klein may be seen as a "bird/beaver" hybrid, in view of the calculational view on complex multiplication and class fields in the 19th century, leading to modular equations etc. Klein's "icosahedron" and Weber's 3 vol. "Algebra" (the best until v.d. Waerden's book and E. Noether's school). link A more modern example may be this, but I know only a part of the story.

I just remember a very nice online docu on the Chudnovsky brothers, an old NY'er article , the "one mathematician in two brains".

I think if you retain the first part minus Zeilberger's quote, and then "correlation" sections on frogs and birds, it would make a stronger piece than if you try to salvage the beaver part. Also, "beavers" seem to be more about rigor/foundations/logic (mathematical methodology), not so much computation.

I think if you retain the first part minus Zeilberger's quote, and then "correlation" sections on frogs and birds, it would make a stronger piece than if you try to salvage the beaver part.

Makes sense - I think I'll follow your suggestion, just briefly indicating that there may be other categories and referencing Zeilberger and Klein.

You go on a lot about birds and frogs, but including so many examples of writing about them seems sort of superfluous to me. I found the two concepts pretty easy to think about, and the added detail didn't do much to increase my understanding.
(You cite them later which is great, but I feel like where the reffered to idea is vital you could just say "**___** said ** __**" and summarize the point you want to use, rather than give us their whole quote to wade through)

Beavers, on the other hand, were introduced abruptly, and then not explained nearly as much. The Beaver classification seemed like an interesting idea to me, but I was sort of disappointed by their coverage.

The article raises some interesting ideas, but I feel like you did a disservice to them by focusing so much on birds and frogs, and so little on beavers, which seem to be the more novel part of your analysis.

I hope that's helpful.

Upvoted, even though I can't really see the appeal to non-mathematicians.

I think you may have gotten beavers confused somewhat. The image I get from Zeilberger's description is that beavers try and build algorithms to solve general problems. I don't think plain computation is enough. I think of them more as birds that like algorithms instead of theorems.

It seems to me that Zeilberger wanted to say computational tools are important, but ended up saying that tool-builders are a distinct personality type. Whyyy? They're kinda out of place in a discussion of mathematics. Okay, Mathematica was built by beavers. How about R? Python? Microsoft Excel? Researchers use all those :-)

Thanks for the feedback.

Upvoted, even though I can't really see the appeal to non-mathematicians.

So, I suspect that the bird/frog/beaver categories are relevant to intellectuals in general, not just to mathematicians; will have to say something about this in the introduction of my revised post.

I think you may have gotten beavers confused somewhat. The image I get from Zeilberger's description is that beavers try and build algorithms to solve general problems. I don't think plain computation is enough. I think of them more as birds that like algorithms instead of theorems.

I think that there's a correlation between interest in algorithms and interest the activity of performing algorithms. Unfortunately, it's hard to explain why I think this without giving examples from personal conversations. I'll think about how to address your point.

The beavers, especially, remind me of a classification of preferred action modes-- people who are happiest developing and following procedures. (The other three were implementers, who want to make something happen, improvisers, and planners.)

The whole system wasn't that people fit one category, but that most people have one mode that they strongly prefer, two that they tolerate, and one that they hate. A few people have equal preferences for at least three.

I'm curious about whether beavers split into bird-like and frog-like, with some wanting to develop theory of algorithms, and others wanting to focus in on improving particular algorithms.

The whole article seems like a counterexample to the idea of g (general intelligence)-- in math, even (especially?) the smartest people aren't generally good at all sorts of math. There is strong specialization.

I think this is interesting. Voted up.

I do agree that there's a bird/frog distinction in intellectual personality types. (I'm less sure about the beaver.) Some people gravitate towards "Big Idea" insights which are essentially metaphorical. "We can describe X in terms of Y" is what I mean by metaphorical thinking -- for example, the idea that we can describe social interactions in terms of graph theory, or that we can describe the shape of a surface in terms of groups. The relevant ability here is seeing one concept in light of another concept. If birds are basically metaphorical thinkers, it makes sense that they have varied interests (the better to see connections) and that they have some feeling for the arts (where metaphorical thinking is common.)

