>>> Introduction

Lately, I have found myself in contentious discussions--generally with theists of one stripe or another--about what the fundamental laws of the universe actually are. It is pretty much unanimously held that there are fundamental laws of Aristotlean logic which are taken to be applied at all levels of this world, and beyond. These laws are assumed to be so sacred that God, Himself could not disobey them, even given that He was omnipotent! In some sense, it is held, these laws are so fundamental that the world cannot be sensibly conceived of or thought about without using and assuming them. Without them, our brains simply cannot operate--and neither could the universe. Or so it is assumed, at any rate.

For reference, by way of the normal definitions, here are some links to standard discussions of these laws:
http://en.wikipedia.org/wiki/Law_of_identity
http://en.wikipedia.org/wiki/Law_of_noncontradiction
http://en.wikipedia.org/wiki/Law_of_excluded_middle

When I encounter a belief system so seemingly incontrovertible, so widely accepted, so ingrained... Well, I can't help but wonder if this might not be a paradigm in need of shifting! Perhaps these laws need to be re-examined. Perhaps they are the proverbial "albatross around our neck" that is somehow in the way of further progress in the fields of ontology, logic, and other areas that seem to me to be kind of stuck in the muck. I ask myself, how fundamental are these laws, really? Is it really true that the universe would "blow up" into utter chaos if we were to stop jealously guarding these rules with such fervor?

In any event, I have recently been reading Feynman's book "Quantum Mechanics and Path Integrals". In the course of this study, it occurred to me that maybe, just maybe, there might be a little wiggle room in at least one of these sacred cows, these so-called "Laws of Thought". Maybe, just maybe, the Law of Noncontradiction is not quite as basic as it might appear: Perhaps it can be derived from a set of looser assumptions. Anyway, that is the position I intend to take in this thread. I want to see how it might be possible to use elements of quantum logic, calculational methods from the Path Integral approach, and a few other considerations to derive this law from even simpler assumptions, or even from its polar opposite--a "Law of Contradiction," so to speak!

This agenda is not an easy row to hoe, I'm afraid. Virtually everyone I have engaged on the topic has deep-seated, intense feelings about the inherent truth of this rule, almost to the point of declaring me a heretic--both scientifically and religiously--for even suggesting such a thing, in the first place. I imagine that I will get a lot of opposition to my program here, as well, assuming anyone pays any attention to this thread at all. All I ask of anyone who wants to participate is this: If you can bring yourself to do so, at least hear me out. I make no claims of expertise in matters of logic, quantum theory, or anything, really. I'm just a guy with a crazy idea that's been nagging me, and I'd like to explore it just to see where it might lead. I just want to be permitted to present the idea. Nobody needs to believe it, or change their belief system. All you need to do is engage in a bit of a hypothetical, for the sake of argument.

Before I get to the meat of the matter, I'd like to set the stage with a few quotes on the topic of open-mindedness from one of its most ardent practitioners. After that, after some sleep, I'll take on the task of explaining my thoughts as best as I can.



(On that note, I am going home to bed. More tomorrow, time permitting.)

Hello Randall, I admire your courage in presenting a set of deeply counter-intuitive, and I believe wrong, propositions.

The law of non-contradiction says a statement cannot be true and false. This means any apparent contradictions conceal an underlying consistent reality. Arguments against this principle of logic rely on sophistry, as there are no real examples of statements that are true and false. Apparent examples such as 'this statement is false' do not refer to anything real.

The law of identity says a thing is what it is and is not something else. It means if a thing appears to be something else, we do not understand it. This can be a very helpful principle to use in critiquing delusory thought, where people imagine that things are other than they really are.

Randall asks if questioning these laws of logic would invite chaos. I would say yes, as these laws are at the foundation of the theory of cosmic order, the view that there is one universe that we live in, and that scientific cosmology provides a unique story about reality. Without logic, there is no cosmology. If our reasoning processes are unreasonable, we have no access to truth. Modern life gives much basis for trust that scientific method provides reliable access to reality and truth.

