Math Is Democracy III: A Short Rant about Voting Theory

I have been full tilt exploring the relationship between mathematics and democracy for the last at least year and a half. E.g., check out my TED talk.

And I’ve been thinking about this relationship, in some form, for the last 8 or 9 years, ever since Jay Gillen said to me, “Everything you love about math is what free people love about democracy.” (See this previous post.)

Given this, it may be surprising that until fairly recently, the well-established mathematical subfield that explicitly addresses democracy, voting theory (aka social choice theory), never grabbed my interest. FWIW, this has changed, but I think my journey around it is indicative of something worth keeping in mind for mathematical people who want to use math to think about how society works. (E.g., me, and hopefully you.)

Voting theory asks: given a large number of people forming a collective (e.g., a nation or state!), and a choice the collective is faced with (e.g., electing a political leader!), what are the possible ways one could aggregate individual preferences into a collective decision? (E.g., everybody vote for one candidate, and the candidate with the most votes is elected, aka plurality vote, aka how most political elections work in practice; but other possibilities too, like instant runoff, the Borda count, approval voting, and score voting.) And what are their properties? (E.g., in plurality voting, there is a potential spoiler effect when similar candidates split the vote of the majority, which is mitigated in approval and score voting since they do not force voters to choose only one candidate to support.)

It’s sort of a puzzle: given how much I’ve always cared about math, and how much I’ve always cared about democracy, why wouldn’t I want to understand this developed and beautiful theory exploring this most fundamental democratic operation — how does a collective make a decision?

Well, I’ll tell you.

The first and loudest thing I heard about voting theory, beyond just what it’s about, was Arrow’s impossibility theorem. This is a “foundational” result in the theory. It is often glossed as asserting that “there is no perfect voting system.” (Do not take this interpretation at face value. More below.)

From this, I got the impression that the point of voting theory was for us to brain out about the possibilities, and then conclude (with mathematical certainty!) that “we the people” can’t win no matter what. I’m supposed to be excited about this?

Was the whole thing just an intellectual exercise? Many years ago, I listened to a mathematician excitedly describe the Borda count to some graduate students. I remember thinking, with some confusion, “Where is the excitement coming from?” It seemed to me that the mathematician was excited only intellectually, about the mathematical properties of this system, and was totally disengaged from the question of whether anybody had used it or would use it or should use it to make any collective decision.

In this context, voting theory seemed almost grotesque to me. I love math for math’s sake — my PhD work is in pure algebra. But taking the urgent moral and political problem of building a system of government that adequately reflects the will of the people, and using it as loose inspiration for some math for math’s sake — ick.

Now, I take full responsibility for this impression. Many practitioners of voting theory believe the system can be radically improved, and are much more engaged with the realities of elections than I was giving them credit for. Steven Brams, now a mentor of mine, was involved in the adoption of approval voting for leadership elections in several professional societies. (Steve’s attempts to push something similar with political elections have mostly been frustrated so far, although advocacy by the Center for Election Science, an organization Steve has advised, played a role in the recent adoption of approval voting by Fargo, ND.) And Warren Smith has been developing a website to spread information about score voting since 2005. Just as two examples.

Now that I understand that voting theory is meaningfully engaged with actual elections, I’ve done a 180, and I’ve been reading up on both the foundations and recent research, and am looking for ways to contribute myself.

That said, I retain the feeling (with hopefully due humility that I could be wrong again) that the field would benefit from greater accountability to the problems it’s trying to speak to. The imperatives of mathematical exploration are different than the imperative of improving the mechanisms by which the will of the people expresses itself, and these differing imperatives can be in tension.

I’m reading one of Donald Saari’s books on voting theory. Saari has developed a beautiful, geometric way of looking at election systems that offers great clarity in making sense of some of the counterintuitive things that can happen. This is beautiful and important math. He also tends to portray the Borda count as kind of the best voting system, and I cannot shake the feeling that this is more because of the mathematical beauty of its properties, when viewed through a particular mathematical lens, than a sober assessment of its value to society.

So. I want to see and be part of a voting theory that is responsive to the problem it involves itself in. We can talk about voting systems using the tools of academic discourse, but we must remember the stakes. This is not an academic exercise.

Appendix

I promised above a little explanation of my comment that Arrow’s theorem should not be taken as stating that “there is no perfect voting system”. Lemme get into that for a sec. Arrow’s theorem says a voting system of a very specific type cannot obey a certain very specific list of prima facie desirable properties all at once. My basic objection is that the theorem is too specific for this broad an interpretation. My more fine-grained caveat is that there is nothing God-given about the particular set of desirable features Arrow chose to analyze, so deviation from them is just one definition of “imperfection”, and there are plenty of others.

