Can we truly prove anything?
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Can we truly prove anything?
I believe Godel once proved that it is impossible for a logical system to prove it's own consistency. If this is so, then doesn't that mean there is a possibility that they, in fact, aren't. If they turned out to be inconsistent, wouldn't that make any proof we've ever made completely useless? (though, it's probably unlikely math is inconsistent. But since we can't prove it, how can we know?) Doesn't this mean that we can't really prove anything at all? And wouldn't this mean that believing in a magical unicorn is every bit of justified as believing in gravity? This doesn't seem right to me. I hope there's something wrong with my reasoning because I'd rather not regard unicorns on the same level of fact as gravity. Any help?
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Re: Can we truly prove anything?
Take a look at the axioms of PA. Are there any of them that you doubt? No? Good. Then you have no reason to doubt any theorem of PA.
Feel free to do the same with ZFC (or simply ZF, if you prefer).
Yes, in mathematics we assume that logic works, and we assume a few basic statements about the objects we study. We have no proof for any of this, but it's necessary to make some starting assumptions or you can't ever know anything, and the starting assumptions of mathematics are pretty benign.
(The same goes for science, by the way, although it makes a few extra assumptions.)
Feel free to do the same with ZFC (or simply ZF, if you prefer).
Yes, in mathematics we assume that logic works, and we assume a few basic statements about the objects we study. We have no proof for any of this, but it's necessary to make some starting assumptions or you can't ever know anything, and the starting assumptions of mathematics are pretty benign.
(The same goes for science, by the way, although it makes a few extra assumptions.)
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Can we truly prove anything?
Or, another example is that one has to assume the definition of a set. You can't really explain what a set is, and it is a core notion to almost everything in mathematics. You can say some of its properties (no duplicate entries), but you can't really define it beyond "a collection of stuff"  which is rather circular. At some point in mathematics, you have to make assumptions, or else nothing really makes sense.
Re: Can we truly prove anything?
Kurushimi wrote:I believe Godel once proved that it is impossible for a logical system to prove it's own consistency.
This is not really what the Incompleteness Theorem says.
Kurushimi wrote:doesn't that mean there is a possibility that they, in fact, aren't.
Yep. It is possible that tomorrow someone could show that ZFC is inconsistent. (Most mathematicians just believe this is highly unlikely.)
Kurushimi wrote:wouldn't that make any proof we've ever made completely useless?
Not really. All that means is that ZFC is a bad foundation and we'd find another one.
Kurushimi wrote:Doesn't this mean that we can't really prove anything at all?
Having a proof of a statement in ZFC isn't the only way to make sense of it mathematically. Plenty of mathematics was done before the development of logic, and that mathematics has led to plenty of advances in science and technology without needing a rigorous logical foundation. Mathematics won't collapse just because one particular way we have of doing it turns out not to work.
Kurushimi wrote:And wouldn't this mean that believing in a magical unicorn is every bit of justified as believing in gravity?
This is not a question about mathematics.
Re: Can we truly prove anything?
It makes no sense to believe a system consistent because it proved itself so; that's just begging the question. In fact, if a logical system could prove itself consistent, that should only hurt one's trust in it:
A system that is consistent might be able to prove itself consistent, or it might not.
A system that is not consistent will be able to prove itself consistent (since anything follows from a contradiction).
So, a system that can't prove itself consistent, is consistent, but a system that can, may or may not be.
A system that is consistent might be able to prove itself consistent, or it might not.
A system that is not consistent will be able to prove itself consistent (since anything follows from a contradiction).
So, a system that can't prove itself consistent, is consistent, but a system that can, may or may not be.
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Re: Can we truly prove anything?
Only tautologies, the rest are just useful, beautiful fictions.
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Re: Can we truly prove anything?
Goplat wrote:It makes no sense to believe a system consistent because it proved itself so; that's just begging the question. In fact, if a logical system could prove itself consistent, that should only hurt one's trust in it:
A system that is consistent might be able to prove itself consistent, or it might not.
A system that is not consistent will be able to prove itself consistent (since anything follows from a contradiction).
Well, the same goes for every other theorem, but seeing a proof of the Lebesgue density theorem didn't damage my faith in ZFC any.
That said, the second incompleteness theorem would certainly make a proof inside ZFC that ZFC is consistent highly troublesome.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Can we truly prove anything?
If you could do infinitely long proofs, there is a proof that PA is consistent in PA. But we're finite beings and proofs are defined to be of finite length.
Re: Can we truly prove anything?
