Question for people who BELIEVE that there are no miracles: is Goldbach's conjecture true?

In every case when specific numbers have been plugged in and a definite conclusion has been reached that those numbers confirm Goldbach's conjecture or a definite conclusion has been reached that those numbers disconfirm Goldbach's conjecture, the calculations have always confirmed Goldbach's conjecture. What gives you reason to weigh the evidence in support of the claim that there aren't any miracles as stronger than the evidence in support of the conclusion that Goldbach's conjecture is true?

Mathematical propositions and solutions are abstract .... Miracles should be concrete realities (if true) .... You are playing semantical games that distort the meanings of the terms and their relationships ....

Non sequitur ....

Mathematics is abstract, but physics (if true) describes concrete reality.

The connection between the two entities has NOT been established, and this puzzle does nothing to solve that question ....

By definition, anomaly/miracle is an event/observable that does not fit the received explanation models (aka theories) and/or the metaphysical belief systems the theories are built on.

For example, miracles/anomalies happen every day in the field of medicine, but sadly the Western psyche is conditioned to take theory (or rather, layman's physics instead of best physics so far) for reality and by cognitive dissonance disregard any and all anomalies/miracles challenging the accepted theory.

I believe that I referred specifically to mathematical physics. How would you describe the role of mathematics in mathematical physics?

You lose.



(If you didn't know, a constitutive relation is a way of mathematically encoding information about a physical system. Goldbach's conjecture doesn't have one, IIRC)

shows that you have no understanding at all of how mathematical conjectures are proven or how they are different from scientific hypotheses. You don't confirm a general mathematical conjecture with "evidence" that is accumulated to sufficient weight. You prove it to a 100% certainty by a deductive argument from agreed upon postulates. No finite number of examples for which a conjecture works is sufficient to "prove" that it's true, (or even to render it more likely to be true) for an infinite number of possibilities. You can only do that with a generalized argument that covers all possibilities.

does the agreement provide 100% certainty that the postulates are actually true?

When sacred and mystical postulates are agreed upon, it is sometimes said that the agreement provides 100% certainty that the sacred and mystical postulates are actually true.

To speak of postulates being "true" is just blatant ignorance.

If some reasoning relies upon some statements that are alleged to be "postulates" to arrive at some particular conclusion (such as Fermat's Last Theorem), then how do we know that the conclusion is actually true?

Statements are not "alleged" to be postulates in a mathematical system. Do you comprehend that or not?

Given that the postulates that are the basis for any such proof cannot be spoken of as being "true" without revealing blatant ignorance, what would be the significance of a proof of Goldbach's conjecture?

Also, do we know that Goldbach's conjecture is a meaningful statement? What if the generally accepted postulates of mathematics aren't a sufficient basis for any proof of Goldbach's conjecture? What if nobody will ever find a counter-example because a counter-example doesn't exist? In this case, it wouldn't be possible to prove Goldbach's conjecture and it wouldn't be possible to disprove Goldbach's conjecture.

Here's an idea. Someone who has superior intuition could tell us whether Goldbach's conjecture or its negation should be a postulate. However, once it is a postulate, we would be revealing blatant ignorance if we spoke of the new postulate as being "true."

It's one thing for us to not know whether or not a particular statement is true. In that case, it's possible that the statement is actually true and we don't know it, or actually false without us knowing that it's false. However, if we go beyond that and consider something that is neither true nor false then surely we are considering something that isn't a meaningful statement. Should we conclude that Goldbach's conjecture might be a meaningless sequence of symbols like the following?
(5 + - +) > (3 + (=2))

What if nobody will ever find a counter-example because a counter-example doesn't exist? In this case, it wouldn't be possible to prove Goldbach's conjecture and it wouldn't be possible to disprove Goldbach's conjecture.

This is what Godel's Incompleteness Theorem is all about - it proves such statements exist and that there's no general way of knowing whether any particular statement is undecidable.

"What assumptions do you rely upon to show that the set of postulates is consistent?" with a one-word message:

"Nothing."

Are you ready to give an example of a proof, based on nothing, that establishes the consistency of some set of postulates?

