A beautiful answer to why pure mathematics matters

Credit goes to Khalid Bou-Rabee, who posted this on MathOverflow, and Benson Farb, to whom the quote belongs.

“Since I am a pure mathematician, Dean Hefley suggested as a possible topic for this talk: “Why the square root of negative 1 is necessary”. I could take up this challenge of justifying pure science on its vast applicability; indeed the square root of negative 1, the basic “imaginary number”, underlies a huge swath of modern technology, from the design of circuits, airplanes and skyscrapers, to the construction of economic and financial models, to robotics. I have decided, however, to take the opposite point of view. I want to defend the value of basic science for its own sake…

…the purpose of pure mathematics, of basic science, is not the quick harvest. It is nothing less than an attempt to bring human thought and understanding to a higher level. It is an attempt to change not just what we think about the world, but how we think about it. The importance of this for human evolution is incalculable. As British physicist JJ Thomson said: “Research in applied science leads to reforms, research in pure science leads to revolutions.”

- Benson Farb, 2012

Some gems from analytic number theory

I am currently attending the Counting Arithmetic Objects summer school at Centre de Recherches Mathematiques in Montreal. During the second day of lectures I heard a remarkable talk from Professor Andrew Granville. He spoke with outstanding clarity on the basic aim of analytic number theory, and I wish to remark on some of the epiphanies I had during the talk.

Without a doubt, one of the most important identities in analytic number theory is the following:

$\int_0^1 e^{2\pi i n t}dt = \begin{cases} 1, & \text{ if } n = 0 \\ 0, & \text{ if } n \ne 0 \end{cases}$

The key insight is that the above identity serves as an “indicator function” for whether a quantity is zero or not. Indeed, this basic observation is behind behind the solutions to Waring’s problem and the proof of Vinogradov’s theorem on the sum of three primes. This “smooth” indicator function allows the use of methods in analysis to deal with arithmetic problems.

A related identity is Perron’s formula. Here again we start with an “indicator” function, defined for $c > 0$

$\frac{1}{2\pi i} \int_ {c-i\infty}^{c+i\infty} \frac{y^s}{s} ds = \begin{cases} 1, & \text{ if } y > 1 \\ 0, & \text{ if } y < 1 \end{cases}$

This allows to handle sums of the shape

$\displaystyle \sum_{n \geq 1} a_n.$

To see this, we start with the observation that for each $n$ in the range of summation we have (provided $x$ is not an integer) that $x/n > 1$. Hence we have

$\displaystyle \sum_{n \leq x} a_n = \sum_{n \leq x} a_n \frac{1}{2 \pi i } \int_{c - i\infty}^{c + i\infty} \left(\frac{x}{n} \right)^s \frac{1}{s} ds.$

The outer sum is finite, but we would not change the sum at all if we included all positive integers $n > x$ because of the indicator function nature. For well behaved $a_n$‘s and taking $c$ sufficiently large to ensure absolute convergence, we have that

$\displaystyle \sum_{n \geq 1} a_n \frac{1}{2 \pi i} \int_{c - i\infty}^{c + i\infty} \frac{1}{s} \left(\frac{x}{n}\right)^s ds = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \left(\sum_{n\geq1} \frac{a_n}{n^s}\right) \frac{x^s}{s} ds.$

We note that $g(s)= \displaystyle \sum_{n \geq 1} \frac{a_n}{n^s}$ is the Dirichlet series associated to the $a_n$‘s. Summarizing, we obtain the equation

$\displaystyle \sum_{n \leq x} a_n = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} g(s) \frac{x^s}{s}ds.$

Now we can apply this to the familiar Von Mangoldt function $\Lambda(n)$ defined by

$\displaystyle \Lambda(n) = \begin{cases} \log p, & \text{ if } n = p^m \\ 0, & \text{ otherwise.} \end{cases}$

where as usual $p$ denotes a prime. Now using Perron’s formula, we have

$\displaystyle \sum_{n \leq x} \Lambda(n) = \sum_{p^m \leq x} \log p = \frac{1}{2 \pi i} \int_{c - i\infty}^{c + i\infty} - \frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s}ds$

Where $\zeta(s)$ denotes the Riemann zeta function. This gives rise, after evaluating the integral on the right using residue theory and noting that the poles of $\zeta'(s)/\zeta(s)$ are 1 and the zeroes of $\zeta$, we have

$\displaystyle \sum_{p^m \leq x} \log p = x - \sum_{\rho: \zeta(\rho) =0} \frac{x^\rho}{\rho}+\frac{\zeta'(0)}{\zeta(0)}.$

This is likely one of the most striking equations in mathematics as the left hand side is a discrete sum while the right hand side involved values of a meromorphic function. Somehow, the zeroes of the zeta function “knows” where the primes are.

