(MiG Tournament) Physical laws must conform to preconceptions of what is logical
Voting Style:  Open  Point System:  7 Point  
Started:  8/23/2012  Category:  Philosophy  
Updated:  4 years ago  Status:  Post Voting Period  
Viewed:  3,780 times  Debate No:  25251 
This debate is part of round 5 of ManIsGood's Debate Tournament. Accepted. As with our last debate, I will speak in scientific terminology that my opponent cannot understand, and my opponent will speak in theological terminology that I cannot understand. This will keep readers alert. 

My position can be summarized in the following two syllogisms. The first syllogism shows that physical laws are contingent upon logic and the second one shows that they must conform to logic. I think either syllogism alone is enough to affirm the resolution, so together, these arguments make a strong case. P1. Mathematics is contingent upon logic.
The reason the qualifications of lawness in physics is so stringent is because these are the very qualities necessary to describe reality. Reality itself is universal, stable, omnipotent, etc. It's not to say that a law must be "simple" to be an actual law of the universe, but that when a theory has all the other qualities but lacks simplicity, it's possible there are multiple smaller laws at work. Nitpicking aside, a physical law is clearly something that describes physical reality. 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.merriamwebster.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 nonlogical.
“Logical axioms are usually statements that are taken to be universally true (e.g., (A and B) implies A), while nonlogical 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 nonlogical axiom is not a selfevident 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 nonlogical 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 nonlogical axioms. Philosophers have wanted to know if arithmetic can be proved to be selfconsistent can be proved to be internally selfconsistent 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 selfconsistency.
So only part of mathematics relies upon universal rules of logic, and once nonlogical 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. 

I presented two distinct arguments in my opening round, and I apologize to my opponent and the readers if I did not clearly communicate that these arguments are separate. The second one does not build upon the first. I had hoped that since both arguments reached the same conclusion, that their individuality would be selfevident. From here on out, I will give them names so as to mitigate further confusion. Are mathematicallyexpressed physical laws complete?
What is this debate about? In R1, Pro said, “I will be arguing that just because there exist theoretical models that work mathematically, they cannot be considered to represent actual reality if in doing so they would violate logic.” In my first response I said, “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.”
The context of the debate underscores the resolution. We had a debate in which modern cosmology, called multiverse theory or string field theory was discussed. Theory is welldeveloped mathematically, but is currently not proved to be a correct description of how reality actually works. The theory includes concepts that seem incredible, such multiple universes that coexist in different dimensions as bubbles of time and space in a timeless backdrop. Pro argues, essentially, that even though the math works, we can rule out the theory on the grounds that reality must be logical and such a construct is illogical. Separately, others have argued that time cannot extend infinitely far into the past because it is illogical to suppose that an infinity can exist in reality. Again, the math works, but we are obliged to impose additional standards of logic.
An old example is interesting because the supposed logic of reality cuts both ways. Is the universe infinite or finite? The logical argument that it is infinite is that if it were finite, there would then have to be something beyond the boundary. However, the argument that there can be no real infinity also applies; if real infinities are impossible, then the universe must be finite. Mathematics allows resolution in ways be cannot imagine: space itself is curved, so the universe can be both finite and yet have no boundary.
Keep in mind that we agree that having a mathematical theory that agrees with present observations does not mean the theory is correct. In principle, many theories can agree with current observations, but might need modification when new data shows them limited. As a theory is found to agree with more and more observations, we become more confident of it's ability to make reliable predictions.
Pro seems to distinguish between scientific laws and theories. There is no formal distinction. We now know that Newton's Laws of Motion are only approximations, while the Theory of Relativity is more broadly correct. All of science is tentative. It works in the domain where it has been tested.
In summary, Pro's position is that a scientific theory can work mathematically, but there is an extra test of logic that lies outside of the math that must be applied before a scientific theory can be accepted. My position is that mathematically correct theories can be accepted without any additional tests, because being correct implies they are completely logical.
Pro's Contentions Support the Con Position
It is critical to my position that mathematical theories must conform to logic. Pro has been contending that, and I fully agree. Reality also conforms to logical axioms. Logical axioms are the part of logic that is taken as self evident, that things cannot simultaneously both exist and not exist and similar basic assertions. In reality there are physical laws that extend well beyond the logical axioms. These include the laws of motion, the laws of thermodynamics, and so forth. In the mathematical expression of physical laws, these are embodied as nonlogical axioms. We demand that both reality and mathematical models of reality be logical overall, meaning that no contradictions can be demonstrated.
At this point in the debate, I believe we have established that no theory can “work mathematically” if it violates logic. Logical axioms are universally true, and if a violation is shown by contradiction the mathematics are invalid. Nonlogical axioms are defined for the particular type of mathematics (like algebra, calculus, or geometry), and if they are found to lead to a contradiction then the math is invalid. Of course, it's possible to make a mistake and formulate a bad theory, but the means of detecting the error is at hand. It suffices to show it contains a contradiction.
