Provability and truth are two distinct notions? — guptanishank

Yes, different. Provability is a syntactic notion. Given some axioms and some inference rules, a given statement either has a proof or its negation has a proof or neither. There is no meaning attached to the symbols.

Truth is a semantic notion. Given an axiomatic system, we choose some interpretation of the symbols, and then we see if the statement is true or false under that interpretation.

Gödel's completeness theorem says that in first-order predicate logic, a statement is true in every model of the system if and only if it has a proof in that system.

How would you know if a statement is true without the proof? — guptanishank

You look at the model in question and see if it's true. If I tell you it's raining, you look outside and see if it's raining.

This distinction between truth and provability is at the heart of ‎Gödel's Platonism. Even though he can't prove the Continuum hypothesis within ZFC, he is certain that "out there" in the actual world of sets, CH has a definite truth value.

The thing is that truth or falsehood of the statement will vary according to the axiomatic framework we consider. — guptanishank

The provability of a statement is a function of the axioms.

The truth of a given statement is a function of which model, or interpretation of the axioms we choose.

But now that you've clarified your ideas I still don't understand your question.

One thing:

How do we know that the statement we are proving outside the axioms is the same statement as inside the axioms? — guptanishank

I don't know what that means. Can you give an example?

A statement is comprised of all the axioms above it. — guptanishank

That doesn't correspond with my understanding of what a statement is. A statement in a formal system is simply a well-formed formula that may be true or false. For example "2 + 2 = 5" is a statement. "2 + 2 =" isn't.

So in a way we are proving different statements to be true, even though they may look the same.
Any thoughts? — guptanishank

Don't know what you mean. Example?

It seems very counter intuitive to me to use two different axiomatic systems for two different proofs of the same statement.
Could it not be that a statement is True under one axiomatic system and False under another then? — guptanishank

Of course. In Euclidean geometry there's exactly 1 line through a given point parallel to another given line. In non-Euclidean geometry there may be zero or many.

ps -- Facts for the fact-challenged.

In July 2000, Taliban leader Mullah Mohammed Omar, collaborating with the United Nations to eradicate heroin production in Afghanistan, declared that growing poppies was un-Islamic, resulting in one of the world's most successful anti-drug campaigns. The Taliban enforced a ban on poppy farming via threats, forced eradication, and public punishment of transgressors. The result was a 99% reduction in the area of opium poppy farming in Taliban-controlled areas, roughly three quarters of the world's supply of heroin at the time.

https://en.wikipedia.org/wiki/Opium_production_in_Afghanistan#Rise_of_the_Taliban_.281994.E2.80.932001.29

Of course that ended with the invasion of Afghanistan by the US in 2002. Now the opium crop is at record levels, thanks to the good old U S of A. And in totally unrelated news, the US is having an opiate crisis. Move along, nothing to see here.

Or maybe the Taliban is in the dope business. — Bitter Crank

But that's an absolute falsehood. By 2002 the Taliban had virtually eliminated the opium trade in Afghanistan.

Are you this seriously uninformed? Or just shilling for the neverending war?

I'm genuinely puzzled by your factually wrong claim.

ps I wanted to make sure I wasn't exaggerating when I said the US guards the poppy fields in Afghanistan. Google served me up this link real quick. I love Google.

Drug War? American Troops Are Protecting Afghan Opium. U.S. Occupation Leads to All-Time High Heroin Production

The date is June 24, 2017. This is happening on your dime (if you're a US taxpayer) and with your pro-rated share of moral culpability.

There's your drug war folks.

https://www.globalresearch.ca/drug-war-american-troops-are-protecting-afghan-opium-u-s-occupation-leads-to-all-time-high-heroin-production/5358053

ps -- I just have to quote this bit. This is a quote from an article in Common Dreams.

The cultivation of opium poppy in Afghanistan—a nation under the military control of US and NATO forces for more than twelve years—has risen to an all-time high, according to the 2013 Afghanistan Opium Survey released Wednesday by the United Nations.

According to the report, cultivation of poppy across the war-torn nation rose 36 per cent in 2013 and total opium production amounted to 5,500 tons, up by almost a half since 2012.

“This has never been witnessed before in the history of Afghanistan,” said Jean-Luc Lemahieu, the outgoing leader of the Afghanistan office of the United Nations Office on Drugs and Crime, which produced the report.

And now in 2017 the crop is the new world's record. That's what the war in Afghanistan about. We're in the dope business.

