Posts tagged with set theory

Walter Ong turns to the fieldwork of the Russian psychologist Aleksandr Romanovich Luria among illiterate peoples [of] Uzbekistan and Kyrgyzstan … in the 1930’s.

Luria found striking differences between illiterate and even slightly literate subjects, not in what they knew, but in how they thought.

Logic implicates symbolism directly: things are members of classes; they possess qualities, which are abstracted and generalised.

Oral people lacked the categories that become second nature even to illiterate individuals [living] in literate cultures…. They would not accept logical syllogisms.

A typical question:

—In the Far North, where there is snow, all the bears are white.

—Novaya Zembla is in the Far North and there is always snow there.

—What colour are the bears?

—I don’t know. I’ve seen a black bear. I’ve never seen any others…. Each locality has its own animals.

….

"Try to explain to me what a tree is," Luria says, and a peasant replies: "Why should I? Everyone knows what a tree is, they don’t need me telling them."

James Gleick, The Information, citing Walter J. Ong and Aleksandr Romanovich Luria

Over time, Blake came to detest Joshua Reynolds attitude towards art, especially his pursuit of “general truth” and “general beauty”. Reynolds wrote in his Discourses that the “disposition to abstractions, to generalising and classification, is the great glory of the human mind”; Blake responded, in marginalia to his personal copy, that "To Generalize is to be an Idiot; To Particularize is the Alone Distinction of Merit".[20]

Yesterday’s #SB5 conflict is an opportunity to talk about the algebra of sets. (Commonly understood through Venn diagrams.)

The algebra of sets is the way ∪, ∩, , and some composite operations work. These words are useful for reminding yourself to logically separate things that are logically separate.

The relative complement of A (left circle) in B (right circle):
$A^c \cap B~~~~=~~~~B \smallsetminus A$

For example:

• Not all feminists are women.

`Feminist ∩ ∁{Woman}`
• Not all women are pro-choice.

`∁{Pro-Choice} ∩ Woman`

Sets can overlap in different ways.

The category of categories as a model for the Platonic World of Forms by David A Edwards & Marilyn L Edwards

• Thales (7th cent. BC) made the first universal statement (proof w/o regard to the gods or mythology, just from pure reason)
• pre-Greek mathematics was essentially engineering maths.
• I owe ya a post on the illiterates in chapter 2 of James Gleick’s The Information. He tells the story of some illiterates in outer Soviet Union. According to the tale, they basically do not abstract at all. No abstract reasoning, no properties ascribed to members of a class, and so on.

It sounds kind of idyllic in the way of NYT tales of the Pirahã or Jill Bolte Taylor’s story of losing the logical half of her brain. I’m not sure if Thales set us on the path to Hell or Heaven.
• Plato set for himself the [goal] of extending geometry [beyond] triangles and circles and such, to all of human thought. He failed, but his vision has come to pass.
• Why did Lawvere succeed where Plato and Whitehead failed?
• Eilenberg & Mac Lane’s notions of category and functor.
• The definition of function for infinite sets is already implicit in the choice of “which set theory”.
• Category theory, unlike earlier formalisations (think Peano arithmetic and Goedel’s proof), is stable to the “meta” step: you do 2-categories, you do n-categories … the abstraction is ultimately a `k → k+1` kind of deal rather than a “And this is the ultimate finality!” kind of deal.

## Holistic

Some say science & mathematics are reductive.

• Galileo showed us how to break apart space into three pieces and that `ẋ`, `ẏ`, `ż` function independently. (The speed the soccer ball falls downward off the cliff is unrelated to how much forward momentum you kicked it.)
• Experimental science tests just one thing and isolates it as perfectly as possible.
• Some say that the entire progress of empiricist science has been the systematic isolation and testing of small parts of reality, combined with “rigourous technical analysis” (by which they mean, theories in the language of mathematics).
• (I’ve seen this view in the CP Snow ish arguments where “literary types” or “critical theory types” want to attack science or scientists en masse, or where “science types” want to attack philosophers/postmodernists/cultural theorists/Marxists/Deleuze/liberal academia en masse.)
• Some philosophers argue that the world is really atomic in some way, not just the particles but the causes and forces in the world are reducible to separable elements.

