The MAP of PHYSICS
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In 1900 the German mathematician David Hilbert stood before the International Congress of Mathematicians in Paris and gave the world's mathematicians their homework assignment for the Twentieth Century. He laid out twenty-three problems whose solutions he felt would do the most to advance mathematics by the year 2000. This series of essays comprises a preliminary solution of Hilbert's sixth problem, which Hilbert titled "Mathematical treatment of the axioms of physics".
Hilbert opened his discussion of the sixth problem by saying, "The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theories of probabilities and mechanics". Further on he described what he had in mind for a solution of the problem by saying, "If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories". In other words, Hilbert sought as the solution of his sixth problem a kind of Euclidean theoretical physics, one in which mathematicians deduce the laws of physics from a small set of axioms just as Euclid deduced plane geometry from his five axioms and five common notions.
I offer nothing quite so elegant. I present these essays as more in the nature of preliminary surveys and plats from which someone, someday, will devise the proper Map of Physics. Someday, in my vision, a modern Euclid will transform my crude first effort into a truly grand flowchart that lays out the logical connections of a properly rigorous axiomatic-deductive system. Eventually others will extend that flowchart to include the Map of Chemistry and the Map of Biology and thence outward to include all of human knowledge in a Rationalist embrace.
In all that follows, in accepting the challenge to deduce the laws of physics from a small set of axioms, we must accept the conditions imposed by the discipline of mathematical logic. Certainly we must strive to found our axiomatic-deductive structure upon some small number of propositions whose truth to Reality we do not doubt. We must conceive those founding propositions, our axioms, with the property of self-evidence; that is, we must so conceive them that they contain within themselves the testimony that confirms their truth to Reality. And to make our axiomatic-deductive structure as simple as possible we must do as Euclid did in creating plane geometry as a logical system and use the minimum possible number of axioms. We seek simplicity in accordance with Occam's Razor (entities are not to be multiplied beyond necessity) because superfluities add nothing to our understanding but will more likely lead us into error.
From our set of axioms we will proceed to deduce the Map of Physics by the usual process of elimination. Combining axioms and/or theorems deduced from axioms, we will lay out all possible consequences of those combinations and eliminate from consideration those possibilities that yield contradictions. I anticipate that what emerges from that process holds out the prospect of freeing us from our dependence upon serendipity for fundamental new discoveries in physics, of allowing us to question it by using logic to extend it in any desired direction (though we will still need experiments to verify that the answers do indeed conform to Reality). Think of a question, any question that falls within the realm of physics, and if you take enough time and apply enough diligence, you will deduce the answer to it. To give you an example of the kind of question that I believe the Map can answer I will end this prologue with one of my questions, one for which I hope to find the answer someday:
Is hyperdrive possible; that is, can we determine that Reality is so structured that we can someday, with accessible technology, build a device that will enable spaceships to feign flying faster than light, presumably through some kind of hyperspace, and thereby travel to the planets of other stars within lapses of time that we regard as reasonable?
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