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A Mathematician's Knowledge

  1. Jan 6, 2007 #1
    Ok, this may sound like a stupid question, but I really, really need to know.

    How much more mathematical knowledge does a typical mathematician have than a math specialist student who has just finished fourth year university math courses?

    Before you criticize my question, let me explain why I ask. I want to become a mathematician, so I need to get a feel of how much knowledge I need to acquire before I can become one. Now, I know that knowledge is not everything. Indeed, it is problem solving skills and generation of ideas that makes a true mathematician. I agree! Nevertheless, one must have immense prerequisite knowledge before they can come up with original ideas and solve open problems.

    If the answer is, say, 3 times as much. Then I can focus on my problem solving skills, read thoroughly the proofs of theorems, etc..., and build my knowledge at the pace of a regular student. If, however, the answer is, say, 100 times as much, then I will know that I have to step up on my reading. So this question, I think, is important in order for me to get a sense of how much and in what manner I should self-study.

    My guess is that a typical mathematician has 50 times as much knowledge as a math student who has just graduated from university. Any other ideas? A mathematician's honest answer would be greatly appreciated (and I won't think you are being arrogant).
     
    Last edited: Jan 6, 2007
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  3. Jan 6, 2007 #2

    Chris Hillman

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    Scaling of knowlege and power

    How do you propose to measure "knowledge"? Is this something like how many pages it would require a research mathematician to write out everything he knows? Note that no-one has ever done this, or ever will!

    I think your question reflects anxiety of the perceived steepness of the learning curve in mathematics. Indeed, most would agree that this is a steep curve. But should not be discouraging; quite the opposite!

    Suppose that knowledge scales linearly with effort, and ask yourself: how does power scale with knowledge? My first guess is that power scales exponentially with knowledge. That is, if A has mastered twice as many notions as B, and in some sense A, B are otherwise "equivalent" in terms of insight and creativity (a dubious assumption in the real world!), then A should have four times as much power as B, and so on.

    IOW, very very roughly, [itex]{\rm power} = \exp( {\rm knowledge}) = \exp({\rm effort}) [/itex] or [itex]{\rm effort} = \log ( {\rm power}) [/itex]. That is, the effort required to attain a given power should scale logarithmically with the desired power. This looks quite steep as [itex] {\rm effort} \rightarrow 0^+[/itex], but gets shallow rather quickly. So if you simply keep increasing [itex]{\rm effort}[/itex], you should find that bye and bye you are acquiring much more power for a given amount of effort.

    In truth, I suspect that knowlege is superadditive, so the learning curve is even steeper than this faux argument suggests!
     
    Last edited: Jan 6, 2007
  4. Jan 6, 2007 #3
    Chris, I agree with your dissertation. But I just want a ball-bark answer. If you were to hypothetically write out everything you know, how much thicker would your book be than that of one of your senior level student's? Even though you are a relatavist, the same principle should apply.
     
    Last edited: Jan 6, 2007
  5. Jan 6, 2007 #4

    Chris Hillman

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    Notebooks

    There is no number large enough to express what I know in senior units.

    Is that what you wanted to hear?

    I may well have known several senior units when I entered math graduate school. Certainly, some of my graduate student colleagues seemed to think that I thought so at the time... My point above was that what I knew by the time I exited grad school was not expressible in senior units. I am trying to say that if you perservere, you might find the same happens for you--- I don't think my experience is terribly unusual.
     
    Last edited: Jan 6, 2007
  6. Jan 6, 2007 #5
    Chris, I'll estimate that your knowledge is 500 (the senior unit will be droped, just as with the constant c=1), in which case I'll have to start giving myself less sleep.
     
  7. Jan 6, 2007 #6

    Chris Hillman

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    But if your knowedge is of the order of one, how could you estimate my knowledge?
     
