My father once told a story about going to a restaurant with some friends, one of whom ordered some soup and was very unhappy with what he got. When the waiter came by and asked what the problem was, my father’s friend said “The soup is bad.”
“I’m sorry the soup did not meet with your approval,” the waiter replied. But this only made my father’s friend angrier.
“It’s not that the soup didn’t meet with my approval!” he shouted. “This is BAD SOUP!”
Like my father’s best stories, this is one that has meant different things to me at different times. Originally I found the story funny because of the patron’s insistence on being not just entitled to his opinion but objectively right, in a matter of taste where there’s no such thing as right or wrong. Later, I realized (okay, my wife pointed out to me) that it is possible for soup to be bad; for instance, it can be teeming with mycotoxins that are as poisonous as they are distasteful. And what’s poignant for me today is that it’s now impossible, all these years later, to determine whether the soup was bad or my father’s friend was fussy. I’m not even sure how I’d find out what the friend’s name was, or where they ate.
But the topic I want to treat ever-so-superficially (before getting back to preparing for the first day of classes tomorrow; I do have a real job) is this desire we often feel to be right not just in our own minds but objectively. Sometimes this is secondary to the petty desire that someone else be wrong and admit it, but at other times, it reflects a deep yearning for something solid to cling to, in a world full of clashing perspectives and contested facts.
In recent years, the formula 2+2=4 has acquired in some circles a talismanic significance — with the plus sign serving as a vampire-hunter’s cross, warding off the bloodsucking forces of unbridled relativism. I think this development can in part be traced back to George Orwell’s “1984”, in which the hero Winston Smith comes to the belief that freedom is the freedom to say that two plus two makes four. (John Stuart Mill might have been more inclined to say that freedom is the freedom to say that two plus two makes five, but we can agree to disagree about that.) Allying yourself with 2+2=4 casts your adversaries as Big Brother, and that’s gotta feel validating.
I’m a little surprised that more people haven’t instead chosen the Law of the Excluded Middle as the hill to fight (and, if need be, die) on. The Law of the Excluded Middle is often phrased as “A or Not-A”, and is a linchpin of dichotomous thinking. The Law seems hard to challenge when enunciated as a general principle of reasoning, but it’s easy to abuse in real-world contexts. Straw-man arguments often follow a template that contains a tacit appeal to the Law: First assert “A or B” as if it were synonymous with “A or Not-A” (hiding the differences between B and Not-A), then show that B isn’t tenable, and then announce that A has been proved by reductio ad absurdum. Used in this way, the Law of the Excluded Middle seems well-suited to the sort of vehement black-and-white arguments one sees in social media, where people try to win arguments by showing that their opponent isn’t being logical.
Meanwhile, math has a kind of prestige that philosophy and logic lack nowadays. So it would make tactical sense for people who claim to speak for Objectivity, Truth and Reason to represent the Law of the Excluded Middle as a core tenet of mathematics and to enlist it in defense of their positions.
But I don’t think mathematics has core tenets. In fact there’s been quite a lot of work on nonstandard logics in which the Law of the Excluded Middle is denied. This hasn’t led to a schism in mathematics because the mathematicians who use the LEM (that’s most of us, most of the time) and those who explore the consequences of denying the LEM are playing the same game.
It’s hard to say what that game is. Two requirements of the game are sensitivity to context and careful expression of ideas. (Well, you don’t always have to be careful, but when things are fuzzy, you’re aware that they’re fuzzy and you value the removal of the fuzz.) There’s a lot more that can be said, but I don’t know how to put it into words. I was going to say something about the pursuit of elegance, but then I realized that math has a wild strain too, where the delight in some mathematical phenomenon comes not from its neatness but from its surprising extravagance. Maybe what’s central is not the elegance or the extravagance, but the delight?
My father was an attorney for many years, and he delighted in the law much as I delight in math. There are obvious similarities between the legal mindset and the mathematical mindset, such as a willingness to attend to minute but crucial details. Another similarity is the centrality of what lawyers call arguments and mathematicians call proofs, and how in both professions one progresses by iteratively finding the weakest link in a chain of thoughts and strengthening it. In math, one preemptively addresses challenges made by an imaginary auditor who may be confused or skeptical; in trial law or contract law, one addresses challenges that an actual adversary would be likely to make.