On the other hand, the statement "This statement is provable in T" is indeed provable in T, by Lob's Theorem.Goplat wrote:It makes no sense to believe a system consistent because it proved itself so; that's just begging the question. In fact, if a logical system could prove itself consistent, that should only hurt one's trust in it:
A system that is consistent might be able to prove itself consistent, or it might not.
A system that is not consistent will be able to prove itself consistent (since anything follows from a contradiction).
So, a system that can't prove itself consistent, is consistent, but a system that can, may or may not be.
Re: Can we truly prove anything?
Black wrote:If you could do infinitely long proofs, there is a proof that PA is consistent in PA. But we're finite beings and proofs are defined to be of finite length.
It's certainly true that we can't write out infinitely long proofs. But what we can do is treat proofs, even those of infinite size, as mathematical objects and write down finite proofs about them. Recall that this is the essence of what Genzten did in his proof of the consistency of firstorder PA. The problem with doing this inside PA is not any impossibility of infinitely long proofs, but rather PA's lack of appropriate machinery (e.g. transfinite induction) for handling them.
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Re: Can we truly prove anything?
t0rajir0u wrote:Kurushimi wrote:wouldn't that make any proof we've ever made completely useless?
Not really. All that means is that ZFC is a bad foundation and we'd find another one.
This got me thinking. Would it be possible that all sufficiently powerful systems (in which we can do that math we want to do; Euclidean geometry doesn't suffice) are inconsistent?
Re: Can we truly prove anything?
Mathematical logic in general, and the definition of inconsistency particularly, is defined in ZF terms. If ZF were inconsistent, then our next step would involve and require a new definition of consistency.
If you're asking in a more general philosophical way if we can ever know that we know anything, then I think the answer is pretty clearly "no", and part of getting out of bed in the morning is having faith that what we know is worth building upon. Personally, I think that even mathematical philosophy requires the same application of Pascal's Wager.
If you're asking in a more general philosophical way if we can ever know that we know anything, then I think the answer is pretty clearly "no", and part of getting out of bed in the morning is having faith that what we know is worth building upon. Personally, I think that even mathematical philosophy requires the same application of Pascal's Wager.
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Re: Can we truly prove anything?
There is also reverse mathematics. While it doesn't answer the fundamental problem, it does (for each theorem) produce the axioms you need to prove the theorem. With enough reverse mathematics, proving that a given axiom is inconsistent simply removes some subset of theorems  and you might even have collections of weaker theorems that don't depend on it ready to go (or alternative axioms).
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: Can we truly prove anything?
Goplat wrote:It makes no sense to believe a system consistent because it proved itself so; that's just begging the question. In fact, if a logical system could prove itself consistent, that should only hurt one's trust in it:
A system that is consistent might be able to prove itself consistent, or it might not.
A system that is not consistent will be able to prove itself consistent (since anything follows from a contradiction).
So, a system that can't prove itself consistent, is consistent, but a system that can, may or may not be.
If a system can prove itself consistent, then it is inconsistent.
Here, Wikipedia sums it up nicely:
"For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent."
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Re: Can we truly prove anything?
Pascal's Wager is shitty even among theological arguments. I'd be reluctant to let it into mathematics in any way. (It'd be circular in any case, seeing as it requires some mathematical reasoning to be able to say anything at all.)Tirian wrote:I think that even mathematical philosophy requires the same application of Pascal's Wager.
Re: Can we truly prove anything?
Tirian wrote:Mathematical logic in general, and the definition of inconsistency particularly, is defined in ZF terms. If ZF were inconsistent, then our next step would involve and require a new definition of consistency.
Wait, what? How is ZF defined, then?
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Re: Can we truly prove anything?
Token wrote:Wait, what? How is ZF defined, then?
It's assumed.
The preceding comment is an automated response.
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Re: Can we truly prove anything?
gmalivuk wrote:Pascal's Wager is shitty even among theological arguments. I'd be reluctant to let it into mathematics in any way. (It'd be circular in any case, seeing as it requires some mathematical reasoning to be able to say anything at all.)Tirian wrote:I think that even mathematical philosophy requires the same application of Pascal's Wager.
I think what he meant was a Pascallike wager:
Either we can know stuff using things like logic, mathematical proof, and scientific induction, or we can't (because our logic is inconsistent, or our axioms are inconsistent, or scientific induction is flawed).
 If it is possible to know stuff using those tools, we can learn about the world, and use our knowledge to help us function in the world.
 If it's not, we pretty much give up all hope of ever knowing anything, and function as people who are blind, deaf, and insensible (because we can't gain any knowledge from our senses).