  8. Jan 6, 2007 #7

    Gib Z

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    It is pretty hard to measure knowledge. Perhaps another way would be to see their position in the field. Do you know everything in your field and researching more, on the cutting edge? Or are you still catching up? Is there alot left to go? Or is the end near? I wish to become a mathematician as well, but from what I see on these forums I'm far too amateur. Knowing Calculus at 11 is no big deal here. And forgetting it by the time your 14 is even worse...Really, I used to think I was a real smart one you know, everyone at school would think so, but then your here on these forums and I'm below average :) Puts it into perspective I guess
     
  9. Jan 6, 2007 #8

    mathwonk

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    the ratio of knowledge of a professional matehmatican to that of a student is like that of the skills of a profesional basketball plkayer to the skills of a college player.

    but that is not how you become a mathematician. as someone once said, if a man can even make a good cup of coffee, already you can talk to him. or an other person said, if you know even one good trick well, you are a magician.

    so just try to understand the math you are studying, and try to extend it a little. eventually you or someone advising you, will suggest a problem you can do that has not been done, anf you will get a thesis and be on your way.

    then every year try to learn more, by reading and running a learning seminar, and go to meetings, and listening to the best people.

    i think it likely i know more than you after my 50 years of learning, but i am also sure you know some things i don't, so dont be shy about your knowledge.
     
  10. Jan 6, 2007 #9
    Mathematics is a vast subject, but it can roughly be broken down into the following fields:

    01. Logic and Foundations
    02. Algebra
    03. Number Theory
    04. Algebraic and Complex Geometry
    05. Geometry
    06. Topology
    07. Lie Groups and Lie Algebras
    08. Analysis
    09. Operator Algebras and Functional Analysis
    10. Ordinary Differential Equations and Dynamical Systems
    11. Partial Differential Equations
    12. Mathematical Physics
    13. Probability and Statistics
    14. Combinatorics
    15. Mathematical Aspects of Computer Science
    16. Numerical Analysis and Scientific Computing
    17. Control Theory and Optimization

    Granted each of these fields is itself quite vast, but the point is that everything seems small from a top down point of view. Learning mathematics is a mix of 80% stategy and 20% time/hard work, most students do not approach the subject this way.
     
  11. Jan 6, 2007 #10
    Matt Grime's knowledge is 500 senior units.

    (waiting for his reply)
     
  12. Jan 6, 2007 #11
    I'm curious as to what you define as a senior unit, since not all schools cover the same material to the same degree of depth in 4 years.
     
    Last edited: Jan 6, 2007
  13. Jan 6, 2007 #12

    mathwonk

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    crosson, i might differ in your percentages, as i think it is more like 90% time/hard work.

    its not a game, its a job, so time/work matter more than strategy.
     
  14. Jan 7, 2007 #13

    Gib Z

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    It doesn't really matter what he defines a senior unit to be, we are looking are the ratios of these amounts. matt grime is probably the smartest guy I know, even if it is over the internet, But I really don't think his knowledge is 500 senior units, no ones here is. Not 500. If one senior year is 4 years, then he has learned 2000 years of mathematics. At least at the rate the schools teach it. I'm not sure how old matt grime is, but lets say 40. For his entire life he has learned mathematics at a rate 50 times as fast as a student. That seems a little out of proportion, no matter how smart matt grime is.

    My goal in life is to know as much math as matt grime does, and its a mammoth goal I tell you :p
     
  15. Jan 7, 2007 #14
    Yes, but the ratio is meaningless without defining what the "senior unit" is. Even if we define this to be 4 years of education at a university this is useless as different universities cover different materials in differing depths.
     
  16. Jan 7, 2007 #15

    matt grime

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    You're aiming in the wrong direction. I know very little maths compared to the people I work with; you have to remember (and this is a little to do with your other thread) that with perhaps one exception per week every question on this forum is answerable by anyone who knows a little of the area (if you look you'll notice I pointedly avoid differential equations as they're tedious and the differential geometry threads 'cos they frequently turn out to be about relativity and I know nothing about that). They just appear hard questions because all maths is hard if you don't know it and easy if you do. Perhaps mathwonk will back me up here, but the reason we (and that is more than just him and me) can answer questions we ought to know nothing about is because we understand how to attack the quesiton and what seems like it will be a fruitful avenue to look at.

    Oh, and frequently the exceptions aren't that exceptional either: anything by Jose under whatever pseudonym is an exceptional post, but mainly in its undecipherability and hostility towards mathematicians.

    Aim to know as much as Terry Tao, that'd be my advice if you really want to think that knowledge is some kind of barometer of mathematical ability. (Terry is both brilliant and knowledgable about many parts of mathematics. He should get some kind of medal.)
     