In this debate, we have also established that new mathematical systems are invented as required to describe reality. Newton invented calculus to formulate the Laws of Motion, and so forth. This important, because it means that mathematical models can be made of anything occurs in reality. As Pro has insisted, math is logical and reality is logical. That's all that's needed.
The interesting part of physical theories relate to the nonlogical axioms. These are the postulated properties of the universe. They are expressed in mathematical form to facilitate testing the theory. It's possible that the axioms lead to contradictions within the theory, in which case the theory fails. It's also possible that the theory will contradict an observation of the physical world, in which case the theory fails because it does not work.
Argument from Contingency
P1. Mathematics is contingent upon logic. Yes, both the logical and nonlogical axioms cannot lead to contradictions. There are many different sets of nonlogical axioms.
P2. The plausibility of physical theories relies upon mathematics. Yes, but this means that both finite and infinite universes are plausible, a multiverse containing bubbles of 3dimensional space plus time is plausible, spontaneous events are plausible, and a steadystate universe in plausible. All are described by logicallycorrect mathematical theories.
C1. Therefore the plausibility of physical theories is contingent upon logic. This only means that theories including logical contradictions are ruled out, and none of the theories at issue in this debate pose logical contradictions. Therefore contention supports the Con position that if a theory is mathematically correct, it is a plausible description of reality. Nothing external is required for it to be proved logical.
Argument from Reality
Yes, reality is logical and therefore nothing need be considered outside of a mathematically correct theory to describe reality. This supports the Con position.
I think the Pro position confuses logical axioms with nonlogical axioms. The logical axioms are the simple rules of logic. They apply universally to math and to reality, but they assert very little of interest. They say that A and B includes A, and other such basic things. They say nothing about whether time or space can be infinite.
As I asserted at the outset, a mathematical theory of the universe is plausible if it is mathematically correct, meaning the combination of logical and nonlogical axioms pose no contradictions, and it agrees with all of our observations of reality. That is all that is required, because it encompasses all that is relevant.
I should clarify that my opening remark about not understanding each other was a joke. But it does reflect the sense of the debate that begs if we are talking about the same subject. I cannot see how Pro's arguments relate to there being something outside of mathematical models needed to describe reality, and Pro claimed I never argued that mathematical models are complete. As I said, it will keep readers alert.


We must not allow ourselves to confuse mathematical plausibility with real plausibility, and this seems to be what my opponent desires to do. He wants to conflate the abstract world (mathematics) with the physical world (the cosmos). This allows him to accept that things like actual infinites are possible, despite the fact that such abstract notions—while mathematically sound—encounter logical issues when applied to physical reality. An imaginary number works in mathematics, but it struggles to find a place in our temporal existence. Accepting preconceptions rejects most of the modern physics proved true The debate is about “preconceptions of what is logical.” For example, Pro believes that it is not possible for an actual infinity to exist because for such to exist would be illogical. Pro claims a scientific theory allows space or time to extend to infinity must be rejected without regard to other evidence of the truth of the theory. I gave several examples of scientific theories that appear to deny logic. “Logic” seems to demands that one can always go faster, yet the speed of light is an upper limit. It seems if space were bounded there would have to be something beyond the boundary, and hence no possibility of space being finite. Science nonetheless proved theories that defy apparent logic are true. So the problem for the Pro position is to determine what appears to defy logic from what actually defies logic. Pro did not respond to my examples with claims of their being within the bounds of our ordinary logic, nor did he deny the wellestablished science of them being true. Pro cites Hilbert's paradox as an example of a preconception of logic that ought to trump any physical theory involving countable infinity. However, Pro's reference [R4.2] says. “[Hilbert's paradox and the like] demonstrate a paradox not in the sense that they demonstrate a logical contradiction, but in the sense that they demonstrate a counterintuitive result that is provably true...” Thus Pro is asserting that if something is counterintuitive but provably true, it must be rejected nonetheless. Much of modern physics –virtually all of relatively and quantum physics is counterintuitive but provably true. Relativity and quantum physics are both mathematically consistent and verified as accurately modeling the real world. Pro makes a distinction between a potential infinity and an actual infinity. There is no such distinction known to mathematics. Mathematics is invented to describe the real world, as Newton invented calculus to derive the laws of motion, yet no mathematician or scientist has invented a mathematics that deal with actual infinities. The distinction is no more than than the bounds of human intuition. Consider calculus, which depends upon the limits of infinitesimally small quantities as they approach zero. We do not insist upon a distinct actual calculus in which the limits are never reached and then demand that the laws of motion be rewritten with actual calculus. Nor do we reject the laws of motion on the grounds that calculus fails a test of intuition. Resolving Zeno's Paradox of Motion depends upon an actual infinity. Ssuppose a very fast runner—such as mythical Atalanta—needs to run for the bus. Clearly before she reaches the bus stop she must run halfway, as Aristotle says. There's no problem there; supposing a constant motion it will take her 1/2 the time to run halfway there and 1/2 the time to run the rest of the way. Now she must also run halfway to the halfway point—i.e., a 1/4 of the total distance—before she reaches the halfway point, but again she is left with a finite number of finite lengths to run, and plenty of time to do it. … And now there is a problem, for this