The reductionistic approach to science has to pair with qualifiers and caveats that “The lab is not the real world” and “We’re just trying to model one phenomenon and understand one thing”: hopefully combining `A` with `B` doesn’t introduce complexity in the sense that `A+B` is more than the sum of its parts — in statistical modelling language, that the interaction terms don’t overwhelm the separable, monic terms.

But is it really true that mathematics is reductionistic? I can think of both separable mathematical objects and not-separable ones. You could argue, for example, that a manifold can be decomposed into flat planes—but then again, if it has a nontrivial genus, or if the planes warp and twist in some interesting way, wouldn’t you be nullifying what’s interesting, notable, and unique about the manifold by splitting it up into “just a bunch of planes”?

Or with set theory: you could certainly say that sets are composed of atomic urelemente, but then again you could have a topological space which is non-decomposable such as an annulus or network or 1-skeleton of cells, or a non-wellfounded set (cyclic graph) which at some point contains a thing that contains it.

How about a sort-of famous mathematical object in the theory of links & knots: the Borromean rings.

The Borromean rings are famous for the fact that they cannot be decomposed into a simpler atom, whilst retaining their Borromean nature. In other words the smallest atom you can find is the 3 rings themselves. If any one or more rings were removed then they would not be linked together.

So not only is it an interesting case for causality (is `ring 1` binding up `ring 3`? No. Is `ring 2` binding up `ring 3`? No. In a way none of the rings is locking up any other ring, and yet they are locked by each other.

It takes the interaction term `[ring 1 ∩ ring 2]`only together do they bind up `ring 3`, but together they do bind it completely. (Did something like this come up in Lord of the Rings or some other fantasy or myth? Like the weakest link or that square battle arrangement with spears, or having a ton of archers in Warcraft 2, but not like Captain Planet, if any member of a group breaks then its entire strength is lost (super convexity) but together they’re nearly undefeatable.)

Also, circularly, as well as `ring 3` being bound-by `[ring 1 ∩ ring 2]`, it also binds — or again, with-its-mate-binds — the other rings.

I don’t know if historically the Borromean rings were a symbol of holism, although one would think so given this picture from the Public Encyclopedia:

Even if it hasn’t been, we certainly could use the Borromean rings now as a symbol of holism, complexity, integratedness, un-separability, irreducibility, and convolution.

## Measure: Sizing up the Continuum

For those not in the know, here’s what mathematicians mean by the word “measurable”:

1. The problem of measure is to assign a ℝ size `≥ 0` to a set. (The points not necessarily contiguous.) In other words, to answer the question:
How big is that?
2. Why is this hard? Well just think about the problem of sizing up a contiguous ℝ subinterval between `0` and `1`.

• It’s obvious that `[.4, .6]` is `.2` long and that
• `[0, .8]` has a length of `.8`.
• I don’t know what the length of `[¼√2, √π/3]` is but … it should be easy enough to figure out.
• But real numbers can go on forever: `.2816209287162381682365...1828361...1984...77280278254...`.
• Most of them (the transcendentals) we don’t even have words or notation for.
• So there are a potentially infinite number of digits in each of these real numbers — which is essentially why the real numbers are so f#cked up — and therefore ∃ an infinitely infinite number of numbers just between 0% and 100%.

Yeah, I said infinitely infinite, and I meant that. More real numbers exist in-between `.999999999999999999999999` and `1` than there are atoms in the universe. There are more real numbers just in that teensy sub-interval than there are integers (and there are integers).

