maths for/about human beings

It’s not always possible to say A ≻ B or A ≺ B. Sometimes

neither A nor B is smaller. A≹B
neither A nor B is more successful. A≹B
neither A nor B is prettier. A≹B
neither A nor B is smarter. A≹B
you don’t love A any more or any less than you love B. ℒ(A)≹ℒ(B)
neither A nor B is tastier. A≹B
neither A nor B is closer. |A−x| ≹ |B−x|
neither A nor B is more fair. A≹B
neither A nor B is better. A≹B

I’ve argued this before using posets. And I intend to argue it further later, when I claim that the concept of Pareto superiority was a major step forward in ethics.

*[The concept of Pareto dominance allows you to make, at least in theory, a valid, fully general comparison between two states of the world. A≻B in full generality iff aᵢ≻ bᵢ ∀ aᵢ ∈ A and ∀ bᵢ ∈ B, by the individual standards of ∀ ≻ᵢ.]

For now, though, I’ll draw some examples of functionals that don’t beat one another. That is, ƒ≹g nor g≹ƒ. (You would probably assume ≹ is 2-symmetric but I’m just stating it for clarity.)

In this drawing, green wins sometimes and purple wins other times. Is it more important to win the “righthand” cases or the “lefthand” cases? How much better for each scenario? (see integrating kernel) Is it better for the L₂ norm to be higher? Or just for the ∫ mass to be greater?

In this drawing, orange wins sometimes and blue wins other times. Is it more important to win the “interior” cases or the “extremal” cases? How much better for each scenario? (see kernel of integration)

How about a function that measures the desirability of a particular boyfriend / girlfriend in various scenarios. How about the function g measures boyfriend B in the various scenarios (domain) and the function ƒ measures boyfriend A in the various scenarios. By measures, I mean the function’s codomain is some kind of totally ordered set where it does make sense to talk about better ≻ and worse ≺.

ƒ(at dinner) ≻ g(at dinner)
ƒ(career) ≺ g(career)
ƒ(in bed) ≫ g(in bed)
ƒ(with your family) ≺ g(with your family)
ƒ(at the beach) ≺ g(at the beach)
…and so on…

So how do you decide whether A≺B or B≺A? Perhaps you have your own priorities sorted so well that you can apply a kernel. Or perhaps A≹B in the final analysis.

I could make a comparable list for

comparing two houses or apartments (well, this one’s closer to the park, but that one has that cozy breakfast nook),
comparing two societies (one where the top marginal tax rate is 41% and one where the top marginal tax rate is 40%),
and on and on.

Sometimes it’s hard to compare. Sometimes — like which of your kids do you love the best — it’s impossible to compare.

Public art works by Katie Sokoler.

“They don’t have money for a gym membership. They don’t have money for a 24-hour gym pass. This is a ghetto pass. They work out in the ‘hood. A lot of these guys are creative, because they’ve been incarcerated. They know how to work out with [whatever’s around]. And you know, these guys are just as toned, just as ripped. They look better than some of the cats at any fitness club around the world.”

Circled #4, oil transfer drawing, 44×30, Glovaski 2009

via planetaryfolklore, oieouio, glovaskicom

One reason polynomials are interesting is that you can use them to encode sequences.

$\begin{matrix} 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + \cdots \\ \\ 5 + 2 \, x + 13 \, x^2 + 87.1 \, x^3 - x^4 + x^5 - 3 x^6 \cdots \\ \\ (\ 5, \ \ 2, \ \ 13, \ \; 87.1, \ -1, \ \ 1, \ -3, \ \ldots\ ) \\ \\ a + b \, x + c \, x^2 + d \, x^3 + e \, x^4 + f \, x^5 + g \, x^6 + \cdots \\ \\ \sum_0^N \mathrm{const}_i \cdot x^i \end{matrix}$

In fact some of the theory of abstract algebra (the theory of rings) deals specifically with how your perspective changes when you erase all of the x^297 and x^16 terms and think instead about a sequence of numbers, which actually doesn’t represent a sequence at all but one single functional.

$\begin{matrix} 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + \cdots \\ \\ (\ 1, \ \ 1, \ \ 1, \ \ 1, \ \ 1, \ \ 1, \ \ 1, \ \ldots\ ) \\ \\ 5 + 2 \, x + 13 \, x^2 + 87.1 \, x^3 - x^4 + x^5 - 3 x^6 \cdots \\ \\ (\ 5, \ \ 2, \ \ 13, \ \; 87.1, \ -1, \ \ 1, \ -3, \ \ldots\ ) \\ \\ a + b \, x + c \, x^2 + d \, x^3 + e \, x^4 + f \, x^5 + g \, x^6 + \cdots \\ \\ (\ a, \ \ b, \ \ c, \ \ d, \ \ e, \ \ f, \ \ g, \ \ldots\ ) \\ \\ \sum_0^N \mathrm{const}_i \cdot x^i \\ \\ (\ \mathrm{const}_0, \ \ \mathrm{const}_1, \ \ \mathrm{const}_2, \ \ \mathrm{const}_3, \ \ \mathrm{const}_4, \ \ \mathrm{const}_5, \ \ \mathrm{const}_6, \ \ldots\ ) \end{matrix}$

