isomorphismes

The jazz educator David Baker had this to say about jazz improvisation:

You start out learning scales; modes; whole songs. You   play along with your favourite records. Then you start breaking it down to pieces—licks, long bits of solos. Gradually as you get more and more mastery of your instrument and over yourself, your control becomes more and more atomic. At the level of full mastery you are feeling, and choosing, every note, every rest. Eventually it’s every fraction of a note, or fraction of a rest, that you’re playing. Actively.

You also want to extend your range. Your body has a wide range of expression at your command. It’s not just your instrument that can make sound. Clomps, stomps, screams, claps, yelps, lip trills, Brooklyn raspberries, exhaling, inhaling, crying out—all of these are tools at your disposal. You also want to become comfortable in every range of your instrument—even very very high, and very very low, have a purpose that in expressing some emotion you may want to utilise.

The bolded part especially rings true for me much more broadly than in music performance. Free will, I feel, can be exercised to varying degrees. If I check my email in the morning, go on Facebook, check my twitter notifications, whatever, I’m yielding up my free will. I’m passively responding to things that I put in front of myself. On better days, or at least the days when I assert more atomic control over my time and choices, I actively spend time in the moment and/or ask myself what I really want to be doing, rather than rolling the wheels through the ruts of habit or letting stimuli lead me to respond.

For me the question of free will isn’t about yes/no deductions—it’s about how much, today?

This week I posted different viewpoints on The Self.

Particularly I’m interested in self as a function of inputs. Just as the size of eyes a fly is born with is a function of the temperature of the eggs, so too, many facets of ourselves are a function of the environment, other people’s behaviour toward us, game-theoretic strategy, incentives, and so on.

Other people’s theories of us can be seen as functions as well. (For example, a hiring manager’s view of employee performance may assume school quality or GPA to be positively related to human capital.)

• Economics: I didn’t get to Jean Tirole’s theory of money-saving as bargains among multiple selves.
• Psychology: Jim Townsend found that self-versus-other dichotomies can be expressed as a negatively curved metric space.
• Personality: I’ve already written that the MBTI is too restrictive a theory of self. It maps from habits to [0,1]^4.
• Douglas Hofstadter’s thoughts on the extension of the pronoun “we”. ‘We’ went to the moon, ‘we’ share a common ancestor with other primates, ‘we’ are overcrowding the planet, ‘we’ have a nice theory of quantum chromodynamics, ‘we’ do not know if ‘we’ are experiencing a simulation or actual reality, ‘we’ don’t really know what makes an economy grow.
• Criminology: My criminal output is a function of the crime level in the neighbourhood I’m raised in. Except when it’s a function of strongly held beliefs.
• Sociology: In contemporary OECD places, ‘we’ are coerced by our cultures to play roles. “There are” certain scripts — modifiable but still requisite or recommended in some sense; at the very least influential, even if only because benefits and rewards are socially tied to role performance.
• The topic of cultural coercion … is something I’ll return to.
• The concept of people-as-functions is one I want to return to later, in discussing history, economics, and a couple different ways of talking about human behaviour mathematically.

I can think of several other mathematics-inspired questions about ourselves. The difference between habit and personality; the yogic metaphor of a river cutting deeper as related to habituation; choice & free will; Markovian and completely-the-opposite-of-Markovian choices (how constrained we are by our past choices); … and a lot more. But you know what, writing is hard. So I do only a little at a time.

A “truly” random, uniform random, completely random sequence might look like

◯◯⨯◯⨯⨯⨯⨯◯◯⨯◯◯⨯⨯◯⨯◯◯⨯⨯◯⨯⨯◯⨯◯◯⨯◯
R code: > xooooo = sample( c("◯", "⨯") , 30, rep = T) 

like the flips of a fair coin. But there are other “random”s as well.

Biased

For example, biased random, like an unfair coin with 4/5 bias, might generate a sequence that looks like this:

◯◯◯◯⨯◯◯⨯◯◯⨯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯

R code: > xooooo = sample( c("◯","◯","◯","◯", "⨯") , 30, rep = T)

 

Self-Correlated

But there’s also autocorrelated, or serially correlated, randomness.

◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯◯⨯⨯⨯◯◯◯◯◯◯◯

For example you feel fine ◯ 80% of the time and 20% you’re sick ⨯ — and of course the sick days are more likely to come one after another. Or 80% of the time you don’t smoke ◯ but then you buy a pack and all of a sudden you smoke ⨯⨯⨯ three days in a row. Once you’ve broken your resolve, you’re more likely to smoke again the next day.

 

Equation-wise, autocorrelation amounts to adding a self-lag term to the other explanatory variables (plus unexplained residual). Besides habit and viral invasion, autocorrelation brings many things under the penumbra of randomness:

• income. The strong gets more, while the weak ones fade. If you made a lot of money at your previous job, your next employer will pay you more either to steal you away or simply because salary history determines compensation in HR’s formula.
• unemployment. Jobless today, jobless tomorrow. Those who are unemployed for more than six months are even more likely to be unemployed for the long term. Also people who take care of their own kids as their job are likely to still be doing so next week and next year rather than working for a company.
• likelihood of cancer. Back to the subject of smoking, your likelihood of getting cancer accumulates faster and faster the more you smoke. I’ve seen claims that there is a kink in the cumulative propensity to cancer rate above one pack / day.
• stock prices. Stocks don’t just jump around in a Cauchy distribution, although maybe the daily change in stock price does. Daily change is a lag term  so that’s serial correlation.

Serial correlation or autocorrelation refers to things that bunch together. When it rains, it pours.