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Brouwer formulates his foundational philosophy using a single a priori notion, that of time; expressing the kernel of his idea thus :

(neo)intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separate in time, to be the fundamental phenomenon of the human intellect.

(All quotes given are taken from “Intuitionism and Formalism” by L.E.J. Brouwer.)

Brouwer gives a name to this concept (that of the falling apart of moments and reuniting them). He calls it two-oneness (and we note this concept is clearly analogous to collectivisation - i.e., set formation).

Brouwer then states his two acts of intuitionism. The first act gives rise to the natural numbers, while the second act gives rise to the continuum. Again quoting Brouwer :

This intuition of two-oneness, the basal intuition of mathematics, creates not only the numbers one and two, but also all finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new two-oneness, which process may be repeated indefinitely; this gives rise still further to the smallest infinite ordinal number ω. Finally this basal intuition of mathematics, in which the connected and the separate, the continuous and the discreet are united, gives rise immediately to the intuition of the linear continuum, i.e., of the “between,” which is not exhaustible by the interposition of new units and therefore can never be thought of as a mere collection of units.

Thus, it is Brouwer’s intuition of betweenness that gives rise to his continuum, and we see that Brouwer’s continuum is not equivalent to the classical continuum.

In particular, Brouwer’s continuum does not fall apart into individual points (“can never be thought of as a mere collection of points”). Yet the intuition of “between” must surely apply to moments of time, so Brouwer's notion of time appears to be that of continuum. But then, how does time fall apart into individual moments (as per the first quote)? What am I misunderstanding here?

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  • I'd suggest thinking of the real line as a mere collection of points isn't sufficient to describe it as the continuum; but adding the standard topology does; but I'm not sure quite how this differs from Brouwers. Aug 13 '15 at 15:41
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    @MoziburUllah Yes, I too, and many others, have considered this view. It is hard to (intuitively) imagine a linear continuum as a collection of points.
    – NWR
    Aug 13 '15 at 16:24
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The experience of the current moment being different from the memory of the previous moment is a discrete phenomenon. Our memory is instantaneous and qualitatively isolated from any other remembered moment. Memory is not continuous, it is itemized. Each event we attend to has the feeling of two-oneness, of becoming an event.

But if we back away from our concrete memory and concentrate on the possible memories we might have had, we find that we might have had memories between any two we actually have. Consciousness is continuous, even if perception is not.

To put this in less vague frame, look at how it plays out in actual phenomenal experience near the limit of our processing resolution.

In "Consciousness, Explained", Daniel Dennett describes the experiment where a red light on one side of our peripheral vision and a green light on the other, when flashed too close together for them to be processed separately, get perceived as a single moving object that changes color.

We perceive the single event of there being a ribbon of light across our visual field which is red on one end and green on the other. But in retrospect we can identify the point where our brain assigns the (illusory) change in color because we naturally force continuity on our discrete sequential experience. So time is experience as continuous, but the experiences are discrete, and mapped onto the continuum artificially.

Since the interpolated continuity is at a different level of processing, and not made up of the actual discrete events, but interpolated from them, one cannot connect them in a way that makes the continuum a collection of points. We are prevented from undoing the illusion, because it is necessary to our narrative arc. From an extreme position, if we want to be intuitively motivated, we should respect this difficulty as natural to our psychology and not pretend we can remove it.

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  • That is a very well formed answer. So, if I understand, our mind is identifying moments in time "from the bottom up" rather than "from the top down", similar, though not identical, to the way Brouwer can identify an individual real number in the continuum using a choice sequence, but cannot identify specific real numbers by separating them from the continuum "top down". Excellent answer. I shall probably accept this, though I should wait to allow others to have a say. (I actually have mild dyslexia, so I'll need to re-read this a few times.)
    – NWR
    Aug 12 '15 at 22:07
  • I think my first reading may have been a bit off base. You are saying that our perception of a continuous flow of time masks the discreet nature of our experience and that we are artificially placing our discreet experiences into the perceived continuum when we access our memory or contemplate our experience. I really like this answer. It is very clever (Mr Smarty-Pants!).
    – NWR
    Aug 12 '15 at 22:48
  • (Right like I need a bigger head...) Both readings kind of come together. The second is what I had in mind, we choose specific points in time to actually perceive, but we know we are calling them out of a sequence that can be arbitrarily refined, so we fill in the gaps artificially. (My own personal theory is that we do so in order to have narrative alignment with other people. We can keep merging our stories, even though the reference points do not align.)
    – user9166
    Aug 13 '15 at 17:21
  • But it leads to the first one. The artificiality means that a real number (like real point in time) other than one given by reference or constructed from stable reference points, is an interpolation or approximation and not a real thing. (We merge narratives into a social reality, the real experiences cited are the bottom, but we learn that the top needs to be there for us all to get along.) I really like Dennett's choice of experiments about perception, and this one makes a lot of sense to me. (Clever makes me happy, even when it escapes praise, and almost as much when it is not mine.)
    – user9166
    Aug 13 '15 at 17:22
  • In the interest of 'mathematics as a branch of experimental psychology', it would be nice to know how built-in the ability to discern "when the color changed", is. I would be very convenient to find out it improves over time.
    – user9166
    Aug 13 '15 at 17:29

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