Logic offers the hope that human intelligence can perceive reality. Questioning the basis of logic destroys the basis of empirical knowledge, casting all claims into radical doubt.

The laws of logic offer a basis for philosophical critique of contemporary thought, especially supernatural thought that conflicts with scientific evidence. For example it is a contradiction to say that Jesus Christ was a man and yet was born of a virgin, because part of the analytical definition of man is carrying a father's DNA, and virgin birth lacks a human father. Without the law of non-contradiction, we have no basis to critique the contradiction between the religious claim and the definition of its term.

Where the law of identity becomes difficult is at the subatomic level. If I understand correctly, what Einstein called 'spooky action at a distance' appears to say a quark has a non-physical connection to another quark, so in a sense it is not just itself but also something else. Logically, from the axiom that we have one self-consistent universe, there has to be in principle a physical reason for the mysterious observations of quantum mechanics. Similar axiomatic knowledge, relying on the laws of logic, can be applied to other difficult scientific discoveries such as the uncertainty principle.

As to God, I would say God should be understood as cosmic force rather than as a supernatural entity. Universal natural force operates omnipotently through the laws of logic, seen in the consistency and universality of the laws of physics. God's power, in this natural sense, is evident in that nothing can ultimately stand against natural processes such as cause and effect that are purely logical outworkings of matter in motion. Our universe operates in accordance with logic, so God can be said to exercise power through logic. If we cannot see the logic, it just means there is a deeper causal process at work that we cannot see. Imagining a God who could act in conflict with the laws of nature is in contradiction to the claim that anything real can only operate through natural law. To anthropomorphise, you could say God chose the laws of physics and logic, and exercises omnipotence through them.

Well, Robert, you have stated relatively accurately the position which I am going to call into question. When I have done so, it goes without saying that you are free to review my arguments, and either accept them or reject them as you see fit.

While I am formulating my next installment, perhaps as an exercise you might like to ask yourself this question:

Is the law of noncontradiction
1a) logically prior to God? (Thus God is forced by circumstance to obey this rule?)
1b) Or do they have the same logical status, being part and parcel with each other?
1c) Or does God freely determine that this rule applies to this universe, "by design"?

If you can think of some other possibility not included here, I'd love to hear about it, and add it to my list.

BTW, thanks for your input on the .mp3 matter. Did you happen to listen to them? Did it help you understand my analogy?

(Apparently, others found them to be somewhat helpful.)

b

Under assumption 1b) we see (I hope) that the Law of Noncontradiction has precisely the same degree of explicability that God has. (Which isn't much, from a theistic POV. God is a given, and so would this law be, in the same way--being that they are "joined at the hip".)

The point being, not that God is assumed by me! But that this law is a basic assumption, or postulate. It has no further explanation that I have seen, but is taken to be a self-evident proposition, in the same way that a theist commonly assumes the existence of God, or others might simply assume the universe exists, and put the universe and this law at the same 'existential' level.

Anyway, to continue:

>>> Working Definitions

As a preliminary, I think it is best to get some semantic issues out of the way, up front:

I. What is a contradiction, anyway? How do we know when we have one?

Are there different kinds of contradiction, and are they all equally verboten? In other words, are there spheres of thought or human endeavor, or perhaps even physics that are more subject to this law, more "sensitive" to contradiction than others? For instance, if the noosphere includes all human thought, then that would presumably include dreams, for example, which seem to be thoughts. But we rarely trouble ourselves over the contradictory aspects of our dream worlds. These dream-thoughts can include all sorts of unphysical (even senseless) ideas that the real world appears to reject out of hand. Furthermore, there are para-consistent logic systems that are explicitly designed to avoid the explosion of provability that would infest any 1st-order logic system that could be shown to involve a contradiction, or an inconsistency, however minor.