It's pretty clear that option two sucks, so we might as well function as if option one is true, even though we have no evidence for this belief. (Essentially this is a Pascal'swagerlike response to Hume's problem of induction, but it works in the context of the trustworthiness of human logic—and, by extension, human mathematics—as well.)
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Can we truly prove anything?
Can we truly prove anything?
No.
http://en.wikipedia.org/wiki/Münchhausen_Trilemma
http://en.wikipedia.org/wiki/Pyrrhonism
Re: Can we truly prove anything?
t0rajir0u wrote:Kurushimi wrote:I believe Godel once proved that it is impossible for a logical system to prove it's own consistency.
This is not really what the Incompleteness Theorem says.
I know this. I just thought it was Godel who also proved this.
Re: Can we truly prove anything?
Nice thread, guys. But I think it needs a link to What the Tortoise said to Achilles.
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Re: Can we truly prove anything?
*nod*, and it comes with the standard trap.
Pascal's wager is often used to "prove the existence" of an eternal reward/god with particular properties. Pascal's wager doesn't tell us if we should believe in the christian god, the jewish god, shiva, budda, a god that infinitely tortures for all eternity anyone who dares believe in a god or a religion, or anything else in particular, a swarm of infovores at the end of time who bring back in an infinite simulation your very soul so long as you don't commit the sin of believing in eternal life...
Ie, the presumption that Pascal makes that the choices are a Christian/Aristotle esque god, or no god at all. And once it restricts the decision space, then plays with infinities, it outputs gibberish.
The same error can be done with the scientific Pascal's wager: if you pick a particular set of axioms, or even particular kind of axioms, as being the ones that you have to believe in if you accept the argument.
We can presume we can know things about the universe (as opposed to not). But if we presume that the things we can know are of any particular kind (even "logic makes sense"  maybe gut feeling is the only way we can ever know about the universe, and once we apply science like techniques the universe warps itself to fool us), we commit the sin of Pascal.
Pascal's wager is often used to "prove the existence" of an eternal reward/god with particular properties. Pascal's wager doesn't tell us if we should believe in the christian god, the jewish god, shiva, budda, a god that infinitely tortures for all eternity anyone who dares believe in a god or a religion, or anything else in particular, a swarm of infovores at the end of time who bring back in an infinite simulation your very soul so long as you don't commit the sin of believing in eternal life...
Ie, the presumption that Pascal makes that the choices are a Christian/Aristotle esque god, or no god at all. And once it restricts the decision space, then plays with infinities, it outputs gibberish.
The same error can be done with the scientific Pascal's wager: if you pick a particular set of axioms, or even particular kind of axioms, as being the ones that you have to believe in if you accept the argument.
We can presume we can know things about the universe (as opposed to not). But if we presume that the things we can know are of any particular kind (even "logic makes sense"  maybe gut feeling is the only way we can ever know about the universe, and once we apply science like techniques the universe warps itself to fool us), we commit the sin of Pascal.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: Can we truly prove anything?
Kurushimi wrote:t0rajir0u wrote:Kurushimi wrote:I believe Godel once proved that it is impossible for a logical system to prove it's own consistency.
This is not really what the Incompleteness Theorem says.
I know this. I just thought it was Godel who also proved this.
I suspect t0rajir0u was objecting to the misstatement of Gödel's Second Incompleteness Theorem rather than its attribution. (It's perfectly possible for an inconsistent logical system to prove  that is, deduce  its own consistency.) The statement quoted above from Wikipedia by Natty is an improvement:
For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.
"The age of the universe is 100 billion, if the units are dog years."  Sean Carroll
Re: Can we truly prove anything?
Yakk wrote:Pascal's wager is often used to "prove the existence" of an eternal reward/god with particular properties. Pascal's wager doesn't tell us if we should believe in the christian god, the jewish god, shiva, budda, a god that infinitely tortures for all eternity anyone who dares believe in a god or a religion, or anything else in particular, a swarm of infovores at the end of time who bring back in an infinite simulation your very soul so long as you don't commit the sin of believing in eternal life...
Ah, I appreciate your complaints. I suppose that I am comparing it to the argument that I choose to believe that I have free will because I'm right if I'm right and I'm not responsible for my choices if I'm wrong. There must be a name for that line of reasoning and that would be clearly distinct from Pascal's Wager.
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Re: Can we truly prove anything?
As mathematicians, I suppose that we like to think that the work we do is closer to actual truth than that of, say, physicists. In a sense that is true since we work at a higher level of abstraction, but in another sense we're really in the same boat that they are.
