Tools for Mathematicians
- “Discussing definitions
- Coming up with counterexamples
- Being wrong often and admitting it
- Evaluating many possible consequences of a claim
- Teasing apart the assumptions underlying an argument
- Scaling the ladder of abstraction
- Discussing definitions”
A primary skill that mathematicians develop is fluidity with definitions…mathematicians obsess over the best and most useful meaning of every word they use. Mathematicians need logical precision because they work in the realm of things which can be definitively proven or disproven…if something can be done “definitively,” it must necessarily be definable.
…the most common question a mathematician has when encountering a new topic is, “What exactly do you mean by that word?”…It’s one of the foundations of the wishy-washy term “critical thinking.”
So now an average citizen who might discard the idea of mathematics is listening to the news and hears a politician say, “We have strong evidence of weapons of mass destruction in Iraq.” If they had a good mathematics education they will ask, “What exactly do you mean by strong evidence and weapons of mass destruction?” And, the crucial follow-up question, does the definition provided justify the proposed response, starting a war? If you don’t understand the definition you can’t make an informed voting decision. (Of course, if you watch the news for entertainment and to be part of a political tribe, the truth is irrelevant)
Everyone has to deal with new definitions…Being able to think critically about definitions is a foundation of informed discourse.
Producing examples and counterexamples
Let’s practice our definitions in an informal setting. By “counterexample” I mean an example that shows something breaks or is wrong. Mathematicians spend a lot of time coming up with examples and counterexamples to various claims. This point ties very closely to the previous about definitions in two ways.
Often, when coming up with a new definition, one has a set of examples and counterexamples that one wants the definition to adhere to. So examples and counterexamples help guide one to build good definitions. When encountering a new existing definition, the first thing every mathematician does is write down examples and counterexamples to help them understand it better.
However, examples and counterexamples go beyond just thinking about definitions. They help one evaluate and make sense of claims. Anyone who has studied mathematics knows this pattern well, and it goes by the name of “conjecture and proof.”
The pattern is as follows:
– As you’re working on a problem, you study some mathematical object and you write down what you want to prove about that object.
– This is the conjecture, like an informed (or uninformed) guess about some pattern that governs your object of study.
– This is followed by the proof, where you try to prove or disprove the claim.
The difference in mathematics is that the “evidence” is a counterexample and it’s only called such if it’s provable. “Evidence” in mathematics is often just a temporary placeholder until the truth is discovered, though for some high profile math problems mathematicians have found nothing but “evidence,” even after hundreds of years of study.
The analogy is also bad because this happens in mathematics on an almost microscopic level. When you’re deep in a project, you’re making new little conjectures every few minutes, and mostly disproving them because you later realize the conjectures were highly uninformed guesses. It’s a turbo-charged scientific process with hundreds of false hypotheses before you arrive at a nice result. The counterexamples you find along the way are like signposts. They guide your future intuition, and once they’re deeply ingrained in your head they help you accept or denounce more complicated conjectures and questions with relative ease.
Again, being able to generate interesting and useful examples and counterexamples is a pillar of useful discourse…This mindset has also had countless applications to physics, engineering, and computer science.
But a subtler part of this is that mathematicians, by virtue of having made so many wrong, stupid, and false conjectures over their careers, are the least likely to blindly accept claims based on a strong voice and cultural assumptions. If, as a collective modern society, we agree that people are too willing to believe others (say, politicians, media “experts,” financial talking heads), then studying mathematics is also a fantastic way to build a healthy sense of skepticism.
Being wrong often and admitting it
…mathematicians, regardless of who is actually right, is not only willing to accept they’re wrong, but eager enough to radically switch sides when they see the potential for a flaw in their argument…Having to do this so often—foster doubt, be wrong, admit it, and start over—distinguishes mathematical discourse even from much praised scientific discourse…here’s just the search for insight and truth. The mathematical habit is putting your personal pride or embarrassment aside for the sake of insight.
