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.

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“Most mathematicians adhere to foundational principles that are known to be polite fictions. For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite fictions without being caught out, but that doesn’t make them right.” (Thurston ed. Hersh 2006, p 48-49)

I remember being greatly troubled, to the point of deep crisis, over how the real numbers were ordered. Can one, for example, say that any two unequal real numbers have a relationship “>” so that we can confidently place them on one or the other side of the “>”? We could if we know what the real numbers are that we are talking about, but there are many real numbers that are undetermined by any easily definable rule. So if I take one of my numbers to be the square-root of 2, and take another number to be just like square-root of 2 except adding one to one of the digits of the decimal expansion of square root of two, the digit being decided randomly from the infinite decimal expansion. Now lets shuffle these two numbers so we don’t know which one is which and compare the two numbers:

Number 1: 1.4142135…

Number 2: 1.4142135…

Do you like my trick? most inspections will yield that the numbers are the same, yet by our rules we know them to be different, do we know one to be greater than the other? Before we answer yes, lets analyze the question: Do we know that Number 1 is greater than Number 2? Do we know that Number 2 is greater than Number 1? So we think an ordering is there, but we can’t apply the ordering to the specific numbers without an arbitrary amount of time to inspect them first.

Another concept of order received much more attention and controversy. The well-ordering theorem asks if we can have a certain kind of ordering, called well-ordering (when a set has a least element) can be created for any set. There were some other details but the point is it was very controversial and eventually proven that no such order existed for the real numbers by Julius Konig in 1904. I am not sure if this was the fiction Thurston is talking about, since according to Mann (https://math.berkeley.edu/~kpmann/Well-ordering.pdf, p2), Konig’s proof was flawed, and the well-ordering principle was proven to be unprovable with the commonly accepted axioms of set theory.

Now, is the fiction that the reals are ordered or that they are not ordered? It seems we don’t know either way, but when I approached my advisor in my M.S. in mathematics program he told me “The Axiom of Completeness orders the Real numbers.” I can see what he meant: the axiom asserts a well ordering of a kind of subset of real numbers: bounded and monotone subsets. There was, however, no hint from my professor that there was any “polite fiction,” and my crisis continued until I rejected the Axiom, and, many years later, found the quote from Thurston today.

The crisis I was having before was not a problem of understanding, but a problem with accepting mathematical theorems as a belief. Peirce argues that the goal of thought, and firstly mathematical thought, is belief, but if belief comes at the cost of understanding, I would rather have understanding.

I would propose that the “…” is not an indication that we know the “rest” of a real number, nor its position in an order, but rather the “…” is an assertion of vagueness about the “rest” of the number and a better symbol to use would be “?” rather than “…”.

The axiom of completeness asserts a kind of empty knowledge of this vagueness. In a sense it “covers” our ignorance with a fact that does nothing for our knowledge, going along for the moment with the Kantian view that arithmetic is the a priori synthetic knowledge of pure time, we have a fact—the Axiom of Completeness—that creates ignorance of our understanding of time.

“Nothing in education is so astonishing as the amount of ignorance it accumulates in the form of inert facts…Before this historical chasm, a mind like that of Adams felt itself helpless; he turned from the Virgin to the Dynamo as though he were a Branly coherer.” —Henry Adams, The Virgin and the Dynamo, 1918

It is my thesis that the tremendous ignorance about time that the Axiom of Completeness creates, metaphorically speaking, is harnessed as fuel for the Dynamo. For an elucidation of the problem of time see “Time, Realism, News” by Kevin G. Barnhurst and me, Andrew Nightingale, in press with Journalism: Theory, Practice and Criticism.

(http://pure.au.dk/portal/en/activities/journalism-theory-practice-and-criticism-special-issue-on-the-shifting-temporalities-of-journalism(63d54d06-0e1a-4512-a237-ae200aafb843).html)

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To Peirce, the clarity of ideas is also important. And clarity is important to possibility. According to Descartes: “To know whether a given idea—for instance, the idea of a circular rectangle—does represent a possibility, one must be able to form a clear and distinct idea of it; if one is able to form such an idea, one has the assurance of God’s goodness that it does represent a real possibility…”(p421 Aune, ed. Edwards 1972)

“…he[Descartes] was further led to say that clearness of ideas is not sufficient, but that they need also to be distinct, i.e., to have nothing unclear about them. What he probably meant by this…was, that they must sustain the test of dialectical examination; that they must not only seem clear at the outset, but that discussion must never be able to bring to light points of obscurity connected with them.(C.S. Peirce ed. Houser and Kloesel 1992, p 125) And the dialectical, involving a deep questioning of a proposition, becomes very important in forming a truly clear idea.