description of her run has her traveling an infinite number of finite distances, which, Zeno would have us conclude, must take an infinite time, which is to say it is never completed. [4. http://plato.stanford.edu...] To simply walk across a room, an actual infinity of decreasing finite distances must be traversed. It is done without a problem. Perhaps calculus students have just not developed the intuition to appreciate the truth underlying the infinitesimals of calculus? I don't think that's the case. Mathematics is accepted because it is both logically consistent and useful. The paradox of Hilbert's Hotel is that all of the infinite number of rooms are full, yet there is room for infinitely more guests. We have no intuition about how that works, but it obeys the rules of logic. Mathematics using the concepts of infinity work to provide results that accurately describe the real world; If we adopt the resolution, we must reject most of modern physics despite it's having been proved true. The correct approach is to not impose preconceived notions of what is “logical,” meaning intuitive, but rather to rely entirely upon actual logic. This allows counterintuitive theories to stand or fall on their merits and not upon limits of human intuition. Any logical theory ought to be allowed We insist that both the real world and mathematical models of the real world obey logical axioms. Logical axioms are basic logical principles, such as (A and B) must contain A. A mathematical system, such as calculus or geometry, combines logical axioms with nonlogical axioms. A nonlogical axiom is like the associative rule for integer mathematics. The combination of logical and nonlogical axioms cannot be proved to be selfconsistent, but it can be disproved by finding even a single case that leads to violation of a logical axiom. Pro spent much effort in R2 and R3 arguing that mathematics has this high level of logical certainty. He's right, it does. That supports relying upon it as the basis of physical modeling. To form a theory of physical science mathematics must be tied real world. Since mathematics is logical and the real world is logical, mathematics can always be derived to model the real world. Mathematics includes statistical models that allow the world to be described without it being deterministic. Mathematics is only a tool for describing the world, but once a mathematical system is formulated it can be tested for logical consistency. With the mathematical systems at hand, physical theories are formed that attempt to accurately model the real world. Physical theories require knowledge of the real world. If the mathematical model of the real world disagrees with an observation, the theory is proved wrong. We have a great deal of data about how the physical world behaves and scientists work to make mathematical models that agree with all that is known. A theory that agrees with all that is known can then be used to predict new results. Validation of the results derives from consistency of new observations with predictions of the theory. The reason that any mathematically valid theory ought to be allowed as plausible scientific theory is because such theories have a high certainty of obeying the logical axioms and also being logically consistent. Rejecting theories on grounds that they are counterintuitive limits science to what humans can imagine. It would reject much of modern physics already proved true. Of course, we agree theories must always be proved consistent with observed data, and proving part of a theory does not prove all of it.
Debate Issues Pro set out In R1 to prove, “.. that just because there exist theoretical models that work mathematically, they cannot be considered to represent actual reality if in doing so they would violate logic.” He then argued in R2 and R3 that mathematical models are logical and that they describe a real world that is logical. In those rounds, he did not give a single example nor did he argue how a theoretical model that works mathematically can fail represent reality. His example, Hilbert's Hotel Paradox, and accompanying argument were presented in R4. But Pro's debate rules given in R1 say that no new arguments can be presented in R4. I rebutted the new argument –being counterintuitive is not a logical paradox but I could have just pointed out that only the Con case had been supported in R2 and R3, and that Pro had not made a prima facie case at that point. The resolution is negated.

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Is KRF using 'logic' to mean intuition/logical preconceptions (as the title would imply) or in reference to formal logic?
My frustration is that GC seems to admit that he has no idea of what was being debated, and so found a technical reason for rationalizing his vote. When the debate topic is "Batman vs. Naruto" the debaters presume significant knowledge of the characters in the debate. I shouldn't be surprised that a debate about the methods of science would suppose substantial knowledge of how science is done. Fewer people understand the methods an concepts of modern science than, say, know how to fix a car. The needed baseline of understanding of science does not exist at DDO to make debates like the current one worthwhile.
In this debate, the resolution depends upon understanding what is possible and what constitutes a logical contradiction. Among contemporary physicists that deal with such things, the opinion is close to 100% that anything that math allows is possible. That is because math is as close to perfectly logical as an tool we have, so if the math works then the theory is prove logical. What eliminates all the untrue theories is not preconceptions of what is logical, but experiments and observations that show what the universe actually does.
KRF is flatly wrong in supposing that for every book accepting the logic embodied in math, there is one opposing it. It's perhaps about the same as opposition to evolution, which has 0.15% of scientists opposed. I suspect that somewhere here is a profession astronomer who believes the earth is flat; the existence of exceptions is not credible. You cannot be a modern physicist and have preconceptions of what is allowed beyond what the mathematics shows to be consistent.
It's not condescending to point out that education may be a defining different here. If I didn't have the knowledge I had gained about logic/mathematics/quantum physics then Pro might have seemed reasonable.
Considering that, Roy has about 100x my knowledge when it comes to logic/mathematics/physics, it's understandable he's frustrated.
Personally, I would have gone at this from a Kantian perspective, but the context of this debate is different (i.e. was previously based on physical theories).