Now, 1st-order logic apparently has a pretty solid definition of what is meant by the word "contradiction", to wit: Any statement that includes or entails the conjunction of terms (A & ~A) has truth value F. But 1st-order logic is a bit more well-defined than most other spheres of human concern, and it is not always clear that logic is speaking about anything "real". Also, we note that different logics seem to apply to different objects in the world and in our imaginations, and so we usually require some demonstration that our momentary "logic of choice" is appropriate for the system under consideration. This problem is what I call "logical grounding": Our atomistic A's and B's and C's have to have some definite referents in the actual world, or else we aren't quite sure how seriously to take a given logical argument. Further, in 1st-order logic, it is perfectly possible to reach erroneous and unreasonable conclusions by arguing perfectly from incorrect premises. Establishing the truth of our premises is often quite problematic, since so many categories in the world at large do not have the absolutely clear dividing lines that we are modeling when we choose to apply 1st-order logic principles.

1. All men are mortal;
2. Socrates was a man;
3. Therefore Socrates is mortal. Q.E.D... Right?

But wait! Some people will say that Jesus was a man, and that he is still "alive", in some sense, and will continue in that capacity. So, if we substitute "Jesus" for "Socrates" in the classic syllogism, we are forced to the unhappy conclusion that Jesus is dead. Done. In Cleesean terms, "definitely deceased"! Or, is it more "reasonable" to assert that premise 1. is false?

Even if we absolutely disallow divine interventions, a further problem arises when we consider that the quantifier "ALL" is hard to establish, as a practical matter. All men? How can we tell? All men are not finished living and dying as of the moment. What if Big Pharma invents the cure for death? Wouldn't that mean we'd have to rethink premise 1. again? "ALL" would have to be narrowed to mean "ALL, so far", or "ALL that we know of" or something slightly less universal than its standard meaning.

It seems to me that the only way one can be sure about the truth of a syllogism like this is if you abstract its terms from the real world. If we say
1. ALL A are B;
2. C is an A;
3. Therefore C is B;
... we seem to be on safer ground. But this only works out if I don't ever tell you what A, B, and C actually are! That is, if I don't "ground" the syllogism in some matrix of actuality; if I let it dangle in some purely formal "world" where A's and B's and C's, and 'ALL's and 'is's don't have any 'existence', except as empty symbols. If we actually specify any "real world" definitions for these terms, the perfect universality gets lost in the process. At best, it can only be approximately true, when taken as a model for the real world. In other words, it becomes more like a rule of thumb than an incontrovertible, immutable "Law of Thought".

<Break for lunch>

<After lunch>

Logic has its utility, but not so much as was originally hoped for by its creators/discoverers. Really, it serves as a set of stepping stones from one purported truth to another. But if those stepping stones don't make it all the way to the river bank, the "ground", then how can we get onto them, in the first place?

But, back to semantics...

Dictionary.com offers this set of definitions:

con·tra·dic·tion   
–noun
1. the act of contradicting; gainsaying or opposition.
2. assertion of the contrary or opposite; denial.
3. a statement or proposition that contradicts or denies another or itself and is logically incongruous.
4. direct opposition between things compared; inconsistency.
5. a contradictory act, fact, etc.

Now, 1. is clearly something which is physically possible. It happens all the time, in politics, and in many human relations. 2. is also possible. Assertions are a dime a dozen, and half of them contradict the other half! And 3. seems indicative of the human condition; most people aren't very logical, and rarely examine all their beliefs in anything like an exhaustive manner. When this does happen, it often transpires that we find inconsistencies in our beliefs and thought processes that need ironing out in one way or another, and we change our minds a little, or a lot, depending.

4. is the first one that even alludes to a physical circumstance. It talks about "things" compared, which could include or refer to physical items. We could, for instance, hire two different surveying teams to measure out a plot of land, or a political boundary. I recently had occasion to hire a surveyor to determine the boundary of my property. And wouldn't you know it? His measurement was something like 4" off of the survey my neighbor had done! So I asked him about it, and he told me that this was very common--in fact, it happens most of the time! But, who is going to go to court over 3 or 4 inches worth of back yard? So, for the most part, these discrepancies just get ignored or glossed over, unless they happen to be large enough to matter.

Now, one might argue that this boundary is not a "physical" thing, but an abstraction, an idea, that can be in conflict with someone else's idea. But there are cases where this position just won't do. Here is a documentary about one such problem:
http://topdocumentaryfilms.com/how-long-piece-string/
(I'll be thinking about my next post while you watch it.)

