Posts tagged with logical circular logic

The purpose of a family is the enhancement of the individual pursuits of happiness … in the overall … preservation of the family as a whole.

—from Family Wealth

Is this a logical sentence, or not?

hi-res

These are my drawings of some simple groups: the “lightswitch group” `ℤ₂`, the “hi-lo-off lightbulb” `ℤ₃`, and the symmetric group `𝕊₃=ℤ₂×ℤ₃`, which is how the letters `{A,B,C}` or the boyfriends `{Pankaj, Nadir, Ajay}` permute.

Edit: Several people have helpfully corrected me that `𝕊₃≠ℤ₂×ℤ₃`. 𝕊₃ needs to be the twisted product of `ℤ₂ & ℤ₃`, not a straight product.

$\dpi{300} \bg_white \large \begin{matrix} \mathrm{ABC} & \xrightarrow{\sigma_1_,_2} & \mathrm{BAC} & \xrightarrow{\sigma_2_,_3} & \mathrm{BCA} \\ &&&& \downarrow ^{\sigma_1_,_2} \\ &&&& \mathrm{CBA} \\ &&&& \downarrow ^{\sigma_2_,_3} \\ & & \mathrm{ABC} & \xleftarrow{\sigma_1_,_2} & \mathrm{CAB} \end{matrix}$

$\dpi{300} \bg_white \large \begin{matrix} \mathrm{ABC} & \overset{\sigma_1_,_2}\longleftrightarrow & \mathrm{BAC} & \overset{\sigma_2_,_3}\longleftrightarrow & \mathrm{BCA} \\ &&&& \updownarrow ^{\sigma_1_,_2} \\ &&&& \mathrm{CBA} \\ &&&& \updownarrow ^{\sigma_2_,_3} \\ & & \mathrm{ABC} & \overset{\sigma_1_,_2}\longleftrightarrow & \mathrm{CAB} \end{matrix}$

GLOSSARY

• unary — takes one argument — like “square x”
• binary — takes two arguments — like “x times y”
• arity — how many arguments a function takes
• function — the target needs to be unique (per source) but not the other way around
• inverse — when you can undo something
• exponents — superscripts representing how many times you apply a verb

Perception is reality. Any beer drinker who is surprised that Guinness has a unique and excellent taste and PBR tastes exactly like Budweiser needs to switch to Guinness because your taste is objectively awful.

That’s why Guinness’ branding is a seal with a ball and Budweiser needs to use bikini babes.

There’s something much deeper going on here, though: a fundamental problem with utility theory and hence, with economic theory. Kahneman & Tversky pointed out that it’s wrong to think of preferences as being read off of a master list. But not only are they constructed in the elicitation process, they’re constructed before as well. You’re looking at experimental proof.

I tried to write about this before in the context of the famous Pepsi/Coke fMRI experiment, but it’s too hard. I want to tie in sardonic Don Draper quips, the invention of diamonds, and my own experiences of my desires and wants and dreams being formed by outside (and therefore, sinister?) forces rather than from truly “within me”  — whatever that might mean. Why do I want what I (think I) want? Even Doug Hofstadter treads tenderly around the topics of free will and one’s own true desires and self-determination and such.

Even though I feel that these things all belong together, I don’t understand it all well enough to put forward a thesis explaining the inchoatia. But even with just the few experimental examples we have, it’s clear that desires can be manufactured, and that there’s a lot of money to be made in doing so. So just with that basic knowledge the Lagrangian model of utility that underlies all of the Edgeworth boxes, welfare theorems, and so on is missing a crucial quality.  Namely, &sym;1% of the global economy is spent on making people want things. That doesn’t bear on “utilitarian” products like oil, shipping, … but it definitely bears on aspiration and retail. I’m talking about circularity in the definition of value. If you can logic that one out, let us know.

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(Source: ocw.mit.edu)

hi-res

Several years ago I sat (after yoga class) with some Zaa Zen practitioners. As I understood the practice from doing it once, Zaa Zen basically consists of sitting in good posture, staring at a blank wall, and clearing your mind.

It wasn’t my favourite meditation I’ve ever tried. (So far my favourite was something that into the continuum introduced me to: Vipassana meditation. The way I did it was to sit outdoors in nice weather and listen to the sounds and stop thinking about my own anxiety or problems. Something much like the John Cage lecture that until a single soliton survives posted. Being aware of the world around you and “listening” or “taking in” rather than “forcing” or “pushing out”.)

But I definitely remember the conversation I had with one of the practitioners (Tony) afterwards. Tony was maybe 20 or 30 years older than me but I felt we instantly connected on some mental level. He told me he had been a failure at pretty much everything he had tried in life. How he was a black sheep of his family; how he tried to be a biologist; there were a few other things he tried and he hadn’t been very good at any of them. But in some sense it didn’t matter (remember, this is the wisdom of years talking. According to economic research people tend to mellow, their aspirations and hopes drop to a realistic level, and they become intimately familiar with the passing of time—whatever you optimise, whatever you read, however much you drink, whatever you earn, however you train, however many relationships you destroy—that passing of time always clicks, click, click, tick, steady.)  and he could always come back to his practice. A different meaning of “return to the breath”.