As I explained before either you understand what I do by consistency and therefore agree on the necessary entailments of this property or you do not and there is nothing that can be discussed.

If you explained something before, then could you please post a link to that explanation that you already provided?

Its possibly true. But as we do not have a means to confirm it we cannot say it is absolutely true. And as this is just an example of an abstract construct conforming to the rules we have specified it does not directly translate to the methodology of collecting evidence and confirming theories in matters of reality.

This is to say that we do not have set rules governing the universe that we created. Instead we determine the rules governing the universe to the best of our abilities and form our hypothesis on the observations.

Based on this claims for a miraculous event are one's which must rise to the degree by which the miracle deviates from our expected norms. Put succinctly, extraordinary claims demand extraordinary evidence.

Conjecture becomes "true" theorem when it is provable (-> proof theory) inside a finite set of axioms. What Gödel proved is that there is allways in Principia Mathematica and other similar "well developed" formal systems that accounts for natural numbers a statement resembling the lier's paradox that is both ("intuitively") true and unprovable in the finite set of axioms of the well developed system.

The new post-Gödelian foundation of math, Category theory and Topos theory, allows mathematicians to construct-imagine-find mathematical toposes with (even wildly) different sets of axioms from the standard set theory and study the interactions of these toposes. This new foundation of math is in its nature dynamical rather than stagnant object. Needles to say, findings of Category theory has allredy very interesting applications in quantum theory - "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html) is as strong as ever (-> mathematical physics -> metaphysics of mathematical platonism).

Conundrums about the prime numbers that still defy mathematicians, such as Goldbach conjecture and even more importantly, Riemann hypothesis, are extremely important for the philosophy of math, proof theory and metaphysics. My feel about Riemann's hypothesis (and I'm not alone) is that it is NOT provable inside axioms of standard set theories, but a "lier's paradox" of sorts. Based on hypothesis that Riemann hypothesis is an observable/intuitive invariant codependent on observable/phenomenological realization of quantum potential a strategy (or rather, many strategies) for physical proof of Riemann hypothesis opens up. I've been thinking that strongest (physical) proof of Riemann hypothesis would involve imagining-finding different mathematical toposes with different qualities of prime numbers (if possible for human psyche, and that is a big if) which would produce "miraculous" or "anomalous" quantum "reality" from this phenomenological one that is observable to observers like us.

This thread poses a question for people who believe that there are no miraculous events. Do you hold the position, until one particular event has been described and you have been persuaded both that it actually occurred and that it was miraculous, that you are justified in assuming that there are no miraculous events?

and it's meaningless even if you do. But go ahead and try to make sense.

If you believe that the thread is now meaningless, then you should try to make meaningful contributions in future.

to try to get you to define what you mean by a "miracle" in your OP. You seem unable to do so, which makes me wonder why you would post such a thing in the first place, other than that you'd rather post nonsense than nothing. Without defining that term, any discussion of your original question is a waste of time.

Do you anticipate that, given a definition of the word "miracle" that is consistent with at least one generally accepted definition of the word "miracle", you might be forced to conclude either that you have no opinion about whether or not there are miracles or that you believe there are miracles?

Just give us the definition of the word "miracle" as you meant it in the OP. Simple, right? I assume you did know what meaning you intended when you used that word?

Are the DU username skepticscott and the associated password being shared by a number of different people?

If you want a definition that uses only words from some list that contains nothing but
a) words that are clear enough to you with no definition required; and
b) words for each of which you already possess a definition that is satisfactory to you...

... then you should identify your word list.

Who knew?

are convinced that there aren't miracles. Directing a question in that manner is quite different from making a claim that there is strong evidence (available to the general public) in support of the view that miracles occur.

Can you explain how your classification of a statement (such as the classification of a statement as an "abstract relationship") is relevant to your decision to accept, reject, or remain undecided about the statement?

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You seem to have a knack for comparing complex mathematics to religious belief.

I accept the notion that Goldbach's conjecture is probably true because all evidence to date suggest that it is, yet I, nor anyone else, can prove it. You could substitute Goldbach's conjecture for "no miracles" and still have a sound statement.

