Two Cultures of Number Theorists

There is a famous distinction in prime number theory between the number theorists who like to multiply primes, and the number theorists who like to add primes. As the primes are very heavily multiplicatively structured, the mathematics of multiplying primes is very algebraic in nature, in particular involving field extensions, Galois representations, etc. But the primes are very additively unstructured, and so for adding primes we see the tools of analysis used instead (circle method, sieve theory, etc.).

Mentioned by Terry Tao in a comment at n-category cafe.

Scott Aaronson on Mathematics

From Luke’s recent interview of Scott Aaronson (theoretical compsci guy at MIT, who blogs here):

Things like linear algebra, group theory, and probability have so many uses throughout science that learning them is like installing a firmware upgrade to your brain — and even the math you don’t use will stretch you in helpful ways.

Categories for the Working Philosopher

Elaine Landry is working on putting together a book, Categories for the Working Philosopher. The list of topics that various contributors are working on is broad, including model theory, special relativity, quantum mechanics and ontology, biology, computer science, foundations, and more.

John Baez has posted an excellent abstract of his planned contribution. It talks about what happens when we attempt to formalize intuitive notions of equality. Some exciting relationships fall out, such as understanding equivalence as a path between two points in a topological space.

In short, by formalizing the concept of sameness using isomorphisms instead of equality, we bring group theory, geometry and topology closer to the foundations of mathematics, giving them a kind of inevitability that they might not otherwise seem to possess. We also see that these three subjects are tightly connected. A lot of important mathematics, and also theoretical physics, flows from this realization.

I recommend checking out the entire thing. John mentions in the comments that there is a (less exciting) paper in the same vein here, and all of this ties into the recent work on homotopy type theory.

Epsilon-Delta Proof Intuition

There’s a nice question on MathOverflow about the mental experience of mathematics. I tend to lean heavily on bodily sensations of motion and mental imagery when working on something mathematical: stretching, compressing, and movement. Proofs and algebraic manipulation often seem like a sort of flowing. (Something I’ve covered before in my post on reading math.)

But, anyways, here’s an excerpt from one of the responses that I think is particularly valuable. On epsilon-delta proof intuition and understanding:

One specific mental image that I can communicate easily with collaborators, but not always to more general audiences, is to think of quantifiers in game theoretic terms. Do we need to show that for every epsilon there exists a delta? Then imagine that you have a bag of deltas in your hand, but you can wait until your opponent (or some malicious force of nature) produces an epsilon to bother you, at which point you can reach into your bag and find the right delta to deal with the problem. Somehow, anthropomorphising the “enemy” (as well as one’s “allies”) can focus one’s thoughts quite well.

Bad at math? Have you tried steroids?

Men with lower T performed better than other groups on measures of spatial/mathematical ability, tasks at which men normally excel. Women with high T scored higher than low-T women on these same measures.1

Our findings are the first that present the relationship between testosterone and the broad range of general IQ in childhood. The boys of average intelligence had significantly higher testosterone levels than both mentally challenged and intellectually gifted boys, with the latter two groups showing no significant difference between each other.2

Deliberately reducing testosterone levels in men, however, harms cognition, as evidenced by testosterone suppression in those with prostate cancer.3 If anything, the relationship seems to be reversed, with increasing testosterone improving cognition:

Significant improvements in cognition were observed for spatial memory (recall of a walking route), spatial ability (block construction), and verbal memory (recall of a short story) in older men treated with testosterone compared with baseline and the placebo group, although improvements were not evident for all measures.4

But wait! Another study found that boosting test levels among healthy men reduced spatial ability while improving verbal skills.5 One study speculates that the effects of testosterone on cognition are fixed after puberty, in which case, alas, there is still no royal road to geometry.6

Sources


1. Gouchie, Catherine, and Doreen Kimura. “The relationship between testosterone levels and cognitive ability patterns.” Psychoneuroendocrinology 16.4 (1991): 323-334.

2. Ostatníková, Daniela, et al. “Intelligence and salivary testosterone levels in prepubertal children.” Neuropsychologia 45.7 (2007): 1378-1385.

3. Green, Heather J., et al. “Altered cognitive function in men treated for prostate cancer with luteinizing hormone‐releasing hormone analogues and cyproterone acetate: a randomized controlled trial.” BJU international 90.4 (2002): 427-432.