I'm a metaphorical thinker (though it's premature to say I have a research style of my own, I do think I can safely say that my brain likes doing some things better than others.) Doing concrete problem-solving work is very important, and I don't think anybody can safely neglect it, but I'm probably in the "bird" zone temperamentally.

I haven't met that many "beavers" and I don't have as clear an idea in my mind of the type. It seems to me that a lot of "frogs" are also interested in puzzles. A puzzle (whether for work or entertainment) is something where metaphorical thinking won't help, and usually neither will an algorithm; a puzzle has a sudden, insightful solution. Codes and languages seem more "beaver-ish" to me, in the sense that you use algorithms to deal with them.

I haven't met that many "beavers" and I don't have as clear an idea in my mind of the type. It seems to me that a lot of "frogs" are also interested in puzzles. A puzzle (whether for work or entertainment) is something where metaphorical thinking won't help, and usually neither will an algorithm; a puzzle has a sudden, insightful solution. Codes and languages seem more "beaver-ish" to me, in the sense that you use algorithms to deal with them.

Your comment is similar to Oscar's I guess. I think (but am not sure) that interest in logic puzzles is correlated with interest in algorithms (perhaps for subtle reasons) even if the process of solving a logic puzzle is different from the process of implementing or analyzing an algorithm.

My previous post on QFT, Homotopy Theory and Ai etc. fits very good to this one, as it is about a special case related to computer science and AI (acc. to their selfdescriptions both a core part of "Singularity"-discussions and fitting to the forum member's areas of specialization/interests), and at the same time part of research programs in which the mathem. mentioned above are active. So I wonder about the strange reactions? (Princeton IAS and Fance's IHES surely are not below the intellectual level here, even if the selfreported IQ's acc. to the survey are above those of Feynman or Grothendieck...)

I upvoted your post and don't know what the people who downvoted it were thinking.

One possible issue is an absence of background. I felt that the content was sufficiently strong so that it deserved to be upvoted, but few people in the audience have the relevant background knowledge, and so maybe they downvoted it because they didn't know what you were referring to when you mentioned QFT and homotopy theory.

Another possible issue was the spelling/grammar/syntax/formatting.

Concerning the spelling, "exiting" should be "exciting," "selfreported" should be "self-reported" and "Eifel" should be "Eiffel."

Concerning the grammar, "analogue to localized categories" should be "analogous to localized categories," "here an example" should be "here is an example," "I wonder what else concepts" should be "I wonder what other concepts," "I myself take them only serious" should be "I myself only take them seriously."

Concerning formatting, it looks like you attempted to format the links in the same way that one formats links in the *comments* on LW and this resulted in the words that you wanted to hyperlink not being hyperlinked. For top level posts one instead uses the html link button in the header.

Thanks for your answer. The html link button did not work when I posted it. As far as AI etc., the possible relevance of homotopy theory is a theme since the 1970's, so it should be not too alien to anyone interested in pattern recogn. and related fields. It is similar unlikely that renormalization from QFT should be an entirely unknown theme, as that sort of dealing with generating series even in cases where they are seemingly ill-defined is a bit issue since long in e.g. combinatorics.

As far as AI etc., the possible relevance of homotopy theory is a theme since the 1970's, so it should be not too alien to anyone interested in pattern recogn. and related fields. It is similar unlikely that renormalization from QFT should be an entirely unknown theme, as that sort of dealing with generating series even in cases where they are seemingly ill-defined is a bit issue since long in e.g. combinatorics.

Here I think you overestimate how common your breadth/versatility is. I've met very few people who have heard of all of the mathematical connections that you have. A point to bear in mind here is the fact that you've spent substantial time in very elite environments like MPI subjects you to a strong selection effect that may have subjected you to highly unrepresentative data concerning what most intellectually curious people know.

