Geometry and theology

Boiled the elements down into the axioms
Mistook a fax for remedial tediums
It seems Ezekiel speaks to some
My mind was hazy and numb
And left hand gripped a clump of palladium
Saw the beast with the wings and the talons
The simple answer but it felt out of balance
Bad news like a blue screen of death
Besides the point, but which hue seems best?

I’ll keep conducting these autistic symphonies
These sentences have sentenced me
Like I didn’t have the sense to sense the mere
The presence grows weird
Doesn’t make sense but I don’t fear, not a damn thing
I live like a man who’s already dead
Like I had a motorcycle but my name is Zed
(I’ll be missed), said another clumsy alchemist
Like he just learned predicate calculus
The existential quantifier, a backwards EX
My rap career is a cataloging of defects
Copy edited by Ryan Seacrest
Like he must be new to this
Milo doesn’t exude hubris, chuuch
Like he must be new to this
Milo doesn’t exude hubris

Modernism in Mathematics

Jeremy Gray makes the case (in here) that modernism applies to mathematics. His modernism consists largely of a move away from representations and towards formal approaches.

So on Lebesgue’s theory of the integral in 1903:

“The axioms specify what the integral is intended to do. They do not start from an idea that the integral is about, say area, or any other primitive concept. It is necessary to show that there is a model of these axioms, but once that is done it is at least possible to prove properties of the integral directly from the axioms and without reference to any model of them. The axioms are sometimes said to define their object implicitly, or to create it. There is no reference to a primitive concept available via abstraction from the natural world.”

And on Kronecker and Riemann:

“Neither man suggested that objects cannot be studied via their representations, but both believed that one must be vigilant to ensure that one establishes properties of the objects themselves and not the properties of merely this or that representation, and to this end it was best to avoid explicit representations whenever possible.”

I didn’t know about the Hausdorff paradox, which feels a lot like Godel. Gray’s summary: “on any plausible definition of the measure of a set there must be non-measurable sets.”

Borel ended up critiquing the use of the axiom of choice to call the paradox into question, but this was another step (apparently) in pushing people to accept that definitions of area are inherently imperfect — pushing us further away from meaning and belief in the representations.

Another interesting point from Gray: you know that thing about the unreasonable effectiveness of math? That wouldn’t have made any sense in the 19th or 18th centuries because math was coextensive with science. Like, there’s nothing surprising at all about the connection between math and the world back then, because math was an attempt to describe the world.

I’m interesting to read more, but I’m feeling as if a question has been answered. Whether we call it modernism or not, this is the time in the history of math when the connection between mathematics and the empirical world was made problematic. If we’re looking for the origins of the idea that math is “useless,” it’s going to be in this movement in mathematics between 1880 and 1920.

Things that I’d like to read: on modernity and mathematics

The world has changed immensely over the past several hundred years. Mathematics has too. Are these changes all related?

Plato’s Ghost looks like a good place to start.

Plato’s Ghost is the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions.

Building on Gray’s work is this presentation by Susumu Hayashi, which introduces (to me at least) the notion of “mathematical secularization.”

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I also came across The Great Rift.

In their search for truth, contemporary religious believers and modern scientific investigators hold many values in common. But in their approaches, they express two fundamentally different conceptions of how to understand and represent the world. Michael E. Hobart looks for the origin of this difference in the work of Renaissance thinkers who invented a revolutionary mathematical system—relational numeracy. By creating meaning through numbers and abstract symbols rather than words, relational numeracy allowed inquisitive minds to vault beyond the constraints of language and explore the natural world with a fresh interpretive vision.

The focus is on early modernity and the shift to algebra, which is an earlier phenomenon than modernism. But maybe it’s part of the same story?

Also in the category of “is this related? maybe??,” there is a working group of philosophers that call themselves the Mathematics, Mysticism and Secularization working group.

There are all these -isms that I learned about in philosophy of math: empiricism, logicism, formalism, fictionalism (wiki). That’s part of this story too.

What I’m attracted to is the idea that math is as much a part of culture as anything else. Over the last few centuries Western society has gotten less and less comfortable with the abstract, invisible realm of religion and spirits. Wouldn’t that have an impact on how that culture thinks about that other invisible, abstract realm of mathematics?

People used to think of x^2 as referring just to a square’s area, but then it was emptied of that meaning. Is the break of algebra from geometry something like the break of philosophy from theology?

People used to think that mathematics was a search for ultimate truths, not just conditional ones. Are we living in a mathematically relativistic world?

Mathematicians sometimes talk — with pride! — of the uselessness of their work. Is that the end result of the sorts of processes described by these authors?

I have no clue, and I have no idea when I’ll be able to read those books. But the questions seem interesting and confusing.

YouCubed, Reviewed

This exponents activity is neither original nor at all an interesting version of the idea. It’s no better than what most teachers would make on their own, if they wanted to teach exponent rules inductively.

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Better versions of this are readily available in practically any textbook, but Illustrative Math has a totally free and online unit on exponents that does this activity better. It’s less tedious and repetitive and it asks questions to push students towards generalizations, rather than asking kids to churn out rows and notice the structure at the very end (“discovery”).

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Yes, it’s at a Grade 8 level, but this lesson is pretty much there too. And if you can wait a few months, you’ll have the high school version available too.

Stop Making Sense

Van Morrison in Rolling Stone, 1978:

But a lot of your most danceable songs have very profound lyrics.
I don’t know, this thing about lyrics – I’m just catching on to this. If you get some of the facts together . . . I mean, I sell records in places where they don’t speak English. And I’ve experienced listening to Greek singers, for instance, and not knowing what the words are, but I get a story and a feeling from it even though I haven’t a clue to what’s being said. So if English-language songs can sell in non-English-speaking countries and people can be touched by them, then we can see how irrelevant the words are.

Sometimes it seems that you let the words dance when you sing them; you release them, and they take off in their own directions.
The only time I actually work with words is when I’m writing a song. After it’s written, I release the words; and every time I’m singing, I’m singing syllables. I’m just singing signs and phrases.

But this can’t be the whole story. Music with incomprehensible lyrics isn’t the same as listening to instrumental music. It’s also not the same as listening to scat or wordless yammering. Somehow the presence of words — maybe it’s just the possibility of meaning — changes the way we listen to a song.