Wednesday, December 28, 2011

From Quora: What is it like to have an understanding of very advanced mathematics?



Someone asked the question above in Quora (a website for generic questions and answers). Someone (Anon) gave a very pretty, detailed answer. I'm keeping the answer here until I read/transform it to my own.
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I'm interested to hear what very talented mathematicians and physicists have to say about "what it's like" to have an internalized sense of very advanced mathematical concepts... As someone who only completed college calculus and physics, and has some basic CS background, but who is very intrigued by mathematics, I've always been curious about this. Does it feel analogous to having mastery of another language like in programming or linguistics? Any honest, candid insights appreciated!
Anon User Replied:

  • 1.You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a few powerful general purpose "machines" (e.g., continuity arguments, the correspondences between geometric and algebraic objects, linear algebra, ways to reduce the infinite to the finite through various forms of compactness) combined with specific facts you have learned about your area. The number of fundamental ideas and techniques that people use to solve problems is, perhaps surprisingly, pretty small -- see http://www.tricki.org/tricki/map for a partial list, maintained by Tim Gowers.
  • 2.You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It's not so much that you can "imagine" the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
  • 3.You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion. More on this in the next few bullets.
  • 4.Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a "fixed point" which does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don't know that they shouldn't be straining.) Instead...
  • 5.When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights. For example, you might imagine two- and three- dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don't have the desired property. Then you think about what was important to those examples and try to distill those ideas into symbols. Often, you see that the key idea in those symbolic manipulations doesn't depend on anything about two or three dimensions, and you know how to answer your hard question.

    As you get more mathematically advanced, the examples you consider easy are actually complex insights built up from many easier examples; the "simple case" you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical illumination about something intractable; you work to reduce it to the things that feel friendly.
  • 6.To me, the biggest misconception that non-mathematicians have about how mathematicians think is that there is some mysterious mental faculty that is used to crack a problem all at once. In reality, one can ever think only a few moves ahead, trying out possible attacks from one's arsenal on simple examples relating to the problem, or using analogies with other ideas one understands. This is the same way that one solves problems in one's first real maths courses in university and in competitions. What happens as you get more advanced is simply that the arsenal grows larger and you have more examples to try, making perhaps somewhat better guesses about what is likely to yield progress.

    Indeed, most of the bullet points here summarize feelings familiar to many serious students of mathematics who are only in the middle of their undergraduate careers; as you learn more mathematics, these experiences apply to "bigger" things but have the same fundamental flavor.
  • 7.You go up in abstraction, "higher and higher". The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves -- that is, you "zoom out" so that every function is just a point in a space, surrounded by many other "nearby" functions. Using this kind of "zooming out" technique, you can say very complex things in very short sentences -- things that, if unpacked and said at the "zoomed in" level, would take up pages. Abstracting and compressing in this way allows you to consider very complicated issues while using your limited memory and processing power.
  • 8.The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques. A theoretical physicist friend likes to say, only partly in jest, that there should be books titled "______ for Mathematicians", where _____ is something generally believed to be difficult (quantum chemistry, general relativity, securities pricing, formal epistemology). Those books would be short and pithy, because many key concepts in those subjects are ones that mathematicians are well equipped to understand. Often, those parts can be explained more briefly and elegantly than they usually are if the explanation can assume a knowledge of maths and a facility with abstraction.

    Learning the domain-specific elements of a different field can still be hard -- for instance, physical intuition and economic intuition seem to rely on tricks of the brain that are not learned through mathematical training alone. But the quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts that make learning other fields easier, as long as you are willing to be humble and modify those mathematical habits that are not useful in the new field.
  • 9.You move easily between multiple seemingly very different ways of representing a problem. For example, most problems and concepts have more algebraic representations (closer in spirit to an algorithm) and more geometric ones (closer in spirit to a picture). You go back and forth between them naturally, using whichever one is more helpful at the moment.

    Indeed, the most powerful ideas in mathematics (e.g., duality, Galois theory [http://en.wikipedia.org/wiki/Gal..., algebraic geometry [http://en.wikipedia.org/wiki/Alg...) provide "dictionaries" for moving between "worlds" in ways that, ex ante, are very surprising. For example, Galois theory allows us to use our understanding of symmetries of shapes (e.g., rigid motions of an octagon) to understand why you can solve any fourth-degree polynomial equation in closed form, but not any fifth-degree polynomial equation. Once you know these threads between different parts of the universe, you can use them like wormholes to extricate yourself from a place where you would otherwise be stuck. The next two bullets expand on this.
  • 10.Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days thinking of a clean argument that completely avoids numbers and strings of elementary deductions in favor of seeing why what they want to show follows easily from some very deep and general pattern that is already well-understood. Indeed, you tend to choose problems motivated by how likely it is that there will be some "clean" insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities. In A Mathematician's Apology [http://www.math.ualberta.ca/~mss..., the most poetic book I know on what it is "like" to be a mathematician], G.H. Hardy wrote:

    "In both [these example] theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way."

