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One of the more consequential kinds of extrapolation happens in the law.
In the case of Islamic law شريعة the Hadith and the Qu’ran contain some examples of what’s right and wrong, but obviously don’t cover every case.
This leaves it up to jurist philosophers to figure out what’s G-d’s underlying message, from a sparse sample of data. If this sounds to you like Nyquist-Shannon sampling, you and I are on the same wavelength! (ha, ha)
Of course the geometry of all moral quandaries is much more interesting than a regular lattice like the idealised sampling theorem.
Evenly-spaced samples mapping from a straight line to scalars could be figured out by these two famous geniuses, but the effort of interpreting the law has taken armies of (good to) great minds over centuries.
The example from this episode of In Our Time is the prohibition on grape wine:
- What about date wine?
- What about other grape products?
- What about other alcoholic beverages?
- What about coffee?
- What about intoxicants that are not in liquid form?
The jurists face the big
N problem—many features to explain, less data than desirable to draw on.
Clearly the reason why cases A, B, or D are argued to connect to the known parameter from the Hadith matters quite a lot. Just like in common-law legal figuring, and just like the basis matters in functional data analysis. (Fits nicely how the “basis for your reasoning” and “basis of a function space” coincide in the same word!) .
Think about just two famous functional bases:
- Polynomials (think Taylor series),
- Sinusoidals (think Fourier series).
Even polynomials look like a
∩ ͡ ; odd polynomials look at wide range like a
/ (you know how
x³ looks: a small kink in the centre ՜𝀱 but in broad distances like /), and sinusoidal functions look like
So imagine I have observations for a few nearby points—say three near the origin. Maybe I could fit a /, or a 𝀱, a 〰, or a ‿.
All three might fit locally—so we could agree that
- if grape wine is prohibited
- and date wine is prohibited
- and half-grape-half-date-wine is prohibited,
- then it follows that so should be two-thirds-grape-one-third-date-wine prohibited—
- but, we mightn’t agree whether rice wine, or beer, or qat, or all grape products, or fermented grape products that aren’t intoxicating, or grape trees, or trees that look like grape trees, and so on.
The basis-function story also matches how a seemingly unrelated datum (or argument) far away in the connected space could impinge on something close to your own concerns.
If I newly interpret some far-away datum and thereby prove that the basis functions are not 〰 but 𝀨𝀱/, then that changes the basis function (changes the method of extrapolation) near where you are as well. Just so a change in hermeneutic reasoning or justification strategy could sweep through changes throughout the connected space of legal or moral quandaries.
This has to be one of the oldest uses of logic and consistency—a bunch of people trying to puzzle out what a sacred text means, how its lessons should be applied to new questions, and applying lots of brainpower to “small data”. Of course disputes need to have rules of order and points of view need to be internally consistent, if the situation is a lot of fallible people trying to consensually interpret infallible source data. Yet hermeneutics predates Frege by millennia—so maybe Russell was wrong to say we presently owe our logical debt to him.
In the law I could replace the mathematician’s “Let” or “Suppose” or “Consider”, with various legalistic reasons for taking the law at face value. Either it is Scripture and therefore infallible, or it has been agreed by some other process such as parliamentary, and isn’t to be questioned during this phase of the discussion. To me this sounds exactly like the hypothetico-deductive method that’s usually attributed to scientific logic. According to Einstein, the hypothetico-deductive method was Euclid’s “killer app” that opened the door to eventual mathematical and technological progress. If jurisprudence shares this feature and the two are analogous like I am suggesting, that’s another blow against the popular science/religion divide, wherein the former earns all of the logic, technology, and progress, and the latter gets superstition and Dark Ages.