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Margaret Wertheim's wide-ranging essay on mathematics, "There’s more maths in slugs and corals than we can think of" covers how mathematics is implemented by humans, animals, or natural processes. It is clever and thought-provoking. Quite long, but well worth the read. Among other topics, the author touches on music, Fourier transforms, tiling, and coral reefs.
What does it mean to know mathematics? Since maths is something we teach using textbooks that demand years of training to decipher, you might think the sine qua non is intelligence – usually 'higher' levels of whatever we imagine that to be. At the very least, you might assume that knowing mathematics requires an ability to work with symbols and signs. But here's a conundrum suggesting that this line of reasoning might not be wholly adequate. Living in tropical coral reefs are species of sea slugs known as nudibranchs, adorned with flanges embodying hyperbolic geometry, an alternative to the Euclidean geometry that we learn about in school, and a form that, over hundreds of years, many great mathematical minds tried to prove impossible.
[...] The world is full of mundane, meek, unconscious things materially embodying fiendishly complex pieces of mathematics. How can we make sense of this? I'd like to propose that sea slugs and electrons, and many other modest natural systems, are engaged in what we might call the performance of mathematics. Rather than thinking about maths, they are doing it. In the fibres of their beings and the ongoing continuity of their growth and existence they enact mathematical relationships and become mathematicians-by-practice. By looking at nature this way, we are led into a consideration of mathematics itself not through the lens of its representational power but instead as a kind of transaction. Rather than being a remote abstraction, mathematics can be conceived of as something more like music or dancing; an activity that takes place not so much in the writing down as in the playing out.
(Score: 4, Funny) by VLM on Tuesday February 14 2017, @09:45PM
I noticed that you can search and replace "organic chemistry" for "math" in the linked article and reads just as well, and now I can't stop. You can also swap in "physics" and "science" in general. Its fun to see what doesn't work as a substitution and re-read. "Theology". "Computer science" don't work so well.
(Score: 2) by AthanasiusKircher on Wednesday February 15 2017, @02:52AM
Indeed. There's another term that comes to mind: Intentional fallacy [britannica.com]. In literature, that is often used to refer to the apparent "fallacy" that we can only judge a work by its author's intentions, rather than recognizing how a work can accumulate meaning based on its audience reception.
But the fallacy can also work in a sort of reverse form -- that is, we can mistakenly ascribe "intention" to a work of art, assuming the author thought the way we do about something, that our abstract model of how the artwork works must have been in the author's mind. But of course that isn't always true. TFA is basically one giant "intentional fallacy" in that regard, because basically the author is arguing that all of these natural things are "performing math" -- hence:
Sea slugs do maths, electrons do maths, minerals do maths. Rainbows do an incredible mathematical performance
NO THEY DON'T. They don't "do math." They do things that we as humans can see patterns in, and one way of modeling some of those patterns is through mathematics. That's the reason you can substitute all sorts of other terms for "math" in the majority of statements in the article -- because the article is mistaking human cognitive MODELS of the world for the actual world. Yes, math is impressive in the way it can be used to model nature, but it's generally only abstracting certain patterns in an idealized form and then formalizing them. It takes a human to create those abstractions, and they're frequently not the only possible models or abstractions that could be used.
As lovely as the idea of "performing math" is, the analogy is fundamentally flawed from the moment music even comes in as a comparison in TFA. Yes, lots of performers can "do music" without reading music notation, but music notation is NOT music. (Well, yes, trained musicians sometimes informally refer to a score as "the music," but they differentiate between a blueprint for performance and the actual sounds created in a performance.) Music notation is a kind of blueprint, a recorded set of instructions on how to play an instrument or sing to produce a set of sounds. But it's optional, because ultimately musicians want to create that set of sounds -- which is the "true" music.
There's no clear analogue in the math world here, at least not in the real world (as opposed to theoretical math or something). Written math is an abstraction, a model, NOT a blueprint to follow for slugs and electrons, minerals and rainbows. At best, perhaps the analogue in music is music theory -- a set of abstractions about chords and scales and whatever or maybe formal grammar in relationship to spoken language. But we don't think of musicians as "performing music theory" or of a great speaker as "performing grammar." Music theory and grammar, like math, are abstract models, usually only picking up on a tiny fraction of the subtleties that make great music or great speeches. For example, music theory is pretty bad at quantifying all the vocal swoops and changes in timbre that often make a pop singer's singing moving, and grammar doesn't do a very good job of tracking emotion conveyed in an effective speech. These are models that take some small part of the real-world practice and attempt to formalize it. That's what math does. And it makes just as little sense to say that a rock rolling down a hill is "performing math" or "performing physics" as it does to say that Martin Luther King or JFK were "performing grammar" when they gave great speeches.
In fact, it makes even less sense, because obviously MLK or JFK probably studied grammar at some point so there's some chance that the abstract model could have affected their actions to some small degree in making a speech -- whereas there's absolutely ZERO chance that the rock rolling down a hill studied physics or math and that that "understanding" or whatever would impact the way it decides to roll down a hill. The math of the rolling rock is only IN OUR HEADS, generally as an incomplete (and somewhat inaccurate and idealized) model in a set of physics equations. It is not being "performed" in any meaningful sense BY the rock.
(Score: 2) by VLM on Wednesday February 15 2017, @01:54PM
I don't disagree with any of that but propose a related alternative that in the OPs culture that LARPing is lower social tier than professional actors.
So people "play minecraft" they don't "go minecrafting" because they're into the LARPing nature of playing and playing minecraft. Yet coal miners "mine" they don't "perform mining" or "work in mines".
So "doing math" is lower tier, LARPing play at math, than theoretical mathematicians which get paid to not "do" things. So sea slugs do math, a Stanford PHD professor IS math, kinda.
In this (possibly spaced out) interpretation, OPs article is more a play on contemporary social hierarchy than telling us anything about math.
(Score: 1) by khallow on Wednesday February 15 2017, @05:07PM
They do things that we as humans can see patterns in, and one way of modeling some of those patterns is through mathematics.
[...]
There's no clear analogue in the math world here, at least not in the real world (as opposed to theoretical math or something).
If the pattern exists in any form (including mere perception by others), even if we aren't aware of it, then the mathematical consequences follow or "play out". So I disagree. They already do math.
(Score: 2) by FatPhil on Thursday February 16 2017, @04:49PM
Great minds discuss ideas; average minds discuss events; small minds discuss people; the smallest discuss themselves