@TuringEquivalent,

TuringEquivalent wrote:Wrong! I care for W as much as i care for other philosophers. The problem with you is that you are so in heat for him that you can` t judge objectively his position.

I take a Wittgensteinian view of mathematics, just as you take a Platonic view of mathematics, but I would hardly describe you as a bitch in heat, I would merely describe you as wrong, and I have enumerated the reasons for this.

Quote:Where did i misunderstood W, moron? I paraphrasing the stanford philosophy dictionary on his view on math. So, i did not "misunderstood". If you do ******* read, and understand the stanford article, you know that his view on math is formalism. If you actually know some philosophy of math, you know that formalism is inconsistent with Godel theorem.

I have read most of Wittgenstein's works, a great deal of the secondary literature, and I have had tutorials with, and been lectured at by, some very eminent philosophers in the field of Wittgenstein scholarship on the matter; I do not take my philosophical opinions from the SEP. In any case, I'm pretty sure that entries in the Stanford Encyclopaedia of Philosophy, as helpful as they often are, do not lay down incontrovertible exegesis.

Formalism is the doctrine that the foundations of mathematics reside in the transformation rules that supposedly govern the move from one string of symbols to the next. In other words, a theorem is true if the transformation rules of a system allow us to go from the axioms of the system to the theorem in question. Thus, formalism shifts the foundations of arithematic from mathematical objects - numbers - to proofs. Now, we might think of the transformation rules as determining what theorems may be "properly" inferred from the axioms, and which theorems may not be. Wittgenstein's rule following passages argue, very convincingly, that the transformation rules of a system do not determine anything. On one interpretation of the transformation rules a certain theorem may be inferred, and on another interpretation it may not be. Moreover, suppose there is a "right" interpretation of a rule. This just hangs in the air with the rule that it interprets! It, too, stands in need of an interpretation if it is to do anything, and thus we get an infinite regress; a rule does not, in and of itself, force us to judge anything to accord or discord with it. We are thus forced to conclude that going by a rule cannot be supplying the right interpretation, otherwise, there is no such thing as going by a rule: anything can be said to be in accord with it. So, why does 2+2=4? Well, we all agree it does, and we do not deem anything else to be a legitimate move in the language-game, that's all there is to it.

Now, it is very easy to see how the incompleteness theorems block off formalism, for if formalism tells us that mathematics is just the application of a set of transformation rules to a set of axioms, then this is not just a philosophical hypothesis, but but it is also a mathematical hypothesis (Hilbert's second problem): All mathematical theorems can be proved from a set of consistent axioms. Godel, of course, proves that if a system is consistent, then there is a statement of its own consistency within the system that it cannot prove. This clearly rules out formalism, as a truth of the system(its consistency, which Godel shows us how to express in the system), cannot be given by the transformation rules. Wittgenstein, however, is not putting forward a mathematical hypothesis, he is discussing the nature of mathematical theorems, and this cannot be refuted by a particular mathematical result.

Quote: No, the issue is not about ******* consistency. It is about completeness! Any deductive system expressive enough to express arithmetic truth is necessary incomplete. This begs the question why there are some math statements that are true, but undecidable. Platonism would give the best answer for this.

Godel's theorem is not a mysterious metaphysical result, it is a rigorous proof-theoretic result showing that

* if* mathematics is consistent, then mathematical truths cannot be reduced to its axioms and transformation rules. This does not add anything positive to platonism. This only begs the question as to why mathematical statements can be true but unprovable if you for some reason thought that provability and truth had to be coextensive in the first place, what makes you think that I (or Wittgenstein) did?

Also, there are more creative ways to add emphasis to a piece of writing than using the word, "*******". It's a bit crude and unsophisticated.