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Wuli mentioned this in passing...

Can someone tell me what is this, and how does it argue against the idea of infinite divisibility?

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Les Sleeth
Gold Member
Originally posted by FZ+
Wuli mentioned this in passing...

Can someone tell me what is this, and how does it argue against the idea of infinite divisibility?
Quoting from a philosophy source book:

"Paradox due to vagueness: With a single grain of sand, you cannot make a heap. If you cannot make a heap with the grains you have, you cannot make one with just one more. So even with 10 million grains you cannot make a heap.

Despite its antiquity, 'heap' may be badly chosen; arguably, you can make a heap of sand with just four or more grains (enough to make a stable heaping-up without adhesive). But the paradox can be recast, e.g.: 1 is a small number, and any number bigger by 1 than a small number is small; so all numbers are small. Responses include: denying the major premise; that is, affirming that there is a sharp cut-off (even if we don't know where); and (alternatively) avoiding the conclusion by revamping classical logic and semantics."

As far as how it refutes infinite divisibility, I don't see it.

I did more than mention this in passing, I went into some detail on the issue already in the thread on Zeno.

Why can you have half a glass of milk, but not a half pregnant woman? When do you decide if a man is bald or hairy? These terms are vague in some respects, but still obviously enormously useful.

If you believe everything is infinitely divisible and words can be refined infinitely by splitting yet again finer hairs ad infinitum, then motion and change are possible. Zeno, however, pointed out that people use a mixture of finite and infinite concepts. At some point in splitting hairs, we no longer have a hair but a carbon atom.

Using the finite concept of an "instant" of time which cannot be further subdivided in conjunction with infinitely divisible space, Zeno showed these concepts led to contradictions. He wasn't so much attempting to "prove" his position, as to demonstrate that his belief in an unchanging universe was no more absurd than its more common alternative.

Such arguments over the absurd were a common way of arguing in ancient Greece and a common form of entertainment and subtle political and religious criticism as well. Some of these of arguments over absurdities formed the basis of the sciences in ancient greece. Likewise, they were quite common in Asia as well and the roots of such an approach can be traced back to shamanistic practices which often strike modern people as bizarre.

Les Sleeth
Gold Member
Originally posted by wuliheron
If you believe everything is infinitely divisible and words can be refined infinitely by splitting yet again finer hairs ad infinitum, then motion and change are possible. . . . Using the finite concept of an "instant" of time which cannot be further subdivided in conjunction with infinitely divisible space, Zeno showed these concepts led to contradictions.
A good point Wuli, and a well-reasoned post.

I guess you are saying that the heap paradox doesn't have to make sense, but rather was just Zeno messing with a fellow citizen (Parmenides). I did find another interpretation of the sorites argument, which is to ask what is the smallest "heap" of sand needed to make an audible noise when dropped. That would seem more related to infinite indivisibilty than that of stacking sand grains.

Edit:

I wanted to add that I like the infinitely divisible riddles for a reason you might not agree with, and that is I believe behind all the divisible structure is an underlying foundation of unity which by its very nature cannot be made "two" or reduced any further. So those intent on reducing to the truth will always be at least one step away from the ultimate nature of reality using that approach to knowing.

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Originally posted by LW Sleeth
A good point Wuli, and a well-reasoned post.

I guess you are saying that the heap paradox doesn't have to make sense, but rather was just Zeno messing with a fellow citizen (Parmenides). I did find another interpretation of the sorites argument, which is to ask what is the smallest "heap" of sand needed to make an audible noise when dropped. That would seem more related to infinite indivisibilty than that of stacking sand grains.
Parmenides was Zeno's teacher and shared his beliefs, mostly Zeno loved making fun of the Atomists.

The relation of the heap to the infinite is the vagueness of both terms. Not only are they vague, but refining their meaning in any natural, obvious way has proven impossible thus far. Of course it is possible to create strained specific definitions, but these are either not natural (i.e. colloquial) or of limited use. For example, a heap need not be of sand, but could be of feathers which would make your definition useless.

As for underlying unity in nature, that's just one of many ways to look at the issue. Whether it is true or not doesn't concern me. What matters is how to constructively build upon the already existing work on the subject in a qualitatively different way. Because linguists and logicians have worked it out to an issue of vagueness, I would put it more in terms of the paradox of existence of course. :0)

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Ivan Seeking
Staff Emeritus
Gold Member

Originally posted by LW Sleeth
But the paradox can be recast, e.g.: 1 is a small number, and any number bigger by 1 than a small number is small; so all numbers are small. Responses include: denying the major premise; that is, affirming that there is a sharp cut-off (even if we don't know where); and (alternatively) avoiding the conclusion by revamping classical logic and semantics."
This strikes me as a proof of infinity. "For any value their exists another much larger value." Perhaps the language problem really just reflects our unsual degree of freedom afforded by an infinite set. This seems a stronger proof of infinity than a proof for finite divisibility.

Originally posted by Ivan Seeking
This strikes me as a proof of infinity. "For any value their exists another much larger value." Perhaps the language problem really just reflects our unsual degree of freedom afforded by an infinite set. This seems a stronger proof of infinity than a proof for finite divisibility.
It could also imply profound unity as Zeno intended to demonstrate. What it most definitely seems to imply is paradox.

Godel's Incompleteness theorem expresses the situation about as rigorously and completely as anything conceived of to date. Before Godel, Russel threw a slew of paradoxes at the foundations of mathematics to see what would emerge, and what came out was just more gibberish, more paradoxes. Godel then showed that what this implied was that at least some of the foundations of math and logic we must take on faith, that to prove the validity of a system we must look for evidence outside of that system.

In the case of formal logic, it's basic axioms are based on natural language which, in turn, are based on human perception. Since we don't know any multi-dimensional alien's with whom we can consult about the validity of our perceptions of the world around us, we must ultimately take our assumptions on faith. The root assumptions that we must take on faith are that some things make sense while others don't.

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Originally posted by wuliheron
It could also imply profound unity as Zeno intended to demonstrate. What it most definitely seems to imply is paradox.

Godel's Incompleteness theorem expresses the situation about as rigorously and completely as anything conceived of to date. Before Godel, Russel threw a slew of paradoxes at the foundations of mathematics to see what would emerge, and what came out was just more gibberish, more paradoxes. Godel then showed that what this implied was that at least some of the foundations of math and logic we must take on faith, that to prove the validity of a system we must look for evidence outside of that system.

In the case of formal logic, it's basic axioms are based on natural language which, in turn, are based on human perception. Since we don't know any multi-dimensional alien's with whom we can consult about the validity of our perceptions of the world around us, we must ultimately take our assumptions on faith. The root assumptions that we must take on faith are that some things make sense while others don't.
Like for instance the assumption that a material reality exists outside of our thoughts, makes sense (but which can not be proven), and the negation of that assumption, does not make sense (which either leads to solipsism -- only I exists -- or the assumption of God).

Originally posted by heusdens
Like for instance the assumption that a material reality exists outside of our thoughts, makes sense (but which can not be proven), and the negation of that assumption, does not make sense (which either leads to solipsism -- only I exists -- or the assumption of God).
Each viewpoint is a positive assertion or negation depending upon how you choose to look at it, and each makes a certain kind of sense and nonsense as well. If I have a road map, I don't try to psychoanalyze myself with it. If I have a diary, I don't try to navigate the big city with it. Nor does the existence of one kind of abstract representation rule out the validity and usefulness of the other.

The finite and infinite define each other which implies the heap paradox may be an unavoidable aspect of language and perception. Rather than denying the validity of either one without proof and against all reason, I prefer to just make use of them.