What is a Theory?

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science does not deal with different magnitudes of falsifiability. Either a theory is falsifiable or it isn't.

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"Magnitude of falsifiability" (MoF) can be defined, and can therefore not be rejected without justification.Given a random theory, from the overall collection of theories T, which has MoF=1, that is, just one possible observation to contradict it, what is the likelihood of that observation contradicting the theory ( P(C | MoF=1) ), in absence of any further information? 1 - 0.5 = 0.5.Given a random theory, from the overall collection of theories T, which has MoF=1, that is, just one possible observation to contradict it, what is the likelihood of that observation contradicting the theory ( P(C | MoF=2) ), in absence of any further information? 1 - 0.5^2 = 0.75.MoF P(C)
1 1 - 0.5^1 = 0.500
2 1 - 0.5^2 = 0.750
3 1 - 0.5^3 = 0.875Therefore, the likelihood that a random theory, from the overall collection of theories T, with MoF=k will be contradicted by the facts is: P(C|MoF=k) = 1 - 0.5^k.What does that say about the likelihood that a random infinitely falsifiable theory will be contradicted by the facts?lim (k -> inf) ( 1 - 0.5^k) = 1 (=always)What does that say about the likelihood that a random unfalsifiable theory will be contradicted by the facts?P(C|MoF=0) = 1 - 0.5^0 = 1 - 1 = 0 (=never)Of course, science does not consist out of randomly constructed theories, by just randomly assembling a set of statements expressed in formal language. On the contrary, the scientific method purposely constructs the collection of statements, constituting any of its theories, in such a way that they are hard to contradict by facts.But then again, every scientific theory is still inevitably member of the overall collection of theories T, for which we can state that the likelihood that infinitely falsifiable theories will be contradicted by the facts is 1 (=always).Therefore, regardless of how well the theory has been constructed, if it is infinitely falsifiable, we may safely assume that it will eventually be contradicted by the facts.In this observation Magnitude of Falsifiability (MoF) is a key concept. Therefore, I must reject the idea that falsifiability should not be quantified. It can be quantified and its quantification is instrumental in supporting the case.Edited by erikp, : No reason given.Edited by erikp, : No reason given.

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If it is a key concept, then why did we have to wait for 100+ messages before it entered the conversation? And I might mention that it wasn't even you who brought it up, but me.

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Indeed. I didn't think of quantifying falsifiability, until you brought up that you thought it had no use, so that I could conclude that it does have use.There is indeed a positive relationship between falsifiability of a theory, that is, the number of potential (future) facts k that could contradict it, and the likelihood that the theory will be contradicted P(C|MoF=k).This relationship is:P(C|MoF=k) = 1 - 0.5^k.And this relationship suggests that infinitely falsifiable theories can most reasonably be expected to be contradicted.All of this supports my case: infinitely falsifiable theories are presumably false. And therefore, supported by the relationship that we established, I can comfortably re-iterate my initial point:Science must be unproven and false in order to be useful.

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Theories are not random variables. They cannot be modeled as random variables. Therefore, attempting to apply statistics to them is a failed experiment.

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Theories can perfectly well be represented by numbers. If theories represent a number, a number can represent a theory. That is exactly the gist of Gdel's (axiomatic) proof. Wiki(Gdel):

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In order to prove the first incompleteness theorem, Gdel represented statements by numbers. Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gdel sentence states that no natural number exists with a certain, strange property. A number with this property would encode a proof of the inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, assuming the theory is consistent (as done in the theorem's hypothesis) there is no such number.

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If a statement can be represented by a number, a theory, that is, a collection (a set) of statements can be represented by a number too:

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a (formal) theory is a set of statements expressed in a particular formal language.

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Therefore, the overall set of theories, T, can be represented as a set of numbers. Like in Gdel's proof, the properties of these numbers translate to properties of the theories they represent.Futher, nothing prevents anybody from taking a random sample of numbers and therefore a random sample of theories, and make valid statistical inferences, based on the sample, about the population of theories.Furthermore, for the population of theories, the observations (facts) that can contradict them, can be observed. The occurrence of these facts is a random process, that can perfectly well be sampled and modeled by statistics.

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Huh? The only way to get to Mars using gravitational theory is if it's wrong? The reason why we had so many failures of probes making it to Mars is because our theory of gravity is actually correct?

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Every theory that goes into building and operating these probes is false, but very hard -- currently impossible -- to prove so. In those circumstances, we can reasonably expect these probes to work, because nobody can tell us why they wouldn't. But that still does not make any of these theories true. They are still false. One day or the other, someone may also finally be able to tell us why exactly they are false.

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The only way to show a theory false is to bring forth the observation.

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Indeed. Bringing forth the observations that will contradict the current body of science in use, is very hard -- currently impossible. So, I repeat, the current body of science is entirely false, but it is currently impossible to know why.

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Theories necessarily conform to all the observations we have ever made.

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In order to be true, it is not sufficient that a theory conforms to all the observations we have ever made. It must also conform to all possible future observations we could ever make.

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Your "measure of falsifiability" claim insists that the "perfect theory" is false. But it isn't false. It's true.

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There is absolutely no way to know if an infinitely falsifiable theory is "perfect", because there are still an infinite number of new observations possible to which this theory also has to conform in order to be perfect.

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Suppose you have a theory that is perfect.

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If that theory exists, it will be infinitely falsifiable, and therefore, there is no way to know that it is perfect.

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Which means that the theory isn't false.

