Internal and external time

I am still thinking about Brightly's objection to my arguments here and elsewhere. Is the impossibility of a Cantorean transfinite consciousness - a conscious moment preceded by an infinite number of conscious moments before that - simply down to the length of time such a consciousness would require? One simply cannot wait for so long. Or is it as I maintain, that the discontinuity between the transfinite consciousness and the finite one is such that the traversal of such a sequence is impossible?

Every conscious moment leads to the next, but that 'next moment' must be finitely far away. For conscious time and physical time are fundamentally different. I cannot 'cheat' infinity by sleeping for an infinite number of days then waking up on the infinitieth day. The physical time is irrelevant, for in my consciousness the passage of an infinite number of days goes unnoticed. The infinitieth day is indistinguishable from any other day in the finite series. Therefore (I would like to argue) in order to get to the infinitieth day, it is essential to be conscious of every one of the previous days. Every conscious moment (i.e. every moment of my consciousness) must be either (a) conscious of some previous moment or (b) conscious of nothing previous, in which case it is my first conscious moment, and nothing of mine preceded it.

Augustine (in the Confessions, e.g., see here) made a similar distinction between 'internal' (conscious) and 'external' (physical) time, holding that only internal time is real.

For if there are times past and future, I desire to know where they are. But if
as yet I do not succeed, I still know, wherever they are, that they are not
there as future or past, but as present. For if there also they be future, they
are not as yet there; if even there they be past, they are no longer there.
Wheresoever, therefore, they are, whatsoever they are, they are only so as
present. Although past things are related as true, they are drawn out from the
memory, -- not the things themselves, which have passed, but the words conceived
from the images of the things which they have formed in the mind as footprints
in their passage through the senses. My childhood, indeed, which no longer is,
is in time past, which now is not; but when I call to mind its image, and speak
of it, I behold it in the present, because it is as yet in my memory. Whether
there be a like cause of foretelling future things, that of things which as yet
are not the images may be perceived as already existing, I confess, my God, I
know not. This certainly I know, that we generally think before on our future
actions, and that this premeditation is present; but that the action whereon we
premeditate is not yet, because it is future; which when we shall have entered
upon, and have begun to do that which we were premeditating, then shall that
action be, because then it is not future, but present. (Confessions XI.18.23).

On this view, however, it seems difficult to explain the passage of time.

Completion and consciousness

Of course the argument I gave yesterday is a blatant fallacy. We have

• Achilles will not reach the tortoise before the sequence is completed
• The sequence is never completed.

but the word 'complete' is being used in different senses in the two propositions, and so the one does not imply the other. In the first sense it means 'happened', and since it seems possible for every event in the infinite sequence to have happened, it is therefore possible for the sequence to be 'complete' in that sense. In the second sense it means something like 'terminated', and in that sense the second proposition seems false. If the sequence were terminated at some point, then Achilles will not reach the tortoise, but we have no argument that it will not be terminated.

But the idea of completion suggests another argument. Every day I wake up from sleep, that 'little slice of death', and become conscious. Imagine the following thought-experiment. I wake up an infinite number of times. Could I have a conscious moment after that infinite sequence? Is it possible that there could be a waking moment belonging to my consciousness such that there are an infinite number of waking moments before that? Surely not. I can't think of an argument to prove it, rather, it seems an irreducible part of my idea of consciousness that I cannot conceive of an actual or 'completed' infinity of conscious moments. (Complete in the first sense of 'already happened').

You object: what if I failed to wake at some point, an infinite number of days passed, and then I woke up? I reply: physical time and conscious time are different. If I go to sleep on Saturday, and an infinite number of days pass, and I wake up, it is no different for me than if I had woken up on Sunday. I cannot conceive of a sequence of infinite waking moments that I can ever 'escape from', in the sense that every one of those moments was behind me. Perhaps there could be some consciousness after such a sequence, but it would not be my consciousness.

Thus any two waking consciousnesses of mine must be connected by a series of finite waking moments. Which leads to the following paradox. Given that every moment of my consciousness is in a sense a waking moment - the only difference being the lapse of physical time which we assume occurs during sleep, and which we assume does not occur when we are waking - and given that any series of conscious moments belong to my consciousness must be finite, how is it that are consciousness also appears continuous. i.e. there are no obvious gaps or 'flickers' in consciousness such as we see in the early movies? How can consciousness be both discrete and continuous?

Completing Zeno

Let's restate the Zeno argument as follows.

(1) At time t2 Achilles reaches a point where the tortoise was at t1, at t3 he reaches the point where the tortoise was at t2, at t4 he reaches the point where the tortoise was at t3, and so on.

(2) Achilles will not reach the tortoise before the sequence outlined in (1) above is completed,

(3) The sequence is never completed.

(4) Achilles will never reach the tortoise.

Unlike the IEP version I referenced below, the conclusion appears to follow logically from the premisses. Furthermore, it does not rely on assumptions like 'cannot do infinitely many things in a finite time'. It relies simply on the definition of an infinite sequence as one which is endless, not terminated, not completed.

Well it appears valid, but is it? More tomorrow, and comments welcome.

More on Zeno

I asked earlier how the four premisses of Zeno's argument given in the IEP imply the conclusion that Achilles never reaches the tortoise.

Clearly there are other assumptions that have to be made. There is the ‘and so on’ of premiss 4. But how does that work? Suppose Achilles aims at the exact spot Y where he is going to overtake the tortoise. Clearly when he has reached that spot, he will have reached the tortoise. If he reaches any spot X before that, he will have not reached the tortoise. So all the ‘argument’ appears to be saying, it seems, is that if we take any point X before Y, then there is some distance to go. And if we take any spot X’ between Y and X, there is still some distance to be ‘and so on’. But this ‘and so on’ doesn’t prove anything. It proves simply (or rather, it assumes that) we can take any distance whatever, and cut it somewhere. How does that prove ‘Achilles will never reach the tortoise’?

Graham Priest has a slightly different version of the argument. He says (I paraphrase) In order to get from a to b, you must first get to each point between a and b. But there are infinitely many points between a and b. Hidden premiss: to get to something = to do something. Therefore to get from a to b in a finite time, you must do infinitely many things. But you can’t do infinitely many things in a finite time. Therefore etc. But there is much to challenge there. Is the hidden premiss correct? Is ‘getting to’ a point the same as ‘doing something’? Can we actually ‘get to’ a mathematical point? How? We can cross such a point, of course. But then the argument amounts to this: there are infinitely many collections of finite distances between a and b, and we can traverse any such collection in a finite time. Indeed, clearly we can, for the total length of any such collection will be the length between a and b.

Aristotle mentions the argument several times in the Physics, arguing that we must distinguish the ‘actual’ from the ‘potential’ infinite, but this distinction is not very clear. I don't have the book to hand, so will post something later.

Is Zeno's Paradox a Paradox?

Brandon Watson has a discussion on Zeno's Paradox here. He refers to a version of the argument given at the Internet Enyclopedia, but I don't follow it. The salient points are listed below. I can see that 1-4 are true. But why is 5 true? How are 1-4 supposed to imply 5?

1. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is
2. But by the time he arrives there, it will have crawled to a new place.
3. So then Achilles must run to this new place.
4. But the tortoise meanwhile will have crawled on, and so forth.
5. Achilles will never catch the tortoise, says Zeno.