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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