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If it is infinitely falsifiable, the theory is presumably false. That has nothing to do with the truth or falsehood of any claim made by the theory, but only with the fact that the number of potential observations is infinite.I stated that:P(k) = 1 - 0.5^kgiving the observation 1 chance out of 2 to contradict the theory (absence of any further information).However, it does not matter how likely an observation will contradict the theory. It could be extremely small:P(k) = 1 - (1-0.00000000000000001)^kFor an infinite number of observations this likelihood will still go to 1 (=always).P(k) = 1 - (1-a)^k, with a the likelihood that 1 observation contradicts the theory.lim (k --> inf) P(k) = 1, on the condition that a is not exactly zero.The perfect theory requires that a would be exactly zero. This perfect theory would have to take everything into account. It would be the Theory of Everything.Concerning the Theory of Everything, Stephen Hawking (Wiki, Theory of Everything):

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Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gdel's Theorem, concluded that one was not obtainable.

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"Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind."

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Rrhain writes:

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{about perfect and other theories}

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What a gem of an argument, thanks! (Not that it will convince Erik, of course...)

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If you can convince Stephen Hawking, I will be convinced too. But then again, you would first have to demonstrate that Gdel was wrong. Wiki:

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Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gdel's Theorem, concluded that one was not obtainable.

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If you do agree, then since the "other theory" necessarily conforms to all observations we have ever made (because it is a theory) and since the "perfect theory" also conforms to all theories we have ever made (because it is true), how do you distinguish between the two of them? The only thing we have to go on are the observations we have made.

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The point is that such "perfect theory" cannot exist. It cannot be phrased.

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Stephen Hawking is a cosmologist, not a mathematician. I would not expect him to understand the field of set theory...I'm saying you don't understand what Godel said...

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Look, we can go on and on and about Gdel, and insist that nobody understands Gdel, except for you (Wiki):

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In his 1966 book The Relevance of Physics, Stanley Jaki pointed out that, because any "theory of everything" will certainly be a consistent non-trivial mathematical theory, it must be incomplete. He claims that this dooms searches for a deterministic theory of everything.

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The fact that you need to introduce a "perfect theory", which inevitably takes all possible factors and influences into account to explain phenomena, and therefore amounts to the "theory of everything", invalidates your argument.

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Now, stop avoiding the question:How do you distinguish the "perfect theory" from the "other theory"?

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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.Furthermore, the idea that Gdel does not apply to gravity, is refuted in "The Relevance of Physics". Now first explain why in addition to Stephen Hawking being wrong, Stanley Jaki is also wrong, before insisting again on the "perfect theory".

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Incorrect. The universe works, doesn't it? Would you agree that it works consistently? Therefore, there is necessarily a theory of it since a theory is an explanation for how processes work in the universe.

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Wrong. That requires a Theory of Everything (TOE), which is presumed impossible.Now, physics is apparently subjected to Gdel's Incompleteness, regardless of what you say (Freeman Dyson, Wiki):

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Gdel’s theorem implies that pure mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. [...] Because of Gdel's theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gdel's theorem applies to them.

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The idea that Gdel applies to physics, is what I originally though anyway, but I couldn't prove it, and that is why I originally conceded the point to you.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 Gdel's theorem applies to them."But then again, I don't need Gdel to demonstrate that science is false.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. If that number is infinite, the relationship says that the theorem will inevitably be contradicted, since a TOE is impossible. Edited by erikp, : properly close quote

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Thus, your claim that all theories are false is also false.

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Only the infinitely falsifiable theories are presumably false. The other theories can be true.But you are right about one thing. "All infinitely falsifiable theories are presumably false." can only be true, if it is itself not an infinitely falsifiable theory. Therefore, the number of infinitely falsifiable theories needs to be finite.It suggests that the number of possible theories in science has a fixed upper bound.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.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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By your own reasoning, isn't your theory that the perfect theory is an impossibility inevitably false?

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No, because I am making a statement that can be contradicted by just one fact and not an infinite number of facts.My statement is therefore not infinitely falsifiable. It is falsifiable only by one, single fact (the appearance in reality of that one Theory of Everything).So, my statement is not necessarily false."Water boils at 100 C" is necessarily false, but saying that the TOE does not exist is not necessarily false.

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Both the perfect theory and the other theory are in concordance with all current observations. The first because it is true. The latter because it is a theory.

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Not only does the "perfect theory" not exist, even the "other theory" does not exist.Both my relationship as Gdel imply that there is only a finite number of scientific theories possible. This means that beyond a certain point scientific theories cannot be improved any further, in order to cover consistently additional observations. At that point, we will have reached the limits of science.Consequently, your "other theory" will not be able to keep up with the "perfect theory" beyond a certain point (beyond a certain number of observations). You simply won't be able to phrase such "other theory".

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Your statement would either be unfalsifiable or false. So it's unprovable.

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It is falsifiable, because the appearance of the TOE would falsify it. It is not necessarily false, because it can only be falsified by a finite number of facts (just one).

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You've equated unprovability with falseness (something which others have told you is wrong)

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Where did I equate that? Please, quote.

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"Water boils at 100 C" is a demonstrably true statement.

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It is false, because water can perfectly well boil at 50 C.

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Stating that a TOE couldn't exist in the future is a different matter. These kind of statements aren't scientific theories, anyway.

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Wrong. The theories about theories are mathematical, and therefore scientific theories, of which the Gdel Theorem is only one example.