Path Integrals, Sum-over-Histories, and the Law of Non-Contradiction

[Randall R. Young]

>>> 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.

“No, no, you're not thinking; you're just being logical.” ~Niels Bohr

“We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough.” ~Niels Bohr

“How wonderful that we have met with a paradox. Now we have some hope of making progress.” ~Niels Bohr

“The opposite of a fact is falsehood, but the opposite of one profound truth may very well be another profound truth.” ~Niels Bohr

“Prediction is very difficult, especially if it's about the future.” ~Niels Bohr

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

[Robert Tulip]

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.

[Randall R. Young]

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.

[Randall R. Young]

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.)

[Robert Tulip]

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.

b

[Randall R. Young]

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".)

[Randall R. Young]

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>

[Randall R. Young]

<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.)

[Randall R. Young]

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.

[Randall R. Young]

>>> 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:

The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude.

Classical action principles are puzzling because of their seemingly teleological quality: given a set of initial and final conditions one is able to find a unique path connecting them, as if the system somehow knows where it's going to end up and how it's going to get there. The path integral explains why this works in terms of quantum superposition. The system doesn't have to know in advance where it's going or what path it'll take: the path integral simply calculates the sum of the probability amplitudes for every possible path to any possible endpoint.

The path integral formulation was very important for the development of quantum field theory. Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time, and are not in the spirit of relativity.

However, the path integral formulation is also extremely important in direct application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space.

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.)