  17. Jan 7, 2007 #16
    One of my relativity professors said that he missed a math course (algebraic topology) that he needed for his field of research. He said that he picked up a standard textbook and read the entire book from front to back in one week. Though he, of course, never took an exam on this crash self-study course, he said that he got straight A+'s in university (taking up to 6 courses each semester) so I trust that he fully understood the entire textbook from that one week of reading.
    At this rate, he could learn a new math course every week and gain the knowledge of one senior unit every 2-3 months.

    Ok, let's say a typical mathematician, of age 50, has the knowledge of 100 senior units. Sounds correct?
     
    Last edited: Jan 7, 2007
  18. Jan 7, 2007 #17

    matt grime

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    Let's not. I'm not even 30.
     
  19. Jan 7, 2007 #18
    As for Matt Grime himself, let's say he learns 10 times faster than the typical senior math student, so he gains a senior unit every, say, 4 months. Since he is under 30 years old, then his knowledge should be about 25 senior units. Sounds right?
     
  20. Jan 7, 2007 #19
    Ok, here's my definition of a senior unit of knowledge. The mathematical knowledge consisting of:


    Calculus I
    A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Taylor’s theorem; sequences and series; uniform convergence and power series.

    Calculus II
    Topology of Rn; compactness, functions and continuity, extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and minima, Lagrange multipliers. Integrals; Fubini’s theorem, partitions of unity, change of variables. Differential forms. Manifolds in Rn; integration on manifolds; Stokes’ theorem for differential forms and classical versions.

    Linear Algebra I
    A theoretical approach to: vector spaces over arbitrary fields including C,Zp. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.

    Linear Algebra II
    A theoretical approach to real and complex inner product spaces, isometries, orthogonal and unitary matrices and transformations. The adjoint. Hermitian and symmetric transformations. Spectral theorem for symmetric and normal transformations. Polar representation theorem. Primary decomposition theorem. Rational and Jordan canonical forms. Additional topics including dual spaces, quotient spaces, bilinear forms, quadratic surfaces, multilinear algebra. Examples of symmetry groups and linear groups, stochastic matrices, matrix functions.

    Ordinary Differential Equations
    Ordinary differential equations of the first and second order, existence and uniqueness; solutions by series and integrals; linear systems of first order; non-linear equations; difference equations.

    Partial Differential Equations
    Diffusion and wave equations. Separation of variables. Fourier series. Laplace’s equation; Green’s function. Schrödinger equations. Boundary problems in plane and space. General eigenvalue problems; minimum principle for eigenvalues. Distributions and Fourier transforms. Laplace transforms. Differential equations of physics (electromagnetism, fluids, acoustic waves, scattering). Introduction to nonlinear equations (shock waves, solitary waves).

    Introduction to Number Theory
    Elementary topics in number theory: arithmetic functions; polynomials over the residue classes modulo m, characters on the residue classes modulo m; quadratic reciprocity law, representation of numbers as sums of squares.

    Groups, Rings and Fields
    Groups, subgroups, quotient groups, Sylow theorems, Jordan-Hölder theorem, finitely generated abelian groups, solvable groups. Rings, ideals, Chinese remainder theorem; Euclidean domains and principal ideal domains: unique factorization. Noetherian rings, Hilbert basis theorem. Finitely generated modules. Field extensions, algebraic closure, straight-edge and compass constructions. Galois theory, including insolvability of the quintic.

    Complex Analysis I
    Complex numbers, the complex plane and Riemann sphere, Mobius transformations, elementary functions and their mapping properties, conformal mapping, holomorphic functions, Cauchy’s theorem and integral formula. Taylor and Laurent series, maximum modulus principle, Schwarz’s lemma, residue theorem and residue calculus.

    Complex Analysis II
    Harmonic functions, Harnack’s principle, Poisson’s integral formula and Dirichlet’s problem. Infinite products and the gamma function. Normal families and the Riemann mapping theorem. Analytic continuation, monodromy theorem and elementary Riemann surfaces. Elliptic functions, the modular function and the little Picard theorem.

    Real Analysis I
    Function spaces; Arzela-Ascoli theorem, Weierstrass approximation theorem, Fourier series. Introduction to Banach and Hilbert spaces; contraction mapping principle, fundamental existence and uniqueness theorem for ordinary differential equations. Lebesgue integral; convergence theorems, comparison with Riemann integral, Lp spaces. Applications to probability.

