Are the rules of logic immutable?

[OccamsRazor]

In the past my philosophical stance was always based on a single overarching truth. This was that the laws of logic and mathematics were immutable and could not be changed for any description of reality that one may provide.

More recently I have been grappling with the question, what if the rules of logic and mathematics are not immutable but subjective or specific to our incarnation of reality?

In Michael Frayn's book The Human Touch he makes the statement:

Logic is just a system we have made up, not an inherent condition of the natural world.

Is this true? Is logic changeable?

In another thread I saw the following statement:

I don't quite know how knowing something about events inside a system from outside of the system is on the same level of impossibility as a logical impossibility. There cannot be a square circle, a rational root of a prime number or the simultaneous existence of an irresistible force and an immovable object. These are logical impossibilities..

Is this true? Could a being outside our own manisfestation of material reality not create such logical impossibilities?

I can see that here many readers of this post would begin to state that logic and mathematics were immutable. That there indeed could not exist a rational root of a prime number and these are objective truths.
This leads on to the question, how may one prove it? Bearing in mind that any proof of the immutability of logic must have its basis in logic. The question is, how can immutable logic prove istelf to objectively exist?

If we then decide that, possibly, logic is not immutable then where does this leave us? Can we ever make a metaphysical argument without firstly assuming that mathematics and logic are immutable?

[zoro]

Confused: In your first post, you state "In regards to your closed system I can add little because most of my information on closed systems is in regards to social evolution and genetics." You meant "open systems": in spite of the efforts of Hitler, Stalin, Mao..., even they couldn't completely close their societies; and as for genetics, all living systems are open systems; when they're closed, they're dead.

Re. entropy, I have little problem with your post until I get to "However, for our universe, the Big Bang started it off in a low entropy state mostly theorized due to gravity [referencing Greene]." In contrast, I'd suggest the obvious reason why entropy started at zero is because there was nothing "here"! Further, if you'll look at the first chapter of my book, you'll see why I suggest that there's still "zero" here (which is part of the reason for the "Zen of Zero" in the book's title) -- but now, the "in-total nothing" has been separated into positive and negative components (e.g., of energy, with some of the "positive energy" now "congealed" into mass and with "space" or "the vacuum" brim full with negative energy). All of which suggests, by the way, that "existence" is not a scalar but a higher-order tensor.

You next state: "However in a closed system, entropy can also decrease to lead to more order with the addition of energy (e.g., sun's energy on universe or the energy to provide to make your bed)." Unfortunately, that statement contains a number of misunderstandings:

1. You mean not "closed system" but "open system", since by definition of a "closed system", no energy (or anything else) is exchanged with its environment.

2. When you add energy in the form of heat, Q, to an open system, you don't decrease its entropy, S, you always increase it [via (delta S) = (delta Q)/T, where T is the (absolute) temperature].

3. In your first example of the "sun's energy on universe", I take it to mean that you're suggesting that the Sun, by radiating energy, increases the entropy of the universe. Actually, though, that's tricky to evaluate. First, notice that no energy has been "added" to the universe, only converted from one form to another (some mass-energy of the Sun, as given by E = mc^2, is converted to electromagnetic [EM] energy). Thereby, the universe may be (the only?) closed system -- although I'm not sure that's so, because I'm not sure what's going on at its "outer boundary". Second, although I agree that some mass-energy of the Sun has in a sense been "degraded" to the form of EM radiation (in the sense that energy doesn't seem "keen" on being transformed in the opposite direction!), which normally is a sign of increasing entropy, yet for me at least, it's difficult to properly account for the time it takes for such degradation to occur and to understand the full scope of the consequences.