I am not comfortable with the comparison of mathematical equations with religious beliefs or in this case the specific notion of the existence of miracles. You will not prove the existence of a miracle anymore than you will prove that matter is made of atoms. You support or disprove. Mathematics is an integral part of science, but the two are not the same and if we are going investigate a miraculous claim science will be the means.

Moreover, phrasing the latter supposition in the negative renders the argument impossible to disprove. The argument that miracles do not exist can be supported but not disproved and therefore is not a scientific question. Neglecting for the moment that the whole notion of a supernatural explanation for a physical occurrence i.e. a miracle has any place in a scientific discussion. So, lets pose the alternative argument that to date no miracles have been conclusively established. Propose your miracle and support your argument, the onus is on the believer to establish a miracle not the skeptic to disprove an insurmountable position in the negative.

I suspect that you will find the task at hand to be very difficult. Hence the reason you have turned to mathematics rather than science as a means toward establishing the existence of miracles. Rather than pose a mental gymnastics routine propose a miracle but note the tag line.

:evilfrown:

Perhaps you could clarify.

For example, the suppositions and/or arguments that you refer to are:

1. Boojatta's suppositions and/or arguments and no others.
2. FM Arouet666's suppositions and/or arguments and no others.
3. Some combination (to be specified) of Boojatta's suppositions and/or arguments and FM Arouet666's suppositions and/or arguments.
4. None of the above.

If you read the title of this thread carefully, you will observe that the question of the existence or non-existence of miracles enters into the thread title only in the attempt to identify a specific audience to whom the question is addressed.

The question is simple: is Goldbach's conjecture true? I think each of "it is probably true" and "it's probably false" gives a misleading impression of objectivity. I would prefer to hear talk about your suspicions or my suspicions on this question.

Moreover, even if we could objectively confirm probable truth or probable falsehood on this question, such confirmation would not necessarily have any practical significance, even presuming for the sake of argument that Goldbach's conjecture itself has practical significance.

Would you donate a lump sum of two thousand dollars to a particular charity if you could somehow know in advance that there's a probability of 4/7 that the money will do much good and a probability of 3/7 that the $2000 will be completely wasted? Would it matter much if those probabilities were reversed to 3/7 that the money will do much good and 4/7 that it will be completely wasted?

"weigh the evidence in support of the claim that there aren't any miracles" your statement, stated in the negative.

Again, what I am interpreting from your post is that you wish to establish an argument in which the skeptic is forced to argue against the existence of miracles. I doubt you will find many on this board willing to accept such an insurmountable task. Nor should they, you seem to support the notion that miracles exist. Please, for the sake of brevity, provide your evidence.

Love the donation bit at the end of your post. Smells of Pascal, and as François-Marie Arouet noted the argument is "indecent and childish."

I have no interest in such arcane discussions. If you support the proposition that miracles exist, provide the thread with one and support it. Simple enough.

O8) :popcorn: :evilgrin:

Wonder what is taking so long. Crikets............

A jesus walking on the water, a burning bush, an image of the sweet lady madonna in a taco chip. Boy, believers are all full of bluster until you ask for a tiny shred of evidence, then Zzzzzz Zzzzzz

You might as well ask me to translate the Original Post from English to Italian. I can't do it. However, reasonable people aren't going to conclude that Italian is an impoverished language not rich enough to express the ideas in the Original Post.

Stop hiding behind obscure arguments draped in logic and mathematics. Get to the point. Do you believe that miracles exist or could exist? If you do, please enlighten me. Provide evidence for such an occurrence.

And yes, mathematics is an impoverished language when tasked with the establishment of the existence of a miracle.

It's hard to miss. Since when does staying on topic constitute a sin of hiding behind something?

You already wrote: "I accept the notion that Goldbach's conjecture is probably true", but that's rather odd. If you stood before a row of infinitely many haystacks, how many haystacks would you check before concluding that there probably isn't a needle in any of those haystacks?

