Wittgenstein argues that mathematics is a language game, that is not only based on language but on “forms of life.” And forms of life are rules of the game that are, perhaps socially constructed, but definitely without doubt. They are given somehow and from who knows where. Proving a theorem is to invent a rule that is also without doubt,
“What is unshakably certain about what is proved? To accept a proposition as unshakably certain—I want to say—means to use it as a grammatical rule: this removes uncertainty from it.” (Wittgenstein 1978, 170)
Proving adds to or informs our forms of life. This means that forms of life are created… can they be destroyed? I’m saying perhaps we built the wrong forms of life, perhaps there is still a nagging wonder about the things supposedly settled. And the question begins the process of changing our forms of life. In my first post I proposed bringing in the “?” into mathematical language. As it was laid out in that post, https://questionsarepower.org/2014/08/ the “?” operation would introduce a controlled retreat from mathematical logic. I showed how the “?” can free us from the Liar Paradox, a language game that is an endless cycle in search of a truth value.
The axiom of completeness is exactly such a form of life that should be questioned. It metaphorically closes off the possibility of leaping from moment to discrete moment. The axiom of completeness is like a poetic spell on the mind that prevents natural movement through time, since moving through time continuously, with out the possibility of a leap, generally means “downhill.” Real analysis is a losing language-game.
I have argued against the “least upper bound;” Dedekind cuts are hardly different from this notion, but it shows the cunning of mathematicians to push their agenda: “Everything is number” by using essentially the same argument in so many different ways. Dedekind cuts are sets of rational numbers with no maximum, that is, if r is in the “cut” called A, then there exists a rational “s” such that r<s, s in A, is similar to saying the number generated by adding another digit to the decimal expansion of r is also in A.
Cuts essentially assert a least upper bound (or the “…” in a decimal expansion) with judicious use of the “<” symbol; it is merely a rewording. Dedekind offered these “cuts” as real numbers and asserted that they exist.
In one of many analysis texts I’ve read,
“The real numbers were defined simply as an extension of the rational numbers in which bounded sets have least upper bounds, but no attempt was made to demonstrate that such an extension is actually possible. Now, the time has finally come. By explicitly building the real numbers from the rational ones, we will be able to demonstrate that the Axiom of Completeness does not need to be an axiom at all; it is a theorem! There is something ironic about having the final section of this book be a construction of the number system that has been the underlying subject of every preceding page…We all grow up believing in the existence of real numbers, but it is only through a study of classical analysis that we become aware of their elusive and enigmatic nature. It is because completeness matters so much…that we should now feel obliged—compelled really—to go back to the beginning…” (Abbot 2001, p 244, Emphasis mine)
Who is being compelled? The book is built so that you have to assume the Axiom of Completeness for a very long and arduous time before they get to the meat of the problem. More importantly, this quote shows the circularity of mathematics, from axiom to theorems back to axioms. Theorems are merely explicit parts of the axioms. With a shuffling of words the Axiom of Completeness turns into a theorem, but certainly the theorem is more explicit and involves more description of what the Axiom is, which is left for the very end of a book devoted to assuming the Axiom.
And how explicit is it really? The cuts are defined as any set of rational numbers with no maximum. Is that explicit? How many sets are like that? And there is an interesting example of a cut: take the set of rational numbers “r” such that when r is positive, else r is in the set (call this set A). Compare with a similar set where (B). This is exactly the sort of thing that mathematicians enjoy, the “almost false.” Dedekind cuts fail to distinguish these two sets, but Dr. Abbot continues with his claim that Dedekind cuts make real numbers explicit. Strangely enough, while here the difference between < and might not matter, elsewhere in the theorem of completeness it matters greatly. First of all many of the cuts can be distinguished by using < as in the set of any rational r < 2 , which is a cut, but r2 is not a cut since 2 is its maximum and 2 is in the cut. How to describe the cut where square-root of 2 is the least upper bound without this ambiguity? We can’t. Both the symbol square-root of two and A, involve algebraic operations without a clear (or even necessarily a single) solution (or lub). We can prove that square-root of 2 is not a rational number, but to say that, whatever square root of 2 is, it is a least upper bound of A is to forget that that is what we are trying to prove.
