The chapter on 'continuum' in Stoic thought brought up many interesting paradoxes. I should note that the Stoics rejected the atomism of Epicureans - that the physical world consisted of indivisible quanta, the building blocks of everything. To a certain extent the Epicureans were vindicated by modern science. But the Stoic continuum, if only applied conceptually, still raises important questions.
Which contains more parts, a body or a finger? The simple answer is the body, for it comprises ten fingers plus much more. But a finger contains infinitely many parts. Even considering the modern scientific understand, a finger can still be conceptually divided into infinitely many parts. The body can be likewise divided. So then, a body and a finger have the same number of parts. Or at least, they are both comprised of infinitely many parts. Obviously this conflicts with our empirical understanding of both things.
What is a limit? The Stoics held it was incorporeal, a mere construct of the mind. The Epicureans were free to envision it as the boundary between atoms, a plane dividing the atoms of one thing from the atoms of another. The importance of this argument is somewhat different today. Take the smallest, most indivisible thing we can postulate. To my knowledge, this would be the single string in string theory; it is the smallest thing that can exist, and nothing can occur at any length shorter than the string's length, for the string has no parts which may interact. Yet, how can a string border another string? Obviously, a whole cannot border a whole. Conceptually it is obvious that a part of the string must border a part of a second string. But how can this be if both are indivisible?
Take a cone and cut it horizontally. Examine the two new surfaces you have created, the upper and lower surfaces that define your cut. Are they equal in magnitude? For if they are, when does the cone change its breadth? If they are not equal in magnitude, the cone was never continuous, but was only planes of material stacked atop one another. The modern understanding of atoms has effectively nullified this argument, but I haven't thought of this before.
Finally, a corollary to Zeno's famous distance paradox. It is clear that when a runner completes a lap around the track, he cannot have run the distance at once. It is obvious to us that it was broken into divisions - one foot was completed, then another, and so on until the lap was completed. But why only divide to a foot? For any distance, however small, can be divided infinitely. First the first inch of the track must be traversed. But wait, now the first micrometer of the track must be traversed. But how can the runner travel even one micrometer, if he has not completed the first half-micrometer? And so on. All motion, conceptually, is hindered by an infinite regression of ever-smaller first distances. When I was studying aeronautics in college, I once had to write a basic computer program which used differential equations. For velocity to increase from zero, acceleration must be infinite - any change from zero to a nonzero number involves an infinite rate of change over a short enough time scale. We of course used constants and workarounds to make the program work - but how does nature really work? For infinite acceleration cannot occur. But if acceleration were to suddenly increase from zero to non-zero, then the derivative of acceleration would be infinite. This, too, is infinitely regressive.
It is a wonder physics works at all.
Which contains more parts, a body or a finger? The simple answer is the body, for it comprises ten fingers plus much more. But a finger contains infinitely many parts. Even considering the modern scientific understand, a finger can still be conceptually divided into infinitely many parts. The body can be likewise divided. So then, a body and a finger have the same number of parts. Or at least, they are both comprised of infinitely many parts. Obviously this conflicts with our empirical understanding of both things.
What is a limit? The Stoics held it was incorporeal, a mere construct of the mind. The Epicureans were free to envision it as the boundary between atoms, a plane dividing the atoms of one thing from the atoms of another. The importance of this argument is somewhat different today. Take the smallest, most indivisible thing we can postulate. To my knowledge, this would be the single string in string theory; it is the smallest thing that can exist, and nothing can occur at any length shorter than the string's length, for the string has no parts which may interact. Yet, how can a string border another string? Obviously, a whole cannot border a whole. Conceptually it is obvious that a part of the string must border a part of a second string. But how can this be if both are indivisible?
Take a cone and cut it horizontally. Examine the two new surfaces you have created, the upper and lower surfaces that define your cut. Are they equal in magnitude? For if they are, when does the cone change its breadth? If they are not equal in magnitude, the cone was never continuous, but was only planes of material stacked atop one another. The modern understanding of atoms has effectively nullified this argument, but I haven't thought of this before.
Finally, a corollary to Zeno's famous distance paradox. It is clear that when a runner completes a lap around the track, he cannot have run the distance at once. It is obvious to us that it was broken into divisions - one foot was completed, then another, and so on until the lap was completed. But why only divide to a foot? For any distance, however small, can be divided infinitely. First the first inch of the track must be traversed. But wait, now the first micrometer of the track must be traversed. But how can the runner travel even one micrometer, if he has not completed the first half-micrometer? And so on. All motion, conceptually, is hindered by an infinite regression of ever-smaller first distances. When I was studying aeronautics in college, I once had to write a basic computer program which used differential equations. For velocity to increase from zero, acceleration must be infinite - any change from zero to a nonzero number involves an infinite rate of change over a short enough time scale. We of course used constants and workarounds to make the program work - but how does nature really work? For infinite acceleration cannot occur. But if acceleration were to suddenly increase from zero to non-zero, then the derivative of acceleration would be infinite. This, too, is infinitely regressive.
It is a wonder physics works at all.