In either case, I don’t blame Arrow for this, but I definitely thought I was being told that making the system better isn’t on the table. My real motivation here is to disabuse you of that misimpression if you had it too.

Math is Democracy II: Math is Democracy!

I announced a series on math and democracy back in October.

It will deal with a lot of concrete areas. Last time I talked about a case that is before the Supreme Court and will influence voting law throughout the land. In the future I’ll be talking about voting, political participation, technology and who has a say over its development, and of course the classroom.

But I want to properly kick things off with a post that is essentially philosophical. I am here to assert the following proposition:

Math is democracy!

What do I mean?

Democracy — from Greek — literally, “rule by the people.” I am referring to the ideal itself, not any particular system of government. Throughout the world we have various systems attempting to implement this ideal. One can ask questions about the degree of success of these attempts, but that’s not what this post is about. I’m just isolating the ideal — democracy — rule by the people.

Mathematics — from Greek — literally, “learning.” Of all the domains of human inquiry, math occupies a privileged place in terms of our confidence in its conclusions. It is the only field where practitioners regularly express unqualified certainty about its results. We sometimes discuss the wisdom it gives us as some sort of celestial gift (as in Wigner’s classic essay on its applicability to the sciences).

I am about to draw a connection. I expect it is still opaque at this point, but hang on.

If math is a miracle, then there is a second miracle: the divine gift was implanted in each of us, since it springs solely from the universal human capacity for rational thought. The wisdom of mathematics was not given us by way of Mt. Sinai, handed down from on high by somebody with privileged access to The Boss. Although many people think back to childhood and recall inscrutable formulas dispensed by a teacher who mysteriously knew the answer (how did they know??), this memory conceals the real truth, which is that the only place mathematical knowledge comes from is a community of peers reaching some kind of consensus after a period of engaged discussion. Furthermore, at least in principle (if not always in practice), anybody in this community has the right at any time to raise good-faith questions about the logic underlying any of our mathematical knowledge, and the matter is not really settled unless these questions have a good answer.

Thus, the only true source of mathematical authority is the consensus of a community of equals.

The principle of democracy is that this is also the only true source of legitimate political authority.

Broadening further, I offer that the principle of democracy holds that the only source of authority (of any kind) over a community is consensus of that community. So math is literally democracy.

This is edited from the version I posted yesterday, where I used the phrase “functional consensus” instead of “consensus.” This was to acknowledge that in a large-scale community such as a nation, or the international community of mathematics researchers, true consensus is not a viable goal. That said, the “functional” didn’t sit well with me overnight, because I thought it could be taken to suggest some sort of majoritarian principle. To me, majoritarianism is a fatal compromise of the principle of democracy articulated here, and it defeats the purpose of the analogy with math.

The thing about math is that, in principle, if an objection is raised to what is regarded as established fact, then that objection needs to be dealt with. Maybe something was overlooked! In actual practice, it may or may not be, because the question of whether you can get people to pay attention to your objection depends on things like if you’re famous, if you’re well-connected, how much work other people have to do to understand it, etc. But mathematicians’ collective understanding of what we’re doing holds that if somebody raises a new objection to something thought to be well-established, we have to answer it, not ignore it, in order to hold onto the established knowledge. This ideal isn’t attained, but it is still how we think about it.

By the same token, it seems to me that the democratic ideal insists that a minority view has the right to be processed rigorously by the community. I am making a high-level analogy so I’m not getting into what that processing might look like. But the failure of a community to take into account minority constituencies in some way is a failure of democracy.

I want to acknowledge some intellectual debt!

In 2008, I went to the Creating Balance in an Unjust World conference and saw a presentation by Sarah Bertucci, Jason Cushner, and several of their current and former students, entitled Consensus is the Answer Key: Empowerment in the Math Classroom. The presentation was on using consensus as the source of mathematical knowledge in the classroom. Later (in 2009?), I visited the school in Vermont where Jason and Sarah were then teaching, and saw Jason’s class. (Random aside: I also met Jasmine Walker!) The ideas have shaped how I saw both mathematics and the classroom ever since. You can see their clear imprint above (and in many of the things I’ve written on this blog over the years).

In about 2010, I was having a conversation with Jay Gillen of the Baltimore Algebra Project. At the time, I was preparing to apply to graduate school in math. Jay asked me many questions about how I thought about the math classroom and the subject itself. At some point he paused and said, “Everything you love about math is what free people love about democracy.” This comment has been continuously blowing my mind for 8 years, and again you can see its clear imprint in the above.