The physical model of our universal is being constantly updated to account for new unexplained phenomena and much of this model is still being hotly debated by physicists. However our axiomatic models of mathematics are just as precariously founded. We construct them in such a way that they behave (mostly) according to our intuitions however it is impossible to grasp the full ramifications of the axioms from the outset. For example the main driver behind the creation of ZFC set theory was Russel's discovery of an unforseen inconsistency in the preceding theory.
As one of my professors put it, "We're all crazy guys living on a knife edge. Who knows, maybe tomorrow some kid working on a commodore 64 in his parents' basement will discover an inconsistency in ZFC and we'll have to start all over."
Of course we are in a better position than the physicists because the consequences of our models can be explored with a very high level of rigor and usually don't involve imprecise experimentation. As people have already pointed out, you can only prove things if you take certain axioms for granted  so you're okay as long as you accept the axioms you're using.
But things also roll the other way! Sometimes we accept axioms that seem intuitive but then have almost upsetting consequences. A prime example is the axiom of choice which has many reasonably sounding reformulations and yet results in a whole slew of bizarre results. The axiom of choice is so weird that (to my knowledge) professors in every field of mathematics always explicitly state whenever it is used since no one can decide whether it makes more sense to accept or reject it.
The physical model of our universal is being constantly updated to account for new unexplained phenomena and much of this model is still being hotly debated by physicists. However our axiomatic models of mathematics are just as precariously founded. We construct them in such a way that they behave (mostly) according to our intuitions however it is impossible to grasp the full ramifications of the axioms from the outset. For example the main driver behind the creation of ZFC set theory was Russel's discovery of an unforseen inconsistency in the preceding theory.
As one of my professors put it, "We're all crazy guys living on a knife edge. Who knows, maybe tomorrow some kid working on a commodore 64 in his parents' basement will discover an inconsistency in ZFC and we'll have to start all over."
Of course we are in a better position than the physicists because the consequences of our models can be explored with a very high level of rigor and usually don't involve imprecise experimentation. As people have already pointed out, you can only prove things if you take certain axioms for granted  so you're okay as long as you accept the axioms you're using.
But things also roll the other way! Sometimes we accept axioms that seem intuitive but then have almost upsetting consequences. A prime example is the axiom of choice which has many reasonably sounding reformulations and yet results in a whole slew of bizarre results. The axiom of choice is so weird that (to my knowledge) professors in every field of mathematics always explicitly state whenever it is used since no one can decide whether it makes more sense to accept or reject it.
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Re: Can we truly prove anything?
silverhammermba wrote:The axiom of choice is so weird that (to my knowledge) professors in every field of mathematics always explicitly state whenever it is used since no one can decide whether it makes more sense to accept or reject it.
No. It gets used without comment quite frequently. When it is used explicitly, the reason for that is almost always not because the speaker is expecting that some people in the audience will reject it. Much more frequently, the reason its use is highlighted is because it tells you something about the object being constructed. Objects that are constructed without the axiom of choice tend to be much more explicitly defined than objects constructed using the axiom of choice, which often has implications further down the line.
This is a bit like the difference between a constructive and nonconstructive proof. Mathematicians almost always care about whether a proof is constructive or nonconstructive, but relatively few mathematicians are constructivists.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Can we truly prove anything?
It depends strongly on the field. Mathematicians in various parts of analysis or topology use the axiom of choice largely without comment, since it's implicit in many of the most useful theorems in the field.
Re: Can we truly prove anything?
Tirian wrote:If you're asking in a more general philosophical way if we can ever know that we know anything, then I think the answer is pretty clearly "no", and part of getting out of bed in the morning is having faith that what we know is worth building upon. Personally, I think that even mathematical philosophy requires the same application of Pascal's Wager.
I'm not going to claim that this gets you very far, but I do think what you said is technically wrong. Of empirical things, certain sensations are immediately apparent to us. We simply know them. We might not know anything else about them, but we do know that certain sensations are happening. Of rational things, many philosophers accept that we know some things a priori. As concerns mathematics, if you intuitively know that the logic you're using works, then you can at least know of the math you're dealing with that it is so far consistent with your logic.
I guess you were looking more at a pragmatic justification of the whole of mathematics, which I've basically ignored here. But it does seem that we can know some things, meager as that knowledge may be.
I came here to read a cool post, a witty dialogue, a fresh joke, but stumbled upon a "bump"...
