Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Objections to Hume's principle

Controversy over Cantor's theory - Objections to Hume's principle

As argued above, many naïve objections depend on implicitly denying Hume's principle, and are therefore question-begging. Wittgenstein explicitly denies the principle, arguing that our concept of number depends essentially on counting. "Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one"

The expressions "divisible into two parts" and "divisible without limit" have completely different forms. This is, of course, the same case as the one in which someone operates with the word "infinite" as if it were a number word; because, in everyday speech, both are given as answers to the question 'How many?'(PR §173)

Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes. In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar. (PR §141).

He argues that the sign for a list of things is itself a list, and that a list is therefore inherently finite ("The symbol for a class is a list ... A cardinal number is an internal property of a list." (PR § 119)

Anti-Cantorians who propose that a "reality criterion" should be added to mathematics are also (in effect) denying that the concept of "number" truly applies to infinite sets. They argue [reference??] that we must take steps to guarantee that formal conclusions reached in the world of abstractions can be translated back into assertions about the concrete world. Now that we have a microscope for mathematics (i.e. the computer), it makes sense to think of the world of computation as real and concrete; infinite sets and power sets of infinite sets (and hence, real numbers etc.) exist only as useful fictions (abstractions) which help us reason about the concrete reality underlying mathematics; axioms and the rules of inference for abstractions should guarantee that any statement about the infinite should have implications for approximations to the infinite. Statements which have no implications observable in the world of computation, are fictions.

They [reference??] argue that it is not clear that anyone has produced a collection of axioms and rules of inference that satisfy these criteria, and are powerful enough to do all potentially useful mathematics. The constructivists have made progress towards that goal [reference??].

Others have argued that the mathematical logic that underpins set theory is essentially mathematical, and therefore lacks genuine logical underpinnings.

...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ..." (Weyl, 1946)

We cannot use the modern axiomatic method to establish the theory of sets. We cannot, in particular, simply employ the machinery of modern logic, modern mathematical logic, in establishing the theory of sets (Mayberry 2000, 7)

If God has mathematics of his own that needs to be done, let him do it himself." (Errett Bishop (19XX))

Others believe that the assumptions of set theory lead to conclusions that are unreal or absurd.

Set theory is based on polite lies, things we agree on even though we know they're not true. In some ways, the foundations of mathematics has an air of unreality. (William P. Thurston)

[The pure mathematicians] have followed a gleam that has led them out of this world... The fact that mathematics is valuable because it contributes to the understanding and mastery of of nature has been lost sight of... the work of the idealist who ignores reality will not survive." (Kline, 1982)

Philosopher Hartley Slater, in a number of papers, has repeatedly argued against the concept of "number" that underlies set theory (see external link below).

In reply, Cantoreans quote Cantor's saying (now inscribed on his tombstone) that "the essence of mathematics lies entirely in its freedom" (Grundlagen §8).

Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real. (ibid.)

Adapted from the Wikipedia article "Objections to Hume's principle", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki