Can We Prove Anything?

durangodawood said:

But who knows what unusual event was a miracle?

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I don´t know. That´s why I personally wouldn´t use the term "miracle" when it comes to matters of epistomology.

The end event of the dinosaurs? The big bang?

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I guess everyone who wants to use the term "miracle" should give a proper definition first.

durangodawood said:

I was clear about what I was asking for: proof that doesnt rest on a system of assumptions (ie axioms).

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I think you better not conflate "axioms" and "assumptions".

Something "ultimate".

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What do you mean by "ultimate"?

durangodawood said:

Can We Prove Anything in some kind of ultimate sense?

I think its not possible in math, as demonstrated by Godel, the overall system being either inconsistent or incomplete. So proofs are kind of conditional. (?) But perhaps his proof about formal systems hits 'rock bottom'. (?) Anyway.....

Is it possible in any other realm?

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Nope. Google Wittgenstein's rhinoceros. Or even Sagan's dragon in the garage.

Then there's the solipsism issue. No matter how much evidence supporting anything you may have, you MIGHT just be a brain in a jar receiving sensory input from a computer, and your entire conceptual model of reality, evidence and axioms included, could be completely false.

durangodawood said:

Can We Prove Anything in some kind of ultimate sense?

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The philosophy you may or may not be looking for is 'Solipsism'. It contains all that you may or may not desire about a subject that may or may not be true.

Can the body do without proofs? i.e. if my oxygen levels are low, I will be impelled to breather faster and deeper. But this system of machinery, can it work without axioms and assumptions?

Is there any system of "proof" in this, or does it just work fine without them?

A rock splashes in a pond. No one is watching. A tree falls and makes a sound when no one listens. All without proofs.

If and when we know, we don't need proofs to know. Knowing happens. Trees fall, eyes open etc.

If it all depended on proofs then nothing would happen at all. I dont see why knowing ought to be "held to ransom".

durangodawood said:

Can We Prove Anything in some kind of ultimate sense?

I think its not possible in math, as demonstrated by Godel, the overall system being either inconsistent or incomplete. So proofs are kind of conditional. (?) But perhaps his proof about formal systems hits 'rock bottom'. (?) Anyway.....

Is it possible in any other realm?

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Don't know anymore who it was, but on a science discussion panel, I remember a scientist saying "Science is not able to prove anything... except what is wrong"

So science can't "prove" what is correct. This would be the reason why the word "theory" is used for scientific explanations. There is always the option, no matter how remote or unlikely it seems, that evidence discovered in the future might refute current ideas / explanations. As long as intellectual honesty dictates such, one can theoretically never "prove" an idea to be correct in the ultimate sense.

However, it does mean that theories can be proven wrong.
When such evidence is encountered that refutes a theory or hypothesis, you have successfully "proved it to be wrong".

Suppose you have a theory / hypothesis about some linke between skin color and eye color from which follows that, if that theory is correct, it would be impossible to find a black person with blue eyes.
The more black people you examine without finding a single one with blue eyes, the stronger that theory will get. But you can NEVER be certain that the next black person you examine won't have blue eyes.

However, it would take only one example of a black person with blue eyes to refute the idea.
At which point it would be proven absolutely that the hypothesis is incorrect. Since there would factually be at least 1 black person with blue eyes.

So, to summarize (and reiterate): Science can't prove anything. Except what is wrong...

durangodawood said:

Not disputable in any way.

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That´s the thing with axioms: They aren´t disputable in a consistent manner.

durangodawood said:

Any force that intervenes from outside nature to change the course of nature.

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Ok. And what´s your question concerning "miracles" again?

durangodawood said:

Axioms may be indisputable. But I dont think they can be proven, strictly speaking. They are accepted as self-evident.

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Yeah, so what?

Cant you have one system of axioms, and then another, where the corollaries of system A are axioms of system B. Like we derive B's Axioms form A's corollaries (i.e. the axioms of B actually are not axioms in system A, but are derived from system A to form the principles of B).

In that case axioms would be both derived and underived, depending on context. This is a genuine question, I am not that skilled in maths and logics.

For instance (I am guessing out of my depth here) a Godel sentence may be treated as an axiom in a system of alternative logic...

e.g. "this sentence is not provable" is an axiom of a formal system with no further proofs, or an axiom of a dialtheic logic (i.e. a logic where there can be true contradictions).



So you may (possiblly, if you dont use a paraconsistent logic (ie one which does not allow for ther principle of "explosion") actually derive the axioms of the first system from a dialtheic logic, because anything follows from a contradiction (according to Bertrand Russls "principle of explosion").

And all formal systems (Godel says?) contain contradictions. So if anything follows from a contradiction, then the axioms of A follow from the axioms of A, indirectly. Via Godel sentence B.

So axioms can be proven in a roundabout way? (And they cant, depending on your use of the Godel sentence.... )

So if we can prove "anything" (ie something like 1+1=2) then we can prove literally anything (ie any statement, form the consequences of incompleteness and explosion, like 1+1=3, and 1+1=2 , and "the moon ismade from green cheese" etc etc etc ad infinitum)....


so - another educated guess - eventually if we treat the axioms of A as the "set of all sets" (is that even possible... google indicates maybe the thought has been thought before, at leats) it ie the set both contains itself (via deduction of godel sentence B) and it doesnt (via an alternative interpretation of B, where the logic halts)? So we would have a restatement of Russels paradox in terms axioms and proofs? Making it a FRACTAL!!!! So in fact axioms both are (and are not....) axioms, in that they can be proven and cannot. Yet if we prove them, we prove "anything..."

So also we have the halting problem. In trying to prove the axioms you either halt t "this sentence is unprovable" (1st uinterpretation f Godel sentence)) or you treat it as the starting point for an alternative system in which anything can be derived (according to the contradiction interpretation of Godel sentence), including the "proof" of the first axioms A....maybe, but also an arbitrarily large system of alternatives from which there is no escape?


If you have deductive closure (ie a logical concept meaning you are commited to what can be deduced from your beliefs) then it is true that axioms (and all else in the system, maybbe) cannot be proven...
the search for proof halts as a Godel sentence; OR: or it continues in another logical guise where infinity rules and the program runs for ever inder a different interpretation of that self same sentence...



Wittgenstein mantioned that logical grammar is (or at least could be) arbitrary:

How many ways to play "rock paper scissors" folks?

quatona said:

durangodawood said:

Axioms may be indisputable. But I dont think they can be proven, strictly speaking. They are accepted as self-evident.

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Yeah, so what?

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Thats contestable, see above...

As well as axiomatic systems there are inferential systems which have just rules without definitions or axioms. Yet even these rules have to be presumed somewhere. Its like a game, you cant really play "zero" ie a game with no moves.

Thats odd, without assumptions you can prove "zero"?

No offence meant... but:

The world is my world: this is manifest in the fact that the limits of language (of that language which alone I understand) mean the limits of my world. - Wittgenstein.