Now, def. 5. is interesting. Can facts be contradictory, in themselves? Or--as is often maintained--is it simply our various conceptions or opinions about what the facts are? I mean, if facts actually contradicted each other then this circumstance would easily account for why the Republicans & Democrats each have their own facts, wouldn't it? They'd simply both be right!

But we all have this idea that somehow, reality itself is contradiction-free; that facts are facts, and if they appear to be contradictory, then that isn't about the facts themselves, but about our perceptions of them. Maybe our perceptions are flawed, but the facts themselves must be internally consistent. (Mostly, it's the other guy's perceptions that are flawed, right? The facts are what I perceive them to be, since I myself am so very perspicacious! I invariably see right through to the heart of things, to "reality"... At least, that's what it seems like to me, just as it seems that way to the selves of all the people I argue with.)

But do we really have the experience sufficient to back up the idea that reality is free from contradiction? It sure doesn't look all that free, what with all these different takes on the "facts". I mean, my reality is clearly not the same place as Rush Limbaugh's or Ayn Rand's. Insofar as their universes are overlapped with mine, I cannot see how these vastly different reports of "the facts on the ground" can possibly be thought to apply to the same "ground".

In spite of all this experience to the contrary, there is something deeply compelling about the position of classical realism. Only by working under the assumption that our perceptions indeed have referents does it make much sense to act on the basis of our perceptions. It only make sense to run away from a tiger that actually exists. And we can only act if we are possessed of physical bodies with legs that can be impelled to run.

>>> Sum-over-Histories and Path Integrals

I will not endeavor to explain all the details of Feynman's approach to relativistic quantum field theory any more than I can help it. There are plenty of resources out there for one to read, if they need to. For the lay person, a good start might be Feynman's "The Character of Physical Law", which presents the implications of the theory in broad outline, without all the math. A more basic understanding can be had from his "Quantum Mechanics and Path Integrals", which gives all the derivations and proofs of how this approach encompasses all the good parts of Schrödinger's Wave Mechanics, as well as Heisenberg's Matrix Mechanics, neither of which took into account the "relativistic" part with much gumption or ambition. As a result, these older, weaker methods have largely fallen by the wayside, and are primarily of interest only for historical reasons, at this point. But this book is very math heavy, and you'd better be prepared to take a year off to really get down to the brass tacks, and another year to learn about partial differential equations, if you don't already have the math. Whatever level of detail one might choose to explore, it is indeed fortunate that Feynman himself has given us these books, for he is not only a principle inventor of these ideas ("the horse's mouth"), but also generally acknowledged to be one of the finest explainers and educators in the business. In addition, he is a very entertaining speaker--a triple threat.

Instead, I will quote a few highlights from WIKI, by way of support for the above contentions:


As you can see, WIKI thinks quite highly of this approach. There are certainly other interpretations of quantum field theory, but this one is very much in use, and is the generator of all those omnipresent stick-figure-like "Feynman diagrams" with the squiggly lines, and loopy circles. These visual aids are the way physicists "count on their fingers", to keep track of all the infinite possibilities and cancellations which one must account for, if one hopes to get those phenomenally accurate answers that they are always (rightfully!) bragging about. With patience, a good calculation can result in agreement between theory and experiment out to 10 decimal places, and narrows down that 11th digit by a lot,--an accuracy unmatched by any other theory in existence. (General Relativity comes close, but still misses this mark by a few digits.) In other words, Relativistic Quantum Field Theory is the truest theory ever invented, if your measure of truth is agreement with experimental results. The Path Integrals formulation is the calculational basis for this remarkable degree of alignment with reality. We should well remember that before the invention of it this level of accuracy was unheard of, even in the old versions of quantum theory.

So, being that it's the best theory ever, fact-wise, it seems to me we probably ought to at least give it half a chance to speak for itself, and take seriously what it has to say about the real world--even if we might not like what it has to say. (Sometimes, the truth hurts... But that doesn't mean we shouldn't hear it, anyway.)

