On Indivisibility

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.

On Knowledge

The Stoic theory of knowledge builds directly on their epistemology, but I find it better conceived and more complete. As Zeno pantomimed many thousands of years ago, the states of knowledge can be traced with two hands. An open palm is an impression. A loosely held fist is assent. A tight fist is cognition, or katalepsis. Finally, a second hand wrapped around a tight fist is scientific knowledge – episteme.

There is also opinion, doxa. Reading that, by the way, cleared up a longstanding question of mine about what ‘doxography’ was. Anyway, opinion according to Sextus Empiricus was weak and false assent. Plutarch, however, held that opinion was assent to the incognitive. It seems the latter is the prevailing Stoic view. The Stoic sage would never opine on anything, for all opinion was false and blameworthy. According to Arcesilaus, then, the Stoic sage must necessarily always suspend judgment, lest he opine. I suppose that’s too lofty of a goal, but not a bad one. A person is right to suspend judgment until he can be sure of his impressions – since one can never be sure, judgment should necessarily always be suspended.

The inferior man, which includes everybody due to the impossibility of becoming a true sage, is always ignorant. Even his assent to true cognitive impressions is ignorance. Since the Stoics typically framed their beliefs in dichotomies, there is no middle ground between the sage’s excellence and the inferior man’s ignorance. Assent to true cognitive impressions comprises scientific knowledge for the sage because he, to use Long and Sedley’s words, has freed himself from “all doubt, uncertainty, falsehood, and instability from his cognitive state”.

Of course, to prevent realizing the futility of man’s quest, one must partially reject the dichotomy of knowledge and strive for a more complete grasp of episteme. Otherwise, why even try at all? Hence, a modern day Stoic handbook should perhaps allow deviations from the scripture, identifying a series of ‘intermediate’ states of knowledge, much like the ‘preferred’ and ‘non-preferred’ neutral objectives in Stoic ethics.

On Dialectic

I was somewhat surprised to learn of the emphasis Stoics placed on dialectic and rhetoric. I had thought even dialectic was relatively unimportant to them, to say nothing of rhetoric. Yet, Diogenes Laertius claims early Stoics valued them both, and texts by Zeno, Chrysippus, and Epictetus partially support this notion. I was quite pleased then, since I recognize the value of dialectic, to see the Stoics attached several virtues to the science of logic and discourse. Non-precipitancy, uncarelessness, irrefutability, and non-randomness.

Zeno of Citium valued both divisions of logic. Dialectic was 'the closed fist' - clarity and brevity. Rhetoric was 'the outstretched hand' - breadth of argument and ability. Chrysippus, naturally, valued them both as well. It seems later Stoics eased off on their support for rhetoric. Certainly it is a useful skill to have, but not essential. Dialectic, however, as far as it comprises definitions and logical analysis is of great necessity to even an amateur philosopher.

I'll end with a strange view on arguments by Zeno. When told not to pass judgement until both sides had spoken, Zeno replied, "The second speaker must not be heard, whether the first speaker proved his case (for the inquiry is then finished) or did not prove it (for that is just like his not having complied when summoned, or his having complied by talking nonsense). But either he proved his case or he did not prove it. Therefore the second speaker must not be heard." I can honestly say I have never heard this take on discourse before, but perhaps there is something to it.