Sorry, not a proponent of the interlocutor as controller of the discussion, I will deviate off your original topic as the tread progresses. Keep repackaging your original argument in as many unrestricted negative analogies as you like. I do not have to prove that there is a needle in some mathematically created fantasy haystack you conjure up, you must provide me, the humble skeptic, with your needle. Otherwise, the discussion is nothing more than background noise.

Your claim is that there probably isn't a needle, but I didn't request a proof. I asked how many haystacks you would check.

Goin to go out and check those haystacks for ya right now. I will be gettin back to ya when you get back to me with that there miracle of yours. ;-)

And now something completely different. I will channel the candidate for vice president Sarah Palin...... :applause:

One last point, in order for you to undermine the position taken by skeptics, that miracles probably do not exist, you also have to prove an unrestricted negative. Namely, that no natural explanation for a miracle exists or will ever exist. Knowing that this is an insurmountable task, you rephrase the question.

You know that you cannot prove a miracle anymore than I can prove that miracles do not exist. If nether one of us is willing to take the bait then the discussion is at an end.

We say it is true in the cases and classes where it has been shown to be true, and don't believe that it is necessarily true in the unproven cases and classes.

In pure mathematics you just prove or disprove something. Not "gather evidence in support of" something.

Note: Here I am using some abiguous terminology. Explanation -
A case is a single instance.
A class is a group of functions.

For instance, before there was proof or disproof of the mean value theorem (and which functions you can use it on), it could be shown to be true in specific cases (eg. 3x^2 +x +4 between x=0 and x=10), so then it would be rational to believe that the mean value theorem was true for that parabola. Then, you could show that it was true for any parabola (a class) (though this might take more work than the original proof itself).

I think it's a matter of being cautious. However, even if one wishes to abandon caution and claim to have absolute knowledge, the claim can be expressed in terms of evidence: an infinite weight of evidence in favor of a given conclusion and zero weight of evidence against it.

and see what you can do with that. Often a lot.

I think it's a matter of ascertaining whether or not a given proposition is correct, and then ascertaining whether or not the method that told us the proposition is correct is itself correct, and so on.

But here's something I don't get.

"However, even if one wishes to abandon caution and claim to have absolute knowledge, the claim can be expressed in terms of evidence: an infinite weight of evidence in favor of a given conclusion and zero weight of evidence against it."

1) Given that "absolute knowledge" means in mathematics "proven for a given set of things under certain axioms", why do you need to abandon caution to claim that it you have absolute knowledge (leaving aside human error in constructing proofs, because that is not what we are talking about yet)

2) Why do we want to express the claim in terms of evidence? Sure, you can distort the meaning to apply it to whatever you want, but for what reason? It's like saying that something has a power level of over nine thousand - you can bastardize definitions and units until it is a correct statement, but then it bears little or no relation to any initial meaning.

3) "an infinite weight of evidence in favor" Why infinite? Why not a weight of 2 of evidence in favour and a weight of cheese sandwhich against? It doesn't carry any more information to rephrase everything in some kind of bastardized notion of evidence that bears no relation to the meaning of evidence in the real world, so why say "evidence" rather than "proven"?

To prove FLT, Wiles didn't introduce a grab bag of assumptions including such things as the Riemann hypothesis. He started from theorems that were available to him because those theorems had already been proved using the ordinary foundations of mathematics. Do you deny that there is such a thing as the ordinary foundations of mathematics? Do you think that things analogous to relative consistency proofs play a major role in all branches of mathematics?


I'm simply introducing a broader concept that includes as a special case the more restricted concept of "evidence" that you seem to prefer. This is done frequently in mathematics. For example, we might apply an "expansion" factor to something in a general sense that includes as possible subcases: expansion, contraction, and no change.


Many people will tell you that there aren't infinitely many different numbers that have exactly the same magnitude. To them, the word "number" is bastardized when used in the phrase "complex number." Furthermore, you can rephrase any theorem involving complex numbers so that you are only talking about real numbers. So you can make a claim analogous to your claim about not carrying more information.

He started from theorems that were available to him because those theorems had already been proved using the ordinary foundations of mathematics.

