[Wittrs] Re: Syntax and Semantics in Mathematics

  • From: "SWM" <SWMirsky@xxxxxxx>
  • To: wittrsamr@xxxxxxxxxxxxx
  • Date: Sun, 28 Mar 2010 02:09:14 -0000

--- In Wittrs@xxxxxxxxxxxxxxx, Joseph Polanik <jPolanik@...> wrote:

> it sounds like you are saying:
> dualism implies that understanding is a bottom line property which
> implies that Searle's third premise is true.
> Searle believes that his third premise is true.
> (therefore) Searle is a dualist.
> this is a slightly more complicated version of the fallacy of affirming
> the consequent. you must really like that fallacy, Stuart!

I am saying:

1) Thinking that understanding is a bottom line (irreducible) property is 

2) To subscribe to premise #1 above is to be a dualist.

3) Searle subscribes to an argument which depends on the assertion presented by 
#1 above.

4) Therefore Searle is a dualist.

Of course my point is that he is implicitly so, not explicitly so while denying 
he is a dualist he actually is one. (But the case for implicitness requires 
additional steps not relevant to the simplified argument above.)

> * * *
> in any case, there are grounds unrelated to the CRA Presumption for
> believing that syntax does not constitute and is not sufficient for
> semantics.

Yes there are. But this is about the CRA and its conclusions and what they 
depend on.

I have NEVER, EVER denied that there are other grounds for claiming the CRA's 
conclusion is true. But Searle's argument is apart from them! He doesn't invoke 
them to make his case. He ONLY invokes his premises (now called axioms!) to 
make the case. Therefore we are entitled to see what underlies these premises, 
i.e., what they imply conceptually.

None of what follows is relevant to the point of Searle and his implicit 
dualism so I'll stop here before we end up with another von Neumannesque 
discourse that ultimately goes nowhere.


> in the early 20th century the mathematician, David Hilbert, advocated
> programmatic formalism. its aim was to reduce mathematics to a set of
> axioms that could be manipulated syntactically and thereby prove its own
> consistency.

> from the wikipedia article:
> "In foundations of mathematics, philosophy of mathematics, and
> philosophy of logic, formalism is a theory that holds that statements of
> mathematics and logic can be thought of as statements about the
> consequences of certain string manipulation rules."

> "For example, Euclidean geometry can be seen as a "game" whose play
> consists in moving around certain strings of symbols called axioms
> according to a set of rules called "rules of inference" to generate new
> strings. In playing this game one can "prove" that the Pythagorean
> theorem is valid because the string representing the Pythagorean theorem
> can be constructed using only the stated rules."

> "According to formalism, the truths expressed in logic and mathematics
> are not about numbers, sets, or triangles or any other contensive
> subject matter ? in fact, they aren't "about" anything at all. They are
> syntactic forms whose shapes and locations have no meaning unless they
> are given an interpretation (or semantics)."

> [http://en.wikipedia.org/wiki/Formalism_(mathematics)]

> the mathematicians of the world woke up from Hilbert's dream when Godel
> showed that a non trivial formal (axiomatic) system would not be
> complete and could not prove its own consistency.

> now, as it happens, Godel is part of the Plato/Penrose axis of platonic
> dualism; but, his mathematical results stand on their own.

> Joe

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