Problems with democracies having different parties

The terms ‘democracy’ and ‘multi-party state’ have almost become synonymous. China, for example, is notorious for being a ‘single party state’, which is a (slightly) more polite way of saying ‘authoritarian regime’ or ‘dictatorship’. Since China’s political system is ubiquitously painted as either bad or outright atrocious, it must mean that a multiple party system is the correct answer. Indeed almost all democracies have multiple parties, including the USA, Canada, Britain, Australia, Israel, Japan, etc. However, the notion that ‘party’ is the defining attribute of a democracy is deeply problematic.

On the eve of the Ontario provincial election here in Canada (which I cannot participate because I missed the deadline to mail in my vote and I am currently out-of-province), we are faced with possibly the lowest voter turn-out in our entire history. This is a grim realization since the current record holder, an abysmal 48%, was set just three years ago in the last Ontario election. To quote this Maclean’s article, would-be voters have “changed the channel” a long time ago. This is because a fundamental aspect of the nature of democratic politics is revealed and amplified by the advance of technology: the most important priority for any politician is to get elected, and the most important priority of any political party is to form government. Ideally, the best way to achieve these priorities is to be good at your job and for your party to be the one to best serve the interests of your citizens. Unfortunately, with the help of modern technology and statistical methods, this is no longer the case. As is painfully underlined in the Eric Cantor case, often the most effective method to stay elected and stay in power is to pander to the loudest, most driven minorities in the electorate at the expense of alienating everyone else.

The fundamental problem is that political parties, small in number, cannot collectively represent every possible person. This is further exacerbated by the unspoken dogma that political parties cannot agree with each other on anything. For a person, it may be perfectly reasonable to be simultaneously pro-abortion but pro-immigration reform, but in the USA such a person would necessarily have to choose between the two issues: since Republicans are pro-abortion and anti-immigration reform, while Democrats are pro-choice and pro-immigration reform. In this case, whoever wins, the person loses.

The tendency for politicians and political parties to serve their own interests even when these do not align with the best interests of the population seems like a major hurdle that no current democracy is equipped to address.

Another problem with political parties is their tendency to be slow to evolve ideologically. For instance, the very notion of ‘left, center, right’ model of thinking in politics may be extremely outdated. Even at a more refined level, it is possible that political parties and their ‘core’ ideologies will simply become irrelevant given sufficient time, but the political parties do not die because they are designed to self-preserve. The very idea that we both Canada and the US have basically the same political parties since 150 years ago is perplexing. In some sense, our politics are stuck in the 19th century.

Why peddle your political system or ideals if you are so discontent with them?

http://www.gallup.com/poll/166838/congress-job-approval-starts-2014.aspx

http://en.wikipedia.org/wiki/United_States_presidential_approval_rating

http://www.washingtontimes.com/news/2014/apr/21/americas-oligarchy-not-democracy-or-republic-unive/

Together, these three links indicate that Americans are extraordinarily unsatisfied with both their elected legislators and generally at best mildly satisfied with their elected Commander in Chief. Even in the best of times, no greater than three quarters of Americans are satisfied with their president at any given time, and usually the approval rating hovers around 50%. More importantly however, it seems that the vaunted constitutions and laws of the United States have failed to protect its status as a democratic republic, according to the third link.

So why do Americans focus so much of their vitriol towards supposedly ‘oppressive’ regimes that probably have much higher approval ratings? Perhaps the American stance would be better perceived if they were at least happy with their system themselves (a much more convincing argument would be if their system actually works the way it should, which it doesn’t).

On so-called human “nature”

Most certainly one of the oldest debates known to civilization is what constitutes human “nature”. This debate is truly ubiquitous with almost every major religion and philosophy chiming in on the matter. In China, this was one of the fundamental differences between Legalists (the philosophy adhered to by the Qin and the First Emperor) and Confucians (the philosophy adopted by Emperor Wu of the Han Dynasty and had since become the de facto state religion ever since). Human ‘nature’ has been used to justify many things, both good and bad, throughout the ages. I will discuss some examples where ‘nature’ or the ‘natural order’ justifies some otherwise difficult to justify stances.