Yeah fentanyl. A few years ago I read about it in the context of it being the drug of choice for medical professionals in hospitals. Doctors and nurses couldn't get enough of the stuff. At least some of them. Of course the vast majority of hospital professionals are not abusing the ambient pharmaceuticals. I hope.

Now suddenly it's this huge drug of abuse.

The owner of a place I used to eat breakfast at died of a fentanyl overdose. I had no idea people in my community are flipping pancakes in the morning and doing that after work. If it touched my sheltered life then it's a lot more prevalent than I thought.

I don't buy the Chinese angle that this is something they're pushing on us. Americans are the world's hugest consumers of illicit drugs by far. The entire world labors to supply the American consumer with drugs. That 's the truth and everything else is the hypocrisy around it. If Americans ever stopped using drugs, the entire global economy would collapse; from the peasant farmers who pick the drug crops to the industrial plants that make the precursor chemicals to the banks who launder the money The DEA, the CIA, and whatever local warlord we want to support that week are the drug business. A lot of mouths to feed. Nobody wants this to stop.

We never ask: What is the sickness in the American soul that needs so desperately to be numbed?

And by the way, why is there a renewed demand for opiates these days? Couldn't have anything to do with our war in Afghanistan, could it? In 2002 the Taliban had virtually eradicated the opium trade. They're against it. The US came in and got it going again. We're for it. The US Army guards the poppy fields over there. True. In the 1980's Reagan ran secret wars in south America and we had a huge coke epidemic, while Nancy Reagan told us to "Just say no" to the drugs her husband's CIA was flying in by the planeload.

Sugar (glucose) is what the brain runs on. It's not just rewarding, it's essential. — Bitter Crank

Ditto caffeine :-)

ps -- On a more serious note, the question is not whether any given drug is a net good for society. The question is, it is less harmful than prohibition? Prohibition inevitably gives power to gangs of violent criminals; causes people to get sick from adulterated product as it did when the US tried outlawing alcohol; and ruins the lives of casual users branded as criminals.

Why am I not surprised... — Agustino

Why aren't you surprised? You wrote:

We weren't designed by evolution to be smoking weed, — Agustino

I pointed out that our brains have receptors for cannabinoid molecules. Therefore you should be surprised. Why ARE you not surprised? Am I being too literal in some way?

We weren't designed by evolution to be smoking weed, — Agustino

Our brains have cannabinoid receptors. Why do you think that is?

"Reality is for people who can't handle drugs."

But a computational process, could go on forever? The symbols might be finite, but they are referring to something quite plausible. — guptanishank

No, computations are required to finish after a finite number of steps. That's part of Turing's definition of computation and it's fundamental to computer science. It's just as proof in math is required to consist of a finite number of steps.

But recursion need not be finite. Surely that is possible. — guptanishank

The strings of symbols that represent recursion are finite. The simplest example is the intuition of the natural numbers 1, 2, 3, ...

Those dots are finite. I used 9 symbols above, not counting spaces. 9. I "represented" infinity but ... what does that mean? It's clever of us humans to have worked out a system of symbols to discuss infinity. But the symbology is finite.

Gödel's theorems are about the properties of certain collections of finite strings of symbols. That's why you can do mathematical proofs on a computer. There's no infinity in the computer but we can use a computer to reason symbolically about infinite sets.

Wouldn't it have to be something like intuition? Or perhaps, on the other hand, new formalizations that are truer to intuition? — t0m

Yes, the work of modern set theory has consisted largely in trying out new axioms that might solve the Continuum hypothesis. I suppose you could say that this is the vision of Gödel. To find better axioms that are natural in the sense of being intuitively right.

As one example, Gödel proposed a model of set theory called L. [Technical definition not important]. In this model, the Axiom of Choice and the Continuum hypothesis are both true. That proves that these statements are at the very least consistent with ZF [Zermelo-Fraenkel set theory].

Now you might think this would be enough. We'd say, we have a model of set theory and AC and CH are both true, so let's all work in L forever and be happy.

However!! It turns out that Gödel himself did not believe that L was the entire universe of sets. We don't work in L, we work in a much more generous model of set theory.

If we call the entire universe of sets V, then the claim that L is the entire universe can be notated as V = L and nobody thinks it's true.

This is perhaps what Gödel is getting at. We can use pure symbolic manipulation to learn more about our axioms. But there is always an "intended" or "real" interpretation out there, and we are not content with a purely symbolic or formal interpretation.

