(MiG Tournament) Physical laws must conform to preconceptions of what is logical

In this debate we are concerned with questions of which physical theories might be true. For example, is it possible that space and time are infinite? Is it possible that space and time are finite? Might time have a beginning? Might it not have a beginning? My opponent maintains that pure logic alone rules out some theories of how the universe works. My position is that the only things that rule out a possible physical theory are (a) a contradiction within the theory itself or (b) a contradiction with observation. For example, one might make a seemingly logical argument that there can be no maximum speed for objects. In ordinary experience, no matter how fast something is going, we imagine it could always go faster. That seems to follow logically from the simple math of addition. But it turns out that reality does not work that way. Special Relativity provides a different way of doing the calculations, and the formulas of special relativity are the ones that turn out to agree with reality. Only observation tells us which theory, as embodied by different mathematics is correct. The speed of light is an upper limit on the speed of an object. Considering Pro's contentions: C1: Contingency upon logic P1. Mathematics is only partially contingent upon logic. Logic is a part of mathematics, so it is “contingent” in the same sense that an automobile is contingent upon wheels. It is a necessary part, but not sufficient. A dictionary definition of mathematics is “the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.” [1. http://www.merriam-webster.com...] The numbers and abstractions are objects of mathematics and statements about how they interact are axioms. In mathematical representations of physical theories the mathematical objects are usually represented by symbols. Axioms are of two types, logical and non-logical. “Logical axioms are usually statements that are taken to be universally true (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory.” [2. http://en.wikipedia.org...] Mathematics is a system for manipulating symbols based upon rules. Pro seems to me to only address the logical axioms in his claim of mathematics being contingent upon logic. That's a small part of a mathematical system. As [2] points out, even a + b = b + a is a non-logical axiom. It is only true because that is the way it is defined for a certain mathematical system. Arithmetic is defined by a small set of non-logical axioms. Philosophers have wanted to know if arithmetic can be proved to be self-consistent can be proved to be internally self-consistent using only the axioms of arithmetic. The short answer is that it cannot. "Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible ..." [3. http://en.wikipedia.org...] In fact, adding an infinite number of axioms is insufficient to allow proof of internal self-consistency. So only part of mathematics relies upon universal rules of logic, and once non-logical axioms are added to usefully apply the math, the guarantee of logical consistency is lost. The properties of mathematical derivations are in any case distinct from the physical objects they describe. Logic is a tool for deductive reasoning, like “All A are B and all B are C, therefore all A are C.” There is no doubt about the result. Mathematics also contains statistical descriptions that describe the probabilities of events. There are mathematical laws that apply to the calculation of probabilities, but the events described are not individually constrained. P2. Physical laws are not contingent upon mathematics. Mathematics is purely a descriptive tool. Physical laws exist whether or not they can be described mathematically. All the laws of nature existed before the invention of mathematics, so it is not possible that they were contingent upon mathematics. We expect anything that can be represented symbolically can be represented mathematically. For example, “x is a random variable” permits a class of statistical data to be treated symbolically. We may then proceed to apply rules to x, and to perhaps establish a correspondence to the physical universe. We might claim that x represents the rainfall in Peoria in a year, and make claims about the average value of x. However, the claims do not constrain the rainfall. In fact, a more accurate representation of the rainfall in Peoria is with fractals, for which there is no average value. The fractal model is that the short term average rainfall varies unpredictably over the eons. Physical laws are also not contingent upon mathematics because we make up new mathematics as required to describe physical laws. Newton invented calculus to describe the Laws of Motion. Calculus proved immensely useful in applying the laws to predict the motion of planets, and observation of the planets verified the Laws of Motion. The mathematics of probability, developed well after Newton, proved essential for describing phenomena of quantum physics, and then for verification of the theory. P3. “What is logical” about physical laws is unknown a priori We know that mathematics cannot be proved consistent, that math is no more than symbolic representation, and that almost anything can be represented symbolically. Mathematical representations of physical laws are nonetheless useful for testing them. Laws are always uncertain, but they can be shown to apply over a large domain. P4. Physical laws describe reality. I agree that is the goal, and that conforming to reality is the test. Consider Multiverse Theory. The goal is to identify a common explanation of the four observed forces in the universe and of the thousands of kinds of subatomic particles that have been observed. Every scientist agrees that the theory is not proved until predictions made by the theory are verified to agree with observation. The first step is to develop a mathematical description of the theory that agrees with all of the known facts about the universe. That's now fairly well along. The next step is to predict new results that can be verified. The Large Hadron Collider supports experiments designed to test predictions. What can we conclude if Multiverse Theory is confirmed? The Theory postulates a universe with eleven dimensions. Confirming the Theory would confirm that we can treat the universe as if it has eleven dimensions. There might be some other way to think of it rather than as dimensions, but the possibility of extra dimensions cannot be ruled out. My point is that the mathematics only works with symbols. How we make analogies to the real world is a separate matter. C2. Physical laws need only conform to reality. Nearly anything can be described mathematically, and mathematics is not guaranteed to be logical. Pure logic is strictly deductive, while nature relies upon the induction found in statistics. The utility of math is to facilitate the testing of conformance with reality.