A question “stimulates the mind to an activity which may be slight or energetic, calm or turbulent.” (C.S. Peirce ed. Houser and Kloesel 1992, p128) And the goal of such action involves “a method of reaching a clearness of thought of a far higher grade than the “distinctness” of the logicians.” (C.S. Peirce ed. Houser and Kloesel 1992, p 127)

It seems, however, that clarity should not be associated with possibility so much as with actuality and belief, and vagueness more closely with the question. But admitting that vagueness is a possibility is to admit a monster into our world. And a dangerous monster it is, because many vagaries are found at the boundaries of every idea’s meaning. If possibility is taken in Diodorus’s sense, vagueness must be real now or in the future. And it seems this is true, since it never seems to go away. Our theories fall short of perfect description, and only partially match observation. And, importantly, observations are vague even without words. For example looking at the world with the naked eye is vague when compared to, say, chemical “reality,” and chemical reality suffers vagueness when compared to quantum observations such as the colors of candlelight. In fact, without vagueness nothing is possible, because the motion across a border from the actual to a possibility, looked at closely, reveals vagueness. There in fact is no limit to the progression, no definable borderline, only a painterly wash of vagueness. If all empirical data suffers from vagueness, how can our imaginations be expected to be clear? (Imagination taken in the sense of Bacchelard’s Air and Dreams where the imagination “deforms” empirical data) Take for example the imagination of a mathematical point. If anything is brought to the mind at all, a space or a dot, we have already departed from a precise imagination of a mathematical point, which is a totally determined and perfectly precise position, neither a dot nor an inhabitable space.

Note that my appeal is not to space but to time: the progression from the actual to a possible future. If we hope to retain some of the meaning of mathematics by asserting that arithmetic is about the a priori synthetic intuition of time, it cannot do so by asserting continuity of time, because continuity is a spacial notion. For example the Intermediate Value Theorem is meaningless without spacial imagination. How can I be sure that at there is an infinitely small period of time, the now, where the past meets the future intermediatly? Such a belief is totally against the experience of moments in time, which are atomic before being divided after-the-fact. The idea of completeness of arithmetical numbers, that there exists a least upper bound, does not reveal itself in time. I cannot be sure of such a short period of time, and normally such sequences are only alluded to with graphs in space. I can, however, find, as I strain my intuitions towards ever-smaller moments of time, that eventually the observations become imprecise. When I strain my mind towards possible futures, I find vagueness in their details, as with my failure of a memory. Vagueness is real and finds its analogy in space as well. The limit or least upper bound does is not as certain when grounded in geometry, and becomes a meaningless formalism; a formalism that falls prey to Godel’s theorems.

It has been proposed that wrestling with the monster of vagueness is worthwhile: The formal representation of vagueness—that of a chain of hypotheticals that eventually breaks down because there is no clear cut of point for class membership to end (there is no “least upper bound”) is exactly where mathematics and observation are in total agreement. I am not advocating probability theory as the way to fill mathematics with content, because there is no necessary reason to believe that probability is the answer to vagueness, especially since there are other logics available since logical consequence itself is vague (Beall, Restall 2006).

Peirce seems to ignore the Skeptics belief in Ataraxia, a peaceful state arrived at by abstaining from rashness in belief as a result of inquiry. Personally I find comfort in the idea that we will never answer all the questions, and that there are some questions, while they may seem answered for a while, continually resurface no matter how long are our investigations. It amounts to a kind of faith that the uncertainty of life will go on and essentially be the same as it always has been. Knowing, like Euclid’s parallel postulate, makes us think we are going straight for a while, but as our path gets longer, we begin to observe curves in our path. People enter crisis and paradoxes arise, non-euclidean geometry is born after thousands of years of the problem of Euclid’s postulate resurfacing. The knowledge paradigm shifts, those in Ataraxia are not perturbed.

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