It’s been almost a year since I last blogged, and I’ve spent much of that time feeling guilty about not blogging enough. So here we are. I was lured out of my state of blog-apathy by a recent post by mathbabe; in fact, this will be the second of my very few blog posts inspired by that blog. If you’re not already reading that blog, you really should – it’s a brilliant mix of math, politics, economics, data, and sex.

In any event, mathbabe was commenting on a video which has apparently been making its way around the internet. In this video, some mathematicians (Or perhaps physicists? What are string theorists calling themselves these days?) attempted to explain the mind-boggling “fact” that

Watch the video if you like, but by now a number of other mathematicians have rightfully pointed out that most of the fishy manipulations in the video amount to fraudulent nonsense which can be used to justify just about anything. This infuriates me, because the people who made the video could have used the opportunity to *legitimately* blow people’s minds by placing the equation above (which does make sense, from the right point of view!) in its proper context and explaining some beautiful mathematics.

I don’t have the apparatus to make a cool video, but I do have a blog. So I’m going to make an attempt to do what I think the video should have done (I am not optimistic that my attempt will get picked up by Slate, of course). Instead of adding up all of the positive integers, I’m going to start by adding up all of the powers of two:

We still get a negative number, so this equation should be just as counter-intuitive as the original one (though admittedly is pretty bizarre). Our strategy for making sense of both equations will be the same:

- Write down an equation which makes sense (both logically and intuitively) in a narrow context
- Observe that the right-hand side of the equation actually makes sense in a much larger context than the left-hand side
- Use the right-hand side as a proxy for the left-hand side in the larger context

This strategy, called *analytic continuation* by mathematicians, is extremely powerful. But the basic idea is really quite simple, and it is even familiar in the context of language. When you “log in” to your e-mail account, your messages are likely organized into various “folders”, some of which are in your “inbox” and some of which are in your “outbox”. Perhaps you have a list of your friends’ e-mail “addresses” in your “address book”. The words that I put in quotes all began life in the narrow context of physical reality but have been extended to the new context of the internet; your e-mail inbox is not a physical box anywhere in the world, nor does your e-mail address refer to an actual place you can go. Someone at some point in the history of the internet realized that physical mail is a good metaphor for the electronic messages people send each other, and thus the language surrounding physical mail actually makes sense in the context of the internet.

The strange equations that I wrote above are mathematical counterparts of taking a word such as “hyperlink” which only really makes sense in the context of the internet and applying it to real world mail. You would end up with a sentence which looks pretty bizarre, but there would nevertheless be a certain logic to it.

Let’s see how this all plays out mathematically. We’ll start with something that isn’t likely to stir up much controversy:

This is the mathematical counterpart of the observation that if you walk across half of a room, then a quarter of the room, then an eigth, and so on then you will have crossed the whole room. (Of course, there are some philosophical questions to be raised by the fact that the phrase “and so on” took the place of an infinite number of actions. Even non-controversial infinite series deserve serious thought.)

You might also convince yourself that

It might not be obvious that the answer is , but this answer is at least plausible: we start with a number which is smaller than and add increasingly tiny numbers to it. And if you plug numbers into a calculator you will get good numerical evidence that this equation makes sense; the further out you go in the sum, the closer you get to . In general, if the absolute value of is a number smaller than , we have:

Of course, the only context in which the left-hand side really makes sense is when ; this ensures that the powers of get very small very vast and thus the settles near a particular value. If then there is no such guarantee: the powers of $s$ do not get smaller, and you can get a number as large as you want by adding up enough terms in the sum.

The right-hand side, on the other hand makes sense in a much larger context: we can plug in any number except ! In particular, we can plug into to obtain . Since the expressions and agree when , it makes sense to use the latter expression as a proxy for the former at other values of , such as . In othe words, it is not entirely stupid to write:

There is some theory which makes this equation even less stupid: is (in a sense which can be made precise) the only sensible way to extend beyond the set . Properly justifying this requires techniques coming from one of the most beautiful subjects in all of mathematics: the calculus of complex numbers. It should not be at all obvious, but in the end this whole discussion is really all about the mysterious powers of complex numbers.

