Why mathematics doesn't need a Mitchell Report

Yesterday, the Mitchell Report, on steroid usage in baseball, was released. A large number of players were named as users or potential users.

The media coverage of this has routinely mentioned that amphetamines seem to be more common in baseball than steroids; one source I ran into said it's believed that half of baseball players use amphetamines regularly.

Now, when I think of amphetamines, I think of Paul Erdos, and the following story: Ron Graham bet Erdos that he couldn't quit amphetamines for a month, cold turkey. Erdos did. Graham paid up. Erdos said "you've set mathematics back a month".

As far as I know, we mathematicians don't have a drug problem. (Unless you count coffee. I'll freely admit I have a coffee problem.) But I don't think that anybody would say that Erdos should be stripped of any award he won because he was using drugs. The difference is that in sports, the players are competing against each other; thus a level playing field is essential. But in mathematics, we like to believe that we are not competing against each other but cooperating to discover truth; thus we may gladly accept all the chemical help we can get.

The meta-Minkowski conjecture

In a talk by Eva Bayer-Fluckiger today at UPenn, I learned about Minkowski's conjecture, which is as follows:

Let K be a number field of degree n, O_K its ring of integers, and D_K the absolute value of the discriminant of K. Then the Euclidean minimum of K is the smallest μ such that for any x in K, there exists y in O_K such that the absolute value of the norm of x-y is less than μ; that is, there's always an algebraic integer "within" M(K) of every element of K. If M(K) < 1 then O_K is Euclidean (this is essentially a rephrasing of the definitin); if M(K) > 1 then O_K isn't Euclidean; if M(K) = 1 we don't know. Then we have

Minkowski's conjecture: M(K) ≤ 2^-n (D_k)^1/2

for totally real number fields K of degree n. (I'm taking this from Eva Bayer-Fluckiger and Gabriele Nebe, On the Euclidean minimum of some real number fields, Journal de Théorie des Nombres de Bordeaux 17 (2005), 437-454, with some rephrasing.)

Curtis McMullen recently proved this for n=6 (link is to a preprint "Minkowski's conjecture, well-rounded lattices and topological dimension", on his web page). This follows proofs:

• for n=2, Minkowski, circa 1900;


• for n=3, Remak, 1928;


• for n=4, Dyson, 1948;


• for n=5, Skubenko, 1976;

and so it seems like we get one higher degree every thirty years! This leads me to formulate the following:

Meta-Minkowski conjecture: Minkowski's conjecture is true, but will never be proven.

(Sketch of heuristic argument: in approximately the year 1840 + 30n, the degree-n case will be proven. So if we wait long enough, Minkowski's conjecture will be proven for any finite value of n.)

Per a question that was asked at the talk, it seems like the approach in each of these papers was substantially different. McMullen gives sufficient conditions for proving Minkowski's conjecture for any given value of n, and it turns out that those conditions were already known for n = 6.

I wonder to what extent it's possible to predict the mathematical future from the mathematical past, in the case of problems such as this where there's a numerical measure of ``incremental" progress. Often I've seen tables where an improving sequence of constants is given for some problem (for example, a sequence of improving constants in some inequality). It's sort of a Moore's Law for mathematical problems.

When would I make a conjecture that a proof for all n would be found? If, say, we had proofs of some statement for n = 1 in 1960, n = 2 in 1990, n = 3 in 2000, n = 4 in 2005, and in general n = k in 2020 - 60/k; then I'd guess that the full problem gets solved in 2020. Of course, in a case like this the proofs would pile on top of each other, and in reality one would be likely to see large numbers of values of n disappearing in one fell swoop sometime in the 2010s.

An exercise for the reader

"Decode" the following equation, seen on a T-shirt:

Fibonacci tattoo!

From Pink Haired Girl, who I apparently share several mutual friends with: a tattoo to generate the Fibonacci sequence, in Scheme.

There are faster algorithms to generated the Fibonacci sequence, but they obscure the recursive definition; her tattoo follows the simple recursive definition F_n = F_n-1 + F_n-2, which takes O(n) arithmetic operations to compute F_n. Wikipedia's article on Fibonacci numbers gives an identity
F_2n+k = F_k F_n+1² + 2F_k-1F_n+1F_n + F_k-2F_n²

Now, note that we get F_-2 = -1 , F_-1 = 1 , F₀ = 0 by running the recursion backwards. Letting k=1 gives

F_2n+1 = (2F_n+1 - F_n) F_n

and letting k=2 gives
F_2n+2 = F_n+1² + 2F_n+1F_n.
Letting m+1 = n, we get
F_2m = F_m² + 2F_mF_m-1
= F_m(F_m + 2F_m-1)
= F_m(F_m+1 + F_m-1).

Thus we can express F_n in terms of Fibonacci numbers of indices around n/2, and thus find Fibonacci numbers in logarithmic time... but that obscures the central fact about these numbers, which is the recursion that produces them. (There are probably other ways to arrange those identities that give more efficient algorithms than the one I'm alluding to, but I didn't try that hard.) There are also identities for F_n in terms of Fibonacci numbers of index around n/3, for example -- see Wikipedia -- which could be even faster.

Look Around You (BBC spoof)

At YouTube: Look Around You - 1 - Maths, an episode of the BBC spoof science program Look Around You. (Why don't we have spoof science programs in this country? As it is, I can't tell which parts I found funny because they're mathematically silly and which parts I found silly because they're British.)

