“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.
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