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Cimode wrote:
> A few comments on the following interesting description of relationship
> between math and other scientific areas.
>
> Marshall wrote:
> > Elsewhere, I commented that:
> >
> > Science : Engineering :: Math : Computer Programming
> >
> >
> > Science is a methodology for study that is intimately anchored
> > to the natural world. Physicists, chemists, biologists, etc. may
> > form their hypotheses, but these hypotheses are not interesting
> > until their usefulness is checked against the actual world we
> > live in.
>
> If I follow your reasonning...you imply that hypothesis are interesting
> (relevant) if useful. *Usefulness* is a consequence of sociological
> and environment context (what is considered *useful* in one context
> (cultural, geographic, historic...) may *not* be considered *useful* in
> another context).
Hrm, well, I wrestled with what word to use there, and settled on the bland "useful". My understanding (I'm not a scientist) is that one determines the utility of a hypothesis by testing its predictive ability. Hypotheses with strong predictive ability give us information about how the universe works, which I would propose is interesting and useful regardless of social context. I did not intend a narrow meaning such as "what will make our stock price go up."
> > Mathematics, in contrast, is much the same kind of methodology
> > as the other sciences, but it is not anchored to the natural world.
> > A mathematical idea may be useful all by itself, without needing
> > empirical verification of any kind. Thus we may derive use
> > from hyperbolic geometry without ever going out in to the
> > natural world and testing whether two parallel lines ever meet
> > or not. Indeed, we would not be able to locate parallel lines
> > in the natural world, because none exist there.
>
> // Mathematics, in contrast, is much the same kind of methodology
> as the other sciences, but it is not anchored to the natural world.//
> Vague. Please clarify *anchored in the natural world*.
One never tests a mathematical idea by conducting an experiment. One tests a mathematical idea by doing more math. It is self-contained in a way that chemistry is not. Chemistry has beakers and flasks and huge vats of bubbling chemicals, and also symbols on the chalkboard. Math has the symbols on the chalkboard, but no beakers or anything like them.
Above I noted the example of hyperbolic geometry. Can one conduct an experiment to determine whether hyperbolic or Euclidean geometry is more "true?"
> //A mathematical idea may be useful all by itself, without needing
> empirical verification of any kind.//
> Mathematics can hardly defined by its usefulness as usefulness is
> context defined and mathematics is not... (relevance would be a better
> term don't you think?). Could you give an example of a mathematical
> idea that meets these criterias of definition?
I was not attempting to define mathematics, merely to describe it. If you like, you can substitute "soundness". The thing is, all the good terms for this are specific to math, and if I used math-specific terms, it would defeat my purpose, which was to show the structural relationships between science, math, engineering, and programming.
Marshall Received on Wed Jun 07 2006 - 10:36:25 CDT