4. Cherrier, M. M., et al. “Testosterone supplementation improves spatial and verbal memory in healthy older men.” Neurology 57.1 (2001): 80-88.

5. O’Connor, Daryl B., et al. “Activational effects of testosterone on cognitive function in men.” Neuropsychologia 39.13 (2001): 1385-1394.

6. Hier, Daniel B., and William F. Crowley Jr. “Spatial ability in androgen-deficient men.” New England Journal of Medicine 306.20 (1982): 1202-1205.

Reading Math: Tips and Heuristics

Reading math is tough. So tough that even Fields Medal winner Bill Thurston wrote about his near-constant confusion. To make it a bit easier, try out these heuristics.

The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.
—G.K. Chesterton

Visualize it: Build a mental image. Lines, triangles, donuts. Add one thing at a time. A significant amount of your brain mass is devoted to imagery. Repurpose it for mathematics. Don’t understand what a function does? It takes an input and produces an output, but that’s too vague. It makes mathematical sausage.

Do I contradict myself? Very well, then I contradict myself, I am large, I contain multitudes.
—Walt Whitman

Metaphorize it: Walt Whitman contains multitudes. So does the number line. Walt Whitman is the number line. The number line is made of numbers. Walt Whitman is made of molecules. Numbers are molecules. Every number can be uniquely factored into a set of primes. Each of Walt Whitman’s molecules can be constructed out of atoms. Prime numbers are atoms.

What did your face look like before your parents were born?
—Zen koan

Kinethesticize it: Grab it, stretch it, tear it, move it, bend it. Zoom in on it. Stack one number line on top of another line. Shift the bottom number line. Compress the line on top of itself. Motion. Feel the equation. What does a summation feel like? What is its original face?

Debug it: Debug yourself. What step does not make sense? What can you not follow? Make it concrete. What’s the simplest case? Plug in numbers. Write it down.

Don’t just read it; fight it!
—Paul Halmos

Fight it!: Unwieldy definitions, holds for all numbers but zero, bah! Why does the author show me something so grotesque? Why can’t I divide by zero? Find out.

What do you know?: You know something like this! What is it?

Great Math Quotes

I don’t collect many things, who needs that junk? But I do have a collection of great math quotes that I’d like to share with you.

Great Math Quotes

We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be: We must know — we will know!
—David Hilbert

If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.
—P. Turan, “The Work of Alfred Renyi”

One must make a start in any line of research, and this beginning almost always has to be a very imperfect attempt, often unsuccessful. There are truths that are unknown in the way that there are countries the best road to which can only be learned after having tried them all. Some persons have to take the risk of getting off the track in order to show the right road to others…. We are almost always condemned to experience errors in order to arrive at truth.1
—Denis Diderot

What is mathematics?

We often hear that mathematics consists mainly of “proving theorems.” Is a writer’s job mainly that of “writing sentences?”
—Gian-Carlo Rota

Last time, I asked: “What does mathematics mean to you?” And some people answered: “The manipulation of numbers, the manipulation of structures.” And if I had asked what music means to you, would you have answered: “The manipulation of notes?”
—Serge Lang

The purpose of computing is insight, not numbers.
—Richard W. Hamming

Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.
—Paul Erdős

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
—John von Neumann

Analysis

I recoil with fear and loathing from that deplorable evil, continuous functions with no derivative.1
—Charles Hermite, on the Weierstrass function

Combinatorics

Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.
—Gian-Carlo Rota

Analytic Geometry

The introduction of numbers as coordinates is an act of violence.
—Hermann Weyl, Philosophy of Mathematics and Natural Science

Non-Euclidean Geometry

Out of nothing I have created a strange new universe.
—János Bolyai

I have made such wonderful discoveries that I am myself lost in astonishment.
—János Bolyai

For God’s sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.
—Farkas Bolyai, quoted in The Mathematical Experience (very recommended, buy a copy)

The assumption that the angle sum of a triangle is less than 180° leads to a curious geometry, quite different from ours but thoroughly consistent, which I have developed to my entire satisfaction. The theorems of this geometry appear to be paradoxical, and, to the uninitiated, absurd, but calm, steady reflection reveals that they contain nothing at all impossible.1
—Carl Friedrich Gauss

