Re: What is scientific method?
From: Will Twentyman (wtwentyman_at_read.my.sig)
Date: 12/22/04
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Date: Wed, 22 Dec 2004 12:55:44 -0500
examachine@gmail.com wrote:
> This is an outstanding view of mathematics as a discipline completely
> detached from reality. I'm not impressed though. I suppose this view
> would not be taken seriously by any philosopher past 1950.
Who cares what the philosophers think? The mathematicians/logicians
seem to hold this view today. It certainly summarizes my view and I
wasn't born until 1973.
> Who grants you the truth of the axioms, may I ask you?
We don't care about the truth of the axioms. What we care about is what
happens *if* they are true.
> Are you aware
> that your description above entails that there is absolutely no
> mathematical *picture* in your head, when you think about a
> mathematical axiom? That you merely grind out the theorems specified by
> axioms and inference rules?
To a certain extent, yes. A picture sometimes helps, but it also
sometimes misleads. You can only rely on the axioms, rules of
inference, and verified theorems.
> That's a rather dull view of mathematics, for it is merely what an
> inference engine accomplishes. A mathematician does much more than
> that, and that is why a mathematician has to be intelligent: e.g. he
> must be able to relate the *meaning* of the axioms with his world
> knowledge.
No, that is what an applied mathematician must do. There are sections
of math that do not have an application (yet).
> That's how we come to axiomatize things like probability theory. If
> what you said were correct it would have been absolutely impossible to
> make mathematics without *first* axiomatizing the said discipline, but
> as history shows that is the case.
The axiomatizing of mathematics is a relatively recent development.
However, the notion of proof from the axioms/definitions goes back
thousands of years. The axioms just weren't always stated as
clearly/precisely as they can be today.
> Having been shown wrong on several accounts, how can you defend your
> formalist, meaning-free and truth-free view of mathematics?
Easy. Formalists care neither about meaning nor truth. They care about
logically consistent. Meaning is imposed by those who wish to apply it.
Truth is determined by the model.
> I'm not certain you will realize your mistake, but if your view of
> mathematics were complete and correct, then it would make new
> mathematics rely on chance! You spoke like a student of mathematics who
> wants to know no more than what is given to him in his calculus books;
> enough formulas to get him past the calculus examination. That I
> believe is a seriously inadequate portrait of mathematics.
And you speak like a scientist who cares only about what can be applied
to describe the world around you.
-- Will Twentyman email: wtwentyman at copper dot net
- Next message: Alfred Z. Newmane: "Re: Is zero even or odd?"
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