Who scribbles meaningless mathematical formulas on the wall, like the kids at the beginning? It often bothers me when meaningless mathematical notation is used for "effect", because it doesn't even look like something a clueless student might scribble; it just takes a bunch of symbols and throws them out there, and I try to understand them but can't. I am incapable of looking at mathematical symbols and not trying to understand them. I suspect this is something like the Stroop effect, which is the experimentally observed effect that it's easier for people to read words which name colors when the ink color matches the colors named by the words than when they disagree. (Incidentally, in this article from the New Republic, excerpted from his new book The Stuff of Thought, Steven Pinker, reporting on an experiment by Don MacKay, says we have even more trouble doing this task when the words are not color words but obscene words; essentially, people can't ignore obscenity. I can't ignore mathematics.)

and I wonder if "Chapter 3.1415926" of the textbook actually exists, since I don't know the context of this. It reminds me of the version number of TeX, which gets one digit longer with each new version; Knuth has requested that after he dies the version number be fixed at π.

There are two people for whom the largest number they could think of was "a hundred thousand" and "nine hundred ninety-nine thousand", which is kind of surprising; why can't you just add one to both of those? At least someone who answered "999,999" is admitting that they don't know the word for the next number. (The program makes light of this by suggesting a very large number as the very largest number... and then adding one to it.)

I also rather liked the following description of a circle: "A uniformly curved line that somehow joins up with itself that science has yet to find a name for. Can you think of a name for it? If you can, the Royal Mathematics Society would like to hear from you, because they hold a competition each year to find a name for this figure." Ah, if only it were so easy. Naming things isn't hard. (Giving them good names, on the other hand, is quite tricky, especially if you restrict yourself to words that are already words existing in natural languages; I want to be sure that the words I pick to name something don't have the wrong connotations. Sure, I can tell people to ignore them, but it's not so easy to actually do so!)

Also, this video is about "maths", not "math". Are other Americans bothered by this, too? I've never been entirely sure what to make of the fact that across the pond, my field is plural.

They're building a science thing!

From The Onion: Scientists Ask Congress To Fund $50 Billion Science Thing.

Another diagram presented to lawmakers contained several important squiggly lines, numbers, and letters. Despite not being numbers, the letters were reportedly meant to represent mathematics too. The scientists seemed to believe that correct math was what would help make the science thing go.

That's right, folks. Mathematics is all about numbers, at least in the eyes of Normal People. But I can't count how many times I've looked at a blackboard during a lecture and realized there wasn't a single number there.

A cultural map of the world

Take a look at the Inglehart-Welzel Cultural Map of the World:

The World Values Surveys were designed to provide a comprehensive measurement of all major areas of human concern, from religion to politics to economic and social life and two dimensions dominate the picture: (1) Traditional/ Secular-rational and (2) Survival/Self-expression values. These two dimensions explain more than 70 percent of the cross-national variance in a factor analysis of ten indicators-and each of these dimensions is strongly correlated with scores of other important orientations.

The resulting "map" doesn't resemble any "traditional" map of the world, but it's interesting. For example, all the English-speaking countries end up near each other, all the countries of Protestant Europe end up near each other, and so on.

I'm not sure why "English-speaking" is its own group, while no other language is given its own group. In particular, all the Spanish-and-Portuguese-speaking countries seem to end up together. (This is obscured by the map, which for some strange reason includes Uruguay in "Catholic Europe" and Portugal in "Latin America". I'm conflating Spanish and Portuguese not because I don't know there's a difference, but because they are fairly similar as languages go.

My instinct is that the "Traditional Values"/"Secular-Rational Values" divide is similar to the ideological conservative/liberal divide in American politics (although I'm not sure how this would be made precise); I want to say that the "Survival Values"/"Self-Expression Values" dimension corresponds to the economic conservative/liberal divide in American politics although that seems like a lot more of a stretch. Apparently countries seem to move from "Survival" to "Self-Expression" (i. e. rightward on the graph) with time.

One sees a lot of these "factor analysis" plots where a large-dimensional space is reduced to just two dimensions in this way; I don't think there's some fundamental reason why two dimensions is the natural way to think about this, but rather that we're just good at drawing two-dimensional pictures. Dave Rusin's Mathematical Atlas includes such a plot (although naming the dimensions is tricky -- I thought they might be discrete vs. continuous and pure vs. applied.) I've also seen a political map like this, based on the voting records of U. S. Senators and Representatives; it's kind of fascinating to watch how the two parties have moved around with time, and how you can explain almost as much variation between politicians by just looking at a single variable as you used to need two variables for. You can almost predict who will vote for a given bill just by lining up the Senators from "most liberal" to "most conservative" and drawing a line somewhere to separate the two sides. Life is more complicated than that.

I wonder if one could use a plot like this (or the data which underlies it) to predict which international borders are likely to create a lot of tension. For example, the U. S. is (according to this plot) much more similar to Canada than it is to Mexico, and there seems to be a lot more tension at the U. S.'s southern border than at its northern one. Perhaps one could predict strife within a country as well, if a survey like this was done for subnational entities. Lumping the entire United States together seems almost ludicrous to me.

Also, how is this sort of thing correlated with language? And if the language spoken in a country changes, for whatever reason, is this correlated with that country becoming more like countries that speak the new language? It seems reasonable that there should be some connection between language and culture, if only because most of culture is expressed through language. But causation is a problem; do countries become more like each other because they speak the same language, or do countries that speak the same language become more like each other?