The feedback by several of the mathematicians of "bird" type told me that this summary of some exchanges fits their mentality quite good. Actually, some of the remarks on poetry I made a few days ago in some other discussion here come directly from those feedbacks.

However, the distinction you draw between "birds" and procedure- or algorithm-mindedness is not so strict: You see this in the way they deal with QFT, leading to (among others) Kontsevich's, Manin's and Voevodsky's previously posted thoughts comes directly from their acceptance that e.g. Feynman integrals are nice ideas, justifyed by their computational power, and that their deficient consistency a topic of minor importance (i.e. you don't need a logical consistent th. physics, because your point of reference is the nature itself, whose consistency is not a reasonable issue; similar with the platonic world of mathematics). A close look to e.g. Manin's and Kontsevich's work shows that they are largely determined by computational issues.

Grothendieck is among those "giants of 20th century mathematics" a very special case, as one sees from the astounded admiration which one senses among those "giants" who met him personally. It is not surprising that Grothendieck thought of himself as "mutant" (after analysing his work and comparing it with that of others). And, as others around him with medical expertise observed, he was different, strangely similar to the novel figure Odd John. There is a very good talk by Yves Andre online and here an other great article by Herreman. A video of Scharlau's talk (in english) at the IHES is here. It may be of interest that acc. to those who knew her, Grothendieck's sister was a genius of comparable power.

And: Somehow my upvote of your post didn't work, some of my links do not too in my QFT-AI post.

However, the distinction you draw between "birds" and procedure- or algorithm-mindedness is not so strict: You see this in the way they deal with QFT, leading to (among others) Kontsevich's, Manin's and Voevodsky's previously posted thoughts comes directly from their acceptance that e.g. Feynman integrals are nice ideas, justifyed by their computational power, and that their deficient consistency a topic of minor importance (i.e. you don't need a logical consistent th. physics, because your point of reference is the nature itself, whose consistency is not a reasonable issue; similar with the platonic world of mathematics). A close look to e.g. Manin's and Kontsevich's work shows that they are largely determined by computational issues.

I'll have to think about this. Certainly there are examples both of bird/beaver hybrids and frog/beaver hybrids.

Grothendieck is among those "giants of 20th century mathematics" a very special case, as one sees from the astounded admiration which one senses among those "giants" who met him personally. It is not surprising that Grothendieck thought of himself as "mutant" (after analysing his work and comparing it with that of others). And, as others around him with medical expertise observed, he was different, strangely similar to the novel figure Odd John. There is a very good talk by Yves Andre online and here an other great article by Herreman. A video of Scharlau's talk (in english) at the IHES is here. It may be of interest that acc. to those who knew her, Grothendieck's sister was a genius of comparable power.

Thanks for the references which I had not seen before.

There is something which one could call the "Pirx paradigm", coming from Stanislav Lem:

The complexity of really nontrivial questions surpasses that of the formalized methods used by the conscious part of the scientist's mind. Only the whole mind's complexity meets the questions in view of complexity and flexibility. Therefore, a great mind/scientist works with his complete personality, which e.g. expresses in the observable unique and personal way big scientists write their work. What one perceives as "humbleness", puzzlement, irrational curiosity, unsecurity or absentmindedness are then actually the marks of essential parts of the personality outside the narrow frame of conscious procedures (personal feelings, memories, associations, Lem stressed explicitely "honesty" too, because that is the absence of conscious "trickyness").

Lem discussed in some of his stories the contrasting side too - the deformation and degeneration of attempted, but conscious and therefore subcomplex, "rationality" into crackpot-science and crackpot-engineering. There, seemingly rational approaches gradually exchange the issues to be tackled (and the parts of nature in which they are embedded) by misfigured echos of the researcher/engineer's neurosis and mental entropy. I recommend to take a look into Lem's stories: The Inquest, Ananke, Test.

BTW, this MIT reserach program looks very much like one of Lem's jokes...

My main question is what your goal is in the post.