>>> So, What Does It Say?

Well, I'm glad I asked! It says a lot of things. But one thing in particular is important here. It says that when a particle goes from place to place, from A to B, it doesn't just go one way. It goes EVERY way. If A is this computer screen, right in the center of this letter 'o', and B is a certain rod cell on your left retina, then the old way of thinking was you could hold a string from that 'o' to your eyeball, and that would closely describe the actual path a photon would take: A straight line. The straighter the better. One single particle, one single path.

But that isn't what the Path Integral model says at all. Instead, we have to add ALL POSSIBLE paths, with their relative phases, and the cancellations they make, as they interfere with each other. That is, one single photon goes an infinity of ways to get from A to B. And these paths aren't just straight lines, bouncing of mirrors and stuff. No, they're all different shapes and lengths and speeds, even--paradoxically--speeds beyond the speed of light! There's paths that go in parabolas, and spirals and curly-qs, and ones that go around the moon, and back to your retina. Every curve and shape that you can imagine--and many more that you probably didn't think of.

Now, for each of these bizarre paths, we note that there is a "mirror image" of it: The function describing any particular path always has an inverse function--the same with the sign reversed, so to speak, such that they pair up, and precisely cancel each other's effect on the final outcome. Except, I can't really say "always", because if EVERY path totally cancelled, there'd be nothing left of the photon to hit your retina. As it happens, there are a relatively few paths that are close-in to the classical trajectory that only partially cancel, since when you get close enough to the "geodesic", the trajectory functions start to constructively interfere, more than the destructively interfere. On path, right down the geodesic has no cancelation at all. But by itself, it is to weak to account for the whole energy of the photon: It needs the paths close in to help it along. Above, I said "relatively few", but even this small spindle of paths actually has an infinity of paths in it--assuming that space is continuous. (If space is also quantized, then this is merely a very large finite number, but it is not yet clear that space is quantized, so we'll gloss over this issue for now, and simply go with the theory we know works, and take our infinities like men.

The upshot is that all these paths, including the one on the geodesic, have precisely equal claims on existence. If you leave any of them out of your calculation, you simply get the wrong answer.

In spite of all this cancellation, it isn't fair to pretend that these paths don't exist, however. You can't just say that they are "calulational aids", because that would make the ones near the geodesic mere "calculational aids", and the one which coincides with the geodesic (the classical trajectory) would suffer the same fate.

If you happen to own a tuning fork, you can see why this must be so. The sound waves that each tine produces have the same thing going on. If you strike it, hold it up to your ear, and twist it around just so, you will notice that at a certain angle, you hear only silence. This is the angle at which the separate waves from the two tines cancel each other out. You hear nothing, but your friend across the room can still hear it, because the waves really are there, radiating away. If one of the tines were to suddenly stop vibrating, you'd hear the wave from the other tine. The only reason you don't hear it is because of the existence of the canceling wave. Both waves have to exist equally for you NOT to hear anything!

In the same way, according to the Path Integral formulation, the required cancelations can only occur by virtue of the existence of the inverse wave arriving out of phase by precisely the correct amount to cancel. If we arrange matters so some of them are missing, we would find that these cancelations fail to occur, and instead of a dot on your retina, you'd get a rainbow smearing of the photons, because you'd have made a diffraction grating, in effect. (Or you might have just sent the photon to the moon, or your friend's retina.) And of course, that agrees perfectly with experiment.

<Time for bed>

(Not that anyone seems all that interested, but I need to finish out my thought, so I will continue.)

>>> Rules From Randomness

Randomness has a reputation for being rather... well, "random", is the best word, I guess. That is, it is unpredictable and seemingly senseless. The quintessential random system that Einstein liked to talk about was the dice game; We all know the famous saying, "God does not play dice with the universe." Of course, no matter how deeply Einstein held this philosophical position, he was unable to get God to help him out by proving it! Apparently, the vacuum of space is one giant crap shoot. It has all these "virtual particles" randomly appearing and disappearing, and knocking each other about without any obvious cause or rhyme or reason. More down to earth, we find it quite difficult to predict many detailed aspects of our lives, like what our shares of IBM are going to do the day they release the earnings report, or whether it will be sunny or rainy this Thanksgiving at Grandma's house.