Now, lets look at the order of Dedekind cuts. For cuts A and B,
AB is defined to mean AB.
Let A be the set defined by , is A a strict subset of B, the set defined by ? It certainly seems like it ought to be. It can be reasoned that A contains less (of what?) than B. We want our different algebraic expressions to have distinguishable numerical values, that is ultimately the motivation for the real numbers, but in this case we don’t have that. The choice of in the definition is of course very careful. If the definition used < we would have a potential counter example. Figuring out the exact difference in certain cases between “=” and “<” is swept under the rug. Luckily mathematicians can add definitions to counter this particular example, but how many other examples of vagueness are there? The definition of a Dedekind cut is so general (not explicit) that there may be many other problems.
Since we don’t know that B is a cut, we cannot claim that it is a real number nor that it represents a least upper bound. How do we know that A is a cut? We know that either way square-root of 2 is not a rational number. We know that there is no least upper bound “next” to B, that if a number is adjacent to square-root of 2, they are so close as to be the same number. How, then, can square-root of 2 be explicit, how do we know that there is a unique and determinate answer to square-root of 2? The truth is, vagueness sets in as r in B get closer to B’s upper bounds. The upper borderline is vague, like any empirical borderline, but somehow we are compelled into believing there is a unique and determinate upper bound to B.
“Wittgenstein argues that logical necessity—be it computing an algorithm, proving a theorem, drawing a deductive inference, or whatever—concerns the following of a rule. Rule following raises the issue of the compulsion to reach a conclusion that is fixed and, if not predetermined, then at least unique and determinate” (Ernest 1998, p 80) Assuming there is a unique and determinate square-root of 2 makes the expression of such a view merely a language-game, not anything profound. The only way to have a philosophical thought about the language game of arithmetic is to resist this compulsion, and with that act, the entire edifice of real numbers crumbles.
Dedekind cuts are based on a rule for determining a very large infinity of sets; so is the square-root of 2 a large calculation. Wittgenstein wrote about rule following, saying that “What you are saying, then, comes to this: a new insight—intuition—is needed at every step to carry out the order..a new decision was needed at every stage” (Wittgenstein 1953, 75). The compulsion for a student to follow a rule such as calculating the value of square-root of 2 is never fully understood, because of the decision making process, we think we understand the rule and give up calculating, but if we were driven to continue calculating square-root of 2, doubts would inevitably pop up, perhaps merely because we have a life to live besides carrying out this rule, but the objections that pop up will be well informed ones, by someone who has done a lot of exploration into the rule. In Wittgenstein:
“…following a rule and agreeing (perhaps implicitly) to its conventional underpinnings…also involves a decision that the new application can be legitimately be subsumed under existing rules, for rules underdetermine their applications.”(Ernest 1998, p88)
We have invented a substance ‘the real numbers’ that is so well ordered that it can obey algebraic rules without any decision. Thus the decision to adopt the ‘real numbers’ is a very important and determining one. Feyerabend argued that a proof resembles a tragedy. It is internally consistent and inevitable. Comedy, on the other hand, is the stuff of continuity. And the place comedy had in the Greek world was:
Everything commonly realistic, everything pertaining to everyday life, must not be treated on any level except the comic…As a result the boundaries of realism are narrow. And if we take the word realism a little more strictly, we are forced to conclude that there could be no serious literary treatment of everyday occupations and social classes…of everyday scenes and places…everyday customs and institutions…of people and its life. (Auerbach 1946, p 31)
Reducing comedy to tragedy has dire consequences for the reducing culture.
Dedekind played on words with “Dedekind cuts,” Now, what does he mean that a “cut” exists? Does he mean the knife that cut, or the space between the two parts that were once one? Or does he mean the ground on which the surgery took place, the “operating table,” as Foucault put it?
If he means the space created by a cut, I have already mentioned the trouble with different “cuts”—that different cuts are different from each other, that there exist at least two different real numbers. This means that , and undermines the notion of identity in mathematics.
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