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In Physics VIII.8, Aristotle refers to his usual resolution of Zeno's paradox of motion:

We should make the same response to anyone who uses Zeno's argument to ask whether it is always necessary to traverse half the distance first, and points out that there are infinitely many half-distances and that it is impossible to traverse infinitely many distances... so traversing a whole distance involves counting an infinite number, which is admittedly impossible.

To which the response is:

originally, we solved these difficulties by taking into account the fact that time itself contains an infinite number of parts.

This when translated into infinite sums is the usual answer. However,

this solution is adequate as a response to the question, it will not do as a response to the actual facts of the matter.

Because we can

ask the same question about time itself, and the same solution would no longer constitute an adequate response.

Aristotle goes on to say:

Anyone who divides a continuous line into two parts is treating the single point at where division occurs as two points - because he is making it both a starting point and an ending point... and this destroys the continuity of the motion.

He concludes by

Though there are infinitely many halves in a continuum, these are potential and not actual... So the reply we have to make to the question whether it is possible to traverse infinitely many parts... is that there is a sense in which it is possible, and which it is not. If they exist actually, it is impossible; but if they exist potentially, it is possible.

Aristotle then is concerned already here, with the classical conception of the continuum as a concept adequate to that of motion.

Question: Does the intuitionistic continuum resolve these doubts of Aristotle? And what does he mean by saying that parts (and not points) exist only potentially?

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There are several notions of intuitionistic continuum, the closest ones to Aristotle's are Brouwer's "fluid continuum", and especially late Weyl’s version of it since On the New Foundational Crisis of Mathematics (1921). We have to keep in mind, however, that Brouwer and Weyl received their view through a major intermediary, Kant. Although Aristotle’s and Kant’s descriptions of motion and continuum are sometimes indistinguishable phenomenologically, what was objective reality to Aristotle was only a phenomenal form of perception to Kant. But all of them did share the most basic premise: continuum is given as a whole, points and parts are imposed on it.

Weyl makes suggestions as to mathematical realization of his fluid continuum and Brouwer built a full blown theory of his somewhat "less" fluid one, but remarks pessimistically that “the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness”, the closer we get to the intuitive continuum the less palatable it becomes mathematically. Predicative continuum of early Weyl, developed by Feferman and applied to physics by Field, is even less fluid, but was shown to be sufficient for all of classical physics at least. In his time Aristotle could afford to be optimistic, for one finds the same conception of “fluid magnitude” in his Physics, as in Euclid’s Elements. In our time intuitionism could only build chain of continua mediating between philosophical insight and mathematical physics.

Weyl concludes that there is a divide between mathematical theorizing and philosophical insight into our experience, of time and motion in particular, which seems to echo Aristotle’s “response to the question” vs. “response to the actual facts of the matter”. But he goes further suggesting that it can not be bridged, which implies that Zeno’s challenge must get different answers on different grounds:”if phenomenal insight is referred to as knowledge, then the theoretical one is based on belief... But where is that transcendent world carried by belief, at which its symbols are directed? I do not find it, unless I completely fuse mathematics with physics and assume that the mathematical concepts of number, function, etc. (or Hilbert’s symbols), generally partake in the theoretical construction of reality in the same way as the concepts of energy, gravitation, electron, etc.

Tieszen gives a nice review of Weyl’s fluid continuum in Philosophical Background of Weyl's Mathematical Constructivism, also his with co-authors Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum compares it to Brouwer’s.


P.S. In On the New Foundational Crisis of Mathematics (1921) Weyl wrote: “an ensemble of individual points is, so to speak, picked out from the fluid paste of the continuum. The continuum is broken up into isolated elements, and the flowing-into-each-other of its parts is replaced by certain conceptual relations between these elements, based on the “larger-smaller” relationship. This is why I speak of an atomistic conception of the continuum”. Weyl speaks here of points “picked out” arithmetically, as in the classical conception or his earlier constructivist one.

To approach fluid continuum we need to commit Brouwer’s “second act of intuitionism” and introduce lawless choice sequences that reflect the fluidity of intuitive continuum. In Philosophy of Mathematics and Natural Science (1949) Weyl explains: “the notion of sequence changes its meaning: it no longer signifies a sequence determined by some law or other, but rather one that is created step by step by free acts of choice, and thus remains in statu nascendi. This ‘becoming’ selective sequence represents the continuum, or the variable, while the sequence determined ad infinitum by a law represents the individual real number falling into the continuum. The continuum no longer appears, to use Leibniz’s language, as an aggregate of fixed elements but as a medium of free ‘becoming’”. This is how intuitionistic points only exist "potentially", in Weyl's view unlike Brouwer's, points in the lawless part of the continuum can not even be individuated.

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I think the solution doesn't have much to do with the intuitionistic continuum. Rather, there are two considerations.

First the mere fact that a continuum can be divided arbitrarily finely doesn't imply that the appropriate measure for how long it will take to cross some particular region is infinite. Whether the time taken to cross a region is finite or not depends on what measure of time the laws of physics respect. The fact that you can imagine measuring the space in question in a different way has nothing to do with anything.

Second the real laws of physics do not allow space and time to be divided arbitrarily finely. In any finite region containing a finite amount of energy, there is a finite number of possible states. That number is typically extremely large by the standards of everyday life, but it is finite: the Bekenstein bound. What changes continuously is the probability of any given state.

  • -1: This misses the point of the question which is about whether the intuitionistic continuum is closer to Aristotles notion; rather than the usual one; you're answering a question I'm not asking. – Mozibur Ullah Feb 2 '16 at 19:41
  • I discovered this article in the IEP useful - which you might also - where they say 'the spirit of Aristotle opposition to infinities ... persists in Constructivism'; and one of the roots of Intuitionism is in such - as they note. – Mozibur Ullah Feb 2 '16 at 23:31
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@conifold- Here is a Spinoza response to a question which tangentially resembles the one here. It concerns ideas which appear to have no reference in reality, in this case a series of rectangles which can be 'pictured' or imagined in the minds-eye' but which have no existence. Not sure I can make any realtime sense of this, any thoughts. Cheers, Charles M Saunders

Prop. VIII. The ideas of particular things, or of modes, that do not exist, must be comprehended in the infinite idea of God, in the same way as the formal essences of particular things or modes are contained in the attributes of God. Proof.—This proposition is evident from the last ; it is understood more clearly from the preceding note.

Corollary.—Hence, so long as particular things do not exist, except in so far as they are comprehended in the attributes of God, their representations in thought or ideas do not exist, except in so far as the infinite idea of God exists ; and when particular things are said to exist, not only in so far as they are involved in the attributes of God, but also in so far as they are said to continue, their ideas will also involve existence, through which they are said to continue.

Note.—If anyone desires an example to throw more light on this question, I shall, I fear, not be able to give him any, which adequately explains the thing of which I here speak, inasmuch as it is unique ; however, I will endeavour to illustrate it as far as possible. The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another ; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists ; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle. Let us grant that, from this infinite number of rectangles, two only exist. The ideas of these two not only exist, in so far as they are contained in the idea of the circle, but also as they involve the existence of those rectangles ; wherefore they are distinguished from the remaining ideas of the remaining rectangles.

Prop. IX. The idea of an individual thing actually existing is caused by God, not in so far as he is infinite, but in so far as he is considered as affected by another idea of a thing actually existing, of which he is the cause.

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