What is a Theory?

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Can you prove that there exist an infinite number of facts that exist in the real world that would falsify this theory?

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You can boil water today, tomorrow, the day after, and so on, ad infinitum. Any of these unbounded/infinite number of facts could conceivably contradict this theory.

But then again, adding a 1 to a finite number, can never produce an infinite number, regardless whether it is a day, an hour, a minute, a second or any other time period. Therefore, there will never be an infinite future point in time.

In this context, "infinite" should probably be understood as "unbounded".

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Are you sure that there aren't an infinite number of numbers one could theorize that water boils at?

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I guess there is no infinite number of temperatures.

The problem is that "infinite" does not really exist outside the realm of mathematical formulas. Projected in to the physical world, we may have to replace the term "infinite" by "unbounded".

What's more, there is already a large body of literature containing theories about "infinite", and I don't want to start making too many blanket statements about "infinite" that could be contradicted by people who happened to have written entire books about it.

I concede the point that the term "infinite" is problematic and should be treated with the necessary care.

Edited by erikp, : typo

In your own words, pay attention to the appearance of the word axiom and its' derivatives in the following quotes..

Gödel's work is very interesting, as it shows the limitations of mathematics and other formal axiomatic systems (the fact that they are necessarily false). axiomatic reduction does indeed not amount to "proof Gödel already proves that all axiomatic theories capable of expression basic arithmetic are incomplete (and therefore false) Mathematical theories are axiomatically reduced, but never proven, because the axioms to which they are (recursively) being reduced, and on which every mathematical statement eventually rests, MAY NOT be proven. Mathematics demands that its entire hypothesis be concentrated in its axioms, which in turn remain unproven.

contrasted with..

Religion uses its core initial axiom concerning the beginning of the universe, in order to phrase rules about what is right and wrong.

And you are telling me that you are not self-contradictory in your understanding? I repeat, your argument is incorrect and destroys the very foundation you are trying to protect..

Here's an atheist's application of the theorem in the same manner you are using it, again incorrectly..

"Godel's Incompleteness Theorem demonstrates that it is impossible for the Bible to be both true and complete."

Here's a quote from a book reviewing the theorem and its' applications:

More reasonable have been attempts to apply the incompleteness theorem to physics. The hypothetical “theory of everything” (TOE) is sometimes taken tobe an ideal of theoretical physics. However, such eminent physicists as Freeman Dyson and Stephen Hawking have invoked Gödel’s theorem to suggest that there is no such theory of everything to be had. Now it seems more reasonable to assume that a formalization of theoretical physics would be the subject of the incompleteness theorem by incorporating an arithmetical component. Nevertheless, Franzénadds,Gödel’s theorem tells us only that there is an incompleteness in the arithmetical component of the theory. Whether a physical theory is complete when considered as a description of the physical world is not something that the incompleteness theorem tells us anything about.

http://www.ams.org/notices/200703/rev-raatikainen.pdf

I think that exploring the limits of science, or the limits of any discipline for that matter, is one of the most important exercises in that discipline. The reason why I am interested in the limitations of science, is because science is often used to attack religion

And the corollary..

I think that exploring the limits of religion, or the limits of any discipline for that matter, is one of the most important exercises in that discipline. The reason why I am interested in the limitations of religion, is because religion is often used to attack science.

What are the practical implications of your views in terms of existing scientific theories? Which of them do you see in a different light as a result of your POV?
What is the significance of what you are stating? Are you stating that a grand unified theory is not possible?
Are you suggesting that other existing scientific theories are way off the mark, or that they may continually need minor modification and tweaking as from Newtonian mechanics to general relativity?

Edited by shalamabobbi, : addition

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I don't find Gödel's language weird at all. He knows the difference between "incomplete" and "false".

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According to the definitions of "true" and "false", incomplete theories are indeed false.

As soon as water has been observed to boil at any other temperature than 100 C, the theory that says "Water boils at 100 C", has been proven to false. It cannot be rescued just by saying that it is "incomplete". The definition simply says that it is false.

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Tell me, can you think of any tools that science uses that are not used in maths and logic, and if so, can you think of ways in which those tools might make a difference to the ways that Gödel's theorems might apply to science as compared to maths and logic?