In this giant casino which is the universe, we note that the House has a certain edge, though. The House knows something your average gambler doesn't: The more random the Game is, the more Lady Luck has nothing to do with it! The human gambler has all these prejudices--lucky numbers, birthdays, etc., that cause him to bet with a certain pattern. These biases are never predictive of actual outcomes, however. Regardless of how many times the dice may have come up snake eyes in a row, the chance of that happening again is still precisely the same--history, notwithstanding. The dice have no memory of history, and have no idea that they just came up snake eyes 10 times in a row, so they never factor this into their 'calculation'. On the next role, the probability of that outcome is still exactly 1 in 36, and the probability of throwing a seven is still 6 in 36.

If we examine only a small number of dice rolls, we will observe that it's quite hard to say what the "pattern" is, or if there is one at all. Here is a set of 10 dice throws:

{9 6 3 5 3 4 4 7 9 9}

Our brains are very good at spotting patterns in data like this, though. So good that our brains routinely find patterns even where none exist! For example, the first three numbers here are all divisible by 3, which might suggest a "theory". (We might also know someone whose telephone number is eerily similar to this number, maybe by transposing a digit or two, or letting one be something else. Our pattern recognition skills are pretty flexible in this regard, and will often be quite tolerant of minor changes like these, in its desperate search for sensibleness in the world.) I could, for instance, add 1 to the '5', subtract 1 each from the '4 4 7', and "see" the divisibility by 3 as a general rule for this set--with exceptions--in the sense that all these numbers are "close to" being divisible by 3. If I have a pet theory like that, it is sometimes easy to forget the mathematical truth that ALL integers are 1 number away (or less) from being divisible by 3; If a given number n isn't a multiple of 3, then either n+1 is, or n-1 is. But my pattern-recognizer / pet-theory-generator may or may not be cognizant of this brute fact, and so may very well see this "pattern", just the same, once it has been suggested.

Other calculations can be performed on this set, as well. We might take the average, for instance, and notice that these numbers are "close to" that number, although not quite as close as they were to the "divide by 3" pattern, above. In this case, the average is 5.3. Suppose I take 100 dice throws and average that? By modeling this process in a spread sheet, I can easily throw virtual dice lots and lots of times. Here are some results for averaging 100 throws at a time:

{6.76, 6.78, 6.94, 6.86, 7.3, 6.68, 7.04, 6.68, 6.84 6.6}

One thing that immediately jumps out of this set is that the numbers are all within spitting distance of the number 7, the proverbial "Lucky Number". There's no 5 point somethings or 9 and a little. Is this just a coincidence? Is it an artifact of my pattern recognizer providing a false positive, like in the multiple-of-3 pattern? Or is it somehow a real property of dice, that they are going to center around 7, on average?

Let's try it again:

{6.94, 6.84, 6.68, 6.9, 6.94, 7.62, 6.82, 7.2, 7.02, 7.04}

Hmmm... Again, the numbers all come out to very near 7. (I think I might be seeing a real pattern developing!)

I performed this experiment 100 times, with a 100 rolls in each test. The lowest number I observed was 6.16, and the highest number was 7.66. (Curiously, test #49 came out to exactly 7, its own square root... What do you think that means? "Coincidence"?) My theory that these average numbers are going to continue to hover around 7 seems very, very sound. Apparently, the universe of dice pairs has a bias in it. This physical system "likes" the number 7, and is quite adamant about sticking to this rule. In fact, I submit that if I average out my averages, I will find that these meta-averages are even nearer to 7. I tried this, and found it was indeed true. The highest value was 7.118, and the lowest value was 6.844... much closer to 7. It's like a conspiracy. The dice just insist on having this number for their average. Every throw, individually, somehow adds its little variations, but taken together, they all work it out amongst themselves to generate this one overall number. How do they know to do this? How do the dice throws "communicate" with each other, and all fall in line to make this "Rule of Sevens" into a reality?

