Re: Programming is the Engineering Discipline of the Science that is Mathematics


Marshall wrote:
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."

I see...you mean *useful* as a charateristics of hypothesis tthat can
represent nature in a thrustworthy and reasonnable manner. Don't you
think *reasonnable* would be a good substitute to *useful* which would
then become a consequence and not a characteristics of science?

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.
I see...
What about mathematical ideas that are generated or invalidated from
observation of computing?
Several mathematical theorems using Reccurence (recursive) Reasonning
to predict integer values in series were proven wrong as a consequence
of observing that they work with small observable values but become
wrong when values become large enough. These demonstrated mathematics
theorems have been invalidated thanks to observation of values large
enough and sufficient computing power. As a result these theorems were
reconsidered as false and new *correcting* theorems emerged.


Above I noted the example of hyperbolic geometry. Can
one conduct an experiment to determine whether hyperbolic
or Euclidean geometry is more "true?"
I do not know. But above is an example that demonstrate the influence
of observation over math. I think there is at least a bidirectional
relationship between math and nature.

//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.
Very interresting indeed...But don't you think that this relationship
is not as unidirectional as you imply?

Marshall

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