A Path Less Taken to the Peak of the Math World
quantamagazine.org
quantamagazine.org
Math, physics, biology, the 'hard' sciences, all have so much more in common with the arts than we give credit. Is a math proof not poetic in its own right? Does a well structured musical composition not engage with our most innate neurology on a microscopic level?
What's sad to me is that we seem to silo people into one of the two camps, which is like locking someone in a room in search of a key that's in the room next door. Businesses do this too - how many engineers get brought on to learn marketing? How many chefs learn the chemistry which serves as a foundation for every recipe they cook?
I say this as a former political-science-major-turned-accountant. I loved tax policy, but struggled to understand the law and treatment of certain issues, so I switched to accounting and found that the same logical, rational work that goes into formulating a political argument can be applied to rationalize control processes or budgeting in businesses. I've tried to un-learn my bias against people who majored in the "soft" sciences and strive to make more of an effort to understand how they think and what drives them.
When mathematicians describe equations as beautiful, they are not lying. Brain scans show that their minds respond to beautiful equations in the same way other people respond to great paintings or masterful music. The finding could bring neuroscientists closer to understanding the neural basis of beauty, a concept that is surprisingly hard to define.
In the study, researchers led by Semir Zeki of University College London asked 16 mathematicians to rate 60 equations on a scale ranging from "ugly" to "beautiful." Two weeks later, the mathematicians viewed the same equations and rated them again while lying inside a functional magnetic resonance imaging (fMRI) scanner. The scientists found that the more beautiful an equation was to the mathematician, the more activity his or her brain showed in an area called the A1 field of the medial orbitofrontal cortex.
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https://www.scientificamerican.com/article/equations-are-art...
From the sound of that, "beauty" evokes a reward response in the part of the brain (medial orbitofrontal cortex) responsible for processing reinforcement learning signals. This is pretty interesting, since it says that the brain can process a reward signal for a very abstract property that probably has very little correlation with direct sensory rewards.
After that bad math test in elementary school, Huh says he adopted a defensive attitude toward the subject: He didn’t think he was good at math, so he decided to regard it as a barren pursuit of one logically necessary statement piled atop another. As a teenager he took to poetry instead, viewing it as a realm of true creative expression.
But then...
By the time he enrolled at Seoul National University in 2002, he had concluded that he couldn’t make a living as a poet, so he decided to become a science journalist instead. He majored in astronomy and physics, in perhaps an unconscious nod to his latent analytic abilities.
Can someone explain how someone who has decided to distance himself from math can major in PHYSICS?!
Exactly that. At my university first-year physics students would effectively take the same courses as mathematics students and Mathematical Methods in Physics on top of that.
This indicates that he views a certain type of worthlessness in math. Then he can justify not being good at math, and distance himself from the pursuit of math on its own grounds.
Of course his perception of him not being good at math and actually not being good at math are two different things. Physics uses math, but not for maths sake. And in reality, I'm sure he was more than competent in math, so the level of math required for undergrad physics was not a problem.
It was only, through his journalism, did he let down his guard enough to enjoy math for maths sake -- and realize that a test you took in elementary school needn't define you.
It's obvious this guy had a love of math and ability in it, from his majors and subsequent events.
But it's also actually not that unreasonable for him to start second-guessing himself when he struggled with math earlier in life.
I do think there's this idea that if you're good at math and have something to offer in it, it will show early on regardless of life circumstances or mentors or role models or whatever, that if it's not immediately obvious that you're a mathematical genius you should forget about it.
"Realizing a test you took in elementary school needn't define you" is actually a nontrivial thing to overcome in today's society, maybe even especially in STEM circles.
Sometimes I wish STEM culture was more focused on sharing the joys of STEM and trying to be as open-minded and inclusive as possible, instead of brandishing it as a competitive tool.
Yes, we definitely treat STEM as a competition to see who can be the closest to "perfect" at set tasks and classwork, rather than as an exploration (or even exploitation) of structures and spaces through strictly logical reasoning.
The pure math world (and certain rarified branches of physics and CS) are quite different from the "measurement-based" sciences. If anything they're almost more similar to endeavors like musical composition (or even poetry) than any "lab-based" science. Particularly when it comes to the fetishization of "genius", "pure talent" and all that -- and especially when it comes to the character model (myth, really) of the lone genius who locks himself in an attic for N years and not only solves some major problem but totally reimagines the field, front to back.
I mean - at least in terms of the culture of pure mathematics, that's the image one can easily fall into the habit of having about one's self, if one attempts to do mathematics. Such that if you were to switch to physics, astronomy, or CS... you actually would feel like you were "leaving math".
One way to explain this might be that physics isn't about doing math, but using math to do computations. Meanwhile, math is about abstract ideas and rigorous proofs. Physics doesn't care about wondering if an integral is a well-defined concept, nor does it care about the real numbers being uncountable. Similarly, maths looks at the formulae in physics and thinks those things are way to complicated.
Our lin-alg course is joined physics and maths. The older physicists complain about all the proofs needed there. Mathematicians lament the lack of rigor in the same course.
Just like mathematicians can turn the results of 'unsolved problems' into axioms and go on with their lives.
It's not the formulas; it's how you look at them. And what they "mean" to you.
Linear Algebra, Functional Analysis, and Abstract Algebra aren't exactly 'measurement-based' subjects.
Right; but those are just the ABCs, is it were.
Physicists and astronomers, by and large, are interested in a broad range of mathematical subjects -- but generally as tools, just enough therein to make sense of what's coming out of their particle detectors and their telescopes.
Pure mathematicians, meanwhile, are into things like the Langlands program, or Inter-universal Teichmüller theory. And they constructions they consider aren't the mere means to some end, grounded in the prospect of obtaining some better understanding of the physical world. They are, rather, the end goal in itself.
Or if you will, the "music".
And it takes a lot for an undergrad to take algebraic geometry class under a Fields Medal laureate. That gives a sight about his math maturity.
I'd wax poetic about most reals being uncomputable, but good luck explaining cantor's diagonal theorem, measure theory, why we need measure theory, what a real number is, why we need real numbers, how sqrt(2) isn't rational, how numbers can be transcendental, what a limit is.
There is so much background needed for maths. People feel like they don't have the talent to get it, whereas really it's a matter of putting in the time to get comfortable with the basics.
At about age 11, I was presented with some relatively advanced math books. My father is a theoretical physicist and he introduced me to calculus. For a while, math was my passion. My parents, however, were reluctant to push me too far, too fast with math (as they saw it) and so it was a long time after that before I was exposed to any math that was really more advanced than basic cal culus. I am not sure in hindsight whether their attitude was best or not. However, the result was that for a number of years the math I was exposed to did not seem fundamentally new and challenging. It is hard to know to what extent this was a factor, but at any rate for a number of years my interest in math flagged.
https://www.sns.ias.edu/ckfinder/userfiles/files/Comemorativ...
Calculus is relatively basic compared to grad level work in that area.
This suggests calculus was all he knew before at least college (for some value of "a long time").
The human interest stuff is also great; just enough to give you a full picture without the padded-out feeling that a lot of long-form journalism has. Great all around!
That doesn't sound right to me.