The Instigator
KRFournier
Pro (for)
Losing
10 Points
The Contender
RoyLatham
Con (against)
Winning
14 Points

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

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Post Voting Period
The voting period for this debate has ended.
after 6 votes the winner is...
RoyLatham
Voting Style: Open Point System: 7 Point
Started: 8/23/2012 Category: Philosophy
Updated: 4 years ago Status: Post Voting Period
Viewed: 3,655 times Debate No: 25251
Debate Rounds (4)
Comments (39)
Votes (6)

 

KRFournier

Pro

This debate is part of round 5 of Man-Is-Good's Debate Tournament.

Synopsis

RoyLatham and I just finished another tournament debate over the implications of the KCA, which you can reference here:

http://www.debate.org...

In this debate, we argued over whether or not something that is mathematically possible can be considered to therefore be physically possible as well. The context was over the possibility of an actual infinite.

Since Roy and I were again paired up for this tournament, Roy asked if I would be willing to instigate the resolution set before you now. He and I will share the burden of proof for our respective positions. 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. Roy will argue the contrary, that logic itself is better embodied in mathematical equations, so if the equations work, then the physics is at least possible.

Rules

The debate shall adhere to the following structure:

Round 1: Acceptance, Rules, and Definitions
Round 2: Opening Statements and Rebuttals
Round 3: New Arguments and Rebuttals
Round 4: Rebuttals and Closing Statements (No new arguments)

Arguments and source citations must appear within the four rounds of this debate only. Arguments and citations may not appear in offsite links, other debates, forums, or comments.

Dropped arguments are not concessions. Arguments should be seen as whole units from opening to closing statements.

RoyLatham

Con

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.
Debate Round No. 1
KRFournier

Pro

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.
P2. Physical laws are contingent upon mathematics.
C1. Therefore, Physical laws are contingent upon logic.

P3. Reality must conform to what is logical.
P4. Physical laws describe reality.
C2. Therefore, physical laws must conform to what is logical.

P1. Mathematics is contingent upon logic.

The most fundamental law of logic is the Law of Identity. [1] The law of identity is a formal title given to a most obvious tautology: what is, is, and what isn't, isn't. The law of identity is essentially the law of reality. It is axiomatic and to deny its metaphysical necessity is to make knowledge impossible.

Likewise, the most fundamental axiom of mathematics is the Axiom of Equality. [2] A = A. At first glance, it appears we have to ways of describing the same tautology. However, mathematics carries with it something that logic does itself does not require: value. While the Law of Identity describes reality of existence ("If you have A, then you have A."), the Axiom of Equality describes number theory ("If you have A number of things, then it is equal to A number of things.") Mathematical logic is the application of pure logic to quantities.

In modal terms, the logical law of identity is modally necessary (could not fail to exist in any possible world). Mathematics could fail to exist if and only if logic failed to exist. Since logic will not fail to exist, neither will mathematics, so it seems they are co-equal. However, the Law of Identity can surely exist regardless of the existence of the Axiom of Equality, but the same cannot be said of the converse. Therefore, the Axiom of Equality is in fact contingent upon the Law of Identity. Ergo, mathematics is contingent upon logic.

To say it in simpler terms, mathematics was developed by reasoning beings that were already reasoning according to logic before mathematics existed. We intrinsically know the law of identity, but we do not intrinsically know the axiom of equality, even though it is certainly intuitive. In fact, it's intuitive only because it closely resembles the law of identity, albeit in a slightly different context.

P2. Physical laws are contingent upon mathematics.

I will venture to guess that this premise will not be contested given the nature of Con's position. So I will rely on its self-evidence nature for now and defend it later if the need arises.

C1. Therefore, Physical laws are contingent upon logic.

If physical laws are contingent upon mathematics, and mathematics is contingent upon logic, it therefore stands to reason that physical laws are contingent upon logic.

P3. Reality must conform to what is logical.

There are three fundamental laws of logic: the law of identity, the law of non-contradiction, and the law of excluded middle. [3] I submit that if any of these do not describe reality is it really is, then we cannot know reality at all. If my opponent wants to argue that reality might contain contradictions, then I would ask how he knows that anything he himself proposes is not a contradiction. And if he gives reasons for his knowing that some things are not contradictions, then I would ask how he knows his reasons are not contradictions. The fact is we need these laws to even begin describing reality.

