01/03/2014

mathemiscellany


(c) Edel Rodriguez (2012)


Those whom heaven helps, call the kin of heaven. They do not learn it by learning. They do not work it by working. They do not come to it with reason. To let understanding stop at what cannot be understood is a high attainment. (Those who cannot do so will be destroyed on the lathe of heaven.)

– Zhuangzi




The thing about doing maths - that is, "higher" maths, the kind of maths that doesn't have or need real-life instances, that in its purest Form doesn't need even a mental instance - is that you're teetering on the edge of senselessness all the time. It is the zero-gravity circus of the mind. Proofs are tightropes, sure, but you can never tell if you're really on one until you get to the assumed other side, and look back, and, in my case, find yourself not smugly bowing to the audience, but crashing to the ground, impacting a surface which is not in fact there. But decelerating painfully all the same.

So people like me, who generate their own gravity, who cannot see where to stand or jump to, depend on others. Acrobats: the magnetic Crouching Tigers who rigged up the ropes and rest platforms that are all the circus is in the first place. (Without whom no circus at all.) Once in a while one of them goes further, shoots the moon, leaps clean over obstacles the likes of me had either not even known were there, or had assumed were innavigable. Leaps like that weave the lot into shapes never suspected, make the wires of the inhuman network sing.


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"The essence of mathematics is its freedom."

– Georg Cantor


(All very well to say if you happen to have wings. Maths is creative, but not in the la-la-la finger-paint expressionist way that the artists' monopoly on the idea would have you expect. One must be humiliated by logical figments for hundreds of bloody hours before one can express oneself, in maths.)


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Is maths really like that, a romp for Great People alone? Uniquely, I think that it is - though it's possible my wounded intellectual pride needs to think so - since if it can deem me congenitally unable to excel, I can't be held responsible for my mathematical mediocrity. Hero worship helps a lot when the only other creatures you meet are the silent and deathless entities called Forms. The fight for understanding is the same everywhere, but it's particularly galling in maths, where geniuses and machines can zip past you at speeds and thus ranges you will never ever reach.


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"Years ago I got off the mathematics train at Quadratic Equations—a neat, airy little station with trellis, ivy, roses, a sunlit platform. There was just a hint of weirdness now and then—stationmaster made clicking noises in his throat, there was an occasional far-off harmonious humming in the sky, strange bells rang; one knew the frontier was not far away, where the line crosses into the vast country of Incomprehensibility, the jagged peaks of the Calculus Mountains standing up, a day's journey over its illimitable plains.

The train thundered off into those no doubt exhilarating spaces, but without me. I sniffed the mountainy air a little, then I crossed the line by the footbridge and went back in a fusty suburban train to my home town. Contemptible Ignorance. This train had no engine; it was simply a train of carriages rolling gently down through the warm orchards of Amnesia Hill
."

– Paul Jennings



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The mathematical fledgling is robbed three times over:

She's robbed of the cheap rebellion and self-definition allowed by other academic fields - she doesn't get to act out, because basically the whole edifice of maths is there for good and stated reasons; and since the past masters can't be avoided, because maths is essentially accumulative: a tower stacked higher by generational strata. Nor even is jumping in at the 13th floor possible (again, for those of us without wings). So she's surrounded by authority of a qualitatively different sort from the soft animal social-psychological influence in English or Sociology (or Climatology): who can face down Proof? The tower wants new storeys, maybe bridges between neighbouring totems, but almost never creative destruction or greenfield construction like others delight in.

Relatedly, she's robbed of the possibility of sustainable bullshit. (Improvised or free-play maths does not end well. Nor begin.)

Worst of all - after being made to work on the same old building site, made to make sense constantly, horribly, she is robbed of schadenfreude: all one mostly sees of others is the perfected work of bloody meritocrats - who are anyway the most intolerable sort of predecessor.


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In real life, all '=' are ≈.

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An exciting thing about stats: its greatest and most constructive critics are also far and away the most readable, artful people in the field. Admittedly this can only be true because statisticians are the dryest culture on earth, with no lightness, allusion, irony, aporia, or humour (in their publications). So it's a duty and a pleasure to start off with the following - and how many pleasurable duties are there in this world?


  • Deirdre McCloskey & Stephen Ziliak. They have for two or three decades pointed out the methodological dishonesty involved in the standard statistical practice of a great many fields: 'null hypothesis testing' uses a constrained and ridiculous idea of 'significance', and genuinely distorts research. Anyway everything McCloskey writes is worth reading - her point is that there is no free lunch.