    Real Analysis II
    Measure theory and Lebesgue integration; convergence theorems. Riesz representation theorem, Fubini’s theorem, complex measures. Banach spaces; Lp spaces, density of continuous functions. Hilbert spaces; weak and strong topologies; self-adjoint, compact and projection operators. Hahn-Banach theorem, open mapping and closed graph theorems. Inequalities. Schwartz space; introduction to distributions; Fourier transforms on Rn (Schwartz space and L2). Spectral theorem for bounded normal operators.

    Point-Set Topology
    Metric spaces, topological spaces and continuous mappings; separation, compactness, connectedness. Topology of function spaces. Fundamental group and covering spaces. Cell complexes, topological and smooth manifolds, Brouwer fixed-point theorem.

    Differential Topology
    Smooth manifolds, Sard’s theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.

    Algebraic Topology
    Introduction to homology theory: singular and simplicial homology; homotopy invariance, long exact sequence, excision, Mayer-Vietoris sequence; applications. Homology of CW complexes; Euler characteristic; examples. Singular cohomology; products; cohomology ring. Topological manifolds; orientation; Poincare duality.

    Differential Geometry I
    Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.

    Differential Geometry II
    Riemannian metrics and connections. Geodesics. Exponential map. Complete manifolds. Hopf-Rinow theorem. Riemannian curvature. Ricci and scalar curvature. Tensors. Spaces of constant curvature. Isometric immersions. Second fundamental form. Topics from: Cut and conjugate loci. Variation energy. Cartan-Hadamard theorem. Vector bundles.


    Have I missed anything?


    Multiply this list by 100, and you have the knowledge of a 50 year-old mathematician. Right?
     
    Last edited: Jan 7, 2007
  21. Jan 7, 2007 #20
    Multiply what by 100? The number of courses? The difficulty of the courses? Are you asking if a mathematician would take a course called "Differential Geometry CC"? Are you asking if someone who is a 50 year old mathematician would have 100 times the understanding of limits than a senior? What does it even mean to have 100 times the understanding of something?

    Your list isn't a set of numbers that can simply be multiplied and make sense. You're attempting to quantify something that can't be quantified. You're going about this the wrong way, if you will.
     
  22. Jan 7, 2007 #21

    matt grime

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    Working definition: a mathematician is someone who knows that somethings are not quantifiable by numbers.... This is just silly. You'd be better off trying to learn maths than spending time inventing numerical scales of ability.
     
  23. Jan 7, 2007 #22

    mathwonk

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    i certainly dont know all that stuff, especially not the pde and diff geom, and i think many profesors i know would omit even more topics.

    I back up Matt's assessment of what we use to answer questions and solve problems, is not so much knowledge of material, as awareness of how to approach questions.

    I have described before here occasions on which i "answered" a question that a much smarter and more knowledgeable mathematican did not answer, simply by saying, from experience or instinct, how to use a certain bit of knowledge to solve the problem. as it happens i did not have that bit of knowledge and the other mathematician did, so in fact the other amtehmaticin took my prescription for how to solve the problem and then "solved" it and explained it to the questioner.

    The learning process is very spotty for a profesional mathematician. Since I have to teach elementary calculus about 2-3 times year, for say 30+ years, I eventually learn that stuff quite throughly. But since i basically never get to teach measure theory, not being an analyst, I only know what I learnt 40 years ago, or can pick up now on my own.

    A student is doing nothing but learning, one course after the other. Every semester thre is a course going in my dept on that students are learning something from that I do not know. I do not have their leisure to sit in on an advanced number theory course, much as I would like to.

    I am not the kind of super learning machine you mentioned, but occasionally I have had intense learning episodes. Like once when was a grad student (student again), I volunteered, or was volunteered, I forget which, to present Kodairas own proof of the vanishing theorem. so I spent the whole thanksgiviing break learning and reading about 60 pages, roughly pages 60-120, of the book by kodaira and Morrow on the topic, sheaf cohomology, deRham theorem, Dolbeault theorem, Riemannian geometry, Laplacians, Harmonic forms, characteristic clases, metrics, curvature tensors, Bochners inequality, and finally the trivial computation that gives the vanishing theorem. I learned it in 5 days, (almost no sleep), and presented it all to the astonished faculty members, at least one professor telling me he himself could not even read the book.

