Both/And – The Prison of a Single Idea

(Another half-thought-out idea that I’m trying to give form to. You’ve been warned. Also will be discussing math while using as little math as possible – just a reckless thrill-seeker, me.)

One of the things it becomes important to know when building models of the physical world turns out to be the slope at a point on a curve. It represents the rate of change at that point for the function that describes the line. (Newton would have made it into the Math & Science Hall of Fame on the strength of having figured out how to determine the slope at a point on a curve if that’s all he did; he and Leibniz share a wing for this.)

How you do it requires a wee leap, because the very logic that compels you to agree that the first derivative is the slope is based on an equation that inconveniently negates itself right when you need it most. Let’s see how that works.

Here is that equation for a secant, a line that connects two points on a curve:

The slope of the secant is the average rate of change over the curve between the two end points of the secant.

Newton and Leibniz noticed that, as you bring the two end-points closer together, you get a slope that’s closer and closer to the slope *at* either of the points – so, if you wanted to know the slope at a point, you could get real close by making the end-points as close as you want.

But you just can’t make the end-points the same point – if there’s no distance between them, then the formula becomes y – y over  x – x, which is just  0/0 – which is not a slope, and in fact is scary enough that mathematicians (at least, the ones I understand)  will say that it is ‘undefined’.

So you just can’t find the slope of the tangent by solving the secant equation for a single point. The math falls apart.

Just so, you can’t solve for free will using God’s omnipotence and omniscience as your formula – that resolves to 0/0, too – undefined.

But saying that God’s omniscience and omnipotence preclude free will for any creatures is a bit like saying that, because we can’t use the slope for a secant formula to solve for the slope at a point, the tangent doesn’t exist. The question should be: does the tangent exist? If it does, then the failure of our formula only means that we can’t use it in this case – we must use some other approach. Formulas do not prove or disprove existence.

Bishop Berkeley (if I’m recalling correctly – it’s been almost 40 years) did argue at the time that the calculus was nonsensical as math, and had no claim on an honest man’s acceptance, because, right where you expect a proof, it starts getting all philosophical and talking about limits and infinitesimals and other mythical creatures. And he’s right, up to a point – the way math had progressed so far seems free of such fantastic doodads. Why we should trust something whose proof requires us to imagine never before imagined things* is at least debateable.

Yet, we know the tangent exists and has a slope – we just can’t get at it directly using that formula.

angle with a tangentSo, how about we show the tangent on a point on a curve exists and has a slope we can know in a special case, then see, once we’re convinced, if we can get to the calculus from there? Here is a tangent to a point on circle.  A circle is just a special curve with an interesting property: all points on a circle lie exactly the same distance from the center. That distance is called a radius.

Here, we show a tangent to a circle, and indicate that it is at a right angle to the radius. Now, a nice reductio proof can be whipped up in a minute or two to show that the tangent must only touch one point, and must be at a right angles to the radius from the tangent point. We’re going to skip that. Instead, I’ll just mention that, for circles anyway, we can use a formula and our knowledge of angles to figure out what the slope is for any tangent to the circle we chose.

The important point: we know that tangents to the special case curve called a circle exist, and we know that we can figure out their slopes with a tidy formula. So, it’s reasonable from what we know about curves and tangents to believe that there might be a tangent with a knowable slope to points on curves that aren’t circles even when we may not have a tidy, non-calculus formula to find them with.

It turns out that the slope of the secant formula used above works fine on circles, and that we can figure out the slope at the tangent independently – 90 degrees to the radius. And, best of all, if we use infinitesimals and limits, we get the same answer we get from geometry.  So, now we have something – a reason to trust Newton and Leibniz when they say the calculus works.

So, do we see reasons to believe free will exists? Yes. It’s certainly the constant subjective experience of mankind. But is free will objectively true? We’ll leave it here for now, with the notion that just because one argument can be presented in such a way as to contradict what we otherwise think is true doesn’t mean it’s the end of the story/

* Well, OK, there’s Zeno. But not imagined in exactly this way. I think.

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Author: Joseph Moore

Enough with the smarty-pants Dante quote. Just some opinionated blogger dude.

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