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The best comparison in this matter, is the difference between math and physics. Both disciplines insist that you reduce theories to underlying theories (axiomatic reduction). The difference, however, is that physics freely accepts new unreduced/unreducable theories, on the condition that they look plausible, that is, not contradicted by existing observations. Mathematics, however, seldom accepts new axioms.

"Water boils at 100C" is a theory that physics would accept (axiomatically), if nobody is able to make water boil at other temperatures. From there on, other theories can be reduced to this accepted (axiomatic) theory and effectively build on it.

Mathematics is way more fussy about these things. You would have to reduce the theory that "water boils at 100 C" somehow with accepted axioms such as "In a point outside a line, you can only draw one line parallel to it."

The difference between physics and mathematics is the easy with which they accept new axioms (physics: postulates). Math, almost never. Physics, very easily, if there is no straightforward reason to reject it, and the new theory seems to be useful.

Every scientific discipline that is readily able to test its theories against observations -- simply because they are massively available -- will operate more along the lines of physics than of mathematics, and massively accept new unreduced, free-standing theories, on the grounds that they seem to be correct based on past observations, and are useful somehow.

But then again, these disciplines will not hesitate to reduce (axiomatically) new theories to existing ones, whenever possible. In the end, it is still preferable to cut down on the number of unreduced, free-standing theories, whenever possible, as it simplifies the discipline, and increases its consistency.

So, the mathematical method of rigorously and systematically rejecting unreduced theories, is more of an ideal to strive to -- unfortunately unattainable -- for the other scientific disciplines.

Edited by erikp, : No reason given.

erikp responds to me:

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Look, we can go on and on and about Gödel, and insist that nobody understands Gödel, except for you

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No, we can't. In order to discuss who understands Godel, you need to know enough about set theory to be able to understand that you don't understand it. It's the practical aspect of the general claim that the more you know, the more you realize you don't know much at all.

I'm not saying I understand Godel. I'm saying I understand more than you. Hint: You need to stop reading Wikipedia and start reading the actual source. There's a reason I mentioned the Continuum Hypothesis. It's pretty much the foundational reason why we have the Incompleteness Theorems in the first place. Godel didn't come up with these things out of the blue. It is because of the work of the set theorists such as Russell who were working to do what Hilbert was envisioning: Create a mechanical mathematics. If you could define your symbology and rules, you could conceivably create a machine that would spit out proofs of theorems.

But the work of Russell and his paradox found that there were some issues with that claim. Self-reference is problematic for a mechanical process. The question of the size of the continuum has long been a puzzle and Godel's work showing that its assumption as true doesn't lead to a contradiction seemed to be a good thing...until Cohen came along and showed that its assumption as false doesn't lead to a contradiction, either. Godel had developed the Incompleteness Theorems before this, but it had been thought that such undecidable questions would be esoteric ones. That something as basic and simple as "How big is the size of the Real numbers?" would be undecidable was a big blow.

This is what I'm saying when I know more about it than you. As a trained mathematician, I know the history. I've had to do some of the foundational aspects regarding the problem. I don't claim to be the world's greatest mathematician. But I do claim to be a better mathematician than you.

Once again, I ask you the question you refused to answer back in Message 30:

What does "incomplete" mean?

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(Wiki):

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You need to stop reading Wikipedia and go to original sources. Stanley Jaki is a physicist, not a mathematician. I do not expect him to understand what incompleteness means. Incompleteness is a trait of axiomatic set theories. At what point does physics become an axiomatic set theory?

Jaki isn't here to answer that question. You are. If you have writings of Jaki that show him demonstrating how physics is an axiomatic set theory, then by all means bring them forward.

Do you?

Since you seem to love Wikipedia, let's see what it has to say about Jaki:

Since any 'theory of everything' will certainly be consistent, it must be either incomplete or unable to prove basic facts about the integers.

That's great. When would a physical theory become an axiomatic set theory? The universe is inherently mathematical in nature and physics is applied mathematics, but that doesn't mean that physics and mathematics are interchangeable.