    But I have not repeated that feat since. Although I did also read Spivaks diff geom book vol 2, in one or two days, that was relatively easy in comparison, and i certainly did not master that material in 2 days.

    As to the kodaira proof, i once heard Raoul Bott, the great Harvard topologist, say in class that he did not have the stamina to read the proof, and so worked out his own, using the principal bundle instead. Now there is a real mathematician, he made up his own proof, instead of slogging through Kodaira and Morrow. I noticed that David Mumford, a Fields medalist, sat in to hear Botts presentation of his own proof.

    So mathematicians are concerned with doing, not learning. In fact Bott once told us to try to find our own proofs of some problems he gave us while we were still young and "before your heads get filled with other peoples ideas".

    I think I know a lot compared to some people, but of the 4 algebraic geometers in my group, I surely know the least, and they are all younger than me. It is very hard for a senior mathematician, at least one at a school with a heavy teaching load, to learn the new stuff that grad students are learning, especially outside ones field.

    I doubt if there is more than a handful of professors at most average schools who would say they "know" all the subjects on your senior list of topics. But ironically, they might be able to work with that material better towards solving a problem, than a senior who has just taken the courses.
    :smile:
    But if you already know all that stuff, and can pass a prelim exam on it, I think you are more than knowledgeable enough to start a research thesis.
     
    Last edited: Jan 7, 2007
  24. Jan 7, 2007 #23

    mathwonk

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    let me give an tiny example of the difference betwen my knowledge and a typical senior's knowledge of linear algebra. Take the clasical cayley hamilton theorem- I can give at least 5 different proofs of it, and explain why it is completely trivial, and use it to prove that a surjective endomorphism of a finitely generated module over any ring is also injective.

    Last semester while teaching diff eq I also applied elementary linear algebra to create two solution methods for linear equations, not in our book, one of which was not in any books I could find, and neither of which was known even to the analysts I asked in the department.

    All I did was apply known methods of analogy to known results in linear algebra, and apply this to the fact that a differential linear operator is a linear map. The difference is that people who think learning is reading instead of thinking, which includes many seniors, only come away from a diff eq course with the same methods that are explained in their book, not the extensions of them that reflection would suggest.

    so the point is to begin to learn actively. no amount of passive knowledge will do you much good unless someone show you how to use it.
     
    Last edited: Jan 7, 2007
  25. Jan 7, 2007 #24

    mathwonk

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    here is a fun topic from elementary calculus: we all know that if f is a continuous fucntion on [a,b] then f has antidertivatives, and any two differ by a constant. this is proved by the MVT and is used to prove the FTC.

    Now what if f is not continuous but only integrable? Its indefinite integral F is still differentiable at most points, namely where f is continuous, ie F has derivative equal to f wherever f is continuous and F is continuous everywhere.

    Now can one generalize the previous characterization of antiderivatives of f? I.e. suppose G,H are two functions that are continuous and both have derivative equal to f where f is continuous. Do G,H differ by a constant?

    This is the kind of question that occurs to a mathematician when teaching calc 1. What does a senior think?

    to learn to ask questions like this, notice what are the hypotheses of theorems. i.e. the FTC has a hypothesis that the function f is continuous. but what about integrable f? one wants to calculate their integrals too. does the FTC work for them? if so, what does it say? ie to compute the integral of an integrable function, what should be the definition of an antiderivative?:smile:
     
    Last edited: Jan 7, 2007
  26. Jan 8, 2007 #25
    Mathwonk, with all due respect,

    Hilbert expressed the opinion that nearly all worthwhile mathematics is created by a small minority of mathematicians. This is undesirable, because all mathematicians write papers and this means that most of those papers are not worthwhile. This is a symptom of a disease called "thinking of research mathematics as more of a job then a game".

    People say of Paul Erdos that his true skill was in selecting the right problems to work on at the right time with the right people i.e. his strategy.

    Unfortunately, the academic establishment both agrees with this statement and bemoans the fact that todays scholars are hopelessly narrow. We need to stop encouraging the production of low quality mathematics by undereducated mathematicians.
     
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