The point is that modern set theory is the search for new axioms that are plausible and seem natural for the world of sets we have in our minds. In our intuition. Yes, it's ultimately driven by intuition. By our intuition about what the Platonic sets must be. Even if we're formalists in the end we must be part Platonist.

For me the finite and the computational are just nakedly "real" or "true." They are more persuasive than the philosophy that might try to ground them. You'll probably agree that it's the infinite that gives us trouble. — t0m

I would say that the infinite is what makes math interesting! Otherwise it's just combinatorics. Balls in bins. Finite sets are boring. Also you need infinity to come up with a satisfactory theory of the real numbers. Which themselves are a philosophical mystery.

It's true that once we allow infinite sets we have paradoxes and strange and counterintuitive results. But that's the fun part! Because when we're doing math, we should think like formalists. That means we just push the symbols and see how much we can prove and if we prove some crazy stuff, well that's fun too. It's a game played with symbols. We do it because it's fun and interesting.

I think that deep down, we're all Platonists. Math is telling us something about the world. But when we DO math, we are formalists. Push the symbols, don't worry too much about what it might mean.

Tentatively this trouble seems to involve the gap between a fuzzy, linguistic concept and a mechanizable concept. There are limits to mechanization (halting problem, for instance), and yet mechanization is as Platonic as it gets? — t0m

Yes. Something that I find interesting is that even though we have all these crazy theories about humongous infinite sets; all of our reasoning is finitistic. Proofs are finite strings. The axioms and theorems are finite strings. The rules of inference are described with finite strings. You could program a computer to check if a proof is valid. This is a huge area of active research these days, they're doing amazing things.

So all we're really doing is playing around with finite strings of symbols. We tell ourselves it's "about infinity," but it really isn't. We are only pretending to be able to deal with infinity. That's one way to look at things.

I did not know that about Godel. — guptanishank

These conversations always make me look things up.

This is from his SEP entry.

In his philosophical work Gödel formulated and defended mathematical Platonism, the view that mathematics is a descriptive science, or alternatively the view that the concept of mathematical truth is objective. On the basis of that viewpoint he laid the foundation for the program of conceptual analysis within set theory (see below). He adhered to Hilbert's “original rationalistic conception” in mathematics (as he called it); and he was prophetic in anticipating and emphasizing the importance of large cardinals in set theory before their importance became clear.

Later in the article in section 3.2, "‎Gödel's Realism," they quote this passage from his writings. These are Gödel's own words.

Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things,” or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.

It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the “data,” i.e., in the latter case the actually occurring sense perceptions.

That's a pretty good description of Platonism. I'm struck by his comment that these objective structures are "necessary to obtain a satisfactory system of mathematics ..." That's a fantastic claim. That the existence of abstract entities is actually necessary to the enterprise of math. You can't dismiss them as a mere formalism.

There's a lot more in that SEP article about all this and more.

To me, ‎Gödel's Platonism is a clue to the meaning of incompleteness. He's not saying that we can't know mathematical truth. He's saying that formal systems are too weak to know mathematical truth. But like they used to say on the X-Files, The truth is out there.

The next question is, if formal systems can't get at the truth, what can? I don't know anything about what philosophers think about that.

Yeah, I meant how do the platonists who do think this way, think about it? — guptanishank

I don't know enough to say. As I mentioned, Gödel was a Platonist and he was pretty smart. There's a lot I don't know about Platonism.

This conversation made me look up this article https://plato.stanford.edu/entries/platonism-mathematics/ . I'll give it a read when I get a chance.

But the world is every changing. How do they account for that? — guptanishank

Not being a Platonist, I couldn't say. I can't imagine that there are mathematical sets that actually exist anywhere except as abstract ideas that behave whichever way the axioms say they do. That's just my personal sense of things. I don't think there's any actual truth about sets beyond trivial observations about finite collections.

One of these days I'll read Tegmark. I only know about his mathematical universe hypothesis and can't really respond to the point you are making about domains of mathematical truth.

The way I understand all this is by example. Say we take Zermelo-Fraenkel set theory, or ZF. We know that the Axiom of Choice (AC) is formally independent of ZF. So we can then adjoin Choice to ZF and study the resulting system (called ZFC) or we can adjoin the negation of ZF and study that resulting system.

A Platonist is someone who thinks that "out there" in the Platonic world there are sets, and that in that world of sets, Choice is either true or else it's false. There's a definite answer. Formalists think neither system is true, we're just describing different conceptions of what sets might be.

