Math is training for the mind, but not like you think
Just a hypothesis:
People have long thought that math is training for clear thinking. Just one version of this meme that I scooped out of the water:
“Mathematics is food for the brain,” says math professor Dr. Arthur Benjamin. “It helps you think precisely, decisively, and creatively and helps you look at the world from multiple perspectives . . . . [It’s] a new way to experience beauty—in the form of a surprising pattern or an elegant logical argument.”
But math doesn’t obviously seem to be the only way to practice precision, decision, creativity, beauty, or broad perspective-taking. What about logic, programming, rhetoric, poetry, anthropology? This sounds like marketing.
As I’ve studied calculus, coming from a humanities background, I’d argue it differently.
Mathematics shares with a small fraction of other related disciplines and games the quality of unambiguous objectivity. It also has the ~unique quality that you cannot bullshit your way through it. Miss any link in the chain and the whole thing falls apart.
It can therefore serve as a more reliable signal, to self and others, of one’s own learning capacity.
Experiencing a subject like that can be training for the mind, because becoming successful at it requires cultivating good habits of study and expectations for coherence.
Math is interesting in this regard because it is both very precise and there’s no clear-cut way of checking your solution except running it by another person (or becoming so good at math to know if your proof is bullshit).
Programming, OTOH, gives you clear feedback loops.
In programming, that’s true at first. But as projects increase in scope, there’s a risk of using an architecture that works when you’re testing, or for your initial feature set, but will become problematic in the long run.
For example, I just read an interesting article on how a project used a document store database (MongoDB), which worked great until their client wanted the software to start building relationships between data that had formerly been “leaves on the tree.” They ultimately had to convert to a traditional relational database.
Of course there are parallels in math, as when you try a technique for integrating or parameterizing that seems reasonable but won’t actually work.
Yep. Having worked both as a mathematician and a programmer, the idea of objectivity and clear feedback loops starts to disappear as the complexity amps up and you move away from the learning environment. It’s not unusual to discover incorrect proofs out on the fringes of mathematical research that have not yet become part of the cannon, nor is it uncommon (in fact, it’s very common) to find running production systems where the code works by accident due to some strange unexpected confluence of events.
Programming, OTOH, gives you clear feedback loops.
Feedback, yes. Clarity… well, sometimes it’s “yes, it works” today, and “actually, it doesn’t if the parameter is zero and you called the procedure on the last day of the month” when you put it in production.
The thing I like about math is that it gives the feeling that the answers are in the territory. (Kinda ironic, when you think about what the “territory” of math is.) Like, either you are right or you are wrong, it doesn’t matter how many people disagree with you and what status they have. But it also doesn’t reward the wrong kind of contrarianism.
Math allows you to make abstractions without losing precision. “A sum of two integers is always an integer.” Always; literally. Now with abstractions like this, you can build long chains out of them, and it still works. You don’t create bullshit accidentally, by constructing a theory from approximations that are mostly harmless individually, but don’t resemble anything in the real world when chained together.
Whether these are good things, I suppose different people would have different opinions, but it definitely appeals to my aspie aesthetics. More seriously, I think that even when in real world most abstractions are just approximations, having an experience with precise abstractions might make you notice the imperfection of the imprecise ones, so when you formulate a general rule, you also make a note “except for cases such as this or this”.
(On the other hand, for the people who only become familiar with math as a literary genre, it might have an opposite effect: they may learn that pronouncing abstractions with absolute certainty is considered high-status.)
Mathematics shares with a small fraction of other related disciplines and games the quality of unambiguous objectivity. It also has the ~unique quality that you cannot bullshit your way through it. Miss any link in the chain and the whole thing falls apart.
Isn’t programming even more like this?
I could get squidgy about whether a proof is “compelling”, but when I write a program, it either runs and does what I expect, or it doesn’t, with 0 wiggle room.
Sometimes programming is like that, but then I get all anxious that I just haven’t checked everything thoroughly!