The same techniques allow us to analyze the sum which got this post started; this time, our starting point is the *Riemann Zeta function*:

This time the sum on the left-hand side makes sense as long as , but the same tools described above imply that the Riemann Zeta function can be “analytically continued” to allow any input except , and its value at can be calculated to be . This calculation could occupy another entire blog post, so I will not go any further than that at this time.

Now that I have explained the sense in which it is not completely stupid to say that the sum of all the positive integers is , I would like to conclude by arguing that it still is pretty stupid. Notice that according to the reasonging described in this post we did not assign the sum a value by thinking about it intrinsically as we can with, for instance ; instead we related the sum to the Riemann Zeta function and analyzed that function. But there are infinitely many other possible functions which have a similar relationship to , and many of them will assign different values to the series following the steps outlined here. In fact, you can use these steps to justify giving the sum any value you want. Still, the Riemann Zeta function enjoys a privileged position in mathematics (and physics) so is a pretty good choice.

## Gender and the Mathematical Community

February 11, 2013

pwsiegel General Audience commentary 12 Comments

I still haven’t posted all that much in this blog, and essentially nothing research-related. I’ve been writing a bit offline, and I’ll probably adapt some of what I’ve been thinking about into blog form fairly soon. In this post I’d like to address some issues related to sexism and gender bias in the mathematical (and perhaps broader scientific) community. I think about these issues rather often, but I’m writing about them now because of recent posts in The Accidental Mathematician (Izabella Laba’s blog) and mathbabe (Cathy O’Neil’s blog).

The thrust of Izabella Laba’s post (entitled “Gender Bias 101 for Mathematicians”) is that gender bias in the mathematical community is not limited to a few grouchy old codgers, but rather that it is a systematic cultural and psychological phenomenon which afflicts everybody. There are two potentially controversial assertions implicit in this statement:

The first assertion is pretty hard to argue with, though I’m sure some people still try. Every math department with which I have been affiliated is *massively* male dominated, and there is ample evidence that hiring practices, salaries, journals, etc. are stacked in favor of men. I’m not going to try to document or justify this in any detail because I don’t have the facts available at my fingertips and because the issue has been argued to my satisfaction elsewhere (e.g. in the Accidental Mathematician).

The second assertion might be more surprising to some, and it’s the one I want to discuss here. Izabella Laba’s post quotes a recent study in which faculty from research oriented universities were presented with applications for a lab manager position with randomly assigned male or female names. The study found that a given application with a male name at the top was consistently rated more highly than the same application with a female name. Interestingly enough, the pattern was independent of the gender of the faculty evaluator: female professors were just as biased as male professors. Cathy O’Neil contributes another study which shows that 15-year-old girls outperform 15-year-old boys in science exams in some countries but not others (not in the United States), indicating that gender gaps in science are cultural rather than biological.

Both of these studies are quite compelling, and I’m sure there are others which point to the same conlusion. My intention is to participate in this discussion subjectively rather than objectively. In short, I am going to use the rest of this post to analyze my own gender-oriented biases. Something feels a bit self-indulgent about this exercise, but I think it will be healthy for me even if it isn’t useful for anyone else.

I will begin by admitting outright that I am biased against women. I consider myself to be a pretty progressive guy – perhaps even more progressive than most – and I think that most people who know me would say that overall I do a good job of treating women with the same respect with which I treat men. But this is not because I don’t have biases, it’s because I work very hard to identify them and eliminate them or at least minimize their impact on my behavior. I am unqualified to generalize my own psychological observations to everyone else, but I suspect that it is neurologically almost impossible for a person socialized in 20th or 21st century American society to avoid gender biases: we are bombarded with overt and covert messages about gender constantly and starting at a very young age. Given what I have been learning lately about how insignificant our conscious thought processes are in comparison to our subconscious psychological machinery, these messages must take their toll.

What forms do gender biases take? There are many answers with varying applicability to me. Here is a non-comprehensive unordered list that I have assembled from reading things online, talking to people, and making my own observations.