But there is a key difference: in the law, one is trying to say what should happen in the real world. Mathematics, at least the kind that I practice, doesn’t say what should happen or even what will happen. It asserts only “IF the world follows such-and-such rules, THEN such-and-such will happen.”
In math, we often clarify an initially vague question inspired by the real world until we’ve turned it into a question that has a unique answer in a specific idealized context. But in doing so, we’ve walked away from the real world, and to the extent that we ignored or rejected other ways in which our initial vague question might have been mathematified, we may have missed a better answer to the real-world situation that inspired the problem.
I would be happy to believe that mathematical education will make you better able to reason about the real world, but I have a hard time convincing myself that it’s true. I think the gulf between the world of mathematics (full of things that are as they are because we have defined them to be so) and the world that we’re born into (full of things that are forever unknowable, especially people) is just too unbridgeable. Instead, I think that mathematical training can give you instincts of thought that help you recognize when other people are applying math-ish reasoning to the real world in bogus ways. (“Wait a minute,” you might think; “is B really the same as Not-A?”) If you’ve developed the skill of finding the weak links in your own reasoning, you can turn the skill around and apply it to other people’s.
I may not be able to prove that things in the real world are indubitably true using math. But I can often spot when someone else is dishonestly claiming to prove real-world assertions with a pretense of mathematical rigor, and I can shout: “Bad soup!”
Thanks to Sandi Gubin.
In memory of Ted Propp, 1923-2022
“Death deepens the wonder.” — Tillie Olsen (“Tell Me A Riddle”)
Maybe what’s central is not the elegance or the extravagance, but the delight?— Such a lovely way of putting it. I am sure that all practitioners of mathematics will agree here.
I would be happy to believe that mathematical education will make you better able to reason about the real world, but I have a hard time convincing myself that it’s true.— Are you taking the rubrics of logic which mathematics trains us to use for granted? Aren’t these rubrics of logic central to reasoning everywhere — the same rubrics of logic which would you are talking about refuting when listening to someone else’s argument. Where else than in mathematics is one trained in logic this way? Maybe I am missing your point here.
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I agree that mathematical training (like legal training) gives lots of practice in careful and systematic thinking. But beyond that, I’m unsure that its value extends beyond math.
I read Jordan Ellenberg’s “How Not To Be Wrong”, which takes the stance that math does help us think better. But I wasn’t convinced by his main example (the Abraham Wald story). Couldn’t the pivotal question “What AREN’T we looking at?” have come from a stage magician (say) rather than a mathematician?
Maybe I’m bending over too far backwards in my eagerness not to be the kind of person who seems to overvalue their profession. Let’s hear from some other people. You who are reading this (including nishantchandgotia): how has mathematical training helped you reason about non-mathematical domains?
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I haven’t read Jordan Ellenberg’s book but I am already looking forward to it.
I do not want to overstate the reach of mathematics either. When I wrote the message I was mostly thinking about how mathematics helps inculcate careful and systematic thinking and not much more. To call for a recent example, when a friend of mine had recently fell into some legal trouble it is precisely this organised thought process which helped me immensely. Once I picked up some jargon and basic laws it was a very similar thought process (at least seemingly).
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I chuckled at your Dad‘s story, about the objectively bad soup. We have a deep cognitive tendency to make our judgments a feature of the objective world.
In this respect, math is similar to ethical thinking. When we argue ethically, we argue that our judgments are not simply expressions of our particular taste, but express what all objective, sensitive observers should endorse. Our cognitive capacity, and tendency, to adopt a perspective independent of our selves makes mathematics possible, as well as ethics. Though we chuckle in the case of BAD SOUP, our natural cognitive groove to objectify our opinions, allows us to adopt a perspective on ethics from which we criticize ourselves.
Do you think that the austere beauty of mathematics comes, in part, from its complete disconnection from our subjectivity?
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Yes. Subjectivity enters math in what we choose to look at, and how we talk about it, but when one is doing pure math research one has the impression of contacting a reality that is completely unmoved by what we wish to be true — an indifference that is both scary and thrilling.
But I think that human mathematics can never fully access the ideal of total objectivity, even though we can come extremely close, because in the end we are only human.