My guess is this has more to do with whether or not you’re doing something basic or advanced, in any discipline. It’s just that you run into ambiguity a lot sooner in the humanities
It helps you to look at the world from multiple perspectives: It gets you into a position to make a claim like that soley based on anecdotal evidence and wishful thinking.
Math is training for the mind, but not like you think
Just a hypothesis:
People have long thought that math is training for clear thinking. Just one version of this meme that I scooped out of the water:
But math doesn’t obviously seem to be the only way to practice precision, decision, creativity, beauty, or broad perspective-taking. What about logic, programming, rhetoric, poetry, anthropology? This sounds like marketing.
As I’ve studied calculus, coming from a humanities background, I’d argue it differently.
Mathematics shares with a small fraction of other related disciplines and games the quality of unambiguous objectivity. It also has the ~unique quality that you cannot bullshit your way through it. Miss any link in the chain and the whole thing falls apart.
It can therefore serve as a more reliable signal, to self and others, of one’s own learning capacity.
Experiencing a subject like that can be training for the mind, because becoming successful at it requires cultivating good habits of study and expectations for coherence.
Math is interesting in this regard because it is both very precise and there’s no clear-cut way of checking your solution except running it by another person (or becoming so good at math to know if your proof is bullshit).
Programming, OTOH, gives you clear feedback loops.
In programming, that’s true at first. But as projects increase in scope, there’s a risk of using an architecture that works when you’re testing, or for your initial feature set, but will become problematic in the long run.
For example, I just read an interesting article on how a project used a document store database (MongoDB), which worked great until their client wanted the software to start building relationships between data that had formerly been “leaves on the tree.” They ultimately had to convert to a traditional relational database.
Of course there are parallels in math, as when you try a technique for integrating or parameterizing that seems reasonable but won’t actually work.
Yep. Having worked both as a mathematician and a programmer, the idea of objectivity and clear feedback loops starts to disappear as the complexity amps up and you move away from the learning environment. It’s not unusual to discover incorrect proofs out on the fringes of mathematical research that have not yet become part of the cannon, nor is it uncommon (in fact, it’s very common) to find running production systems where the code works by accident due to some strange unexpected confluence of events.
Feedback, yes. Clarity… well, sometimes it’s “yes, it works” today, and “actually, it doesn’t if the parameter is zero and you called the procedure on the last day of the month” when you put it in production.
Proof verification is meant to minimize this gap between proving and programming
The thing I like about math is that it gives the feeling that the answers are in the territory. (Kinda ironic, when you think about what the “territory” of math is.) Like, either you are right or you are wrong, it doesn’t matter how many people disagree with you and what status they have. But it also doesn’t reward the wrong kind of contrarianism.
Math allows you to make abstractions without losing precision. “A sum of two integers is always an integer.” Always; literally. Now with abstractions like this, you can build long chains out of them, and it still works. You don’t create bullshit accidentally, by constructing a theory from approximations that are mostly harmless individually, but don’t resemble anything in the real world when chained together.
Whether these are good things, I suppose different people would have different opinions, but it definitely appeals to my aspie aesthetics. More seriously, I think that even when in real world most abstractions are just approximations, having an experience with precise abstractions might make you notice the imperfection of the imprecise ones, so when you formulate a general rule, you also make a note “except for cases such as this or this”.
(On the other hand, for the people who only become familiar with math as a literary genre, it might have an opposite effect: they may learn that pronouncing abstractions with absolute certainty is considered high-status.)
Isn’t programming even more like this?
I could get squidgy about whether a proof is “compelling”, but when I write a program, it either runs and does what I expect, or it doesn’t, with 0 wiggle room.
Sometimes programming is like that, but then I get all anxious that I just haven’t checked everything thoroughly!
My guess is this has more to do with whether or not you’re doing something basic or advanced, in any discipline. It’s just that you run into ambiguity a lot sooner in the humanities
It helps you to look at the world from multiple perspectives: It gets you into a position to make a claim like that soley based on anecdotal evidence and wishful thinking.