Since I expect that my last sentence was confusing, let me try to say it differently. Since nothing (including time) can travel faster than EM radiation, therefore from the perspective of the Sun's EM radiation (or of an observer assumed to travel with it), then "instantaneously" (i.e., no time advance), the radiation not intercepted by nuisance factors such as the Earth (!) reaches the "outer boundary" of the universe. Then, what happens there? Does the EM radiation from our Sun "instantaneously" contribute to expanding the universe? Would any such expansion increase (or decrease) the entropy of the universe? I don't know the answer. And I don't think the answer is obvious, because the usual calculation of entropy ignores the entropy of "space" (or "the vacuum"). Yet, I wouldn't be surprised if the answer is that the total entropy of the universe is (and always has been) zero -- but we've made the mistake of focusing only on the entropy associated with what we're pleased to call "positive energy". And to give you an idea of my confidence in that idea, I'm willing to bet you a full dollar on it! -- even including postage!! (but only to avoid the nuisance of my sending coins).

4. In your second example of entropy decrease ("the energy [you?] provide to make your bed"], let me say that, as a minimum, it's not stated well. As I already mentioned, normally when energy is applied to a system, its entropy increases. In the case of the bed of one particular grandchild, it's easy to argue that his bed possessed substantially more (gravitational potential) energy than after he's forced to make it! [That is, I'm sure that he must have expended substantial energy to pile the blankets in such a heap!] Therefore, his job, in fact, is to remove some of the gravitational potential energy from his bed -- to increase its entropy, in the sense of increasing its order, decreasing the randomness. That he complains that such an activity requires an enormous amount of his energy is, I maintain to him, not germane to how much energy his bed possess -- since most of his energy almost invariably seems to be expended just bickering.

5. Re. your statement "In application to logic, we see that using logic, entropy should increase however, by manipulation, we see that entopy decreases", I don' know what you're getting at. I'll assume that you're utilizing Shannon's idea that the entropy of a system can also be viewed as a measure of our lack of knowledge (viz., ignorance) of the system, e.g., we know very well the state of a beam of molecules, and with his formulation, our ignorance of such a beam (the entropy) is small; in contrast, for molecules at equilibrium in a container, our ignorance about the system (the entropy, our ability to specify positions and momenta of each molecule) is large -- in fact, it's at a maximum, in the sense of our complete ignorance. If that's what you mean, then when you state "we see that using logic, entropy [our ignorance] should increase", I would object that the correct statement is exactly the reverse. You then add "however, by manipulation, we see that entropy decreases." I don't know what you mean by "by manipulation".

6. With respect to the rest of your first post: I trust that I no longer need to define open and closed systems. Also, I'd point out that your statement that "deductive logic belongs in the realm of empirical while inductive logic in the realm of metaphysical" is exactly opposite from the usual classification.

Finally, in your final post you stated "Now, in regards to me being a hypocrite, I am unsure where in any of my posts I alluded to that. Perhaps you might point it out." In response, I quote from your second post: "Are you sure you weren't Zorro. Your arguements seem to be taking on the same demeaning and rude tone with the same irrationality." My view is this. I consider such a statement to be "demeaning"; therefore, notice the necessary (and even humorous) hypocrisy of someone "demeaning" a person by saying that his comments are "demeaning".

But enough of that and enough of these communications. It's time (way past time!) for me to get back to work. Besides, Google is finally helping to get my message to kids: with my site in Google for only a few weeks now (and still a long way from being completely indexed), 37,000 hits so far -- including even some hits from Egypt and Saudi Arabia. Things are lookin' up!

[Furrowed Brow]

I don't know if this is true or not; it all depends on what you mean by "axiom".

Euclid, or Hilbert for example. Ok Euclid’s axiom of a straight line are now known not to be universal in the sense they only apply to a flat surface. However, they are still a universal rule for flat surfaces.

However I now realise that rather than axioms, it is the rules of derivation that have the kind of self evidence I really have in mind. Modus Ponens for example, is a rule of argument that is just self evidently true. Also logical theorems that can be proved to be without assumption like |- ~(P & ~P) or "it is not the case that P and not-P".

I'm not sure I understand what this means. Fuzzy logic doesn't deal with binary truths or falsehoods; it deals with values between 0 and 1, AFAIK.

Ok I’ve though about this and I’m going to have to shift stance. Fuzzy logic tries to do away with the rule of excluded middle. Thus 100% true, and 100% false or just two results on a spectrum. I think this is the wrong way to frame the problem.

