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Philosophy of Mathematics is Stupid, Here's Mine (And Why Math is Storytelling) - Pattern Recognition
May 25th, 2013
09:31 pm

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Philosophy of Mathematics is Stupid, Here's Mine (And Why Math is Storytelling)
No philosophy of math that I've encountered seems to say much of meaning about mathematics as I've actually done and understood it, and most of it seems ridiculous to me. So let me set down my own meaningless and ridiculous understanding, and say a bit in particular about why mathematics as it's actually done and experienced is a type of narrative. I'm going to have to use a lot of attack quotes in what follows; attack quotes will denote a word that applies in some senses and not others. (That's all words, really, but...)

I perceive mathematics in two aspects: it has a meaningless/formalist face and an empirical or quasi-empirical face.

The formalist aspect of mathematics is the axiomatic face: like choosing rules for a game. Imagine a set of rules, when fixed, instantly spinning out a universe of consequences. Most of the actual work of math is exploring that universe. Think of the rules of chess as 'creating'--more accurately allowing the creation of, opening a space for--every specific game of chess ever played, and astronomically more games that never have been played. Think of the axioms of Euclid's Elements--each one agonized over as an understanding of reality, but once fixed and joined together freed of that relation and 'creating' (allowing, opening up) the conceptual world of geometry.

Any such universe, any such web of consequences, is 'objective' inasmuch as it's unyielding to any influence but itself. But I can churn out these universes forever, by picking different axioms (rules, games): exploring some will be trivially boring, others will be complex and interesting, and still others will be overwhelmingly, impossibly difficult. As such 'universes' can be 'created' in unlimited profusion, and discarded if I find them dull or annoying, it's odd to claim that they 'exist'. Sort of like creating a hologram and then peering through it to see the 'universe' it depicts. The image isn't 'real' in most senses that we understand that word, but it's real-ish as an emergent phenomenon defined by the structure of the film. (You made the film itself, so that's real, but arbitrary.) Nevertheless, you might see something interesting and unexpected in the image. Might see a whole ghostly world that you could never have conceptualized yourself. Think of the Mandelbrot set, defined by a few simple rules and the equation zn+1 = zn2 + c, and the infinitely complex image it generates.

The quasi-empirical face of mathematics is where we can see mathematics as narrative--the exploration of these ephemeral universes is something that we experience over time, and encoding that exploration in a coherent temporal/symbolic/conceptual way is precisely the art of narrative. Every chess game is a story, with characters, a journey, conflict, and resolution, and so is every theorem. (Most of them are very badly told stories, but that's Sturgeon's Law.) Choosing axioms (rules, games, universes) is a tiny fraction of what mathematicians actually do. Most of a mathematician's work consists of journeying into these ghost-worlds, and telling stories about the expedition.


tl;dr: Most philosophy of mathematics overloads the word 'exists' until it becomes useless.

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From:luagha
Date:May 26th, 2013 06:02 pm (UTC)
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Well told. I'll be stealing it on occasion. :)
From:littlebbob
Date:June 5th, 2013 01:45 pm (UTC)
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Hardy has a bunch of stuff to say about this.

There's a dimension of "interestingness" that plays into that narrative side very well. We can spin out true results all day long, it's the interesting ones that matter, though, and interestingness appears to be almost entirely a social construct. If it's not purely social, the objective aspects of it are extremely hard to get a handle on.
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From:st_rev
Date:June 5th, 2013 02:37 pm (UTC)
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I think 'interesting' is partly just being at the sweet spot between 'trivial' and 'impossible'.
From:littlebbob
Date:June 5th, 2013 03:06 pm (UTC)
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There's that, and the obvious network effect -- is related to other "interesting" stuff.

Just as side note something I have been mulling over is prime numbers. Given the completely insane complexity of pretty much everything to do with primes, one wonders if the problem here is simply that we've gotten ahold of the integers by the wrong handle. It's a super *interesting* handle, it's made a whole lot of PhDs, and it even has some applications. Still, the question can be asked: Is this stuff actually the guts of the integers, are are we just chasing some absurd glob of complexity?
From:(Anonymous)
Date:July 30th, 2013 12:32 pm (UTC)

Underrating the quasi-empirical

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I don't entirely understand why narrative is on the quasi-empirical side -- and why empirical things like change, motion, space, shape, causality, etc. are missing.
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From:st_rev
Date:July 30th, 2013 05:59 pm (UTC)

Re: Underrating the quasi-empirical

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Empiricism is exploratory; narrative is a favored human means of organizing and communicating the data of an exploration. Experiments are also narrative: We did this, and this happened.

I'm not sure what you mean by 'why empirical things...are missing'.

Edited at 2013-07-30 06:00 pm (UTC)
From:(Anonymous)
Date:July 31st, 2013 03:36 am (UTC)

Re: Underrating the quasi-empirical

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I guess I meant something like "modeling" as a driver of which universes we explore. For example, I understand natural numbers (and the associated axioms) as an abstraction of real-life counting; Euclidean space (and its axioms) as an abstraction of the space in which we live, etc. Where does applied maths fit in your scheme?
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From:st_rev
Date:July 31st, 2013 02:34 pm (UTC)

Re: Underrating the quasi-empirical

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From:st_rev
Date:July 31st, 2013 05:28 pm (UTC)

Re: Underrating the quasi-empirical

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In more detail: there's certainly some connection between sets of rules that seem interesting, neither trivial nor impossibly difficult, and models of our experience. This may be because we live in a universe that's neither trivial nor impossibly difficult, I dunno.

But I was talking about mathematics as I've done and experienced it, and I'm not an applied mathematician by training.
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