Re: Epistemology 201: The Science of Science

From: Neil W Rickert (rickert+nn_at_cs.niu.edu)
Date: 02/01/05 

Date: Tue, 1 Feb 2005 05:53:19 +0000 (UTC)

"Jason" <jasonstevensNOSPAM@free.net.nz> writes:

>> >What makes maths different from any other random formal system is its
>usefulness
>> >in science.

>> Mathematics isn't a formal system. I'll grant its usefulness to
>> science. However, the attempted comparison with "any other random
>> formal system" is bogus.

>> > Science is about the world, therefore so is mathematics, by
>proxy.

>> That doesn't follow.

>But maths is a formal system, or at least there is a formal system of
>mathematics. That is, a formal grammar that describes the language of
>mathematics, and rules of inference that describe legal moves from one
>mathematical statement to another. Rules of inference from the empty string are
>the axioms of maths. You are correct in that it has not always this way, but at
>present, formal system theory is well established. This may change too of
>course.

You are perhaps referring to First Order Predicate Calculus (FOPC).
And indeed, mathematicians do use FOPC. However, mathematics is not
FOPC, and FOPC is not sufficiently expressible to allow it to be used
exclusively.

Given a particular system of axioms, say PA (the Peano Axioms),
mathematicians could in principle use FOPC applied to those axioms.
But mathematics is not confined to working within a particular axiom
system. Moreover, the discussion axiom system itself is part of
mathematics.

>There are the various properties of formal systems, but what makes maths special
>to us is not so much these properties but what we use it for. What bridges the
>gap between maths as a formal system and maths as useful to us, is the semantics
>we give it, our interpretation of the formal system of maths. But this is not
>part of the formal system itself, which is just syntax, so a comparison to other
>random formal systems is justified.

I agree that what is important is the semantics, and not just the
syntax. But that already argues that mathematics is more than just a
formal system.

>Since mathematics has evolved along-side science and plays a large part in
>describing and predicting how the world works, then as a formal system goes, it
>seems to be on the money as far as capturing something about the world.

That's your opinion. As a mathematician, I have a different
opinion. I consider it important that mathematics is not about the
world. Roughly speaking, mathematics is about what would happen if
reality did not intrude. We discover a lot about reality by seeing
how it differs from the mathematical ideal.

Relevant Pages

  • Re: Epistemology 201: The Science of Science
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    ... Mathematics is much older than FOPC, ... Mathematics is a subset of differences and differences between ... The formal system of maths is ripe for exploration. ...
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  • Re: Epistemology 201: The Science of Science
    ... Mathematics is much older than FOPC, ... Mathematics is a subset of differences and differences between ... The formal system of maths is ripe for exploration. ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... >mathematics, and rules of inference that describe legal moves from one ... formal system theory is well established. ... You are perhaps referring to First Order Predicate Calculus (FOPC). ... Given a particular system of axioms, say PA, ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... >mathematics, and rules of inference that describe legal moves from one ... formal system theory is well established. ... You are perhaps referring to First Order Predicate Calculus (FOPC). ... Given a particular system of axioms, say PA, ...
    (sci.physics)
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