P4. Physical laws describe reality.

Physical science, like all sciences, works in the realm of scientific theory. It is the goal of the scientist to test these theories to eventually expose law-like understanding of reality. It is difficult for theories to advance into law, since a law as the following properties as identified by Davies (1992) and Feynman (1965):

  1. True, at least within their regime of validity.
  2. Universal. They appear to apply everywhere in the universe. (Davies, 1992:82)
  3. Simple. They are typically expressed in terms of a single mathematical equation. (Davies)
  4. Absolute. Nothing in the universe appears to affect them. (Davies, 1992:82)
  5. Stable. Unchanged since first discovered (although they may have been shown to be approximations of more accurate laws—see "Laws as approximations" below),
  6. Omnipotent. Everything in the universe apparently must comply with them (according to observations). (Davies, 1992:83)
  7. Generally conservative of quantity. (Feynman, 1965:59)
  8. Often expressions of existing homogeneities (symmetries) of space and time. (Feynman)
  9. Typically theoretically reversible in time (if non-quantum), although time itself is irreversible. (Feynman)

The reason the qualifications of law-ness 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.

Theoretical physics is an important field, as the multitude of theoretical equations and models serve to bring us ever closer to discovering more laws about reality. However, theories—while based in mathematics—are never guaranteed to describe reality, which is why they remain theories until science can determine its veracity as law. Here we must resist conflating theoretical physics—as vital a science as it is—with physical law—the accurate description of reality as we know it.

Just because physicists can develop mathematically sound models of the universe before planck time does not make them representative of reality. Mathematical soundness tells us whether or not we are on the right track, but by itself it cannot tell us if we are correct. In fact, the only reason why so many theoretical models exist is because our lack of complete knowledge of reality leaves room for them. As more physicals laws are discovered, we will have fewer physical theories about them. The reason is that theories are put to rest once they logically contradict reality regardless of their mathematical soundness. In other words, physical laws trump theoretical models because they conform to reality.

C2. Therefore, physical laws must conform to what is logical.

If reality must conform to logic and physical laws describe that reality, then it stands to reason that physical laws must conform to logic.

Conclusion

I have offered two straight forward arguments to affirm the resolution. However, there is one thing I have admittedly taken for granted: the laws of logic. I anticipate that my opponent will want to debate my preconception of logic and I presume he might provide his own. If so, I will aim to defend the absolute and transcendental nature of logic as needed. In the meantime, I present the above two opening arguments to my opponent and invite his critique.

Sources

  1. http://en.wikipedia.org...
  2. http://en.wikipedia.org...
  3. http://en.wikipedia.org...
  4. http://encyclopedia.thefreedictionary.com...
RoyLatham

Con

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.

Debate Round No. 2
KRFournier

Pro

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 self-evident. From here on out, I will give them names so as to mitigate further confusion.

I concede that my first argument as it was stated is invalid, so I am going to change it. Since this is not the last round, this "new argument" should satisfy the rules of this debate:

Argument from Contingency
P1. Mathematics is contingent upon logic.
P2. The plausibility of physical theories relies upon mathematics. (Formerly: Physical laws are contingent upon mathematics.)
C1. Therefore, the plausibility of physical theories is contingent upon logic. (Formerly: Therefore, Physical laws are contingent upon logic.)

Argument from Reality
P3. Reality must conform to what is logical.
P4. Physical laws describe reality.
C2. Therefore, physical laws must conform to what is logical.

Argument from Contingency

P1. Mathematics is contingent upon logic.

It is true that mathematics has non-logical axioms, but it also has logical ones. Moreover, mathematics would be incomplete without those logical axioms. Therefore, mathematics is incomplete with logic. Ergo, mathematics is contingent upon logic.

Con's focus on non-logical axioms is a red herring unless he intends to convince us that mathematics could be complete without logical axioms. In the meantime, if mathematics needs even one logical axiom, then P1 stands.

P2. The plausibility of physical theories relies upon mathematics.