In other words, if you filled a set with all of the things between `.99999999999999999999` and `1`, there would be infinity things inside. And not a nice, tame infinity either. This infinity is an infinity that just snorted a football helmet filled with coke, punched a stripper, and is now running around in the streets wearing her golden sparkly thong and brandishing a chainsaw:

Talking still of that particular infinity: in a set-theoretic continuum sense, ∃ infinite number of points between Barcelona and Vladivostok, but also an infinite number of points between my toe and my nose. Well, now the simple and obvious has become not very clear at all!

So it’s a problem of infinities, a problem of sets, and a problem of the continuum being such an infernal taskmaster that it took until the 20th century for mathematicians to whip-crack the real numbers into shape.
3. If you can define “size” on the `[0,1]` interval, you can define it on the `[−535,19^19]` interval as well, by extension.

If you can’t even define “size” on the `[0,1]` interval — how do you think you’re going to define it on all of ℝ? Punk.
4. A reasonable definition of “size” (measure) should work for non-contiguous subsets of ℝ such as “just the rational numbers” or “all solutions to `cos² x = 0`(they’re not next to each other) as well.

Just another problem to add to the heap.
5. Nevertheless, the monstrosity has more-or-less been tamed. Epsilons, deltas, open sets, Dedekind cuts, Cauchy sequences, well-orderings, and metric spaces had to be invented in order to bazooka the beast into submission, but mostly-satisfactory answers have now been obtained.

It just takes a sequence of 4-5 university-level maths classes to get to those mostly-satisfactory answers.

One is reminded of the hypermathematicians from The Hitchhiker’s Guide to the Galaxy who time-warp themselves through several lives of study before they begin their real work.

For a readable summary of the reasoning & results of Henri Lebesgue's measure theory, I recommend this 4-page PDF by G.H. Meisters. (NB: His weird ∁ symbol means complement.)

That doesn’t cover the measurement of probability spaces, functional spaces, or even more abstract spaces. But I don’t have an equally great reference for those.

Oh, I forgot to say: why does anyone care about measurability? Measure theory is just a highly technical prerequisite to true understanding of a lot of cool subjects — like complexity, signal processing, functional analysis, Wiener processes, dynamical systems, Sobolev spaces, and other interesting and relevant such stuff.

It’s hard to do very much mathematics with those sorts of things if you can’t even say how big they are.

## Complement

Here’s an inside-out thought: The air around us is a 3-manifold with 3-holes where solid objects are, and the 2-boundary is the ground. Or if you think of all the sky, it’s a spherical 3-shell (with one 3-hole, the Earth) floating in empty space.

I wish I could draw what I’m thinking of. Something like this.

From a child’s-eye view, “the air” is the complement of everything that’s “actually there” (solid or liquid things).

$\large \dpi{200} \bg_white \text{air} = \complement (\text{what's there})$

$\large \dpi{150} \bg_white \{\text{what's there} \} \equiv \{\text{solids}\} \cup \{\text{liquids} \}$

You could correct that child by bringing up outer space (another 2-boundary on the air), or the fact that air is made of particles as well.

$\large \dpi{200} \bg_white \partial \; \text{air} = \partial \{\text{outer space}\} \cup \partial \{\text{the ground}\}$

But wouldn’t you rather be sitting in the middle of a field imagining yourself, the trees, the grass, the clouds, the birds, and the wood chips being cut out by a Photoshop lasso?

Paul Finsler believed that sets could be viewed as generalised numbers. Generalised numbers, like numbers, have finitely many predecessors. Numbers having the same predecessors are identical.

We can obtain a directed graph for each generalised number by taking the generalised numbers as points and directing an edge from a generalised number toward each of its immediate predecessors.

It has been shown that these generalised numbers can be “added” and “multiplied” in a natural way by combining the associated graphs. The sum a+b is obtained by “hanging” the diagram of b onto that of a so the bottom point of a coincides with the top point of b. The product a·b is obtained by replacing each edge of the graph of a with the graph of b where the graphs are similarly oriented.

Paul Finsler, David Booth, Renatus Ziegler in Finsler set theory: platonism and circularity