Randall, I have read your posts and found them interesting. However, I still consider you have said nothing that seriously challenges the logical law of non-contradiction.

The example of Jesus Christ as both man and immortal only shows that religious thought is able to maintain antinomies, ideas the hold opposites together, in ways that contradict observation. The belief is intended to import a miraculous dimension into thought that flatly rejects the scientific method. So if we want to use science to understand who Jesus was, we have to consider the impossible claims of faith as allegory for something possible.

Feynmann's theory that photons travel on all paths simultaneously is highly counterintuitive. If I put something in between my eye and an object I can no longer see the object. Light does indeed travel in straight lines, bent only by gravity. This is why we see stars where they are and not somewhere else.

I am interested in the contradiction between quantum mechanics and general relativity, although I do not pretend to begin to understand it. You might use this as an example, regarding how the calculation of the cosmological constant differs wildly between GR and QM.

I remain basically Einsteinian in my views on God and dice, except that I do not discount the possibility of some fundamental randomness and unpredictability at the quantum level. Einstein used the logic of necessary truth derived from Kant and Aristotle. It starts from the axioms that the universe exists and all facts are consistent with each other. Apparent inconsistency means we do not understand the facts properly. Truth is a necessary property of nature.

Your observation of contradiction in political beliefs operates at the level of values, beliefs about what ought to happen, not facts, knowledge of what actually occurs. Value contradiction is a daily event, as people have differing ideas of the good. Where this becomes interesting is when we seek the evidence for different values, and ask what the consequences of different actions may be. Actions do have consequences, and these can be measured empirically as a way to base values on facts.

I'm not sure what your point was in showing that long term dice throws average 7. This is simple statistics. It proves there is an algorithm built into the structure of dice, such that the most likely number and long term average is 7. This can be proven both deductively, through algebra, and inductively, through testing. It is necessarily true, for perfect dice thrown randomly.

What the dice example illustrates to me is that Feynmann's path integral ideas are a very long way from bringing together deductive and inductive reasoning into a unified understanding as necessary truth. In the absence of a theory of everything, speculation about non-linearity of photons looks more like a problematic staging post on the path towards a coherent theory.

<Robert, I didn't see your post above until this one was submitted. I'll see about responding to it later. Thanks for your interest.>

>>> Contradictions and Averages

Well, the answer is that dice have no teleology. They don't need it. The only "conspiracy" involved works as follows: For every 1 that shows up in my sample set for a single die, in all likelihood there is a 6 somewhere to partner it with. Add these, and you get a 7. Likewise, for every 2, there's a 5, which also adds to 7; and the 3's partner with 4's. The symmetry of this situation is not perfect. If it was, then all dice throws would equal 7. But the more dice I throw, the more likely it is that my results will have this canceling symmetry. The average of the faces on a single die is 3.5, the two dice are summed in our game, so the net for 2 dice is 3.5 + 3.5 = 7, our "Lucky Number".

That is, for each face, there is an opposite face to contradict it. It is "opposite" across the "axis" of 3.5, the average of the six faces. Any axis through the center and face-center hits two faces that total to 7. For each data point in my set of 20,000 single die throws, there is one that came up the opposite side, for virtually every data point. Once I am done canceling out these "contradictory" results, I am left with a very small number of non-matching pairs, which have a very small effect on the average, in total.

Now, is it proper to call the opposite faces of the die "contradictory"? Is it fair to say that one die physically contradicts another? Clearly, one die, when thrown, comes up with 1 face at a time, and only one. But a two (or more) die system, or else one die, thrown repeatedly, CAN have this "contradictory" situation. In the sense of definition 5. above, "5. a contradictory act, fact, etc.", the "fact" of one die face landing up has a contradictory fact either elsewhere, or "else-when" in the data--provided only that the data set is big enough.