My goal was to raise awareness of the cognitive diversity present among very high caliber thinkers in order to help people find others like themselves and see the value in those who are different from themselves.

It wasn't clear to me upon a perusal, and I think it would be good to state it up-front regardless.

Will do.

Here is a draft of a potential top-level post which I'd welcome feedback on. I would appreciate any suggestions, corrections, additional examples, qualifications, or refinements.

Birds, Frogs and BeaversThe introduction of Birds and Frogs by Freeman Dyson reads

Dyson is far from the first to have categorized mathematicians in such a fashion. For example, in The Two Cultures of Mathematics British mathematician Timothy Gowers wrote

Similarly, Gian Carlo Rota's candid Indiscrete Thoughts contains an essay titled

Problem Solvers and Theorizerswhich draws a similar dichotomy:I believe that Rota's characterizations of problem solvers and theorizers are exaggerated but nevertheless in the right general direction. Rota's remarks are echoed in Colin McLarty's: The Rising Tide: Grothendieck on simplicity and generality

In addition to the sources cited above, Grothendieck discusses a dichotomy which resembles that of birds and frogs in the section of

Recoltes et SemaillestitledThe Inheritors and the Buildersand Lee Smolin discusses such a dicotomy in The Trouble With Physics Chapter 18.In his Opinion 95, Doron Zeilberger added a supplement to Dyson's classification, saying:

Zeilberger's statement that beavers are more important for the process of science than birds and frogs is debatable and I do not endorse it; but I believe that Zeilberger is correct to identify a third category consisting of people whose primary interest is in algorithms. Indeed, as Laurens Gunnarsen recently pointed out to me, Felix Klein had already identified such a category in his 1908 lectures on Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra and Analysis. In the section titled

Concerning the Modern Development and the General Structure of Mathematics, Klein identified three plans A, B, and C roughly corresponding to the natural activities of frogs, birds and beavers respectively:The three categories described above appear to have correlates of personality traits, mathematical interests and superficially nonmathematical interests. Below I'll engage in some speculation about this.

Correlates of the bird categoryMy impression is that birds tend to have high openness to experience, be anti-conformist, highly emotional sensitivity, and interested in high art, history, philosophy, religion and geometry. Here I'll give some supporting evidence. I believe that the thinkers discussed would identify themselves as birds.

1. Dyson's article discusses Yuri Manin as follows:

2. In The Trouble With Physics Lee Smolin writes of "Seers" who have something in common with Dyson's "Birds":

3. Thomas remarks that Yuri Manin has written about how "mathematics chooses us" and "emotional platonism" which are characteristic of shamanism and that the number theorist Kazuya Kato writes "Mysterious properties of zeta values seem to tell us (in a not so loud voice) that our universe has the same properties: The universe is not explained just by real numbers. It has p-adic properties … We ourselves may have the same properties" which fits into the shamanistic way of thinking ("knowing something is becoming it").

4. In his autobiography titled The Apprentice of a Mathematician, Andre Weil wrote about how he was heavily influenced by Hindu thought and studied Sanskrit and mystic Hindu poetry.

5. According to Allyn Jackson's article on Alexander Grothendieck:

Grothendieck's interest in music is corroborated by Luc Illusie who said:

According to Winifred Scharlau's Who is Alexander Grothendieck?

and

6. According to Frank Wilczek's Introduction to Philosophy of Mathematics and Natural Sciences by Hermann Weyl,

7. In Robert Langlands' Lectures on the Practice of Mathematics and Is There Beauty in Mathematical Theories?, Langlands discusses the history of mathematics at length and quotes Rainer Maria Rilke, and Rudyard Kipling.

8. Some examples of famous birds who identify as geometers in a broad sense are Bernhard Riemann, Henri Poincare, Felix Klein, Elie Cartan, Andre Weil, Shiing-Shen Chern, Alexander Grothendieck, Raoul Bott, Friedrich Hirzebruch, Michael Atiyah, Yuri Manin, Barry Mazur, Alain Connes, Bill Thurston, Mikhail Gromov.