    [...]

    "[A solution to a difficult chess problem] is quite genuine mathematics, and has its
    merits; but it is just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly) which a real mathematician tends to despise."
  • 11.You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles. Mathematicians don't really care about "the answer" to any particular question; even the most sought-after theorems, like Fermat's Last Theorem [http://en.wikipedia.org/wiki/Fer..., are really only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing. The accomplishment a mathematician seeks is finding a new "dictionary" or "wormhole" between different parts of the conceptual universe. As a result, many mathematicians do not focus on deriving the practical or computational implications of their studies (which can be a drawback of the hyper-abstract approach!); instead, they simply want to find the most powerful and general connections. Tim Gowers has some interesting comments on this issue, and disagreements within the mathematical community about it [ http://www.dpmms.cam.ac.uk/~wtg1... ].
  • 12.Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: "First I will lay this foundation, then I will build this framework using these familiar pieces, but leave the walls to fill in later, then I will test the beams..." All these steps have mathematical analogues, and structuring things in a modular way allows you to spend several days thinking about something you do not understand without feeling lost or frustrated.

    Andrew Wiles, who proved Fermat's Last Theorem, used an "exploring" metaphor:
    "Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them." [ http://www.pbs.org/wgbh/nova/phy... ]
  • 13.In listening to a seminar or while reading a paper, you don't get stuck as much as you used to in youth because you are good at modularizing a conceptual space and taking certain calculations or arguments you don't understand as "black boxes" and considering their implications anyway. You can very often make statements you know are true and have good intuition for, without understanding all the details. You can often detect where the delicate or interesting part of something is based on only a very high-level explanation.
  • 14.You are good at generating your own questions and your own clues in thinking about some new kind of abstraction. One of the things I've reliably heard from people who know parts of mathematics well but never went on to be professional mathematicians (i.e., write articles about new mathematics for a living) is that they were good at proving difficult propositions that were stated in a textbook exercise, but would have no idea what to do if presented with a mathematical structure and asked to find and prove some "interesting" facts about it. This kind of challenge is like being given a world and asked to find events in it that come together to form a good detective story. Unlike a more standard detective, you have to figure out what the "crime" (interesting question) might be, and you have to generate your own "clues" by building up deductively from the basic axioms. To do these things, you use analogies with other detective stories (mathematical theories) that you know and a taste for what is "surprising" or "deep". This is the hardest thing in this answer to articulate precisely but also the thing that I would guess is the "strongest" thing that mathematicians have in common.

    Concretely, this amounts to being good at making definitions and formulating precise conjectures using the newly defined concepts. It is interesting that one of the things one learns fairly late in a typical mathematics education is how to make good, useful definitions.
  • 15.You are easily annoyed by imprecision in talking about the quantitative or logical. This is mostly because you are trained to quickly think about counterexamples that make an imprecise claim seem obviously false.
  • 16.On the other hand, you are very comfortable with intentional imprecision or "hand waving" in areas you know, because you know how to fill in the details. The mathematician Terence Tao is very eloquent about this here [ http://terrytao.wordpress.com/ca... ]:

    "[After learning to think rigorously, comes the] 'post-rigorous' stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the 'big picture'. This stage usually occupies the late graduate years and beyond."

    In particular, an idea that took hours to understand correctly the first time ("for any arbitrarily small epsilon I can find a small delta so that this statement is true") becomes such a basic element of your later thinking that you don't give it conscious thought.
  • Before wrapping up, it is worth mentioning that mathematicians are not immune to the limitations faced by most others. They are not typically intellectual superheroes. For instance, they often become resistant to new ideas and uncomfortable with ways of thinking (even about mathematics) that are not their own. They can be defensive about intellectual turf, dismissive of others, or petty in their disputes. Above, I have tried to summarize how the mathematical way of thinking feels and works at its best, without focusing on personality flaws of mathematicians or on the politics of various mathematical fields. These issues are worthy of their own long answers!
  • 17.You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer. This makes it more comfortable to be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.

Friday, October 7, 2011

Ada Lovelace Day 2011: Christine Ladd-Franklin


Posting about women in mathematics, a reminder from Ada Lovelace Day...