Example 1 – stance against homosexuality/gender reassignment.

The argument: certainly this is “unnatural”, since the basic goal of any organism is to reproduce. Since humans reproduce sexually, in order to reproduce we must seek a mate of the opposite sex. Certainly gender reassignment surgery never existed in ‘nature’, and even if it did, all it does is to make a specific person unable to reproduce at all. Thus this kind of state of existence is undesirable and thus there must be something wrong with it.

A possible justification: aiming specifically to refute the argument above without introducing any other loftier ideals of equality and human rights, a possible explanation for the continued existence of homosexuals and other persons with distinct gender and sexual identities in our collective gene pool is that while for individual organisms reproducing and passing down their specific genes is obviously optimal, evolution tends to favor entire species which can pass down their genes collectively. From that perspective, individuals that are not able or unwilling to reproduce themselves but willing to rear young may be useful to the collective. Indeed, even in nature, orphans will exist; and often cannot survive on their own. Sometimes orphans are taken care of by their clans, but if there are individuals that are willing to take in orphans because they have no offspring of their own, then it helps preserve genetic diversity and perhaps even pass down some of the best genes. Thus, from this perspective, there is no “natural” problem with the existence of homosexuals or persons with distinct gender identities.

Example 2: People (males specifically) are more violent by nature, and so the basic aim of reducing forms of violence that tend to be perpetrated by men (i.e. sexual violence) is fundamentally futile.

Argument against: There is a lot to be said of sense of ‘violence’. Speaking broadly, most dystopian works of fiction and almost all apocalyptic films focus on this concept that humans are fundamentally violent, chaotic beings and as soon as the established order is broken, they will kill each other. One only needs to look at any zombie movie or even the new Planet of the Apes movie to get a sense of this. However, as was pointed out here, it was human nature in the first place that built civilization.  And despite the recent spate of news about sexual violence (not just the Eliot Rodger incident neither, even in India and Sudan we hear related news all the way over in North America), the cause isn’t that this is happening more often; it’s just being reported more as it enters into the public consciousness. Without any statistics on-hand (this type of thing tends to be under reported anyway, so I am not sure how much statistics can prove), one can guess that sexual violence as a whole is going down. This has many reasons. Among them is that through better education and overall social progress, more men understand that sexual violence is morally wrong (and not just wrong because it might land you in prison). However, a more subtle and definitely less lofty reason is the increased access to sexually explicit material. Much of the ‘nature’ of men’s tendency towards violence (against women) is from a physical need. When that need is dealt with by ‘Pamela Handerson’, the urge to commit violent acts decreases. At that point it becomes psychological and surprise, psychology mostly comes from social construction and not inherent from birth.

However, as I’ve stated in other media before, the women’s rights movement will always have a tough time compared to other similar causes (arguably, the LGBT movement has made much more progress in the same time period than feminism) because a lot of men fail to see the fundamental concept in feminism as it is divergent from their daily experience. I forgot where the quote comes from, but I believe this is a brilliant way to summarize why a lot of men don’t feel like they are the ‘dominant, aggressive’ one in their lives: “When I was a child, my mother bossed me around. When I became a man, my wife bossed me around. When I became a father, my daughter bossed me around. How am I oppressing women?” This is not unfounded. Using another ‘this is nature’ argument which I have advocated against thus far, men aren’t really needed in the grand scheme of evolution. There is no inherent problem with keeping a small male to female ratio in a population; indeed that’s exactly what happens with livestock. This is the most logical reason why men have traditionally been used to fight wars or to work the most dangerous jobs; they are expendable. Thus far men have tried to justify their existence through intimidation and violence, but this time has come to an end. To continue to justify our existence in the future, we have to do the fair share or better.

Some corrections for Bourgain-Gamburd-Sarnak’s paper “Generalizations of Selberg’s 3/16 theorem and affine sieve”

I am currently reading the paper

J. Bourgain, A. Gamburd, P. Sarnak, Generalizations of Selberg’s $\frac{3}{16}$ theorem and affine sieve, Acta Mathematica, 207 (2011), 255-290

and I noticed some problems with Section 3. Namely, there are two errors. The line “taking $z = w = i$ in (3.2)” is plainly wrong; since that would yield $u(i,i) = 0$ which is not desired. Further, equation (3.3) is wrong as stated; it should be $\displaystyle \lVert g \rVert^2 = 4 \frac{u(gi,i)}{\det(g)} + 2$.