>>> The Reality of Contradictions

Now, it seems obvious that Reality as a whole is composed of large numbers of small events, some of which work out one way, some of which go the other. At the smallest scales, it may even be the case that "quantum dice" don't really land upon one and only one face! They could be in a superposition of states, like Schrödinger's semi-unfortunate cat, both alive and dead, simultaneously. That would equate to a quantum die landing on a "face" that was 1/6(1) + 1/6(2) + ... +1/6(6), until someone chose to "open the box" and take a look.

The thing to remember is that our eyes and senses don't sense just one quantum event at a time, as a general rule. Unless the experiments are done very, very carefully, what we get at the point of sensation is the result of large numbers of events which have been averaged out. That is, if the dice face pairs could only be read out a million at a time, as an average value, then every time we read this average, it would be so close to 7 that our frail human sensory apparatus could never distinguish the results from a "hard seven". Hence, we are all under the impression that this is a Law, and apparently a "universal" one, at that. (It is certainly true of everything we ever see with the unaided human eye.) This human eye is the same human eye that Aristotle used when he formulated his Laws of Thought, including the one which concerns me in this essay. He didn't think in terms of what was going on at these very small scales, and how the "Law of Noncontradiction" might be just a statistical result, based upon many, many small contradictions canceling themselves out of the Big Picture, which is the "classical" result. To see that this isn't "really" the case, Aristotle would have had to have access to very fancy equipment that was still 2300 years from being invented.

Since Aristotle got his idea out there early, it has had lots of influence on the Group Mind of humanity. Most people simply accept this so-called "universal law of thought" without much question. After all, it works so well at the human scale that there wasn't any reason to consider other possibilities for millennia. Only when it became apparent that something about it wasn't kosher at the quantum scale, that reality had a statistical basis, did there arise any need to rethink this "law" that worked so very well. Just like Newton's law of gravity seemed so predictive and perfect, until Einstein came along to throw it into doubt, Aristotle's law has an even longer history of virtually flawless application, and a stellar reputation of irrefutability congruent with that history.

>>> Contradicting the Law of Noncontradiction

But refute it we must! Here, I am suggesting that the ACTUAL universal law is the exact opposite! Reality is almost nothing but contradictions! But these contradictions occur at such small scales, in such great numbers, that the average effect of all of them becomes indistinguishable from the situation that Aristotle thought was the case. Because every event which has a contradicting, symmetrical event is washed away by these cancelations. Contradictory events work against each other, partnering up to contradict themselves out of the net result, just like all the alternate histories in the Sum-over-Histories interpretation that fits with the Path Integrals formulation ends up by defining an average path for a photon that (it so happens) looks suspiciously like the "classical" path along a geodesic. Just like the vacuum of space is filled with pair creations that have the correct symmetries to be eliminated from the human-scale view, "in the large", and which makes the space appear empty to our eye. If our eyes were better instruments, and could make out details billions of times finer than we can see, we wouldn't have made Aristotle's mistake.

So my formulation goes like this:

A & ~A is almost always true. But whenever it is, the A and the ~A remove themselves from the calculation in a mutual suicide pact, and we only observe what's left over: The few times when there is no ~A to be found in near enough proximity for it to shoot its lover.

>>> The End

Well, that's it then! I was expecting this thread to encounter some derision and contradiction. So far, nothing like this has happened, except for Robert Tulip's concise statement of the standard philosophical position. I can only surmise that my title was so obscure that nobody bothered to read the thread! (Oh, well. C'est la vie!)

Well, in that case, I guess the Law of Noncontradiction isn't exactly a "Law of Thought", is it? At any rate, it isn't a general one. (Which, after all, was the point of my bringing it up.)




Uhh... How so? What about refraction and diffraction? Why are short-wave radio broadcasts bending around the earth?

How do you account for the Young's double slit experiment? Don't you count A={"Electron passes through slit A"} as contradicting A={"Electron passes through slit B"}? If not, can you please cite what constitutes a contradiction, in your view? Or are you simply disallowing any physical result at all, bases on the side constraint that physical reality cannot contain contradictions? In which case, I submit that your view is simply a tautology based on defining "contradiction" as "that which cannot occur in physical reality".

But then, what do you call these things, instead? 'Physical antinomies'?