Notes on what I'm reading

Sunday, March 24, 2013

HoPwaG: Zeno and Melissus

Recall from last week: the Eleatic Zeno to vindicate Parmenides' metaphysical doctrines about impossibility of change. Method of paradoxes. Para (against) doxa (belief), i.e. contrary to what we believe. Refuting options: can show something wrong with the reasoning in the paradox, or give up what we believe.

Zeno's dichotomy paradoxes: aim to have listener give up commonsensical beliefs about motion.

Another way Zeno argues: if things many, then finite; and if things many, then infinite; but things not both finite and infinite; therefore, things not many, i.e. being is one, a la Parmenides.

The dichotomy (i.e. motion) paradoxes: question of what the problem is:
  • can't do infinite number of things; or 
  • would take infinite amount of time to get from A to B.
Refuting Zeno: introduce idea of limit, so that the geometric series .5, .25, .125 adds to 1. So the distance sums to 1. Problem: Zeno knows that; in fact the paradox relies on it! He's asking a question about time, space, and motion, not just the mathematical model of the situation. That is, can still ask about the relation of the mathematical model to reality.

Aristotle's response: divisions apply both to time and distance, so time reduces as distance does; so wrong to think that it takes an infinite amount of time; so the paradox doesn't show that motion is impossible. Problem: assumes second interpretation of paradox. On first interpretation, it's the infinite number of partial motions that make up the full motion that's the problem, not the time taken to do them.

The arrow paradox: at any moment during flight from archer to target, arrow at rest, since time must pass for any motion to take place; but true at any instant; so at all instants the arrow is motionless; but if at all instants it's motionless, it can't ever move.

Aristotle's response: time not made up of durationless instants, but divisions of time are divisions into (maybe very short) periods, which denies one of Zeno's premises:

Another of Zeno's paradoxes: between any two distrinct things, must be an infinity of other things. Given two things A and B, they must be separated from each other, because A and B not continuous (since distinct). So there must be a third thing C between A and B, distinct from both. But if C distinct from A, there must be a fourth thing D separating it from A, and a fifth thing E separating C from B, and so on. So there is an infinity of things separating A and B, if A and B are distinct.

Aristotle's response: tries to account for how two things can be contiguous without being continuous. Note that the responses need subtle distinctions to get around the Eleatic problems; Aristotle's positive ideas come as a response to the paradoxes.


Now, Melissus, another Eleatic. Follows Parmenides' method rather than Zeno's, i.e. positive argument rather than undermining paradoxes:
  • applies Parmenides' reasoning about the impossibility of being beginning to space; so get that being is unlimited; for, if anything were beyond being, it would be non-being, but that's ruled out by PPMI, so being unlimited (vs. Parmenides' conclusion that being spherical). Note that this is rather like Anaximenes' conclusion about what there fundamentally is.
  • argument against motion: there cannot be emptiness, i.e. a place with nothing in it, i.e. a void, by PPMI; but if no void, then no place for anything to move to; so motion impossible (HoPwaG analogy: think of being as a packed train compartment).
Response options to argument against motion:
  • agree that no void, but say motion nevertheless possible (e.g. by continual displacement of everything)--this Aristotle's response;
  • there is void, so motion possible--this something like current scientific beliefs, but first developed by Democritus and Atomists.

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