From Algebraic Logic to Optics and Psychology

By hook and by crook Christine Ladd managed to attend graduate courses in mathematics at Johns Hopkins University despite the fact that the university was not open to women. Jacob describes how she managed to do that:

[T]he university first announced its fellowship program in 1876, and one of the first applications to arrive was one signed "C. Ladd." The credentials accompanying the application indicated such outstanding ability that a fellowship in mathematics was awarded to the applicant, site unseen, and was accepted. When it was discovered that the "C." stood for Christine, several embarrassed trustees argued she had used trickery to gain admission, and the board immediately moved to revoke the offer. They failed to reckon, however, with the irascible Professor James J. Sylvester, stellar member of the first faculty. In 1870 Sylvester had been named the world's greatest living mathematician by the Encyclopedia Britannica, and his presence at Hopkins was a real coup for the struggling university. He was indispensable and knew it, in an ideal position to insist on virtually anything he wanted; in this case, he had read Christine Ladd's articles in English mathematical journals, and he insisted upon receiving the obviously gifted young woman as his student. Miss Ladd was admitted as a full-time graduate student in the fall of 1878. Though she held a fellowship for three years, the trustees forbade that her name be printed in circulars with those of other fellows, for fear of setting a precedent. Dissension over her continued presence forced one of the original trustees to resign.

From http://www.agnesscott.edu/lriddle/women/ladd.htm

Monday, May 9, 2011

Louise Hay: Roles Models in Logic...

Louise Hay: 1935-1989, Robert I. Soare. Association for Women
in Mathematics Newsletter 20:1 (January-February 1990) 3-4.


The life and work of logician Louise Hay are described in this article.
Born in France in 1935, she came to America in 1946 and received a B.A. from Swarthmore College and her M.A. and Ph.D. from Cornell University.

Her doctoral dissertation dealt with recursive function theory. Initially on the faculty at Mount Holyoke College, she moved to the University of Illinois at Chicago in 1968 where she combined an active research career with teaching and administrative work. She succumbed to cancer in 1989.
Her research dealt with index sets connected with recursively enumerable sets
as well as with computational complexity theory. She was very influential in
encouraging women to pursue mathematical careers.

Apparently the Logic Seminar in Chicago is called the Louise Hay Seminar, if I'm not imagining things..

Tuesday, April 19, 2011

Problems and materials...

I've discussed so far with the classes:

Andrew Abbott's `Aims of Education'

the future of Philosophy, either via Ken Taylor post or Jung's article and

Wigner's `The Unreasonable Effectiveness of Mathematics in the Natural Sciences'. There is also Hamming's ``The Unreasonable Effectiveness of Mathematics" and perhaps the unreasonable effectiveness of data might be mentioned...

The unusual effectiveness of logic in CS is also good, but doesn't pack the same punch. (And the contrarian `The unreasonale effectiveness of data' is another nail in a perforated coffin, added much later, Sept 2015.)


There are questions that I did not find a good source like `what is a reasonable classification of the sub-areas of philosophy?' Mind you I also have not found one about the sub-areas of logic: I'm told that the classification into proof theory, model theory, recursion theory and set theory was Sack's, but I don't know where he did it...

Anyway I want to keep here a list of other questions and other articles that I think are cool enough to try to force the students to read. Must re-check the Canon of Tech, they had some good suggestions, like Vannevar Bush's `As we may think' article.

Tuesday, March 29, 2011

Phil50 First lecture

NEW (03/31/11): Since I now believe the course is set up in SUNET's CourseWork, I will stop posting materials here. Email me if you have problems.

Here are the slides, the questionnaire (for Friday) and the first real assignment.

Monday, March 28, 2011

PHIL50 Basics, for the time being...

Stanford University, Spring 2011
PHIL50 Introduction to Logic

Professor: Valeria de Paiva (PhD Cantab)
Email: valeria.depaiva@gmail.com
Lectures: MW 10:00-10:50 Room 206 EDUC
BOOK: Language, Proof and Logic,
John Barwise, John Etchemendy
University of Chicago Press
ISBN 157586374X


WARN­ING! Do not buy a used copy of the text! The copy of the soft­ware that comes with the book can only be reg­istered once. If you can­not reg­is­ter the soft­ware, you can­not sub­mit so­lu­tions to home­work ex­er­cises of take-home exam prob­lems or have the cor­rect­ness of your so­lu­tions au­to­mat­i­cally checked for you prior to sub­mit­ting them.

HOME­WORK/QUIZZES: Each week, there will be ei­ther a home­work as­sign­ment due or a quiz. In ad­di­tion, there will be a midterm exam and a final exam. Each exam will con­sist of an (open-book, open-notes) take-home part and a (closed-book, closed-notes) in-class part.

TOP­ICS TO BE COV­ERED:

1. Why logic? Computational thinking for philosophers? (1 lec­ture):

2. Propo­si­tional Log­ics (approx 10 lec­tures):
• The syn­tax and se­man­tics of propo­si­tional log­ics
• The log­i­cal con­nec­tives.
• Build­ing truth ta­bles to test for­mal va­lid­ity, both "by hand" and using Boole.
. The dis­tinc­tion be­tween im­pli­ca­tion and im­pli­ca­ture
. "Fitch" and formal proofs.

