Saturday, April 26, 2008

Mathematics Platoism and Natural Law

One of the most interesting and vexing questions in (and about) mathematics is the so called discovery question. Is mathematics discovered or invented? The U.S. patent office, in the face of the common wisdom of antiquity, and I would say common sense, has declared that mathematics is invented, and therefore patentable. So is created the situation where one must pay a corporation thousands of dollars to use an algorithm which is the result of the flow of mathematical relationships.
Science News has an article which quotes a paper from last June's issue of the Newsletter of the European Mathematics Society which speaks against the discovery theory of mathematics, called the Plato theory of mathematics.
Platonist note that mathematical statements are either true or false independently of the personal beliefs of the mathematician. In base ten 1+1 will always equal 2. That makes mathematics independent of human belief and existence.
In his article in EMS Davies states:
Platonists believe that our understanding of mathematics involves a type of perception of the Platonic realm, and that our brains therefore have the capacity to reach beyond the confines of the physical world as currently understood, albeit after a long period of intense concentration. If one does not believe this then the existence of the Platonic realm has literally no significance. This type of claim has more in common with mystical religions than with modern science.

In that Davies makes the mistake of many secularist who ignore the theory of Natural Law and many physicists who while realizing that the universe is written in the language of mathematics fail to see the hand of the divine within that writing.
I remained unconvinced that calculus, which circumscribes the behavior of fluids in a piston or electron containment in an accelerator is a happy accident of human invention as opposed to a deep extension of divine will revealed to human knowledge by the grace of God.


Frank said...

Mathematics can be reduced to the study of sets because all mathematical objects are representable as sets. Mathematics assumes only the existence of the concept of "set" satisfying a finite number of axioms (in Godel-Bernays set theory). Mathematicians do this because the danger of adding a false assumption to a deductive system is that the deductive system can then be used to "prove" false propositions.

Natural Law philosophy is considered disreputable by scientist because it "cheats" by continually adding new, untested assumptions and because Natural Law makes predictions about the natural world without accountability for the accuracy of these predictions.

TerryC said...

Not all mathematicians agree with the use of sets as the foundation for mathematics, so stating the concept as a fact is incorrect. It might be correct to say that most mathematicians believe that sets can be used as the foundation of mathematics, but that only a small number of actual mathematical theorems have actually been derived using set theory.
As for Natural law being considered disreputable by scientist, I would think that depends very much on the scientist.