Yes, things that had already been proven from the axioms, or things that are taken to be axiomatic. Like I said, when we say "know for certain" (in mathematics) this is what we refer to. Proven just like that.

Do you deny that there is such a thing as the ordinary foundations of mathematics?

Uh, what? Well, no. Not that it has anything to do with anything I've been saying at all.

"I'm simply introducing a broader concept that includes as a special case"

Ok, I hereby broaden the term "democracy" - in the special case that one vote has infinite weighting and the other zero, we get the special case of democracy known as a dictatorship.

And now I'm not using democracy to mean anything like its original meaning, and have added nothing to our understanding of a dictatorship.

I deserve a medal.

Back to evidence, the fact that you can apply it to more things doesn't mean that you have meaningfully "generalised" the concept, rather that you have just tacked on a new special case and declared it part of the whole.

For instance, if I generalise the mean value theorem from parabolas to all polynomials, then the special case of parabolas is easily shown to be part of a complete framework. In this "evidence in mathematics" thing you are going on about, we have ideas about evidence and then you say "well, we're certain about this. We'd also be certain about something if we had infinite evidence for it, so let's rename 'certainty' 'infinite evidence'".

What's missing is any idea of what evidence is. To 'generalise' the concept you have to lose the meaning of evidence and that is why it is not meaningful, to be tautological about things.

Many people will tell you that there aren't infinitely many different numbers that have exactly the same magnitude. To them, the word "number" is bastardized when used in the phrase "complex number." Furthermore, you can rephrase any theorem involving complex numbers so that you are only talking about real numbers. So you can make a claim analogous to your claim about not carrying more information.

Wow, so you're telling me that sometimes you have to be careful about your definition of 'number'. And also, it is not possible to rephrase any theore involving complex numbers so that you are only talking about reals. For instance, De Moivre's theorem.

What you can do is rephrase any problem that is inherently real in terms of complex numbers, and then rephrase it again so you are only talking about reals. In fact, I do it all the time as it is very useful, even if you don't carry any more information.

So yeah, sometimes we rewrite things without adding more information. In maths, there isn't even a name for it, and in science it means "a model". Neither cases are anything more than useful.

What is involved in proving or disproving something? There are at least two different projects that one may pursue:
1. Try to prove it.
2. Try to disprove it.

Assume that you are close to completing a clear, well-organized, and correct proof one way or the other. In other words, you're close to producing something that won't require any more editing before it is ready for a tough audience. Then you will probably have already arrived at a definite opinion about which of the two projects listed above will succeed. You will probably be devoting more time to one of those two projects than to the other.

Perhaps I'm mistaken, but I'm getting the impression that you are reserving the word "evidence" exclusively for documentation that goes to an audience that includes critics, but that doesn't include anybody who is actively working on settling the question.

Suppose that you are devoting equal amounts of time to each of the two projects and making little progress. If someone who has a deep understanding of the problem makes a presentation that persuades you to abandon one of the two projects and quickly bring the other project to a successful end, then you seem to be committed to denying that your decision to concentrate entirely on one of the two projects was based on "evidence." You seem to be committed to claiming that you "just proved it" or that you "just disproved it."

So, going through things one at a time.

1) In mathematics, you may choose to devote more time to the project of proving than disproving, especially if you have a valuable lead (that is, the ability to move things around, solve related problems, et cetera). This does not, however, confer any measure of evidence for one or the other to be correct, only that you are able to 'do stuff' better when pursuing one avenue.

When you actually prove it, that is the important bit.

2) When I said it didn't count as evidence, I was talking about something being real or not. Used to investigate relations between real variables, as it were, rather than purely abstract objects. (ie. objects than never form or are real operators on constitutive relations)

Note:

Constitutive relations are information about the system encoded mathematically that we always use. For instance, Hooke's Law that Force = -k.x is a constitutive relation.

A real operator is simply one that corresponds (eventually) to a real physical quantity. In the example above, Newton's Second Law would be an operator.

(There is a theoretical backing to that/way to test if something is real. The requirement is in fact that in fact operators be hermitian on wavefunctions, but that is quantum mechanics and we are using classical approximations here)