Intelligence and Competence Bias:This is simply the assumption that women are less intelligent or less competent than men. I have heard numerous stories in which Andrew launches into a lengthy explanation to Barbara about a subject in which Barbara is more of an expert than Andrew. Here is a particularly cringe-inducing example of this. I tend not to offer unsolicited explanations to men or women very often, and when providing solicited explanations I usually make an effort to identify my audience’s background, so I don’t think I am terribly guilty of this particular behavior. Instead, I notice this bias in myself when I am seeking an expert on a particular subject and I am presented with a male option and a female option. Sometimes I catch myself behaving or thinking according to the assumption that the male expert is more knowledgable or more adept than the female expert even if I have no particular reason to make such a judgement. I have to force myself to think deliberately about what I know and don’t know when making these sorts of comparisons.Experience Bias:Lately I have started noticing a disturbing pattern in my judgements about a person’s age, experience, education level, etc.: my estimates are consistently lower than reality for women and higher than reality for men. I have heard many stories from women in which they are demoted from faculty member to graduate student or from graduate student to undergraduate by a male interlocutor, and I am embarrassed to admit that I have done this before. I have also heard stories in which a female graduate student or faculty member has been assumed to be a secretary or staffperson; I don’t necessarily consider this to be a “demotion,” but I doubt I would appreciate it if it happend to me. These days I try to avoid guessing somebody’s position or experience level at all, and if I do make a guess it’s generally “faculty” regardless of gender (in a university setting). Still, it requires conscious effort on my part.Common Ground Bias:This is the assumption that, all things being equal, I will have more in common with a male than a female. This bias is fairly understandable – there are, after all, real biological and social differences between men and women – but I think it has unfortunate consequences in an academic setting. Few of my mathematical conversations with my peers begin, “Hi, my name is Paul. Would you like to have a conversation about elliptic cohomology?” Instead, they typically begin with the typical introductory social graces and lead into mathematical territory after a basic rapport has been established. This rapport is more difficult to establish with a person with whom I assume I will have a harder time identifying with before the conversation even begins, and consequently I am more likely to engage in mathematical conversations with my male colleagues than my female colleagues. I don’t know how socially isolated women in math departments feel, but I suspect that it’s more of a problem than I realize. My plan for reducing the impact of this bias is to simply be more bold and less awkward about engaging people in conversation, but this isn’t always easy.Sexual Biases:I am a heterosexual man who is attracted to intelligent and ambitious women, and the women that one finds employed in a math department often fit this description. Sexual attraction is firmly rooted in extremely powerful subconscious processes, and I am certain it affects my interactions with my female colleagues in ways that I don’t fully understand. If nothing else, it consumes some measure of my mental energy that is liberated when I’m interacting with men. It seems very hard to deal with the subconscious aspects of this bias, but I long ago adopted a mechanism which at least helps me manage the factors that are under my control. I decided early on in graduate school that I would categorically avoid romantically pursuing anyone in my own department. This allows me to sidestep the hazards associated with workplace romances in general, but mainly it helps me ensure that I treat all of my colleagues as professionally as possible. I don’t know how often the average woman in a math department is forced to deal with romantic overtures from her male colleagues, but given the highly skewed gender ratios I’m guessing it’s more than I imagine. I am also largely ignorant of the consequences of this behavior.I’m sure there are other biases worth mentioning, but this list feels like a good start. One interesting supplementary observation about biases in general is that thinking about them leads to an unfortunate feedback loop: worrying about biases against women affects my behavior toward women. I think this effect is fairly minimal in comparison to the consequences of ignoring my biases and failing to monitor my behavior at all, but it’s there all the same.

My final remark about this subject is that there are many other bias issues which are also largely ignored by the mathematical and scientific community. I have encountered some discussion of racial bias in science, but I have heard almost no discussion about biases related to sexual orientation. If anyone reading this is aware of any studies or references about these issues, I would be interested in seeing them. Also, in this post I have focused on the effects of bias on my interactions with my colleagues, but the way my biases manifest themselves in my teaching is a whole other subject which I might take up in the future.