Math is Democracy I: The Citizen and the Gerrymander

I am intending a series of my typically long, elaborate blog posts entitled Math is Democracy. The ideas have been brewing for years although they have been rapidly expanding and taking on new urgency since January. I alluded to this intention previously.

I wasn’t ready to start it yet, but I feel I must. I was reading the oral arguments in Gill v Whitford, the Wisconsin partisan gerrymandering case currently before the Supreme Court. I had to stop and have a moment when I read this:

CHIEF JUSTICE ROBERTS: Mr. Smith, I’m going to follow an example of one of my colleagues and lay out for you as concisely as I can what — what is the main problem for me and give you an opportunity to address it.

I would think if these — if the claim is allowed to proceed, there will naturally be a lot of these claims raised around the country. Politics is a very important driving force and those claims will be raised.

And every one of them will come here for a decision on the merits. These cases are not within our discretionary jurisdiction. They’re the mandatory jurisdiction. We will have to decide in every case whether the Democrats win or the Republicans win. So it’s going to be a problem here across the board.

And if you’re the intelligent man on the street and the Court issues a decision, and let’s say, okay, the Democrats win, and that person will say: “Well, why did the Democrats win?” And the answer is going to be because EG was greater than 7 percent, where EG is the sigma of party X wasted votes minus the sigma of party Y wasted votes over the sigma of party X votes plus party Y votes.

And the intelligent man on the street is going to say that’s a bunch of baloney. It must be because the Supreme Court preferred the Democrats over the Republicans. And that’s going to come out one case after another as these cases are brought in every state.

And that is going to cause very serious harm to the status and integrity of the decisions of this Court in the eyes of the country.

Now, there’s a lot here one could react to.[1] But the main thing I reacted to was this:

The Chief Justice of the highest court in the land thinks Americans don’t feel empowered to judge an argument on the merits if there’s math involved.

You know what? He’s probably right about that.

But this situation is very, very wrong.

Math is being used increasingly to make decisions governing our lives, for good or ill. Increasingly sophisticated math.[2] The instance most familiar to readers of this blog is probably teacher value-added scores, but the many various uses share this: they are not accountable to the public.

One reason the Wisconsin case is so hot is because the process that led to the map currently being challenged included a lot of fancy mathematical modeling intended to make the Republican legislative majority as bomb-proof as possible — an effort that appears to have worked really well. That the map was drawn with this goal and these tools is not a controversial point in the case. This was a use of math by legislators aimed at becoming less accountable to the public.

What I’m getting at: math is a species of power, and it’s a species that multiple antidemocratic forces are using, very effectively. And it’s a kind of power that citizens, by and large, totally lack.

So, the game is unfair. We the People are supposed to be able to participate in public decision-making. That’s the heart of democracy. But math is increasingly becoming a kind of secret key to power that, if the Chief Justice is right, We the People mostly don’t have. As soon as there’s math involved, we can’t even participate in debates about the very consequential choices that are being made. In which case, nobody who wants to use the power of math (for good or ill!) needs to be accountable to us.

I mean, this was true before the explosion of data-science driven business and governmental practices Cathy writes about, or the computer-assisted 2010-11 legislative redistricting.[3] But now it is more intensely true than ever.

What this leaves me with is that doing our jobs well as math educators is completely urgent for democracy. Every kid we leave traumatized and alienated from formulas and data analysis is a citizen that doesn’t have a voice.

Don’t let anybody tell you it doesn’t matter.

Notes:

[1] For example: This author at ThinkProgress thinks Roberts has a lot of nerve claiming to be concerned with the perception that the court is partisan when he has so consistently voted along partisan lines in landmark cases. This author at WaPo thinks it’s not legitimate for the Court to be considering its public perception in the first place. I am personally inclined to believe that Roberts is earnestly concerned about the court’s reputation and that his question was earnest (mostly because of his surprising and apparently similarly-motivated vote in NFIB v Sebelius), although I do think that the fact that he doesn’t appear to be equally concerned with the perception of partisanship if the court does not “allow the claim to proceed” reflects a rather striking partisan limitation in his image of the “intelligent man on the street.” I know plenty of intelligent men, and women, who would be inclined to conclude that he himself is a partisan hack on the basis of the above quotation alone.

[2] Shout out to Cathy O’Neil.

[3] This seems like a good moment to acknowledge the deep debt of my thinking here to Bob Moses, who has been on this tip for a long time. Also, there is some relationship to the work of math educators in the Freirean tradition such as Marilyn Frankenstein and Rico Gutstein, though I can’t take the time now to figure out exactly what it is.