When you put that together with observations about polynomials

Every sequence is a functional. (OK, can be made into a functional / corresponds to a functional)
So plain-old sequences like 2, 8, 197, 1780, … actually represent curvy, warped things.
Sequences of infinite length are just as admissible as sequences that finish.
(After all, you see infinite series all the time in maths: Laurent series, Taylor series, Fourier series, convergent series for pi, and on and on.)
Any questions about analyticity, meromorphicity, convergence-of-series, etc, and any tools used to answer them, now apply to plain old sequences-of-numbers.
Remember Taylor polynomials? There’s a calculus connection here.
Derivatives and integrals can be performed on any sequence of plain-old-numbers. They correspond (modulo k!) to a left-shift and right-shift of the sequence.
You can take the Fourier transform of a sequence of numbers.
How about integer sequences from the OEIS? What do those functions look like? How about once they’re Taylored down? (each term divided by k!.)
Sequences are lists. Sequences are polynomials. Vectors are lists. Ergo—polynomials are vectors?!
Yes, they are, and due to Taylor’s theorem sequences-as-vectors constitute a basis for all smooth ℝ→ℝ functionals.
The first question of algebraic geometry arises from this viewpoint as well. A sequence of “numbers” instantiates a polynomial, which has “zeroes”. (The places where the weighted x^1192 terms sum to 0.)

So middle-school algebra instantiates a natural mapping from one sequence to another. For example (1, 1, −2, −1, 1) ⟼ (−1, 1−φ, 1, φ). Look, I don’t make the rules. That correspondence just is there, because of logic.

Instead of thinking sequence → polynomial → curve on a graph → places where the curve passes through a horizontal line, you can think sequence → sequence. How are sequences→ connected to →sequences? Here’s an example sequence (0.0, 1.1, 2.2, 3.3, 4.4, 0, 0, 7.7) to start playing with on WolframAlpha. Try to understand how the roots dance around when you change sequence.
Looking at sequences as polynomials explains the partition function (how many ways can you split up 7?) As explained here.
Also, general combinatorics http://en.wikipedia.org/wiki/Enumerative_combinatorics problems besides the partition example, are often answered by a polynomial-as-sequence.
Did I mention that combinatorics are the basis for Algorithms that make computers run faster?
Did I mention that Algorithms class is one of the two fundae that set hunky Computer Scientists above the rest of us dipsh_t programmers?
There is a connection to knots as well.
Which means that group theory / braid theory / knot theory can be used to simplify any problem that reduces to “some polynomial”.
Which means that, if complicated systems of particles, financial patterns, whatever, can be reduced to a polynomial, then I can use a much simpler (and more visual) way of reasoning about the complicated thing.
I think this stuff also relates to Gödel numbers, which encode mathematical proofs.
You can encode all of the outputs of a ℕ→ℕ function as a sequence. Which means you may be able to factor a sequence into the product of other sequences. In other words, maybe you can multiply simple sequences together to get the complicated sequence—or function—you’re looking for.

This is an example of when the kind of language mathematics is, is quite nice. Every author’s sprawling thoughts coming from here and going to there while taking a detour to la-la land, are condensed by uniformity of notation. Then by force of reasoning, analogies are held fast, concrete is poured over them, and eventually you can walk across the bridge to Tarabithia. Try nailing down parallels between Marx & Engels, it’s much harder.

All of these connections give one an archaeological feeling, like … What exactly am I unearthing here?

The idea of art as free expression is romantic folly. Artists are problem-solvers. They are working for a living. They produce, play music, and dance for others for money. My artist friends, the successful ones, are taken up much of the time considering how to perfect or extend their craft and how to sell more product.

…While…taking a design course [my instructor] passed by me as I was laboring over a design project…. “Have you solved it yet?” he asked. That was when I realized that the essence of art was applied problem-solving…

[Let me point out how] completely erroneous [many popular] ideas about success in the arts are: as if one somehow either was born with the ability to play the violin or not. Talent plays a role, but time-on-task is the great determiner of achievement in playing an instrument and in doing mathematics. These arts are mastered at the cost of sweat, and their practice is not easy.

The last decade’s debt record for several rich countries.

3-month Bond Yields owed by some of them: (SOURCE: Bloomberg)

Japan   .10%
UK      .41%
Germany .28%
US      .04%

And here’s one of the yield curves (US’):

(Remember, higher yield means the debt costs more to service for the country that’s borrowing.)

3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 … modulo 89

An application of group theory to Navy sonar — can you generate a sequence whose 1-differences are random? (whose 1-autocorrelations are nil)
Also the patterns look a bit like low-discrepancy sequences:

According to some rough estimates, world living standards grew less than 50 percent … from A.D. 1 until the Industrial Revolution. By contrast, they grew a whopping 1,000 to 2,000 percent in the 19th and 20th centuries.

Strawberries by Asobi Seksu

[A] wall of fifty or sixty glass demijohns, wired tight against earthquakes, exhibit creatures from the [United East India] Company’s once-vast empire.

A pickled dragon of Kandy…a slack-jawed viper of the Celebes…A baby alligator from Halmahera…The alligator’s umbilical cord is attached to its shell for all eternity….the jar of a Barbados lamprey…[Its] mouth is a grinding mill of razor-sharp V’s and W’s.

Preserved from decay by alcohol, pig bladder, and lead, they warn not so much that all flesh perishes—what sane adult forgets this truth for long?—but that immortality comes at a steep price.

(Source: danielrecinos)

Isomorphismes

see things differently

≹

Polynomials as Sequences

BROWSE