Message 166 of 249 (494433) 01-16-2009 1:31 AM

Reply to: Message 162 by Modulous 01-15-2009 8:32 PM

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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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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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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

Message 167 of 249 (494434) 01-16-2009 1:40 AM

contrasted with..

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..

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

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

And the corollary..

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

Message 168 of 249 (494435) 01-16-2009 2:09 AM

Reply to: Message 167 by shalamabobbi 01-16-2009 1:40 AM

Only use the big "Gen Reply" button if you are not replying to a specific message.

You did no crime, but things work better if you follow the above guidelines.

Before using the "Submit Reply" button, using the "Preview" button is also a good thing (maybe you are using it). That way you see the (tentative) final results, which might include coding errors which will be flagged in red. Then you can make changes without having to do edits (a stack of "edited by" messages tend to look dumb).

NO REPLIES TO THIS MODERATION MESSAGE.

Adminnemooseus

Message 169 of 249 (494444) 01-16-2009 5:11 AM

Reply to: Message 163 by bluegenes 01-15-2009 9:44 PM

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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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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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"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.

Message 170 of 249 (494445) 01-16-2009 5:28 AM

Reply to: Message 140 by erikp 01-15-2009 5:51 AM

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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:

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:

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:

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:

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:

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

Of course, Jaki then screws up his own statement:

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:

And then there's this:

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:

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:

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.

Message 171 of 249 (494446) 01-16-2009 6:10 AM

Reply to: Message 141 by erikp 01-15-2009 6:02 AM

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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.

Hell, even your own source makes the point:

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:

Feferman continues in his response to Dyson:

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:

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

Message 172 of 249 (494447) 01-16-2009 6:10 AM

Reply to: Message 167 by shalamabobbi 01-16-2009 1:40 AM

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

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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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Message 173 of 249 (494449) 01-16-2009 6:19 AM

Reply to: Message 143 by erikp 01-15-2009 6:16 AM

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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.

Message 174 of 249 (494450) 01-16-2009 6:24 AM

Reply to: Message 145 by erikp 01-15-2009 6:37 AM

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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?

Message 175 of 249 (494451) 01-16-2009 6:37 AM

Reply to: Message 165 by erikp 01-16-2009 1:21 AM

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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?

Message 176 of 249 (494453) 01-16-2009 6:51 AM

Reply to: Message 166 by erikp 01-16-2009 1:31 AM

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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.

Message 177 of 249 (494455) 01-16-2009 7:02 AM

Reply to: Message 169 by erikp 01-16-2009 5:11 AM

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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?

Message 178 of 249 (494471) 01-16-2009 9:48 AM

Reply to: Message 175 by Rrhain 01-16-2009 6:37 AM

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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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Message 179 of 249 (494472) 01-16-2009 9:52 AM

Reply to: Message 169 by erikp 01-16-2009 5:11 AM

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.

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!

Message 180 of 249 (494473) 01-16-2009 9:56 AM

Reply to: Message 176 by Rrhain 01-16-2009 6:51 AM

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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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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 !?