I did not consider the implication of the original wording of P2, and Con was right that it was false. Physical laws themselves are not contingent upon mathematics. However, we rely heavily upon mathematics to formulate these laws and then test them. Indeed, mathematics is usually the initial testing condition. While mathematics never guaranteed that a theory will become a law, it always eliminates theories as being possible in the first place.

C1. Therefore, the plausibility of physical theories is contingent upon logic.

Given the change in P2, C1 is not as strong an argument in favor of the resolution. I can admit that much. However, it is still evidence. Seeing as my opponent has not offered any positive arguments in negation of the resolution (offering only rebuttals so far), I am content that this argument still makes a positive case albeit weaker than I had originally envisioned.

Argument from Reality

P3. Reality must conform to what is logical.

My opponent's rebuttal is straw man, as it refutes an entirely different premise. He even uses the heading, "'What is logical' about physical laws in unknown a priori." P3 is not concerned whatsoever about physical laws or mathematical representations. This premise is about reality. I'm making a blanket philosophical claim that reality—whatever it is—must conform to what is logical.

That is to say, reality will not exhibit something logically impossible. For example, it will never be reality that I am sitting in my chair typing this while simultaneously not sitting in this chair typing this. If we were ever to observe such a phenomenon, then we would rightly assume it has a logical explanation. This is exactly what led to the development of wave–particle duality. [1] While something cannot simultaneously be wave and particle (as we had defined them), it is logical to have something entirely new that is something like a hybrid of the two. We developed a new mental category to describe something that brought pre-exsiting mental categories into logical contradiction thus reinforcing the notion that reality is always logical though our understanding of it might end up otherwise.

In fact, the whole reason wave–particle duality was pursued in the first place is precisely because we expect reality to be logically coherent. If one denies P3, then why wasn't the issue just dropped and the antinomy accepted as is? If we presume that reality itself can contradict logic, then there is no value to scientific pursuit. If reality can be illogical, then even the statement, "The earth is in orbit around the sun" cannot be known because we'd always have to consider that it is also not in orbit around the sun. Science becomes a vain endeavor in such circumstances.

P4. Physical laws describe reality.

As far as I can tell, Con doesn't really dispute this. He says that this "is the goal, and that conforming to reality is the test." In other words, a physical theory becomes a physical law when it has been thoroughly tested to conform to reality. I agree.

Con brings up the multiverse theory, which is an interested case study of how theories live or die by their predictive qualities. I don't have a problem with the multiverse being a useful model, but the multiverse is not a physical law. In fact, there are many theoretical models regarding the state of our universe prior to the singularity. None of these, however, are physical laws, so it doesn't matter which ones do or do not describe reality. They will all be tested until one or none are discovered to conform to reality. So, the existence of the multiverse theory doesn't really dispute this premise either.

C2. Therefore, physical laws must conform to what is logical.

This conclusion follows from the premises, so Con's comment that mathematics can describe anything even though it is not guaranteed to be logical is superfluous. C2 logically follows from P3 and P4. Since my opponent did not seem to succeed (in my estimation) in refuting P3 or P4, C2 stands in affirmation of the resolution. The fact that logic is deductive rather than inductive seems to be red herring. I'm not sure how it affects any portion of this argument.

Conclusion

When Con approached me with this topic, he implied that we would share the burden in this debate. In fact, I mentioned in my opening round that he would "argue the contrary, that logic itself is better embodied in mathematical equations, so if the equations work, then the physics is at least possible." This sentence came directly from our private communication, and he didn't dispute it when accepting this debate. I am disappointed not to see that argument manifest directly, though perhaps Con feels that the argument is natively present in his rebuttals. If so, I will give him the benefit of the doubt and respectfully ask that he present that case in a way that I can rebut more directly.

In the meantime, I have two arguments in favor of the resolution. The first one, revised from my previous round, is evidential. It's not logical proof, I admit, but it still supports the resolution. The second argument is very powerful as it directly affirms the resolution so long as its premises stand.

Sources

  1. http://en.wikipedia.org...
RoyLatham

Con

Are mathematically-expressed 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 well-developed 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 non-logical 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. Non-logical 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 non-logical 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 non-logical axioms cannot lead to contradictions. There are many different sets of non-logical 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 3-dimensional space plus time is plausible, spontaneous events are plausible, and a steady-state universe in plausible. All are described by logically-correct 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 non-logical 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 non-logical 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.

