Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 02/07/05 

Date: Mon, 7 Feb 2005 10:12:51 -0500

In nresponse to Jason and Neil's discussion:
> > >> >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.
>
> Maths is an extension of FOPC, like PA. The ZFC axioms are conventionally used
> and assumed, as far as I am aware. If another system is used in maths then
> people need to know about it. The ZF system without the axiom of Choice for
> example, can lead to the creation of two spheres out of one in topology.
>
> The study of axioms don't take place in maths. It is meta-logic or meta-maths
> that deals with this. Godels theorem for example is a meta-mathematical proof.
>
>
> > >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

People confuse mathematical truth with the language used to express it.
All languages are formed to represent things in the real world,
ultimately, whether physical, mental, or abstract. While natural
language concentrates largely on qualitative aspects of the world,
mathematical language is developed to represent abstract quantitative
aspects of the world, or measurements. These measurements are not always
numerical in the sense of being integers or reals. In formal logic,
which is clearly a branch of mathematics, we are measuring a truth value
that, in the broadest sense as a probability, varies from zero (false)
to 1 (true). In other words, I think math is most succinctly and
accurately described as: a system for describing measures and their
relationships. Science would then be the application of math to the real
world. As far as human discovery is concerned, science and math go hand
in hand. 

-- 
Tony

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... >> And indeed, mathematicians do use FOPC. ... >> But mathematics is not confined to working within a particular axiom ... You talk about "the formal system of maths", ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... >> And indeed, mathematicians do use FOPC. ... >> But mathematics is not confined to working within a particular axiom ... You talk about "the formal system of maths", ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... >> And indeed, mathematicians do use FOPC. ... >> But mathematics is not confined to working within a particular axiom ... You talk about "the formal system of maths", ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... >> And indeed, mathematicians do use FOPC. ... >> But mathematics is not confined to working within a particular axiom ... Science would then be the application of math to the real ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... >> And indeed, mathematicians do use FOPC. ... >> But mathematics is not confined to working within a particular axiom ... Science would then be the application of math to the real ...
    (sci.physics)
Quantcast