Viewed that way, I think this is all much less mysterious than it's sometimes made to seem. It's more of a syntactic problem. Sufficiently complex formal systems will always have well-formed statements that can neither be proved nor disproved by the system. There's less than meets the eye. It's not any kind of cosmic mystery. It's just a limitation of formal symbolic systems.

Yes, I understand that much. But, having objects that can only exist in infinite Hilbert space isn't a mystery to you? — Posty McPostface

No, why? Perhaps you can explain your point of view to me. I didn't study much physics but I've studied functional analysis. Hilbert spaces aren't very mysterious at all. In fact when I learned that the mysterious bra-ket notation is nothing more than a linear functional acting on a vector, I felt enlightened, as if perhaps QM isn't that far beyond me. If you can explain to me what you're thinking I'm sure I'd learn something. Maybe there's a mystery I'm not appreciating.

A la, Penrose, if one believes in mathematical Platonism and such, then there seems to be a final system that could account for all proofs in it, no? — Posty McPostface

‎Gödel was in fact a Platonist. He believed in mathematical truth. His incompleteness theorems show the limits of formal systems in finding that truth.

It seems intuitively obvious if you consider QM in infinite Hilbert Space or anything in infinite Hilbert Space. — Posty McPostface

Not sure what that means. Hilbert space is easily modeled within standard set theory. Infinite dimensional spaces aren't very mysterious. The set of all polynomials has basis {1, x, x^2, x^3, ...} That's an infinite dimensional space that's accessible to the understanding of a high school student. Hilbert space in general is just a function space; that is, a collection of functions with pointwise addition and scalar multiplication. There is no mathematical mystery to Hilbert space.

So you're saying that left to themselves, neural networks will spend their time looking at cat videos?

Well, when they came out they were a massive 'fuck you' to the Hilbert Program — fdrake

Indeed, historians have recently discovered a cache of previously unpublished letters between Gödel and Hilbert. I take the liberty of summarizing them here.

Godel: Fick dich!

Hilbert: Oh no mein freund. Fick DICH!!

Gödel: Fick dich to the n-th power!

Hilbert: Und deine Mutter auch!

etc.

Astronomers assume — Banno

Assume. Assume. Assume.

what you quoted was just a phrasing of the principle of relativity, simplified for the audience. — Banno

ok

The laws of physics are universally applicable. That’s what makes them laws. — Banno

Really? How do we know the laws of the physics are valid everywhere? We have a very small sample of local observations.

And that's what makes them laws? That's really not a good argument. Are Newton's laws universally valid? Were they universally valid and "laws of the universe" in 1900 but not 1920? I hope you can put your claims into context because as it stands they're just wrong.

Are you not aware that the universality of physical law is an assumption? There's no way to know if it's true. There's no way to know if there are any physical laws in the first place. What we call physical law is the historically contingent output of humans.

that the speed of light is constant no matter what the frame of reference, is a very flimsy principle, not verified, nor verifiable from human beings' present technological condition, but quite likely not at all acceptable as a universally applicable law. — Metaphysician Undercover

Isn't that true of all physical law? The universality of gravity is an assumption for which we have no evidence. When we say all swans are white it's only because black swans are so rare. All knowledge of the world is inductive and subject to refinement and outright refutation.

Thank you, fishfry. It is apparent I had misunderstood you. My apologies.

One is a principle of classical logic; and the other is a principle of modern physics.
— fishfry — Banno

Glad this is clear. I mention only in passing that I think this is a more general problem. I am not sure anything at all in the real world is subject to classical logic. What statement can be said to be either true or false? If I point to a red apple and say, "The apple is red," someone can argue that redness is a subjective experience, not something inherent in the apple. Maybe your red isn't my red. Inverted qualia, Mary's room, and so forth.

Formal logic is an abstract model for how we reason. But the world is much more complicated and nuanced and sometimes contradictory. We are not rational creatures. The world is not a rational place. Formal logic is a useful tool, but the world isn't a formal system.

You can do this by contriving your definition of "true", such that truth is relative to the observer, — Metaphysician Undercover

That's not how sentential logic works. A proposition P is true or false. There are no observers.

I am not aware of any effort within physics to rewrite the rules of logic to account for this.

For this reason I don't quite understand what you're saying here.

Just a general idle comment here. It's always struck me as interesting that when the God of the Christian Bible created he universe, he said, "Let there be light." And in modern physics, the most fundamental thing in the world is electromagnetic radiation. Light.

Whatever the universe is, however it got started, the first thing to know about it is that there is light.

I have no idea what light is. Does anyone?

