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The calculus of choice

October 25, 2012

1. If making a choice means changing something over time, then calculus, the science of change over time, ought to be able to talk about choices.

1.1. Let us imagine a person p as a function of time t, such that for each time there is a corresponding location in choice-space, which we imagine as roughly analogous to physical space.

1.2. If we can look at the entirety of p(t) from the vantage point of eternity, and can isolate a specific point, say t=0, at which p makes a choice, what will p have to look like at and around that point in order the word “choice” to be appropriate?

2. There are several intuitive answers, all of which are gravely disappointing, ways that can help us understand what it is we want a choice to look like.

2.1. We can define a piecewise function p1(t)={1 if t<0, -1 if t≥0}, but then p1 lacks continuity at t=0, and the fore and aft halves have nothing in common but their name.

2.1.1. A slightly modified piecewise function p1.1(t)={-t if t<0, t if t≥0} preserves continuity, but lacks differentiability at t=0, so the only thing it has in common before and after is name and location, and there is nothing internal to p1.1 that is preserved; interestingly, this would perhaps not be considered a problem prior to the invention of calculus.

2.2. A function p2(t)=t makes a change from positive to negative at at t=0, but it is difficult to call this a real change, since p2 never changed direction, and so this cannot satisfy us if we think it important that p2 be in some sense a “different” person before and after t=0.

2.2.1. We can define instead p2.1(t)=t^2, and there will be an inflection point at t=0, but then p2.1′(t)=2t, p2.1”(t)=2, and p2.1”'(t)=0 [the ‘ symbol being used to indicate derivatives], meaning the supposed choice was determined solely by the pre-determined parameters of the function, with nothing about p2.1 itself changing at t=0; so this too cannot satisfy us, unless we are satisfied with making choices before the creation of the world and spending our lives simply playing them out like actors with a script.

3. The above examples suggest two qualities a choice must have for the p-person making it to be free: p must have an “unforced” “difference” between its values at t<0 and t>0; and p must be “substantially” “the same” before and after the choice; but it is far from clear what we mean by any of these words.

3.1. By “unforced” “difference” we could mean any number of things:

3.1.1. That the value of p immediately after t=0 follows from the value and state of p at t=0: this is attractive if we want our choices to come from “who we are.

3.1.2. That the value of p immediately after t=0 not be predictable based on the value and state of p for t≤0: mutually exclusive with 3.1.1, this is attractive if we want our choices to be not only unforced, but also unpredictable.

3.1.2.1. That the function not be able to be described given a finite number of words: this variant is attractive if we not only want unpredictability, but also want it to be impossible to describe the outline of our life.

3.1.2.2. That the function not be able to be described given a countable number of words: This variant adds to 3.1.2.1 uncountability, which we may find reassuring.

3.2. By “substantially” “the same” we could mean any number of things:

3.2.1. That p is smooth: this is attractive if we care about physical continuity and being able to give reasons for why we do things.

3.2.1. That p be expressible as a non-piecewise function: this is attractive if we think it important to be able to sum up “who we are” given the choices we have made.

4. Functions exist that fit criteria sets A=(1.1)(2.1) and B=(1.1)(2.2); and C=(1.2)(2.1), D=(1.2.1)(2.1), E=(1.2.2)(2.1); by definition no function can fit F=(1.2.1)(2.2) or G=(1.2.2)(2.2); I do not know if any functions exist fitting H=(1.2)(2.2).

4.1. (A) and (B): The function q1(t)=cos(t): Although we know a priori that cosine waves change directions at t=0, the function is defined not by the value of its derivatives, but by its identity with its fourth derivative, q1””(t)=cos(t)=q1(t), suggesting that the change in a sense comes from “within the moment,”  making q1 less like an actor reading a script (p2.1) and more like a computer executing a piece of code; there are reasonable arguments that this is an acceptable model of freedom (it is essentially the compatibilist view; also I suspect one could make an argument here involving the halting problem).

4.2. (C): There do exist functions that are everywhere smooth but non-analytic (some are even nowhere analytic), and if we define q2(t)={q2a(t) if t<0, q2b(t) if t≥0}, selecting q2a and q2b (as is quite easy to do) such that q2a’n'(0)=q2b’n'(0) for all n [their nth derivatives are equal at zero for all n], then q2 will be smooth but unpredictable in sense (1.2); if we can bring ourselves to tolerate piecewise definitions, this seems to me the best possible account of c-choice, though perhaps either the infinite or the uncountable variant should be considered superior for whatever reason.