And sure enough, going beyond Wikipedia and reading the actual sources, you find that Jaki and Hawking weren't actually saying incompleteness applies directly. They were using it as a metaphor:

What we need, is a formulation of M theory, that takes account of the black hole information limit, but then our experience with supergravity and string theory, and the analogy of Goedels theorem, suggest that even this formulation, will be imcomplete. -- Stephen Hawking, "Gödel and the end of physics"

Hmm..."Analogy." What might that mean? From reading the rest of it, the claim is that Hawking thinks that physics cannot be reduced to a finite series of statements. This is analogous to the Incompleteness Theorems, but it is not the same. If Hawking is right and physics is not reducible to a finite description, it is not directly because of incompleteness. It is, instead, analogous to incompleteness.

Now, Jaki writes:

Therefore any theory of physics, which contained more than a trivial form of mathematics, was subject to the restriction of Gödel's theorem. -- Stanley Jaki, "A Late Awakening to Gödel in Physics"

Now this, unfortunately, is an error. Physics doesn't "contain" mathematics. Physics uses mathematics and is applied mathematics, but it is not simply mathematics. That is, the objects of physics behave mathematically, but they are not the objects of mathematics. Therefore, we cannot apply the traits of mathematics to physics out of hand.

Trivial example: Numbers are infinitely divisible. Objects in physics are not. One of the wonderful tricks of math is that you can take a sphere and by dividing it into parts, you can reassemble them into two complete spheres of the exact same size as the first, no holes, no gaps. But this requires you to be able to divide it down into infinitely many infinitesimal pieces.

You can't do that with an actual object.

So given that physics is a limited subset of mathematics, why would we expect it to have all of the problems of mathematics?

Physics is not axiomatic set theory. Therefore, why should we expect incompleteness to apply? Why do we expect physics to depend upon those statements that in current mathematics are undecidable? The size of the continuum is undecidable. Is there something in physics that requires the size of the continuum to be known? From what we can tell, reality isn't continuous.

Jaki continues:

Then there is a book with the title The End of Science by John Horgan, a senior member on the staff of Scientific American. The book begins with the declaration that "Gödel's theorem denies us the possibility of constructing a complete, consistent description of physical reality."11 This, is, of course, not what the theorem denies. -- Stanley Jaki, "A Late Awakening to Gödel in Physics"

But the thing is, that's exactly what you're claiming: That the Incompleteness Theorems deny the possibility of a complete, consistent description of physical reality.

But wait, there's more:

Gödel's theorem does not mean that physicists cannot come up with a theory of everything or TOE in short. -- Stanley Jaki, "A Late Awakening to Gödel in Physics"

So if your own source denies your claim, where does that leave your claim?

Of course, Jaki then screws up his own statement:

But in terms of Gödel's theorem such a theory cannot be taken for something which is necessarily true. -- Stanley Jaki, "A Late Awakening to Gödel in Physics"

Why not? The condensed version seems to be that because physics uses simple arithmetic, that necessarily means that all the incompleteness of describing simple arithmetic applies to physics.

But that isn't true on its face. If all you need out of simple arithmetic is to be able to say that 1 + 1 = 2, then what is the problem? Since you seem to love the argument from authority so much:

It’s indeed the case that if the laws of physics are formulated in a formal system S which includes the concepts and axioms of arithmetic as well as physical notions such as time, space, mass, charge, velocity, etc., and if S is consistent then there are propositions of higher arithmetic which are undecidable by S. But this tells us nothing about the specifically physical laws encapsulated in S, which could conceivably be complete as such. -- Solomon Feferman, "The nature and significance of Gödel’s incompleteness theorems"

And then there's this:

Nothing in the incompleteness theorem excludes the possibility of our producing a complete theory of stars, ghosts and cats, all rolled into one, as long as what we say about stars, ghosts and cats can’t be interpreted as statements about the natural numbers. -- Torkel Franzén, Gödel's Theorem: An Incomplete Guide to its Use and Abuse

Hint: Franzén takes on Hawking to show that his attempt to model numbers in physics as blocks of wood is inappropriate.

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The fact that you need to introduce a "perfect theory",

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Incorrect. Do you not understand how indirect proof works? You start by assuming to be true that which you are trying to show to be false. By leading yourself to a contradiction, you show that what you assumed to be true is actually false.