One thing to note here is that contradiction is not inconsistency. Or, as a matter of terminological precision: inconsistency is a function of contradiction within a formal system: a system is said to be inconsistent if it contains contradictions. The claimed inconsistencies between relativity and QM are not of this kind: the entire point is that there are inconsistencies between two systems. — StreetlightX

Point well taken. I'm aware of what inconsistency means in the context of the study of formal systems, ie axiomatics. Of course you and @Banno are correct that I'm using the word inconsistency when I should be saying incompatibility.

There's no axiomatic basis for physics in the first place, so we can't be meaning inconsistency in the technical sense. But I can see that I've been confused myself on this point. The right word is incompatibility.

The nature of the incompatibility between QM and Relativity is in their predictions. There are situations where in a given situation, they predict different outcomes. If we wanted to express this formally it would be "Theory X predicts P and theory Y predicts not-P". Does this relate to modal logic?

Note also just how strict the criteria are to meet the standard of contradiction: X and its opposite must be 'true' in order for contradiction to hold: a proposition must say that X AND ¬X is true. — StreetlightX

Picky picky, you mean provable, not true. An axiomatic system is inconsistent if it allows a formal proof of both X and ¬X for some proposition X. It's purely syntactic. The idea of truth belongs to semantics, where we put an interpretation on the symbols.

But of course there is no theory that claims any such thing. As it stands, the operator between the apparently 'contradictory' statements between QM and relativity is - I think assumed to be - a XOR operator (exclusive or): X ⊻ ¬X , not X ∧ ¬X. — StreetlightX

I'm trying to understand this. In the first place, neither QM nor R are axiomatic theories. They're not formal systems at all. They're a collection of heuristics and differential equations. In other words when a QM or R theorist wants to calculate the expected output of an experiment, they don't apply axioms or theorems. Rather, they plug their numbers into models, which are essentially differential equations [I'm way over my depth here physics-wise]. In other words there are no "propositions" in the sense of logic.

That's my understanding, anyway. That the conclusions of both QM and R are not deductions, they're approximations to some theory that's not axiomatized. So the entire field of logic doesn't really apply the way we're trying to apply it.

Ok now I see your point. There's some experiment for which QM and R predict different results. So it's an XOR. Yes I see that!! Ok got it. Yes, good point.

There is no 'contradiction' here, in the logical, intra-systemic sense. Things might be confusing because science writers are not logicians, and they are apt to use terms in ways that are not the technical terms of logic. This is to be expected, but it is also to be watched out for when trying to draw conclusions. — StreetlightX

Yes I think you're right.

How is this not a violation of the law of non-contradiction, when what Beth believes is clearly contradictory to what Angie believes? — Metaphysician Undercover

Wait, that's not right. X believes P, and Y believes not-P. That's not a contradiction.

A contradiction would be, X believes P and X believes not-P. Actually I'm not sure that's a contradiction. A contradiction would be, X believes P and X does NOT believe P. It's possible that X may have no beliefs about P at all one way or the other.

But why do some of these stories give correct predictions and others don't? — litewave

But they don't!

They give approximately correct predictions, to the limits of our experimental apparatus.

Newton wasn't correct, nor is Einstein. They get closer and closer to something that may or may not be there.

Back in the day we looked up at the stars and said, "Oh, there's Orion the hunter with his mighty bow." And they took those stories every bit as seriously as we do our stories.

There is no question that science and rational inquiry have been very handy, creating bridges and cellphones and all this wonderful stuff we have around us. The question of why science is so useful is the proper inquiry of the philosophy of science. That's a good question, to which nobody has a conclusive answer.

Another point is that the laws of physics that we've been able to come up with so far are due to the very limited perspective of where we are in time and space. If we lived a long time ago. or farther out in the universe we might find new laws. The uniformity of physical law throughout the universe is an assumption, not a proven fact.

ps -- Ok I went Googling. But as I said earlier, we're just talking Googling here, I'm not telling anyone anything they couldn't find with the same Google search themselves. I'm not supplying physics knowledge here, just typing.

But check this out. Article titled: The Inconsistencies Between Relativity and Quantum Therory [typo in original. Maybe that's a bad sign].

http://www.quantumtemporaldynamics.com/background-physics/the-inconsistencies-between-relativity-and-quantum-therory/

Relativity tells us that if we were to capture a cube of matter exactly one Planck length to a side, the matter inside that cube will the as dense as any heart of any black hole. Quantum theory tells us that the same cube can only contain a single quanta of energy. Each statement is inconsistent with the other.

