isomorphismes:

“the most important part of a principal components analysis is naming the axes”

—

William Kruskal

(source: I heard this second-hand but I don’t know if it’s written down anywhere)

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(For those not in-the-know,

Principal components are composite dimensions, like 5×faculty pay + 3×library size + … ÷ 10 or 3×vote on bill 3 + 8×vote on bill 12 + … ÷ 100

The hope is that,

by using linear algebra

, you can present many things as fewer things. [beta vs p examples]

The reasons this hope might have some possibility of working are two: (1) covariation and (2) small contributions. If two of your data fields do similar things (1), you can combine them into one dimension which acts pretty much the same. (in mathematical terms, by rotating the basis). If many of your data columns make a small contribution, why not smush them into one composite dimension (eg irrelevant_1 + irrelevant_2 + … + irrelevant_N)

You want dimension names to look like this

but how do you get them from reality = messy data which wasn’t gathered well, isn’t necessarily defined how you want, isn’t defined the

You lie.

Product idea: instead of giving people what they want, lie—saying you’re giving them what they want—then deliver something else.

— isomorphismes (@isomorphisms)

May 3, 2017

Cartesian functions send {A}→{B} with exactly one tail a↦ per a∈{A} connecting to each head ↦b∈{B}.

In other words B has to be equal size or smaller than A.

This is true mapping rings to rings, groups to groups, sets to sets, vector spaces to vector spaces,

… it’s just a property of arrows really.

When mathematicians want to talk about “one-to-many” (using the database lingo) or “multimaps” (some stupid word I heard on Wikipedia which absolutely nobody anywhere ever thought was a good term), though, they’re not left outside.

If you’ve got a bundle of arrows ⇶ with tails from {a₀, a₁, a₂, a₃} ⇶ {b₁₄}, then that’s a bundle of tails all heading to the same place. If you “grab them all by the head”

So when mathematicians want to talk about a multimap, they use a preimage ƒ⁻¹. Let’s say the kernel for example–it’s “everything that gets thrown in the trash”—so if multiple things get thrownin the trash, 

(linear subspace / quotient / ring morphism kernel)

So this is how they can associate a bunch of stuff, to one point. For example every point on a manifold gets a tangent space. Maybe this is a vector space for example–which is a lot bigger than just one point.

That would be a problem for 1-to-≥1 functions, so the mathematicians need to turn the arrows around. That’s why they define the projection map π:E→B to send a ton of things e∈E onto that one point b∈B i.e. p∈M.

isomorphismes:

“The … conception of the person as a bounded, unique, more or less integrated motivational and cognitive universe, a dynamic center of awareness, emotion, judgment, and action organized into a distinctive whole and set contrastively both against other such wholes and against its social and natural background, is, however incorrigible it may seem to us, a rather peculiar idea within the context of the world’s cultures.”

—

Clifford Geertz, From the Native’s Point of View

Grateful Geertz has gathered these assumptions together in one places where they can be conveniently interrogated.

• bounded – where did my thoughts come from? environmental factors - parents - chauvinism
• unique – no, there are others like me
• integrated – ignores situational psychology - stimulus/response – so without the chance to be kill I won’t do it (nad won’t have that future life history) – without the chance to work at an investment bank – without the chance to be brave
• self-motivated – then where do tropes come from? why are so many comic book authors putting women in refrigerators? Is it because they really individually wanted to do that? Or is there some kind of interpersonal causality at work.

isomorphismes:

“At the turn of the century, the Swiss historian Jakob Burckhardt, who, unlike most historians, was fond of guessing the future, once confided to his friend Friedrich Nietzsche the prediction that the twentieth century would be “the age of oversimplification”. Burckhardt’s prediction has proved frighteningly accurate. Promising a life of bread and bliss, just after the war to end ask wars. Philosophers have progressed daring reductions of the complexity of existence to the mechanics of elastic billiard balls. Others, more sophisticated, have held that life is language, and that language is in turn nothing but strings of marble -like units held together by the catchy connective of Fregean logic. Artists well dished out in all seriousness checkerboard patterns in red wine and blue fetch the highest bus at Sotheby’s. Biologists are mesmerized by the prospect that life may be reduced to a double helix.”

— Gian-Carlo Rota, Husserl and the reform of logic

isomorphismes:

“As telling examples of the views Sokal satirized, one might quote some other statements. Consider the following extrapolation of Heisenberg’s uncertainty and Bohr’s complementarity into the political realm: “The thesis ‘light consists of particles’ and the antithesis ‘light consists of waves’ fought with one another until they were united in the synthesis of quantum mechanics. …Only why not apply it to the thesis Liberalism (or Capitalism), the antithesis Communism, and expect a synthesis, instead of a complete and permanent victory for the antithesis? There seems to be some inconsistency. But the idea of complementarity goes deeper. In fact, this thesis and antithesis represent two psychological motives and economic forces, both justified in themselves, but, in their extremes, mutually exclusive. …there must exist a relation between the latitudes of freedom df and of regulation dr, of the type df dr=p. …But what is the ‘political constant’ p? I must leave this to a future quantum theory of human affairs.” Before you burst out laughing at such “absurdities,” let me disclose the author: Max Born, one of the venerated founding fathers of quantum theory [3]. Born’s words were not written tongue in cheek; he soberly declared that “epistemological lessons [from physics] may help towards a deeper understanding of social and political relations”. Such was Born’s enthusiasm to infer from the scientific to the political realm, that he devoted a whole book to the subject, unequivocally titled Physics and Politics [3].”

— Mara Beller, The Sokal Hoax: At Whom Are We Laughing? (via isomorphismes)