Classic example: Is there a largest prime number? The answer is no and the common way to do so is by assuming that there is one. It doesn't actually exist, but you assume it to exist in order to show that its existence leads to a contradiction. That's the entire point.

Assume there is a perfect theory. Your "measure of falsifiability" would declare it to be false. But that's a contradiction. Therefore, your "measure of falsifiability" is false.

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which inevitably takes all possible factors and influences into account to explain phenomena, and therefore amounts to the "theory of everything"

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Incorrect. Presburger arithmetic is a perfect theory. It is complete, consistent, and decidable. It takes into account all possible factors and influences to explain phenomena.

But it is hardly a "theory of everything."

Note, your "measure of falsifiability" would declare Presburger arithmetic to be false. But it is true. Therefore, your "measure of falsifiability" is false.

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There is no need to distinguish between the "perfect theory" and the "other theory", because your perfect theory is an impossibility.

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Presburger arithmetic doesn't exist? Then how did we manage to define it?

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Furthermore, the idea that Gödel does not apply to gravity, is refuted in "The Relevance of Physics".

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Where? Chapter, page, and full quote in complete context, please.

Remember, your own source denies you:

Gödel's theorem does not mean that physicists cannot come up with a theory of everything or TOE in short. -- Stanley Jaki, "A Late Awakening to Gödel in Physics"

What happens to your claim now?

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Now first explain why in addition to Stephen Hawking being wrong, Stanley Jaki is also wrong

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I already did: They are physicists, not mathematicians. We cannot expect them to understand the details of set theory. Jaki makes that exact comment, himself:

The paper ["Formally undecidable propositions of Principia Mathematica and related Systems I."] could not be easy reading for most physicists, or even for most mathematicians for that matter.

As for Hawking, he doesn't say what you claim he says. He simply uses the incompleteness theorems as a metaphor for the problems facing physics.

At any rate, they are not here. You are. It would help if you would stop using Wikipedia and started using original source material.

I'm still waiting for you to answer the question I asked back in Message 30:

What does "incomplete" mean?

Edited by Rrhain, : Added references of responses to Jaki's claim and Hawking's misunderstanding.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

erikp responds to me:

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Wrong. That requires a Theory of Everything (TOE), which is presumed impossible.

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Why? Because you say so? Why should we believe you?

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Now, physics is apparently subjected to Gödel's Incompleteness, regardless of what you say

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Except there is no evidence of such, no matter how many Wikipedia quotes you care to pull forward.

Hint: Stop reading Wikipedia and start referring to original sources.

It’s indeed the case that if the laws of physics are formulated in a formal system S which includes the concepts and axioms of arithmetic as well as physical notions such as time, space, mass, charge, velocity, etc., and if S is consistent then there are propositions of higher arithmetic which are undecidable by S. But this tells us nothing about the specifically physical laws encapsulated in S, which could conceivably be complete as such. -- Solomon Feferman, "The nature and significance of Gödel’s incompleteness theorems"

Nothing in the incompleteness theorem excludes the possibility of our producing a complete theory of stars, ghosts and cats, all rolled into one, as long as what we say about stars, ghosts and cats can’t be interpreted as statements about the natural numbers. -- Torkel Franzén, Gödel's Theorem: An Incomplete Guide to its Use and Abuse

Hell, even your own source makes the point:

Gödel's theorem does not mean that physicists cannot come up with a theory of everything or TOE in short. -- Stanley Jaki, "A Late Awakening to Gödel in Physics"

Now, believe me, it pains me to say this: Dyson is wrong. Dyson is a mathematician. He ought to know better. [And wait for it: I'm coming to that.] Feferman's comment above is a direct response to Dyson's claim.

Dyson's comment as you quoted it above was from a New York Times Review of Books review of Brian Greene's book, The Fabric of the Cosmos. Alas, Dyson puts words into Greene's mouth, claiming that Greene said, "When we know the fundamental equations of physics, everything else, chemistry, biology, neurology, psychology, and so on, can be reduced to physics and explained by using the equations."

As Greene pointed out in his rebuttal, he said no such thing. In fact, he believes the exact opposite:

I can't imagine making such a statement as it runs thoroughly counter to my long-held beliefs. While there is apparently no transcript available, my views on this issue were expressed in The Elegant Universe (1999), page 17, where I write that finding the fundamental equations of physics "would in no way mean that psychology, biology, geology, chemistry, or even physics had been solved or in some sense subsumed."