I also found:

https://www.theguardian.com/news/2015/nov/04/relativity-quantum-mechanics-universe-physicists

http://nautil.us/issue/29/scaling/will-quantum-mechanics-swallow-relativity

https://www.quora.com/Why-are-Quantum-Mechanics-and-General-Relativity-incompatible

https://io9.gizmodo.com/why-cant-einstein-and-quantum-mechanics-get-along-799561829

http://www.askamathematician.com/2009/12/q-howwhy-are-quantum-mechanics-and-relativity-incompatible/

https://physics.stackexchange.com/questions/387/a-list-of-inconveniences-between-quantum-mechanics-and-general-relativity

https://arxiv.org/pdf/1704.02587.pdf

I hope this is helpful. I haven't read any of the links except the one I quoted.

I had a tooth pulled last week. It was easier than this. Quicker, too. — Banno

But I don't understand why I'm in this conversation. I originally said something that I thought was very innocuous. I'd sooner retract whatever it was I said than have to defend something I thought everyone already knew.

It's non-locality, isn't it. Relativity implies that nothing can travel faster than light; QM says it can. — Banno

I don't know enough physics. However I do not believe that QM lets anything go faster than light. Perhaps you have a link or some context.

Or, alternately, it is not possible for us to construct a coherent account of reality. — Banno

This is actually something I believe.

There are two different things. One is the laws of physics, which are historically contingent works of man. Aristotle, Newton, Einstein, etc. The collected body of physics papers. The stories we tell the freshmen, the stories we tell the grad students, the stories physicists tell each other.

The other thing is "reality." It may or may not have laws at all. If it does, they may or may not be knowable to us. And even the very question of whether "reality" exists as some external thing to be studied is arguably a meaningless question.

That's why when @litewave tells me what "reality" is like, I simply say that's a metaphysical belief without evidence. We can go to the university library and read the latest edition of a physics journal. We can NOT know "reality."

Contradictions occur between statements. a contradiction is an indication that one of the statements is wrong. — Banno

There are no "statements" in physics because we do not have an axiomitization of physics. That's Hilbert's sixth problem, which I linked earlier.

You are thinking there are a set of propositions in a bag called QM, and another set in a bag called Relativity, and we will find P in one bag and not-P in another.

I don't think it exactly works that way. Perhaps that's the point you're trying to make. That there is an incompatability, but not necessarily a direct contradiction. I don't think that's right. I think there are direct contradictions. But I'm perfectly willing to admit that I'm way out of my depth here.

I'm standing by for supporting evidence that QM lets things go faster than light. I know about quantum tunneling, in which a photon inside of a black hole can suddenly appear on the outside. I believe it's called Hawking radiation. Black holes give off huge amounts of energy via this magic process. I'm not sure if that's faster than light travel or not.

Set it out, man. — Banno

You have the same access to Google that I do. If QM and relativity are consistent with each other then why have physicists been trying to hammer out a unified theory for over 100 years?

Please. I'm done here. I made a very simple point to start with and really have nothing more to add.

there was some aspect of quantum mechanics that was directly contradicted by general relativity — Banno

There is. I've given references. IMO this discussion is far past the point of diminishing returns. I have nothing to add.

Identity of a thing is determined by its properties. — litewave

Ah, identity of indiscernibles. But "properties" are imposed by sentient observers. A thing would still be a thing even if there were no people around to enumerate its properties. You're confusing physics with metaphysics again.

Well I can't argue with you about this anymore. I don't think you've made your case. And you keep making claims about "reality." As I've noted, those are statements of metaphysical beliefs. I can't argue with you about your articles of faith any more than I could argue with the Pope about matters of theology.

Ah! You mean it is incompatible. — Banno

Yes I can see that might have been a better word. But "X is incompatible with Y" and "X is inconsistent with Y" seem to be a distinction without a difference, especially in the context quantum theory and relativity.

Well, if a ball that is not a ball makes sense to you, I have nothing else to say. — litewave

Again you are claiming that a single inconsistent aspect of the universe [an entirely metaphysical notion] implies denial of the law of identity. I have stated that I do not follow your logic and do not agree with the claim.

An inconsistent aspect of the universe is nonsense. — litewave

Unprovable and evidence-free metaphysical claim.

We can only speak rationally if we don't insist that such an aspect of the universe exists — litewave

Already falsified, with your agreement.

And if we abandon the law of non-contradiction completely, we cannot speak rationally about anything. — litewave

False as noted, reputable links supplied, and your agreement secured.