Debate Round No. 3
KRFournier

Pro

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.

Hence, my thesis. Mathematics is useful in separating physical theories with potential from those with none whatsoever, but it does not mean that we should blindly consider such theories as "fatal blows" to competing philosophical arguments. That is what happened in our previous debate. I argued that the KCA points toward a personal creator and Con relied on mere mathematical soundness to flatly reject those implications. This seems to me to be rather dishonest intellectually.

What we have is two views on what can and cannot be real, and that is a pretty big deal. So which view is more rational? The view that looks for only mathematical veracity or the view that looks for both mathematical and logical soundness? Con seems to indicate that mathematical veracity is equivalent to logical veracity, but that can't be the case given the reliance on so many non-logical axioms in many physical theories. Indeed, if mathematical soundness were universally accepted as having automatic logical soundness, then we would not be debating the possibility of an actual infinite to begin with.

I argue that intellectual honesty demands we consider the whole spectrum of veracity when it comes to major claims about reality, particularly issues about the origin of our existence. It is one thing to create mathematical models of transcendent multiverses but another thing entirely to say that, because the model works mathematically, we can go ahead and reject theistic arguments. Mathematics alone cannot tell us what can be, only what cannot be. My first argument—the Argument from Contingency—supports this. Logic must be applied independently from mathematics, not subordinately, to determine what can be. This is what my second argument—the Argument from Reality—asserts.

To see our differences at work, consider the problem of actual infinites. We both agree a physical theory that relies on an infinite set is sufficient enough to make the theory worth exploring. We disagree, however, on the likelihood of such a theory being representative of reality. For him, the mathematics is "all that is required, because it encompasses all that is relevant." I insist that actual infinites reach a philosophical (logical) showstopper that cannot be surmounted.

There are two notions of infinity: potential infinity and complete infinity. [1] Potential infinity "refers to a procedure that gets closer and closer to, but never quite reaches, an infinite end." This is the kind of infinity Con appeals to when discussing the expansion of the universe. I agree that the universe exhibits a potential infinity of time. A "completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done." This kind of infinity is what is postulated in a beginningless physical model such as the multiverse.

The problem is that complete infinities result in a contradiction. Using time as an example, we observe an event A and ask, "How did we get to event A?" The answer lies in an event previous to that, event B. Then we ask, "How did we get to event B?" and so on. We cannot answer the question regarding event A until we've answered the question to event B. But if time is infinite, we will never answer any question about any event prior to event A, thus the question is impossible to answer without a beginning. In fact, it's much worse, because what happens in a complete infinity of time is that we have the case in which the following statement is made: "It is true that event A already happened and has yet to happen at the same time in the same relationship." It leads to philosophical absurdity.

The paradox of Hilbert's Hotel [2] makes this clear as well. Hilbert's Hotel is mathematically sound. However, it cannot happen in reality due to logical constraints. Remember, Hilbert's Hotel is a complete infinite, not a potential infinite. All hotels have potential infinity so long as new rooms can be added. Hilbert's Hotel is a complete infinite, and it leads to the contradiction that "all rooms are empty and full at the same time and in the same relationship." It is illogical. I know my opponent will disagree, but watch his rebuttal carefully to see when and where he equivocates a complete infinity with a potential one.

I hope I have successfully communicated precisely where our epistemologies diverge. Moreover, I hope I have successfully convinced the readers that my epistemology is more comprehensively rational. Mathematics is useful and meaningful in helping us discover the physical laws of our universe, but in the end, those laws must conform also to logic.

Sources

  1. http://www.math.vanderbilt.edu...
  2. http://en.wikipedia.org...
RoyLatham

Con

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 well-established 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 counter-intuitive result that is provably true...” Thus Pro is asserting that if something is counter-intuitive but provably true, it must be rejected nonetheless. Much of modern physics –virtually all of relatively and quantum physics-- is counter-intuitive 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 half-way, as Aristotle says. There's no problem there; supposing a constant motion it will take her 1/2 the time to run half-way there and 1/2 the time to run the rest of the way. Now she must also run half-way to the half-way point—i.e., a 1/4 of the total distance—before she reaches the half-way 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 counter-intuitive 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 non-logical axioms. A non-logical axiom is like the associative rule for integer mathematics. The combination of logical and non-logical axioms cannot be proved to be self-consistent, 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 counter-intuitive 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 counter-intuitive 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.