Anyway, we were talking about various I guess spiritual things. More like a mixture of the mental-ethereal and the sense-grounded. He was telling me how Zaa Zen was so great and I would really like it and I should read this book and so on. You know how people always do that—they’ve read a book and then they say you would love it. Well, no, I think just you liked it and I have my own stack of stuff that’s my to-read list already. So normally I would just keep that kind of thought to myself but since Tony and I had an unusual level of honesty and directness for perfect strangers who just met, I brought up what I see as the circular-logic problem of picking up any book.

• When deciding whether I want to read a book or not, I am acting on incomplete information—and not just random incomplete information, marketing and Ising-spin-ish hubbub. I have a hazy idea of what the book is going to be like.
• As I read the book it is going to change me.
• "Be very, very careful what you put into that head, because you will never, ever get it out." —Thomas Cardinal Wolsey (c.1475-1530)
• I can’t unread the book and I can’t unthink or unknow whatever ideas it gives me.
• So even before I know what it is I have already consented to be changed.

This is why, I said, I won’t read the book you’re telling me I will like so well. From my outsider’s perspective I don’t trust enough in the Zaa Zen idea. Not to say that it is some hokey New Age crystals or whatever, but I don’t sense—from standing on the threshold—that this is a house I want to get comfortable in.

(This is also why I started reading so much mathematics. From an outsiders’ perspective it seemed like “This is where the truth is. Following Wolsey’s idea, with a hungry reification of Plato’s philosopher-kings, if I put in only veracity and earnest labour, the result should be something good.)

Tony told me this attitude was actually quite Buddhistic or Zen of me. So I felt very proud that in avoiding looking at the Zaa Zen I had apparently picked up something of it—and it’s a nice geometric shape now that I reflect on it.

` `

So it’s a logical circular logic and a higher modal order than the standard model of choice—and it relates to two other themes I want to talk more about later:

1. So many economic decisions are just like this. Beyond just knowing my edge, I need to decide whether quantitative finance is actually a thing (and not just the subject of a book by Emanuel Derman) before enrolling in an MFE. (There are various signals on the interwebs that suggest MFE’s are not a good idea. I wrote out my reasoning more fully when I was making this decision, google “DIY MFE”.) And say I spend half a decade training to be a lawyer or engineer or doctor. Then what if I don’t like it? Since young people don’t intern or work in hospitals / law firms / alongside engineers before choosing their course of study, their decision is based on folderol, disinformation, heresay, and outer appearances. If I would have loved a career in X I’ll never know it because I couldn’t possibly sample.

On the hypothesis that most people don’t know what they want most of the time (nor do most corporations know what they’re doing or why it works, except by accident), I’d rather look at economic agents as operating at some higher order level, away from all the information. The most I feel I can do as a rational maximiser is try a lifestyle and sample how it makes me feel (although…again, I am changing both with time and changed by my own choices as I do this). Sampling from my own utility function rather than knowing it beforehand. (Or with a corp sampling from revenue & other responses.)
2. "Dug like a river" / "Hebbian history". One of the famous models of brain development is “Neurons that fire together, wire together”. Yogis (need a link, sorry) draw the analogy to a river—as water flows from tributaries to deltas, the act of doing so cuts a deeper and deeper channel along the same course.

These are the same idea and I think juxtaposing habit (in mathematical terms, bien sûr!) alongside personality, mood, preference, desire, intent, pleasure, happiness, goals, rank, and free will is going to lead somewhere interesting. I’ll write more about how I can exercise “second order” free will more easily than first-order.

For example if I close this laptop and hide it from myself I will waste less time on the internet than if I leave it open and tempt myself. (On the other hand—back when I had much better time discipline from running my business I was quite better at focussing whilst at the computer. But from doing more computer stuff since then the “edges of the water” “eroded” the “sides of the channel”—and now my computer time management is spilled out like a floodplain. So very Hebbian in that story itself.) Some people pay a personal trainer so that they’re committed to work out (but couldn’t they have saved money and just worked out?). And a married man may stay away from strip clubs, red light districts, and too many drinks with attractive coworkers—and would we consider his desire to steer clear of temptation a form of infidelity?

The jazz educator David Baker described the progression of jazz improvisational creativity this way: first you learn to copy long licks, scales, pre-formed patterns. Second you start playing with these, so that you have a coarse level of control (free will, in my “interpretation”)—splicing together the known parts. As you progress to higher levels of mastery, your control, focus, creativity become ever more atomic. A true improvisational master is present—deciding, thinking—in every millisecond of the notes, rests, articulation, and consciously chooses every aspect of what s/he’s doing and why.

I’ve found this pattern to hold for me in areas besides jazz improv (and it even holds a lesson for maths explanations—to remember that your audience is probably not at such a fine-grained level) and I want to juxtapose as well whatever this view of personal development is pointing to, against the Lagrangian utility concept.

## 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.

Life is meaningless, unless you believe otherwise.

The theory of universal algebras was well-developed in the twentieth century. [It] provides a basis for model theory, and [provides] an abstract understanding of familiar principles of induction, recursion, and freeness.

The theory of coalgebras is considerably [less] developed. Coalgebras arise naturally, as Kripke models for modal logic, as automata and objects for object oriented programming languages in computer science, and more.

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