Wittgenstein's Perspectives on Mathematics: A Critical Examination

[disregardthat]

I recently read "Wittgenstein's lectures on the foundations of mathematics", a book written from notes of students of wittgenstein during his series of lectures on mathematics. I found it interesting, and was wondering if anyone has read it and their thoughts of it, as well as if anyone are familiar with his view on mathematics.

He challenges the commonly understood notions of mathematics, and particular what a mathematical statement is. He has definite constructivist and finitistic tendencies, and spends a lot of time arguing against formal frameworks such as - and in particular - set theory as a foundation of mathematics.


Any thoughts or objections to his point of view?

 

[Willowz]

What do you mean by 'finitistic tendencies'?

 

[disregardthat]

He argues that a mathematical statement should be verifiable by finitistic means. By verifiable he does not mean deducible as in deducible from a set of axioms. Rather; for example, mathematical statement of natural numbers can only say something about a finite amount of natural numbers. This is the essence of his finitism. It doesn't mean that he don't agree with talking about anything but a specific finite set of integers however.

Example: He does not consider "the decimal expansion of pi contains the sequence 777" a mathematical statement because of this reason.

Example: Fermats last theorem states that for all integers x,y and z, x^n+y^n != z^n where n is an integer larger than 2. He does not consider this a mathematical statement. Rather, he looks to the hypothetical proof of this, and says that any mathematical statement we could make in this instance is that we have specific integers x, y and z and n > 2, THEN we can say without calculation that x^n + y^n != z^n. The theorem adds something to our calculus, we can conclude that two numbers are unequal in a different way than before (in this example; x^n+y^n and z^n).

Example: Proof by induction does not prove a statement for all integers. Rather, for a specific integer n, the proof by induction works as a scheme for the proof of this particular instance, namely the statement for n. We take the proof of induction as an addition to our calculus, a new way of concluding something about a number.

Example: There is no ordering by value of Q in a sequence. He stresses that this says nothing about the collection of rational numbers, he argues there is no such thing as a collection of the rational numbers. Rather, he takes a proof of this, (by contradiction; for any two consecutive numbers in the sequence we can find an number which is in between), and says that what this proves is that any specific sequence of elements of Q will lack the property of being an ordering by value. We add this thereom to our calculus, a new way of determining a fact about a specific sequence of rational numbers.

 

[Willowz]

He argues that a mathematical statement should be verifiable by finitistic means. By verifiable he does not mean deducible as in deducible from a set of axioms.

By those standards, does he propose a new set of axioms or what?

 

[disregardthat]

By those standards, does he propose a new set of axioms or what?

As I said, he fundamentally opposes any fundamental framework of axioms as a groundwork of mathematics. Read

http://plato.stanford.edu/entries/wittgenstein-mathematics/

if you are interested. Maybe we can have a discussion.

Given that mathematics is a “MOTLEY of techniques of proof” , it does not require a foundation and it cannot be given a self-evident foundation . Since set theory was invented to provide mathematics with a foundation, it is, minimally, unnecessary.

(I have removed the excess of references in the quote, but you can find it in the link I provided)

Note that I don't claim to undestand his philosophy of mathematics in great detail, but I do find it compelling.

 

[Willowz]

if you are interested. Maybe we can have a discussion.

Quite interested. But, I have no idea where to start. Early, Intermediate, or latter W?