I did pretty well in calculus as long as we were dealing with problems that had definite methods to find solutions. When we got to partial differential equations, I was a deer in the headlights. It was humbling. Fortunately, you didn’t suggest anyone explain calculus, just its relevance to the matter at hand, and I think I can contribute there.
Calculus provides mathematical ways of dealing with puzzles like Zeno’s paradoxes. Later Greek mathematicians like Euclid and Archimedes actually realized it was possible in principle to calculate the sums of infinite series of numbers long before Newton and Leibniz formulated infinitesimal calculus (independently of one another, but both building on the work of others). Mathematical series can be either divergent or convergent. The sum of a divergent series (e.g. 1+2+3+4+5+…) will continue to grow without limit as it approaches infinity. The sum of a convergent series, on the other hand (e.g. 1+1/2+1/4+1/8+1/16+…) will approach a finite limit, 2 in the case of the example. Note that just because a series of numbers gets smaller each time, it does not automatically follow that it’s convergent. The harmonic series, 1+1/2+1/3+1/4+…, is an example of one that is not.
All that is to say, despite Zeno’s paradox of Achilles and the tortoise, if you are driving down the road at 60 mph you will travel one mile in one minute even if that length is infinitely divisible. It will take you half a minute to travel a half mile, a quarter minute to travel the next quarter mile, and so on. As the remaining distance approaches zero, so does the time required to travel it. Discussion question: if the signs on the approach to Antelope Freeway were placed to mark the remaining 1 mile, then 1/2 mile, 1/3 mile, 1/4 mile, 1/5 mile, etc., would you ever be able to reach Antelope Freeway, and why or why not? Bonus points if you understand that reference from listening to nice analog vinyl rather than digital.
Obviously, calculus covers a lot more than just the sums of infinite series. It allows you to calculate tangents on the curves of mathematical functions and the area under those curves. Newton invented his system for the specific purpose of working out his theories. It’s indispensable in modern physics and especially useful for understanding motion.
Just because it’s useful, though, doesn’t mean it solves all the philosophical problems of motion. We can invent mathematical systems to do whatever we want regardless of whether they have any relevance to the physical world. I know, for instance, there are various systems for assigning finite numerical values to sums of divergent series. I don’t know why those systems exist, but I assume they must be useful in some esoteric branch of math or another. We have math that can deal with infinitely subdivided space, but we have also discovered real limits to how small a space we can actually observe. To resolve an object of Planck length (about 10^-35 meters), we would have to bombard it with so much energy it would form a black hole, which would immediately hide it from observation. Is this actually the smallest possible unit of length? We don’t know. Smaller things might well exist, but we’re unable to discover anything about them even in theory. Some questions about our universe can’t be answered by anyone who’s a part of it. The Creator of the universe, though, is under no such restrictions.
This is an excellent introduction/review (depending on the reader!), but I still require an explanation of how this sheds light on the “state” vs. “process” theories of change. You almost got there…
Challenge accepted! Let me start by explaining basic calculus in more detail and I’ll see if I can then get closer to what you’re looking for. I can’t make this entirely math free, but I’ll try to keep it accessible to people who haven’t studied calculus or who studied it a long time ago and don’t really remember it very well now. Partly because I’m a nice guy, but mostly because I kind of fall into that latter group myself. If you’re a complete math phobe, don’t worry too much about it. Aristotle didn’t understand calculus either. It’s relevant, but it’s not vital to understanding our text.
I am going to assume everyone reading this is at least familiar with the idea that math formulas can be plotted on graphs. If you draw the graph for y=x^2, for instance, you’ll get a parabola that opens upward and has a vertex on 0,0. Now, every point on this curve has a slope, the direction that the curve is moving at that point. If you draw a line that just touches the curve but doesn’t cross it, that line is the tangent, and it has the same slope as the curve at the point where it touches.
The first thing you learn in calculus is how to find the derivative of a mathematical function. The derivative is a formula that will tell you the slope of the function at every point. For instance, if the function equals x^2, the derivative is 2x. If you plot y=2x on a graph, you’ll get a straight line that crosses the axes at 0,0 and moves up 2 units on the y axis for every unit it moves to the right on the x axis. The y value of that line for any x value is exactly equal to the slope of the parabola from the previous equation.
The second thing you learn in calculus is how to find the integral of a function. The integral will tell you the area between the curve and the x axis for a given section of the curve. For some simple curves, you can accomplish that without calculus, but others are too complicated to use only pure geometry. It’s really easy, though, to find the area of rectangles. If you draw a bunch of rectangles under a curve that reach from the x axis up to the height of the curve, you could add up their areas and find a value that’s close to the value of the area under the curve. It won’t be exact, of course, because you’ll have lots of corners on the rectangles that don’t quite fit, but you can always get a better approximation by making the rectangles thinner. If you could somehow make the rectangles infinitely thin and add them all up, you would be able to get an exact answer. As I mentioned in my precious post, calculus gives you tools that can enable you to essentially do just that, at least in certain cases. Archimedes actually used a method not unlike this to find the area of parabolic segments, except with triangles rather than rectangles. The Greeks came pretty close to inventing calculus. I remember reading an Asimov short story in which he suggested that if they’d had our modern decimal system, they would have succeeded. He might have been right.
The reason Newton and Leibniz are credited with inventing calculus, though, is that they both realized that derivatives and integrals are fundamentally the inverse of each other. The derivative tells you the rate of change of a function, and the integral tells you how much change has occurred. If you have a function that describes the velocity of an object, the derivative will tell you how much the velocity is changing at any given moment (i.e., the acceleration). and the integral will tell you how much it’s position has changed during any given span of time.
In one of his other paradoxes, Zeno argues that because an arrow is not moving in a single instant of time and it also is not moving in any of the following single instants of time, it therefore cannot be moving over any period of time. To put it another way, if the infinitely thin rectangles in our area calculation have no width, then they should have no area, and so, it seems, the curve should not have an area either. But the whole reason the calculus of Newton and Leibniz was originally called the ‘calculus of infinitesimals’ was because it introduces the concept of quantities that are infinitesimally small, and yet not zero. Infinitesimal is not the same as nonexistent in this framework. This, I believe, is essentially what Aristotle is getting at with his comments on continuity and infinite divisibility.
Calculus then, provides a unified method, at least within the realm of mathematics, for dealing with the state a thing is currently in and the processes it is undergoing. We can look at an arrow’s movement over time and say that at any given moment, it has a particular position. We can chart it’s position over time, find the derivative, and then say that at that same given moment it has a particular velocity. We can do that again and find it’s acceleration, and again and find what physicists usually call its jerk. Incidentally, the next three derivatives are informally but popularly known as ‘snap’, ‘crackle’, and ‘pop’. I didn’t make that up, but I’ll confess I would be unreasonably proud of myself if I had,
Tom, thanks very much for this! Well explained!
‘To be capable of health’ and ‘to be capable of illness’ are not the same, for if they were there would be no difference between being ill and being well. Yet the subject both of health and of sickness-whether it is humour or blood-is one and the same.)“
It’s quite possible I’m stretching this way beyond what Aristotle intended, but is this in some ways applying to the problem of evil:
– rejecting evil as an illusion (clear difference between good and evil)
– rejecting dualism (same humor or blood, not two distinct bodies one good and one bad)
Some of this sounds like Augustine (even if from what I remember he was more greatly influenced by Plato)
I think Aristotle’s point was simpler than you might wish. He’s just saying that the same body, with its balance of bodily “humors,” is potentially healthy and potentially ill, but just because it’s the same body, that does not make such potential the same. If it has an application to the problem of evil, it would be unintentional and indirect. But, I mean…maybe you’re onto something.
I think you’re correct. I might be reading too much into it 🙂
– Please keep showing the Greek words, I find them really illuminating.
– Your explanations brought clarity and simplicity. Why does Aristotle write in a difficult to follow style? Is it an ancient Greek thing, a formal logic syntax thing or just an Aristotle thing?
– I also did a similar mental translation of “motion” into “change” as I was reading. Since we think of motion in Newtonian terms, to do with, position, velocity and acceleration. But Aristotle meant it much more broadly. i.e. A child growing into adulthood is also motion. The whole potentiality moving toward actuality concept implies a design and purpose. And hence a Creator God. As opposed to the material viewpoint of randomness and survival.
I’m glad that the Greek notes help somebody! Thanks for the feedback.
The explanation often given is that Aristotle’s works are not written by him but are notes written by students; as such, they are abbreviated and full of unconnected and unexplained details (that would have been connected and explained in the original). Another reason Aristotle is difficult is that he does not bother to explain things that would have been well understood by students of philosophy. Just as today, he was working in a tradition, one that seemed established and old to him (and it was indeed at least a few hundred years old). So to learn Aristotle is to a certain degree to learn to think like a Greek, particularly because he frequently addressed himself to the details of his opponents’ views.
Quite right re motion and change.
Also, Aristotle becomes much more interesting when you think of his argument as aiming to establish an unchangeable changer.
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