  • Taleb. Makes many cool points about misapplication and misassumption, but the big lesson is: "Bell-curve statistics are sometimes lethal and immoral, most obviously in 'high' finance." Some of his recent extensions over-reach, are maybe guilty of the centralisation and big-picture generalisation he so deftly identifies in others held to be scientific.

  • William Kruskal. Fairly conventional, super-clever, dry stat bloke, until that is his 1982 American Stat Association address, a rollicking philosophical indictment of basically all applied linear modelling via Hume and Jesus Christ. (Yes, ok I do get excited when non-philosophers talk philosophy, and yes ok it's hardly hard to - but they get things done when they do it!)

  • Richard Berk & David Freedman. Again: standard career in stats, though being less boring, then BOOM an eruption of methodological whupass. Lesson: trust no social scientist until they have explained just how it was that their convenience sample of convicts were magically picked up by the criminal justice system totally independently of each other.

  • Paul Meehl. His "Tabular Asterisks" is philosophically deep, psychologically dismissive, and animatingly angry.


You could argue that it's only non-statisticians who make such crass Procrustean errors with stats - that the project of Robust statistics is well underway and universally welcomed - that these people are not mavericks but tenured and tolerated court-jesters - and that since only statisticians read about the methodology of stats the above papers are doomed to irrelevance. But the errors live on all over the place, making a mockery of science - and one big reason they do is that reading the average stats text is a revolting experience. Everyone should read Taleb and McCloskey - and, delightfully, most people can, because they're genuine stylists.

(Special mention for Andy Fields, who writes the funniest introductory stats. A man unafraid to display a proud dripping-blood gif in 2014.)


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The scalpel brackets out the person to get at the body: there's no other way. The statistical model brackets out the person to get at your ideal analytic Unit: there might be.


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Why are statistical methods so central to science? And given that's so, why is the good kind of Bayesianism still not used? Is the answer to each of these that statistics is there to offer empirical enquiries a veneer of utter objectivity - and because subjective Bayesianism freely admits it is just a veneer?

"The laudable view that science should embrace subjective statistics [in the Bayesian sense] falls on deaf ears: scientists come to statistics because they wish it to provide objective validation of their science. I mention the superficiality of this sociological rationale because it does not reflect the depth of what subjective Bayesian statistics actually achieves - a readily understandable communication of the observed data's information for any scientific question that is posed."

James Berger



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Ineliminable patches of subjectivity (and so bias, and so perhaps prejudice) in applied statistics:
  1. Choice of study subject
  2. Model choice.
  3. Choosing the final p-value.
  4. Defining a 'population' in the first place.
  5. Isolating variables from the phenomenon? - nah, we have Factor analysis now.
  6. Choice of test stat? (Since all have countervailing assumptions) - nah, we have Robust tests now.
  7. Status quo bias?

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The infinite! No other question has ever moved so profoundly the spirit of man."

– David Hilbert


Hippies, monotheists, and mathematicians love trying to face infinity. In so doing they always come face to face with their own impotence instead, but it doesn't stop them. Half of them are seeking out this metaphysical failure - as if being bad at seeing God was evidence of Him, as if (what they hope is) apophasis were proof. "Yay incapacity!" say Dante, Leopardi, Bruno, and millions upon millions of pseuds. "More things on heaven and earth" say middlebrows all over this happy world. (This passivity and intellectual sleep makes sense: Infinity itself is macho - come-and-have-a-go-if-you-think-yer-large-enough. Yet gentle Cantor made it bleed.)



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A pep talk for myself:

Humans summarise reality. We can't avoid it: it's wired-in. But even if it wasn't, reality's too big, our attention too short & too narrow, and our need for action too basic for us to do anything much else.* (Imagine acting without summary: you can't imagine it, but if you could you'd see death, a hastily hastened death.)

When we summarise badly we lose touch with the world. And this we all surely do: we call this 'error', 'delusion', 'stereotype'. (Those things people on the Other Team have.) When we fail to summarise at all, as some unfortunates with unusual neurology do, the world becomes a terrible chaotic flow, beyond category, absurd. (You will have had an inkling of the perspective on a comedown from certain compounds, or from your really really close reading of French philosophy.) And yet we have hold of something, most of the time.

The formal method of summarising well is called statistics, & despite the exciting preamble I just gave, it is commonly maligned, & very commonly abused, & hard, & dry as a pint of sand. When - when - you falter and despise it & cannot even derive joy from the irreplicable insight it gives you into modern physics** - try to remember that without it you are totally doomed to be a bigot, a dupe and a throwback.





* Monks and nuns of various sorts have a good old go at doing something else. But it amounts to trying to get used to the world-as-terrible-chaos - and that's not on in a world which can still be improved by action.

** The world as it weirdly is, all down there and all up There.







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