3. First-Or­der Log­ics (approx 10 lec­tures):
• The syn­tax and se­man­tics of first-or­der log­ics
• Ex­press­ing your­self in first-or­der log­ics
• Build­ing struc­tures to demon­strate for­mal in­va­lid­ity, by hand and using Tarski's World
• Con­struct­ing for­mal de­duc­tions to demon­strate for­mal va­lid­ity, by hand and using Fitch


4. Modal Propo­si­tional Log­ics (3 lec­tures):
• Deontic, epistemic, tem­po­ral, dy­namic, and gen­eral modal propo­si­tional log­ics

5. Wrap-up (1 lec­ture):
• Spe­cial em­pha­sis on the ques­tion of how much sup­port a for­mal de­riva­tion of a propo­si­tion pro­vides for be­liev­ing it to be true

Saturday, March 26, 2011

COEN260 Basics

Santa Clara University, Spring 2011
COEN 260 Truth Deduction and Computation

Professor: Valeria de Paiva (PhD Cantab)
Email: valeria.depaiva@gmail.com
Lectures: MW 7:10-9:00 Room 106 Bannan
BOOK: Language, Proof and Logic,
John Barwise, John Etchemendy
University of Chicago Press
ISBN 157586374X



Cat­a­log De­scrip­tion: In­tro­duc­tion to math­e­mat­i­cal logic and se­man­tics of lan­guages for the com­puter sci­en­tist. In­ves­ti­ga­tion of the re­la­tion­ships among what is true, what can be proved, and what can be com­puted in the for­mal lan­guages for propo­si­tional logic, first order pred­i­cate logic, el­e­men­tary num­ber the­ory, and the type-free and typed lambda cal­cu­lus. Pre­req­ui­site: COEN 19 or AMTH 240 and COEN 70. (4 units)

WARN­ING NOTE ON BOOK: WARN­ING! Do not buy a used copy of the text! The copy of the soft­ware that comes with the book can only be reg­isted once. If you can­not reg­is­ter the soft­ware, you can­not sub­mit so­lu­tions to home­work ex­er­cises of take-home exam prob­lems or have the cor­rect­ness of your so­lu­tions au­to­mat­i­cally checked for you prior to sub­mit­ting them.

HOME­WORK/QUIZZES: Each week, there will be ei­ther a home­work as­sign­ment due or a quiz. In ad­di­tion, there will be a midterm exam and a final exam. Each exam will con­sist of an (open-book, open-notes) take-home part and a (closed-book, closed-notes) in-class part.

TOP­ICS TO BE COV­ERED:

1. What is Computational Thinking? How does logic fit into it? (1 lec­ture):

2. Propo­si­tional Log­ics (7 lec­tures):
• The syn­tax and se­man­tics of propo­si­tional log­ics
• The log­i­cal con­nec­tives.
• Build­ing truth ta­bles to test for­mal va­lid­ity, both "by hand" and using Boole.
. The dis­tinc­tion be­tween im­pli­ca­tion and im­pli­ca­ture
. "Fitch" and formal proofs.

3. First-Or­der Log­ics (7 lec­tures):
• The syn­tax and se­man­tics of first-or­der log­ics
• Ex­press­ing your­self in first-or­der log­ics
• Build­ing struc­tures to demon­strate for­mal in­va­lid­ity, by hand and using Tarski's World
• Con­struct­ing for­mal de­duc­tions to demon­strate for­mal va­lid­ity, by hand and using Fitch

4. Lambda-Calculus (3 lec­tures):
• The syn­tax and se­man­tics of lambda-calculus, typed and untyped. Curry-Howard isomorphism.
• A quick look at some real-world tools for ap­ply­ing higher-or­der log­ics to soft­ware en­gi­neer­ing prob­lems

5. Modal Propo­si­tional Log­ics (1 lec­ture):
• The syn­tax and se­man­tics of tem­po­ral, dy­namic, and gen­eral modal propo­si­tional log­ics

6. Wrap-up (1 lec­ture):
• Spe­cial em­pha­sis on the ques­tion of how much sup­port a for­mal de­riva­tion of a propo­si­tion pro­vides for be­liev­ing it to be true

Thursday, March 24, 2011

A simple list of Logic teaching materials

ProofWeb Courses, provers from Radboud University Nijmegen, NL. Preprints in Publications

CMU Open Learning Pittsburgh

Logic in Action Amsterdam

Gateway to Logic, logic teaching software from Vienna, Austria.

Peter Smith, An Introduction to Formal Logic, slides.

Curtis Brown, lectures on Logic.

A whole lot of logic books...

Computational Thinking (slides by Brigitte Pientka)

Computational Thinking (oped and slides by J. Wing)


Less relevant to me:

Paul Teller's Logical Primer

Twootie and Bertie software for natural deduction

JAPE (Oxford) seems to be dead?