Foundations and Certainty

I shall persevere until I find something that is certain or, at least, until I find for certain that nothing is certain.1
—René Descartes

[This] science is the work of the human mind, which is destined rather to study than to know, to seek the truth rather than to find it.
— Évariste Galois

Persist and faith will come to you.
— Jean le Rond d’Alembert

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics; it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But.. the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
—Bertrand Russel, Portraits from Memory

The splendid certainty which I had always hoped to find in mathematics was lost in a bewildering maze….It is truly a complicated conceptual labyrinth.
— Bertrand Russel, My Philosophical Development

But does mathematics need absolute certainty for its justification? In particular, why do we need to be sure a theory is consistent or that it can be derived by an absolutely certain intuition of pure time, before we use it? In no other science do we make such demands. In physics all theorems are hypothetical; we adopt a theory so long as it makes useful predictions and modify or discard it as soon as it does not. This is what happened to mathematical theories in the past, where the discovery of contradictions has led to modification in the mathematical doctrines accepted up to the time of that discovery. Why should we not do the same in the future?
—Haskell B. Curry, Foundations of Mathematical Logic

We are less certain than ever about the ultimate foundations of mathematics and logic. Like everybody and everything in the world today, we have our “crisis…” We have had it for nearly fifty years. Outwardly it does not seem to hamper our daily work, and yet I for one confess that it has had a considerable practical influence on my mathematical life; it directed my interests to fields I considered relatively “safe,” and has been a constant drain on the enthusiasm and determination with which I pursued my research work. This experience is probably shared by other mathematicians who are not indifferent to what their scientific endeavors mean in the context of man’s whole caring and knowing, suffering and creative existence in the world.1
—Hermann Weyl

The method of postulating what we want has many advantages; they are the same as the advantages of theft over honest toil.1
—Bertrand Russel

The recent research on foundations has broken through frontiers only to encounter a wilderness.
—Morris Kline, Mathematics: The Loss of Certainty, (recommended, buy a copy)

We have put a fence around the herd to protect it from the wolves but we do not know whether some wolves were already enclosed within the fence.
Henri Poincaré

Applications

This science [mathematics] does not have for its unique objective to eternally contemplate its own navel; it touches nature and some day it will make contact with it. On this day it will be necessary to discard the purely verbal definitions and not any more be the dupe of empty words.
—Henri Poincaré, The Foundations of Science

It would be necessary to have completely forgotten the history of science not to remember that the desire to understand nature has had on the development of mathematics the most important and happiest influence… The pure mathematician who should forget the existence of the exterior world would be like a painter who knows how to harmoniously combine colors and forms, but who lacked models. His creative power would soon be exhausted.
—Henri Poincare, The Value of Science

The mathematics of our day seems to be like a great weapons factory in peace time. The show window is filled with parade pieces whose ingenious, skillful, eye-appealing execution attracts the connoisseur. The proper motivation for and purpose of these objects, to battle and conquer the enemy, has receded to the background of consciousness to the extent of having been forgotten.
—Felix Klein, Development of Mathematics in the Nineteenth Century

Nature does not offer her problems ready formulated. They must be dug up with pick and shovel, and he who will not soil his hands will never see them.1
—John L. Synge

Platonism

God eternally geometrizes.1
—Plato

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.1
—Johannes Kepler

Formalism

A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from the definitions and postulates that must be consistent but otherwise may be created by the free will of mathematicians. If this description were accurate, mathematics could not attract any intelligent person.1
—Richard Courant

Further Reading

  • There are more math quotes here and here (with comics!). This is a good list, too.
  • For computer science quotes, try here. The nazis at StackOverflow deleted a great collection of quotes, but you can find an archive here.
  • For more general quotes, this page is my favorite.

Sources


1. Quoted in Mathematics: The Loss of Certainty.

Math Art: Picasso as a Mathematician

picasso-and-the-bull

Math art.

I wanted to live deep and suck out all the marrow of life, to live so sturdily and Spartan-like as to put to rout all that was not life, to cut a broad swath and shave close, to drive life into a corner, and reduce it to its lowest terms.
Henry David Thoreau, Walden

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
John von Neumann

Picasso drew math art: a bull in full and then simplified the figure to a few lines, to the essence of a bull. This is the role of mathematics in the sciences. She takes reality and distills it to its essence. She drives life into a corner and reduces it to its lowest terms.