Correlates of the frog categoryMy impression is that frogs tend to be highly detail-oriented, conservative (in the sense that Rota describes), have a good memory of lots of facts, high technical prowess, ability to focus on a problem a very long time, and be interested in areas of math like elementary and analytic number theory, analysis, group theory and combinatorics. Here I'll give some supporting evidence. I believe that the thinkers discussed would identify themselves as frogs.

1. The conservative quality of frogs that Rota alludes to is negatively correlated with openness to experience. For an example of a conservative frog, I would cite Harold Davenport:

Davenport's frog aesthetic comes across in his remark

2. The Odd Order Theorem in finite group theory is a seminal result which was proved by the two frogs Walter Feit and John Thompson. One of my friends who did his PhD in finite group theory said that understanding

a single lineof their 250 page proof requires a serious effort. In a 1985 interview, Jean-Pierre Serre saidClaude Chevalley was an outstandingly good mathematician. I read the fact that somebody of such high caliber had much trouble as he did with the proof as an indication of Feit and Thompson having unusually high technical prowess and ability to focus on a single problem for a long time even

relative to other remarkable mathematicians. This is counterbalanced by the mathematical output of Feit and Thompson was essentially restricted to the topic of finite groups theory in contrast with that of many mathematicians who have broader interests.3. The identification of combinatorics as a mathematical field populated by problem solvers comes across in Gowers' essay linked above. The subject of elementary number theory was very heavily influenced by Erdos who has been labeled a canonical problem solver. In the introduction to a course in analytic number theory, Noam Elkies wrote

Two of the founders of the mathematical field of analysis, namely Cauchy and Weierstrass were frogs. Klein writes

Many of the prominent contemporary analysts like John Nash and Grigori Perelman are problem solvers.

Correlates of the beaver categoryMy impression is that beavers tend to be interested in jigsaw puzzles, word puzzles, logic puzzles, board games like Go, sorting tasks, algorithms, computational complexity, logic, respond best to a stream of immediate feedback in the way of tangible progress and can have trouble focusing on learning mathematical subjects in ways that require a lot of development before one engages in computation. Many computer scientists seem to me to fall into the beaver paradigm. Here I'm on shakier ground as I've seen little public discussion of beavers and most of what I've observed that supports my impression is born of subjective experience with people who I know, but I'll try to give some examples that seem to me to fall into the beaver paradigm:

1. Doron Zeilberger's focus on algorithmics, ultrafinitism and constructivist mathematics.

2. Harold Edwards' focus on constructivist mathematics (which comes across in his books titled

Higher Arithmetic: An Algorithmic Introduction to Number Theory,Essays in Constructive Mathematics, Galois Theory, Fermat's Last TheoremandDivisor Theory) is in the beaver paradigm.3. Terence Tao's interest in logic puzzles like the blue-eyed islanders puzzle and his interest in ultrafilters and nonstandard analysis.

4. The work of Jonathan Borwein and Peter Borwein computing a billion digits of pi.

5. The computational work of historically great mathematicians like Newton Euler, Gauss, Jacobi, and Ramanujan.

6. Don Zagier's remark in his essay in Mariana Cook's book

7. The focus on explicit formulae in the area of q-series.

8. Interest in Newcomb's Problem and its variations.

9. Scott Aaronson's interest in computational complexity, algorithms, and questions between logic and algorithmics as reflected in his MathOverflow post Succinctly naming big numbers: ZFC versus Busy-Beaver.

To Be ContinuedIn a future post I will describe superficial similarities and superficial differences between the three types and misunderstandings between different types which arise from generalizing from one example and cultural differences. Regarding his mastercraftspeople/seers dicotomy, Lee Smolin says

Timothy Gowers writes about how there's a schism between his two categories of mathematicians and says

As Dyson and Zeilberger said, all three types are important to scientific progress. I believe that intellectual progress will be increase if the three types can learn to better understand each other.