4.2.1. (D): Though we cannot explicitly define these functions, since that would directly violate (1.2.1), we do know that such functions exist; I could even tell you how to pick one fitting it, though it would require infinitely many choices before you had defined the function: for each natural number t pick a random value for q2.1(t), then connect adjacent points using smooth transition functions.

4.2.2. (E): Finally, I cannot even begin to tell you how to pick a function, and I suspect this follows directly from the definition: if I could outline such a process, then completing that process would require only a countably infinite list of my arbitrary choices, which violates (1.2.2); still,  I am almost certain that they exist (indeed that an infinite number of such functions exist, indeed, that they comprise the overwhelming majority of all smooth functions).

4.3. (F) and (G): By definition such functions do not exist, since (2b) requires the function to have a formula defining it, while (1c) and (1d) forbid not only such formulas, but also unformalizable functions that can nevertheless be described with absolute precision.

4.4. (H)?: Since q2 was defined piece-wise, it cannot succeed here; I rather doubt any function p3 can succeed here, but I do not know for sure; such a function would, of course, be the best of all possible worlds.

5. What do I intend the above to demonstrate?

5.1. That q2 and its variants perhaps resemble free will more than we might have expected of a mathematical function, suggesting free will and the like may be salvageable in unrelentingly reductionist accounts (by which phrase I mean to indicate less a philosophical principle and more an aesthetic, one of overwhelming flatness).

5.2. That our intuitions about free will as being like movement become very strange when we start talking about a one-dimensional choice-space; which in turn ought to suggest that we don’t live in a choice-space with simply more dimensions, but rather we don’t live in anything like a choice-space at all; so the entire metaphor is, perhaps, worthless as philosophy, but in any case is intended more in the spirit of Flatland (or the equally excellent Sphereland).

5.3. How mathematical and philosophical history bear certain similarities to one another, indeed, how mathematics is philosophy’s Ophelia:

5.3.1. How just as the flaws with p1 and p2 would have been clear to the Greeks, the basic dilemma regarding free will (how to have same yet different) have always been fairly clear.

5.3.2. How just as the issues with p1.1 and p2.1 only become clear with the advent of calculus, the issues with the classical account of free will only become clear with the advent of early modern philosophy.

5.3.3. How just as q1 only becomes possible with late 18th century theoretical advances casting trigonometry in terms of differential equations, the first real move past the early modern failure to account for free will only happened in the late 18th century.

5.3.4. How just as q2 and its variants are of dubious value and only become possible with modern innovations in analysis, the contemporary debate on free will in academic analytic philosophy is bizarre to learn about, technically adequate, and in no way satisfying.

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6 Comments leave one →
  1. Joseph Simmons permalink
    October 25, 2012 1:45 am

    Thanks to Nathaniel Helms for his helpful feedback.

  2. October 25, 2012 7:56 pm

    Actually, (E) is not possible. Any continuous function can be described by countably many choices: for every two rationals p and q, decide whether f(p) < q. This determines the value of f at every rational, and then continuity determines the value of the function at every point.

    I think (H) is impossible, although it depends on exactly how it is formalized. Certainly, if you just consider the functions that you can write using the elementary functions, their inverses, and the standard arithmetic operations, then everything you can write will be analytic, so it will not satisfy (3.1.2). In general I think it will be hard to come up with any way of defining what it means for a function to be "non-piecewise" such that the set of non-piecewise functions is not a subset of the set of analytic functions.

    • October 26, 2012 12:38 pm

      Good call with (E). I don’t know how I overlooked that. Though we can still give criteria stronger than for (D), namely, that it takes infinitely many words not only to describe the function, bit also to describe any non-point interval on the function. So the choices can’t be uncountable but at least they can be dense.

    • October 26, 2012 12:47 pm

      Regarding (H), I agree there are problems with formalizing the “non-piecewise” definition. I think this if because our intuitions are confused. The basic idea is, it should be doing the same thing at all times, not something different at different times; but this doesn’t mean it has to be composed of elementary functions. In fact I suspect that the absolute value function defined geometrically, rather than as a piecewise function, would satisfy us. Maybe even something fractal like the Weierstrass function would. It might be that we’re not demanding anything mathematically formalizable at all, but something vaguely like “elegance”.

      • October 26, 2012 1:12 pm

        Hmm… you can construct a bump function by applying the Banach contraction principle to a certain integral operator which acts on the space of nonnegative continuous functions with compact support and integral 1. Perhaps this could be considered “elegant”…

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