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I am reading Hume's A Treatise of Human Nature and was struck by his arguments against infinite divisibility.

In Part II - Of the ideas of space and time; Section I - Of the infinite divisibility of our ideas of space and time Hume expands on his earlier assertion that the mind has a finite capacity.

Tis universally allow’d, that the capacity of the mind is limited, and can never attain a full and adequate conception of infinity: And tho’ it were not allow’d, ’twou’d be sufficiently evident from the plainest observation and experience. ’Tis also obvious, that whatever is capable of being divided in infinitum, must consist of an infinite number of parts, and that ’tis impossible to set any bounds to the number of parts, without setting bounds at the same time to the division. It requires scarce any induction to conclude from hence, that the idea, which we form of any finite quality, is not infinitely divisible, but that by proper distinctions and separations we may run up this idea to inferior ones, which will be perfectly simple and indivisible. In rejecting the infinite capacity of the mind, we suppose it may arrive at an end in the division of its ideas; nor are there any possible means of evading the evidence of this conclusion.

But my understanding of Zeno of Elea is that he argued differently, that finite bounds, i.e. where I am standing now to 10 feet away, are infinitely divisible. If you take half of where I need to be (5 feet), and again, (2.5 feet), again and again, you just keep getting halves, to infinity.

Was not this Zeno's proof that motion or change itself was an illusion?

I am not a mathematician, so I was looking for clarity on this. Is Hume right, or Zeno, or am I misunderstanding something?

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The idea that Hume expresses here concerning the limitations of the human mind to grasp the infinite was an idea that was widely held in his day among philosophers such as a Descartes, Locke and Arnauld. In order to understand Hume's argument, it is important to take notice that he is not speaking of an abstract conception of infinity but of a conception which is rooted in phenomenal primitives accessible through human experience. In other words, the idea of an infinitesimal part is beyond what we are able to experience and therefore cannot provide an adequate grounding for the concept. This idea is elaborated in the following:

"What Hume says is that we 'can never attain a full and adequate conception of infinity'. He is in fact not claiming that we have no idea of infinity, only that we have no 'full and adequate idea.' These are in fact two different things. The notion of an 'adequate idea' is defined in Locke's Essay: adequate ideas are those which 'perfectly represent those archetypes which the mind supposes them taken from; which it intends them to stand for, and to which it refers them. Inadequate ideas are such, which are but a partial or incomplete representation of those archetypes to which they are referred'." (Fred Wilson, The External World and Our Knowledge of It)

Wilson goes on to clarify that Hume never denied that we able to form abstract concepts of infinity:

"In fact, Hume never argues that we have no idea of infinite extension or of infinite divisibility. What Hume proposes is that 'if it be a contradiction to suppose, that a finite extension contains an infinite number of parts, no finite extension can be infinitely extended', and what he argues is that it is indeed a contradiction to suppose that a finite extension contains an infinite number of parts." (Fred Wilson, The External World and Our Knowledge of It)

Although Zeno also employed the idea of the infinite in his arguments, he was also trying to show that paradoxes result from some of the ideas that we associate with infinity:

"[Zeno] attempted to show that equal absurdities followed logically from the denial of Parmenides' views. You think that there are many things? Then you must conclude that everything is both infinitely small and infinitely big! You think that motion is infinitely divisible? Then it follows that nothing moves!" ("Zeno's Paradoxes", SEP)

  • I understand that Hume did not mean to say that we do not have abstract ideas of infinity, my question regards the mathematics of the infinite divisibility of finite extensions. Hume says (according to your second quote and the primary text) that a finite extension cannot contain an infinite number of parts. Zeno argued the opposite, did he not? Who is right? – Summer Developer Feb 26 '17 at 18:59
  • For example, if I understand Cantor's first set theory article correctly, Hume was incorrect, was he not? – Summer Developer Feb 26 '17 at 19:08

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