However probability theory, as I understand it, is able to produce the same results as Fuzzy logic. However, in probability theory, it is the probability of something that is being calculated. That something being a propositions - which itself is true or false. The probability comes into consideration because there is uncertainty about that truth or falsity. But uncertainty is a problem of knowledge, and not a problem of how we argue and think about that knowledge.

(As I type this I realise there is way more to be said. I am opening a can of worms here that needs to be gone into, but I’ll let you come back at me with some criticism or question, otherwise you are going to get a ten page rambling post.)

Well, if I were to look at your brain, I wouldn't see logic there. I wouldn't see any propositions, conclusions, modus ponens, or binary true/false values. Instead, I'd see a bunch of neurons connected in parallel, affecting each other's firing rate. That's not logic at all.

My brain? Nothing between the ears here at all according to Zoro.

An electrical circuit is not logic either. But wired up in the form of a logic gate and it suddenly embodies relationships like ‘P & Q’ or ‘P v Q’. I think Penrose makes an attempt to show how in principle cells could be wired up, to form logic gates. I think this is fool hardy, and I suspect the kind of logic we use will not reduce this way. How does it reduce? Dunno. But if you brought George Boole back to life and showed him the circuitry of a computer and asked him to point out where his algebraic principles were being embodied - I’d suggest he’d struggle to tell you.

What if we meet some culture that uses rules that we have never come up with at all ? Can you demonstrate why such a culture cannot possibly exist ?

Well no. And maybe there is some stuff we are till to learn about logic. But I will give you my personal guarantee these aliens - if they have reached a similar level of intellectual achievement and are not cave dwellers - will use and understand modus ponens. If they don’t we would not call these guys thinking beings.

[Furrowed Brow]

The whole argument about adding holes or silly putty chunks, etc., is starting to resemble (at least, to me), ye olde argument about angels dancing on the head of a pin. I think it illustrates quite handily what happens when you assume that numbers in particular, and math in general, are some sort of an independently existing entity. You run into problems like these.

Yeah. It was getting that way. But unlike angels on the heads of pins, the stakes were higher. If Zoro were right then there would be important implications for true/false logic, theory of reference, and the foundations of logic.

[Confused]

Though I disagree with 90% of what zoro said, QED and Bugmaster, you have both just gone way over my head so I will enjoy the conversation now.

[Bugmaster]

Huh, that's odd, I thought I'd answered already, but my comment is not showing up...

Modus Ponens for example, is a rule of argument that is just self evidently true. Also logical theorems that can be proved to be without assumption like |- ~(P & ~P) or "it is not the case that P and not-P".

Ok, but is modus ponents self evidently true because it reflects some universal principle, or is it a made-up (by us consciously, or by our evolution, or both) basis for our made-up (ditto) system of logic ? How would you go around proving it, one way or the other ?

However probability theory, as I understand it, is able to produce the same results as Fuzzy logic. However, in probability theory, it is the probability of something that is being calculated. That something being a propositions - which itself is true or false.

Well, it depends on how you look at it. Fuzzy logic is able to deal with continuous values, not just probabilities. For example, your height is not a boolean; it is not the case that you're either the tallest person on Earth, or the shortest person. Instead, your height falls somewhere within that range, and fuzzy logic can deal with this; it can also quantize your height into multiple "truth" values; f.ex. "short", "average", and "tall" (3 values). I don't know if this is mathematically equivalent to probability theory or not.

An electrical circuit is not logic either. But wired up in the form of a logic gate and it suddenly embodies relationships like ‘P & Q’ or ‘P v Q’.

Yes, but this is not the only way to wire a circuit. In fact, in ye olde days we had analog computers, which used analog circuits to model the problem that needed solving (typically, ballistic calculations for shooting down airplanes). Analog computers are not as popular now as they used to be, basically because digital computers are cheaper to build and easier to program. Our brain is sort of a cross between an analog and a digital computer, as far as I understand.

If they don’t we would not call these guys thinking beings.

That's the question, though. Could a culture that does not use logic ever build flying saucers ? I grant you that we would not be able to communicate with them, most likely, but I'm still apprehensive at the thought of rejecting the very possibility of their existence out of hand.