Feferman continues in his response to Dyson:

In practice, a much different picture emerges. Beyond basic arithmetic calculations, the mathematics that is applied in physics rarely calls on higher arithmetic but depends instead mainly on substantial parts of mathematical analysis and higher algebra and geometry. All of the mathematics that underlies these applications can be formalized in the currently widely accepted system for the foundation of mathematics known as Zermelo-Fraenkel set theory, and there is not the least shred of evidence that anything stronger than that system would be needed.

And here's where I get to that point I told you to wait for above:

Dyson admits he was wrong. After Greene and Feferman's response to his review, Dyson writes:

Each time I publish a book review in The New York Review of Books, I receive a bimodal set of responses. First come the responses from nonexpert readers who write to tell me how much they like the review. Second come the responses from expert readers who write to correct my mistakes. I am grateful for both categories of response, but I learn much more from the second category. It is inevitable that I make mistakes when writing about fields in which I am not an expert, and I rely on the experts to set the record straight. I am especially grateful to Brian Greene for correcting my misrepresentation of his views. I apologize to him for the misrepresentation, and for not checking with him before publishing the review. I am grateful to Solomon Feferman for explaining why we do not need Gödel's theorem to convince us that science is inexhaustible. I am grateful to many other readers who have written to me privately to correct other mistakes. [emphasis added]

You need to get your nose out of Wikipedia and start referring to original sources.

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But now I have to retract that concession, because: "the laws of physics are a finite set of rules and include the rules for doing mathematics, so that Gödel's theorem applies to them."

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Since Dyson has retracted that claim, will you now retract your retraction?

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But then again, I don't need Gödel to demonstrate that science is false.

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Then why don't you do so because you haven't managed to do it using the Incompleteness Theorems.

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It is sufficient to demonstrate the relationship between the probability that a theorem will be contradicted by a fact, and the total number of such potential facts.

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But this "measure of falsifiability" of yours would declare a true theory to be false. This is a contradiction which means your "measure of falsifiability" is false.

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If that number is infinite

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What does infinity have to do with it? If the theory is true, it doesn't matter how many scenarios there are: The theory describes them. You are essentially claiming that there cannot be an infinite string of heads from a tossed coin. While such an event is highly unlikely, it is not impossible.

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the relationship says that the theorem will inevitably be contradicted

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But the theory is true, so how can it be contradicted? This is a contradiction, therefore your claim is false.

By the way: There are an infinite number of theories. Therefore, your own "measure of falsifiability" requires that it be tested against them. Since your "measure of falsifiability" requires that the result of an infinite number of scenarios be that it fails at least once, this necessarily means that your "measure of falsifiability" is false at least once, meaning that there is a perfect theory.

QED

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Godel's Incompleteness Theorem demonstrates that it is impossible for the Bible to be both true and complete.

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Religion does not massively use axiomatic reduction nor does it use falsifiable statements, that can be contradicted by future facts. Therefore, it is not incomplete; nor will any future fact be able to point out incompleteness (falsehood).

This also means, indeed, that religion cannot be used to predict the future.

By sticking to past facts (proven, true) and unfalsifiable statements (unproven, true), according to the definition, religion can never be false. Religious statements are necessarily unproven (but scientific statements are too).

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Nevertheless, Franzén adds,Gödel’s theorem tells us only that there is an incompleteness in the arithmetical component of the theory.

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The entire edifice of physics rests on that arithmetic component ...

I think that exploring the limits of religion, or the limits of any discipline for that matter, is one of the most important exercises in that discipline.

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The reason why I am interested in the limitations of religion, is because religion is often used to attack science.

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Religion cannot readily be used to predict a stream of future events. Therefore, it does not even have the same purpose as science.

Religion phrases rules about right and wrong, and invites the believers to obey those rules. What are the limitations here? Well, since the rules are unfalsifiable, we can't readily validate them against a stream of future events. But then again, by staying clear of infinite falsifiability, religion stays clear of being necessarily false.

