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Can a sound mathematical formula become a science theory if it is constructed using a pattern creation process from sense-data, applied to observations by an inductive mapping, in contrast to deductively 'prove' ?

Or in other words: Can a sound mathematical formula become a science theory if it is not deductively derivable from any axioms (there are no axioms) ?

I have read on an internet forum: "Mathematics does work from axioms, but basic perceptions do not. Mathematics has axioms and the calculations made from those axioms (using the symbols and operations that are defined by the system) can be considered to be theorems of the system. They are deductively 'provable' within the system. But the mathematics system is, in its pure form, entirely abstract. It has no explicit correlation with anything in the real world. In contrast, perception is an inductive process whereby the conclusions are not deductively derivable from any axioms. (There are no axioms.) Instead they are constructed using a pattern creation process from sense-data. In science, mathematics is applied to observations by an inductive mapping. If it is seen that the data can be described by using sound mathematical formula then that becomes a 'theory'. This theory can subsequently be used to make predictions and be tested against further observations."

So, my physics question now is:

Can the sound mathematical formula 'the ear differentiates and quadrates', inductively be applied to provide in dedail descriptions of perceptions of the hearing of beats by the auditory sense, so that the mathematical formula 'the ear differentiates and quadrates' is an candidate to become a 'theory', so that it can subsequently be used to make predictions and be tested against further observations ?

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+1 for being an author. –  Mew Jan 31 '13 at 0:31
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1 Answer 1

up vote 2 down vote accepted

Sure. You're describing the generation of an inductive law, which is done constantly by experimentalists. Theorists will then try to generate as many of these laws from as few axioms (which are usually grounded in very basic sets of these inductive conclusions) as possible.

This is, of course, a wild simplification, but the idea is there--

observations -> rough laws

small subset of laws -> axioms -> mathematical model/theory -> new predictions to test

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