In Kant’s Critique of Pure Reason he relies heavily on the continuity of time. For Kant, arithmetic is the a priori form of time, and so it is of utmost importance that arithmetic can describe continuity, or else it is not known whether time is continuous. The function of the Axiom of Completeness is to ensure that all positions in a continuous progression can be expressed with an arithmetical number, thus arithmetic can describe mathematical continuity. What follows is the various ways that Kant depends on mathematical continuity.

1. Kant’s famous question “How are synthetic a priori judgments possible?” (first edition xvii) is given credence because of (to him) examples of such judgments:  geometry and arithmetic.
2. Continuity of time. Arithmetic, to Kant, is the a priori form we impose on phenomena in time. If Arithmetic cannot describe continuity, time cannot be continuous.
3. Application of his categories to phenomena. How can a category, for example, existence, or an a priori form for time, be applied to experience? They can, according to Kant, because of “a “transcendental schema,” which is a “transcendental determination of time.” (Second Edition 181) The schema is “properly, only the phenomenon, or sensible concept, of an object in agreement with its category” Thus being phenomenal means being in time, Kant’s worlds are phenomenal, and the categories find their sense because of primarily our a priori sense of time.
4. Induction over time. Kant with Hume sees that connecting experiences-in-time together (synthesis) is required for empirical knowledge. They both notice that this principle “induction” is not an analytic nor empirical principle. Kant argues against Hume’s belief that experiences-in-time are discrete, instead saying that without continuous (or connected) experience of time we would not be able to synthesize events.
5. Causation, which is just a particular kind of connection between events (a connection according to rules, rules determined by our categories, categories given sense because of the a priori form for time) is possible because events are, in general, connected and continuous.
6. Synthesis and Knowledge. “Synthesis in general is the mere result of the power of imagination, a blind but indispensable function in the soul; without which we would have no knowledge whatsoever, but of which we are scarcely ever conscious.”  (Second Edition 103) When asking how synthesis becomes knowledge, it is in the intuition of data according to the a priori forms of space and time. To Kant, knowledge is not possible without being connected by a unitary consciousness, which in turn requires that time is not discrete.

“For this unity of consciousness would be impossible if the mind in knowledge of the manifold could not become conscious of the identity of function whereby it synthetically combines it into one knowledge. The original and necessary consciousness of the identity of the self is thus at the same time a consciousness of an equally necessary unity of the synthesis of all appearances according to concepts, that is, according to rules, which not only make them necessarily reproducible but also in so doing determine an object for their intuition…”(First Edition 107-108)

7. The self, or unity of consciousness depends on a background of external things at rest or unchanging. Having a “permanence” that requires the a priori form of time for resting to continue.
8. Conservation (“Permanence”) of substance—no total generation nor destruction. Kant again uses the unity or continuity of time, saying that objects at rest, not changing but passing through time continuously, provide the background for grasping change.
9. A realist world. The background provided by this permanence is the basic way we perceive the external world.
10. Because the form of time is an a priori, we have the result that the external world and the ideal world are interdependent.
11. Descartes seems to believe that mind is the first thing we are away of, while it is necessary to infer the existence of external things. Thus external things are open to doubt (indistinct?). Kant argued that because we need permanence of external things to infer a self, external things are not open to doubt. In the “Anticipation of perception” section Kant asks how sensations can have a determinate degree (not indistinct).

Geometry and logic were also generally considered perfectly settled fields in Kant’s time, and were important to Kant’s philosophy. This has changed drastically, however, with the emergence of nonEuclidean geometries and other logics.

To see my argument against the Axiom of Completeness, and against a continuous passage of time, see https://questionsarepower.files.wordpress.com/2016/03/many_roads_from_the_axiom_of_completenes-2.pdf

If we hope to retain some of the meaning of mathematics by asserting that arithmetic is about the a priori synthetic intuition of time, we cannot do so by asserting mathematical continuity of time, because mathematical continuity is a spacial notion. For example the Intermediate Value Theorem is meaningless without spacial imagination. How can I be sure that there is an infinitely small period of time, the now, where the past meets the future intermediately?  Such a belief is totally against the experience of moments in time, which are atomic before being divided after-the-fact. Trying to interpret “intermediate value” by referring to non-discrete “real” numbers has already departed from the immediate intuition of time. The idea of completeness of arithmetical numbers, that there exists a least upper bound, does not reveal itself in time, and I believe Brouwer agrees on this point.

I think that the feeling of “flow” from, say, a breeze or putting your foot in a stream is the experience of temporal continuity, but separating its temporal aspect from its spacial aspect is not so easily done, and maintaining ones focus on the experience of purely temporal continuity is quite difficult. Usually one’s concentration breaks and with it, the moment as well. No matter the ability to concentrate, one cannot remain awake forever. Unifying atomic moments after the fact is a feat of the intellect, but it is not the primal intuition framing an event. I am saying that an atomic moment is unbroken and continuous, but a purely temporal intuition of continuity is not captured by mathematical continuity.

Brouwer’s constructive reconstitution of the continuum from the memory of life-moments that have “fallen apart” seems to acknowledge that the intuition of continuity cannot be completely defined. I also feel a deep scruple in pretending that continuity can be defined; such a definition would mean the “end” of time. It is a marvel that mathematicians, who should be less pretentious than philosophers, make such a pretense. The feeling of continuous time should be left to mysticism. For philosophy, atomism of moments is the best we can expect.