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Suppose I take any finite length and subtract half of it continuously. So the size of the remainder, after each subtraction, is equal to its original length multiplied by one half taken to the nth power, where n is the number of subtractions. (e.g. 1/2, 1/4, 1/8, 1/16, 1/32,...).

Now with each subtraction of a half, the size of the remainder decreases, and as the original length is finite, it should eventually diminish to nothing. That would agree with the concept of a mathematical limit, because the limit of this sequence or exponential function, as the number of subtractions tends to infinity (i.e. n), is zero.

Here is the difficulty I'm having, each subtraction always leaves exactly one half of the previous remainder, thus it cannot actually disappear altogether or the finial subtraction would be complete removal and not subtraction of a half.

This is, loosely, Zeno of Elea's argument against motion.

In this case, I'm asking when does the remainder of something, after many subtractions from it, still exist?

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    What does "sound" mean in this context? (Also, what do you believe is specifically philosophical about this question?) – virmaior Jul 5 '14 at 13:52
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    " it should eventually diminish to nothing" -- false. "one over infinity is zero" -- false. Back to math class for you. – user4894 Jul 5 '14 at 17:03
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    @MichaelLee I thought we were talking about mathematics, not gasoline. If you keep dividing a quantity of gasoline in half, you'll eventually have to split a molecule into hydrogen and carbon and you'll no longer have gasoline. This has nothing to do with the mathematical theory of limits. If I take the sequence of real numbers 1/2, 1/4, 1/8, ... then for any n, the n-th term is greater than zero. That's an essential fact to understand about how limits work. There is never any moment or point where the sequence is zero. – user4894 Jul 6 '14 at 0:21
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    @MichaelLee Good grief. This morning someone negged my answer to another question because they were ignorant of basic classical logic. This place is depressing. – user4894 Jul 6 '14 at 3:22
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    @user4894 As I'm not a psychiatrist, I cannot help you with your depression. I suppose you are right, since infinity is not a number, you can not do algebraic operations on it, but I'd find your answers more convincing if you would develop a bit more tact than saying "back to math class for you." – Michael Lee Jul 6 '14 at 10:54
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historical context

This essentially, in philosophical terms, broadly alludes to the paradoxes of Zeno which originally aimed at the defending the notion of Parmenidian Monism; it lead on to the development of Atomism.

In mathematics it was harnessed in defence of the Calculus after the criticisms of Berkeley; the idea of a limit was developed by Cauchy - though the key ideas was already in Leibniz.

To place it in the modern context, the question admits of two answers; the geometrical and analytical with an interlude.

analytical

The analytical answer is the notion of a limit. This is the construction of new ideal points. For example the sequence 1,1/2,1/4,1/8... are all rational fractions that are indexed by the natural numbers; for any finite number n we have the number 1/2^n; hence we always find that it is non-zero; the limit of this sequence is not found in any of these fractions; it lies outside; that is it is the number that is indexed not by any finite n;

To be clear here, these numbers that we are using to index the fractions are not numbers in the usual sense since we do not care that we can add, subtract or multiply them; we care only for their ordering properties; there is a class of numbers that do exactly this; the ordinals; so to repeat - for the nth ordinal we have the rational fraction 1/2^n; now the finite ordinal have a limit - called the first infinite ordinal omega;

so we can ask what is the fraction indexed by 1/2^(omega); the only good answer here is zero; this is a new ideal point, even though it looks like one that is already kicking about; we say this because if we do what we have just done for any arbitrary sequence of rationals we get many more, in fact uncountably many more fractions that do not lie within the rationals.

It is this schema that is used construct the modern notion of the Reals from the Rationals.

interlude

This relies on the notion of an infinitesimal - which is nonzero & yet smaller than any fraction. It is either constructed formally by the tools of mathematical logic; or posited axiomatically; all this relies on work by Robinson.

geometrical

This brings us back to the original notions of calculus which wasn't a purely analytical manouver; but a geometrical one; one is after all interested in geometrical notions like the tangent to a curve; which links to the physical notions of motion; velocity as a tangent, say;

Here one finds the notions of a non-zero, infinitesimal rod can be made rigorous (using Topos theory); so a curve is simply just made up of these rods lined up end-to-end; and to find the tangent at a point; one simply picks the unique rod at that point.

It relies on work by Lawvere & Kock amongst others.

More can be said: An alternative axiomatic construction relies on the theory of dual numbers; these are numbers that square to zero; but the theory can only be made sense of if we move from classical logic to intuitionistic logic - the logic where the law of the excluded middle no longer holds; it is through this logic that it is the bridge to Topos Theory.

Its also worth noting that Deleuze wrote in Difference & Repetition:

“it is a mistake to tie the value of the symbol dx to the existence of infinitesimals; but it is also a mistake to refuse it any ontological or gnoseological value in the name of a refusal of the latter. In fact, there is a treasure buried within the old so-called barbaric or pre-scientific interpretations of the differential calculus, which must be separated from its infinitesimal matrix. A great deal of heart and a great deal of truly philosophical naivety is needed in order to take the symbol dx seriously...”

Taking the symbol dx seriously is exactly what the geometrical notion of the infinitesimal is about, in that it aligns itself with the geometrical naivety that first brought the calculus to fruition.

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You can take this in a mathematical context. In a mathematical context, you have a sequence with a limit of zero, where all the inidividual members of the sequence are not zero. That's not a problem in mathematics at all. Just read up on the definition of a limit, and you'll see why.

In a physical context, your assumption is wrong. You can't take any length and take half of it away, repeatedly. Well, first you'd need to tell us what you mean by "taking half of the length away". For example, if your object is a sausage, cut it in half and throw one half away. Now invest a tiny bit of money in a sausage and a good knife, try it, and tell us how often you managed to "take half of it away".

  • I'm thinking of objects of thought, that is mathematics and logic. There is no limit to the number of times one can divide a real number by two. The ancient Greeks thought subtraction is removal of a part of another body, thus such equations as 2-2=0 meant nothing to them as that is called complete removal and not subtraction of half of the previous remainder. – Michael Lee Jul 10 '14 at 21:13

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