Death is a discrete phenomenon. It is a creature of units. Vegetation relies on sunlight for life. A plant can die, however, only because it is a unit of vegetation. When the amount of light falls from 3.481 to 2.377 on some scale, there is death only because each plant requires a minimum of 0.05 of light, or 0.3, or 0.4.
So, why? Why this discreteness?
The complex feelings of lawyers and humanist scholars with respect to quantitative subjects, and particularly the quantifization of the social sciences, ought to give them greater empathy for the illiterate and uneducated. The humanist scholar is to the scientist as the illiterate are to the literate.
The illiterate view books with distrust, for books are used to undermine their most heartfelt positions in ways against which they are unable to mount a defense. But this is precisely how the lawyer feels when her nuanced doctrinal argument is demolished by a mathematical model of the economy that shows that regardless of the substance of the legal rule, the same economic outcome will obtain.
“It’s just mathematical mumbo jumbo,” says the lawyer. “These economists don’t know how things work in the real world.” But what the lawyer cannot do is to beat the economist at her own game. She can’t show that the economic model cannot withstand close scrutiny; all she can do is try to delegitimize the entire method. But the illiterate levy the same charge on the literate: “it’s just book learning,” they say. They cannot defend themselves in writing; but they can try to delegitimize writing itself.
It is particularly bitter for the humanists that they have been socialized to occupy the power position. For millennia, since the invention of writing, they have been the ones who use their learning to lord it over others. But now these merely-literates, these innumerates, must know what it means to be crushed by ideas. A very bitter position indeed.
I do not mean to say that the mathematicians have any better claim on the truth. But if the humanists think the mathematicians don’t, then it should perhaps worry the humanists to think that maybe they don’t either, in relation to the illiterate. Or maybe we are marching forward, after all, from one stage of intellectual progress to the next!
When optimization arrives, either others will optimize against you or you will optimize against others. Business against you or you against business. There will be either corporate planning or central planning.
The Law of Large Numbers is completely meaningless to me when it is phrased as “the probability that the sum of results, divided by the number of independent samples, equals expected value gets very high as the number of independent samples goes to infinity.” mu=(sum of ys)/n, in which mu is the expected value, the ys are the results of independent samples, and n is the number of samples. Why should anything converge to the expected value?
But it is very meaningful to me when it is rephrased as “the probability that the expected value times the number of independent samples equals the sum of the results gets very high as the number of independent samples goes to infinity.” mu*n=sum of ys.
Yes, as the number of samples gets high, you know better and better exactly what your aggregate results will be.
As you multiply your expected value by larger and larger sample sizes, expected value goes from being totally fictitious and unhelpful to completely real. If I get $100 with a 50% chance and zero otherwise, it is meaningless to tell me that my expected value is $50. I will never have $50. But if you tell me that I will face this chance 1000 times, then I can tell you with great confidence that I will have $50 times 1000 equals $50,000.
The extraordinary thing about algebra is that it provides accurate solutions in advance of intuitive understanding, rather than after it. Normally this only happens with the observation of empirical phenomena. You see that a thing happens and then you try to explain it. But with algebra, too, sometimes you see that a result pops out of your equations, and then you try to explain it. But algebra is pure thought! Herein the facticity of thought.
In what way are mathematical models that validate our intuition different from specious etymologies? History is sexist because it is his story. Dogs are backward gods. Etc.
For the humanist, the mathematical rubber hits the road of thought when she understands that almost all of her supposedly qualitative thought, particularly as it relates to the economy, involves ranking or statements of magnitude. “Invading Iraq set the country back to the stone age,” for example, is the statement that Iraq now has fewer cars, or dollar bills, or hospitals, or whatever, than it had before. Or consider “a dictatorship is better than an occupation.” In the first case you are counting, even if you don’t realize it. And in the second you are ranking, and any ranking can be represented as a counting of units of preference. So implicitly you are doing math.
The rub is that if you don’t actually write down functions and equations to describe your implicitly mathematical arguments, you end up doing very rudimentary and crude math. You are stuck with general mores versus lesses. Once you add the tools of math to your statements, you can multiply and divide your mores, integrate and differentiate your lesses, optimize them all, and so on. You explore your theories with much greater precision and insight than if you stick to just > and <. Moreover, you can compare your statements and harmonize them in ways that you cannot do when you lack the compact mathematical notation that allows you to include many complex ideas on a single line of text. And perhaps most beautifully and powerfully, at least for me, you can go out into the world, and get actual numbers that can be input into your mores and lesses, so that you can say, with extraordinary magic, precisely how much more and how much less, precisely how badly Iraq was injured.
The frightening thing for the humanist is that math, far from being an obfuscation and a superficiality, is something one has been doing crudely all along. It is a humbling that a true humanist should welcome to discover that her humanism is just an ignorance, or perhaps an illiteracy. Indeed, an innumeracy.