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What are the practical implications of your views in terms of existing scientific theories? Which of them do you see in a different light as a result of your POV?

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Science is just an instrument, replete with limitations, massively abused to justify questionable political decisions, and benefiting from an aura of infallibility. It is about time that people realize that science is absolutely not an infallible instrument, and that it cannot be used as a final argument in political decision making. Then, science simply becomes a dictatorial ideology. Scientific/unscientific does not equate with right and wrong.

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Are you stating that a grand unified theory is not possible?

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Does unifying the basic forces amount to phrasing a theory of everything? I don't know. Could be.

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Are you suggesting that other existing scientific theories are way off the mark, or that they may continually need minor modification and tweaking as from Newtonian mechanics to general relativity?

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Science should, of course, continue its tweaking and keep refining the instruments in order to predict future events better. Nobody questions the usefulness of science.

erikp responds to me:

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Only the infinitely falsifiable theories are presumably false.

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But there are infinitely many theories. Therefore, the claim that "infinitely falsifiable theories are presumably false" is infinitely falsifiable.

And thus by its own terms, is false.

QED

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Therefore, the number of infinitely falsifiable theories needs to be finite.

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But there are infinitely many theories. It is trivial to generate them.

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It suggests that the number of possible theories in science has a fixed upper bound.

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But reality is the exact opposite. There is no upper bound. There are infinitely many theories.

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In other words, there cannot be an infinite number of scientific theories. On the contrary, the number of scientific theories that could ever be phrased, is (potentially large but) countable.

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You don't know what "countable" means, do you? It does not mean finite. The integers are countable, but they are not finite. Finite sets are countable, but countable sets can be infinite. All squares are rectangles but not all rectangles are squares.

You need to stop reading Wikipedia and start looking at original sources.

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The collection of numbers representing these theories has the same upper bound. This means that all past and future science can be represented by a fixed, finite series of numbers.

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Incorrect. There are infinitely many theories. It is trivial to generate them and it can be mechanically done.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

erikp responds to me:

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Not only does the "perfect theory" not exist, even the "other theory" does not exist.

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Huh? We don't have theories right now?

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Both my relationship as Gödel imply that there is only a finite number of scientific theories possible.

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But there are infinitely many theories. They are trivial to generate and can be mechanically done. Your "measure of falsifiability" is thus infinitely falsifiable and, by its own claim, is false.

QED

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Consequently, your "other theory" will not be able to keep up with the "perfect theory" beyond a certain point

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"Beyond a certain point"? What is this "beyond a certain point"? You're referring to these fantasy observations that haven't been made yet, again. That isn't going to help you because we haven't made those fantasy observations. Therefore, right here and right now, how do we distinguish between the perfect theory and the other theory?

If you can't, how do you justify claiming that the theory that we have isn't the perfect one?

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You simply won't be able to phrase such "other theory".

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Huh? We don't have theories right now? What on earth are all those scientists doing?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

erikp writes:

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According to these definition the theory "Water boils at 100C" is false.

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But that's not a theory. That's simply a statement.

Do you even know what a theory is?

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What if there are simply no observations (facts) possible for a theory?

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Then it isn't a theory.

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Then the theory as well as its anti-thesis are both true.

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Incorrect. It isn't a theory at all. Theories refer to observations and if there are no observations, there is no theory.

A -> B
~B -> ~A

You don't know what a theory is, do you?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

erikp writes:

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In this context, "infinite" should probably be understood as "unbounded".

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Why? Do you even know what "unbounded" means?

There are infinitely many theories. Therefore, your "measure of falsifiability" is infinitely falsifiable. Thus, by its own measure, it is false.

QED

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I guess there is no infinite number of temperatures.

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Why not? There are an infinite number of numbers between 1 and 2. Since we can calibrate the temperature scale to include a "1" and a "2," then this would necessarily mean there are an infinite number of temperatures, right?

Hint: I'm setting you up. I don't like playing "gotcha" games, but I think I have to at this point. You're the one saying that physics can model math, so here's your chance to do so. Math says there are infinitely many numbers between 1 and 2.

So are there infinitely many temperatures or not?

If there aren't, then what makes you think physics can model mathematics since math includes infinity.