Evaluating many possible consequences of a claim
In math, very often the way you discover a bug in your argument is by realizing that the argument gives you more than you originally intended — vastly, implausibly more…Indeed, exploring the limits of a claim is the mathematician’s bread and butter. It’s one of the simplest high-level tools one has for evaluating the validity of a claim before going through the details of the argument. Indeed, it can also be used as a litmus test for deciding which arguments are worthwhile to understand in detail.
Sometimes, the limits of an argument result in an even better and more elegant theorem that includes the origin claim. More often, you simply realize you were wrong. So this habit is a less formal variation on being wrong often, and coming up with counterexamples.
Teasing apart the assumptions underlying a claim
One perhaps regrettable feature of mathematics is that is fraught with ambiguity. We like to think of math as rigor incarnate. And I would even argue that’s a reasonable idea, once the math has been built upon for a hundred or so years. But even so, the process of doing math—of learning existing ideas or inventing new ideas—is more about human to human communication than stone-cold rigor.
As such, when someone makes a claim in mathematics (out loud), they’re usually phrasing it in a way that they hope will convey the core idea to another human as easily as possible. That usually means they’re using words in ways one might not expect, especially if the conversation is between two mathematicians with a shared context and you’re an outsider trying to understand.
When you face a situation like this in mathematics, you spend a lot of time going back to the basics. You ask questions like, “What do these words mean in this context?” and, “What obvious attempts have already been ruled out, and why?” More deeply, you’d ask, “Why are these particular open questions important?” and, “Where do they see this line of inquiry leading?”
These are the methods a mathematician uses to become informed on a topic. The unifying theme is to isolate each iota of confusion, each assumption underlying a belief or claim. It’s decidedly different from the kinds of discourse you see in the world.
For example, in the contentious 2016 US presidential election, who has tried to deeply understand Donald Trump’s worldview? Most liberals hear, “I’m going to build a wall and make Mexico pay for it,” and deride him as insane. Treating the claim mathematically, you would first try to understand where the claim came from. To whom is Trump appealing to? What alternatives to the immigration problem has he ruled out, and why? Why is immigration such an important issue to his supporters, and what assumptions on his part causes him to answer in this way? What is he tapping into that makes his bid successful?
Scaling the ladder of abstraction
The last habit is a concept…It’s the idea that when you’re reasoning about a problem, there are many different resolutions at which you can think (“rung”)…At a higher level, you can control different parameters of the algorithm (and time) with a slider, elevating one algorithm into a family of algorithms that can be tuned. And you can further generalize which parameters and behaviors are tunable to find a way to search through the space of all possible algorithms. As you go, you’re looking for high-level patterns that can help you achieve your end goal, designing a great car-driving algorithm back down at the bottom rung (the coarsest resolution).
Mathematicians regularly scale this ladder, particularly in the later stages of graduate school when you need to learn to read a lot of research papers. Here, you don’t have time to deeply understand every bit of notation and every claim in every paper (just those papers that are important enough to read in detail). Instead, you develop a ladder of abstraction: the lowest rung is individual definitions and theorems and examples, the next level up is the high-level flow of a paper, next up is a paper in relation to other papers and the broader mathematical context, and above that is the overarching trends in the field, what’s considered important and fashionable, etc.
You might start at the lowest rung of the ladder, understanding some examples of a definition to get your bearings, then jump up to the main theorem of the paper and understand how its perceived as a huge improvement over previous work. You might see they use some technique from the 50’s in a field you’re not familiar with, but you use that idea as a black box and try to understand the high-level proof of the main theorem, stepping down one rung. Then maybe you go to the open problem section to see what work is left to do, and if it seems enticing enough you can prepare yourself to do that by reading the rest of their paper in detail.
Indeed, mathematicians have to exercise this ladder-scaling muscle whenever they give a talk on their own work. There are many kinds of audience, each of which can appreciate the content of a mathematical idea at a different resolution…
Part of the big struggle of mathematics is synthesizing all of the information in all of these ladder rungs into a coherent world-view that you can personally scale up and down at will…