Debate Round No. 4
39 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Enji 4 years ago
Enji
If I were to vote (I can't), I think I would vote Con. Con showed that plausibility of mathematical explanations for the universe are not dependent on what is preconceived to be rational, and that results that initially seem counter-intuitive can be logically explained. I felt that Pro's argument that reality must conform to what is logical fell short in demonstrating that reality must conform to what is logical in both the intuitive sense (as counter-intuitive things can be shown to be logically, mathematically, and realistically correct) and in the formal sense (as Con pointed out, the non-logical axioms are needed just as much as the logical ones in order to be applied to reality).
Posted by Enji 4 years ago
Enji
"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"

Is KRF using 'logic' to mean intuition/logical preconceptions (as the title would imply) or in reference to formal logic?
Posted by RoyLatham 4 years ago
RoyLatham
@MIG, It's my job in the debate to convince GenesisCreation and everyone else that my position is correct. He has the right to not be convinced and to vote as he pleases.

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.
Posted by RoyLatham 4 years ago
RoyLatham
Larry the Cable Guy poses the problem: "Y'all think us rednecks are stupid, until ya need us to start your car." The common error error of college professors is that they suppose that because they know a lot about one thing, they have the knowledge problem licked and therefore must know a lot about everything. I know a hundred times more about science than the average young earth creationist, but the average young earth creationist knows a hundred times more than I do about raising children. That's because I've never raised any children, and many of them have. Moreover, I recognize that teaching a child to be honest is immensely more important than knowing how old the earth really is.

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.
Posted by Man-is-good 4 years ago
Man-is-good
@RoyLatham, what issue do you find with GenesisCreation's vote?
Posted by Man-is-good 4 years ago
Man-is-good
Only as an indication that the debate is wide enough in scope to permit interpretations that are the subject of heated discussions. However, I am sometimes attracted to the more theatrical side of the matter...so to speak.
Posted by Man-is-good 4 years ago
Man-is-good
As the moderator of this tournament, I must confess that I like that there is some controversy brewing here...
Posted by Wnope 4 years ago
Wnope
Seriously though, Roy's dead on when he says a lot of y'all are using englightment-era thinking about science not, say, 19th century and up.
Posted by Wnope 4 years ago
Wnope
Btw, Roy has a point.

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).
Posted by KRFournier 4 years ago
KRFournier
Just check the con box and click Cast My Vote. You can change a vote at any time.
6 votes have been placed for this debate. Showing 1 through 6 records.
Vote Placed by Wnope 4 years ago
Wnope
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Reasons for voting decision: Pro reformulated his arguments after seeing how weak they were, but his reformulations did not a dent in Con's case. Pro showed extreme lack of understanding when it comes to the application of logical/non-logical axioms to "reality." As Con demonstrated, Godel and Hilberts paradox show that Pro's ideas on the constraints of mathematics are misguided.
Vote Placed by TUF 4 years ago
TUF
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Reasons for voting decision: I did find this debate a little repetetive, as 5 round debates often do. I believe conduct, sources, and S/G are all a tie in this debate. What it came down to for me was the argument Roy based on reality, and how they contrasted to those of KRF's. I had a hard time buying the mathematics pre-conceived notion of obtained logic, in comparison to highly logical, and plausible theories on physical (multi-verse theory, etc), that Roy mentioned. What is logical, is contingent on what we know.
Vote Placed by GenesisCreation 4 years ago
GenesisCreation
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Reasons for voting decision: RFD in comments
Vote Placed by FourTrouble 4 years ago
FourTrouble
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Reasons for voting decision: RFD in Comments.
Vote Placed by larztheloser 4 years ago
larztheloser
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Reasons for voting decision: See comments.
Vote Placed by The_Fool_on_the_hill 4 years ago
The_Fool_on_the_hill
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Reasons for voting decision: RDF