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The problem is that "infinite" does not really exist outside the realm of mathematical formulas. Projected in to the physical world, we may have to replace the term "infinite" by "unbounded".

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You don't know what "unbounded" means, do you? Infinite things can be bounded. They can be unbounded.

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What's more, there is already a large body of literature containing theories about "infinite", and I don't want to start making too many blanket statements about "infinite" that could be contradicted by people who happened to have written entire books about it.

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Ahem.

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I concede the point that the term "infinite" is problematic and should be treated with the necessary care.

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Indeed. Since it is clear you don't understand what the word means, I would agree that you shouldn't use it.

But replacing it with another word you don't understand doesn't solve the problem.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

erikp writes:

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According to the definitions of "true" and "false", incomplete theories are indeed false.

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Incorrect. It's called "incomplete," not "inconsistent." ZFC is incomplete, not inconsistent.

I'm still waiting for you to answer the question I asked back in Message 30:

What does "incomplete" mean?

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As soon as water has been observed to boil at any other temperature than 100 C, the theory that says "Water boils at 100 C", has been proven to false.

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Incorrect. "Water boils at 100 °C" is not a theory. Even as a statement, the observation of water boiling at a temperature other than 100 °C is not a refutation of it for I can easily show you water boiling at that temperature.

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It cannot be rescued just by saying that it is "incomplete".

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You don't know what "incomplete" means, do you? I'm still waiting for you to answer the question I asked back in Message 30:

What does "incomplete" mean?

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"Water boils at 100C" is a theory

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Incorrect. It is a statement, not a theory. The observation of water boiling at a temperature other than 100 °C is not a refutation of it for I can easily show you water boiling at that temperature.

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The difference between physics and mathematics is the easy with which they accept new axioms (physics: postulates).

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You don't know what a "postulate" is, do you?

Question: Why is it called the "parallel postulate" in geometry if that's a physics term?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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But that's not a theory. That's simply a statement. Do you even know what a theory is?

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We've been through that problem before, quoting the various alternative definitions for the term "theory" and establishing that "Water boils at 100C" is a theory. I am not going to go through all of that again, just because you did not read that part of the thread.

erikp writes: bluegenes writes: I don't find Gödel's language weird at all. He knows the difference between "incomplete" and "false". According to the definitions of "true" and "false", incomplete theories are indeed false. As soon as water has been observed to boil at any other temperature than 100 C, the theory that says "Water boils at 100 C", has been proven to false. It cannot be rescued just by saying that it is "incomplete". The definition simply says that it is false.

Here's where I quibbled with your chosen phrase before. Had you chosen "Water only boils at 100C", the statement would have been falsified, and is clearly false. As water frequently does boil at 100C, the statement isn't false, merely of little or no use to us in explaining anything about water. Here's an example of incompleteness:

"Biological evolution proceeds by variation combined with natural selection."

That's observably true. However, it also proceeds by other means, like genetic drift (neutral evolution), so:

"Biological evolution proceeds by variation combined with natural selection and by variation alone" is an improvement."

Neither statement is a comprehensive theory of all biology, but both are definitely true. Scientific procedure presumes the incompleteness of theories. They play a different role from the theorems of maths.

erikp writes: So, the mathematical method of rigorously and systematically rejecting unreduced theories, is more of an ideal to strive to -- unfortunately unattainable -- for the other scientific disciplines.

In the rest of your post, you did bring in observations, correctly, and that's what I was looking for. It is observation and experimentation in the physical world that makes science radically different from maths, you'll agree, and that's perhaps why the effect of Godel's theorems will be different (but not non-existent).

You've got far too much to deal with from others, so I won't pile on!

Good luck!

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Do you even know what "unbounded" means?...Since it is clear you don't understand what the word means, I would agree that you shouldn't use it...

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If that is your pov, quote the definition, and then demonstrate that I used the word inappropriately.

You've already argued that Stephen Hawking is some kind of an idiot who doesn't understand anything about mathematics and that you are the one who should receive the Nobel prize in his stead.

How unfortunate for you that nobody seems to agree with you. If you are so much smarter than Stephen Hawking, how comes nobody is aware of that?

Everybody is obviously unjustly underestimating your amazing intelligence! Why would that be !?