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Way to go, jerk... ~CordlessPen
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Re: Can we truly prove anything?
skeptical scientist wrote: Much more frequently, the reason its use is highlighted is because it tells you something about the object being constructed. Objects that are constructed without the axiom of choice tend to be much more explicitly defined than objects constructed using the axiom of choice, which often has implications further down the line.
My point is that the axiom of choice is the only axiom to receive such special treatment (unless you're studying set theory or mathematical logic maybe).
Re: Can we truly prove anything?
silverhammermba wrote:My point is that the axiom of choice is the only axiom to receive such special treatment (unless you're studying set theory or mathematical logic maybe).
The Boolean prime ideal theorem and the HahnBanach theorem are of interest to plenty of people who don't study set theory or logic, and they're both independent of ZF but strictly weaker than AC.
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Re: Can we truly prove anything?
silverhammermba wrote:skeptical scientist wrote: Much more frequently, the reason its use is highlighted is because it tells you something about the object being constructed. Objects that are constructed without the axiom of choice tend to be much more explicitly defined than objects constructed using the axiom of choice, which often has implications further down the line.
My point is that the axiom of choice is the only axiom to receive such special treatment (unless you're studying set theory or mathematical logic maybe).
Yes, and I explained why that is, and the reason is not because "no one can decide whether it makes more sense to accept or reject it." The axiom of choice is the only axiom of ZFC which says that an object exists, when that object is not uniquely specified by the axiom. (Exceptions arguably include the axiom of infinity and the axiom of collection, but both of those can be made specific by requiring a minimal set satisfying the existential, which results in equivalent statement, over the remaining axioms.) So in that sense, the axiom of choice is less constructive than the other axioms, so proofs which use it are less constructive than proofs which do not.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Can we truly prove anything?
I kind of hate to be bringing this up, but I need to make 5 posts since I'm starting to post links.....
So
Lewis' observation about Godel was that the mere consideration of the Godel theorems strongly implies that consciousness is not so constrained. I believe he also felt (and the consensus on this is...) that consciousness does not have to be rigorous and so, Godel's theorems do not apply.
On the one hand, the whole mind body issue is raised by this. Also, personally, I feel the mind/computer issue is SETTLED by it (personally, I believe that current conceptions of computers by being bound by Godel's theorem mean they cannot be conscious nor can they model consciousness).
On the other hand, the cogito would say you can prove some things........
Jigo
So
Lewis' observation about Godel was that the mere consideration of the Godel theorems strongly implies that consciousness is not so constrained. I believe he also felt (and the consensus on this is...) that consciousness does not have to be rigorous and so, Godel's theorems do not apply.
On the one hand, the whole mind body issue is raised by this. Also, personally, I feel the mind/computer issue is SETTLED by it (personally, I believe that current conceptions of computers by being bound by Godel's theorem mean they cannot be conscious nor can they model consciousness).
On the other hand, the cogito would say you can prove some things........
Jigo
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Re: Can we truly prove anything?
"gmalivuk cannot consistently assert this statement."
I cannot make that statement and remain consistent, and yet you can see that it's true. Does this mean that your mind is somehow fundamentally different from mine? No? Then Godel's theorems don't show any of our minds are fundamentally different from computers.
I cannot make that statement and remain consistent, and yet you can see that it's true. Does this mean that your mind is somehow fundamentally different from mine? No? Then Godel's theorems don't show any of our minds are fundamentally different from computers.
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Re: Can we truly prove anything?
gmalivuk wrote:I cannot make that statement and remain consistent, and yet you can see that it's true. Does this mean that your mind is somehow fundamentally different from mine? No? Then Godel's theorems don't show any of our minds are fundamentally different from computers.
Another reason the Gödel antimechanist argument is bullshit:
By Gödel's second incompleteness theorem, the Gödel sentence G(F) for a reasonably strong formal system F is equivalent to the statement that F is consistent. In fact, F can actually prove that G(F) is equivalent to Con(F). So the statement that humans can recognize the truth of G(F) while computers can't (presumably for the formal system F which represents the inner workings of the computer) is really the statement that humans can recognize that said system, which completely represents an arbitrarily complicated piece of hardware and software, is consistent. Considering the number of times humans have been wrong about whether a given set of axioms was consistent, one cannot reasonably claim that humans have superior powers in this regard.
Scott Aaronson has a good exposition of this argument.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
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Re: Can we truly prove anything?
skeptical scientist wrote:Yes, and I explained why that is, and the reason is not because "no one can decide whether it makes more sense to accept or reject it." The axiom of choice is the only axiom of ZFC which says that an object exists, when that object is not uniquely specified by the axiom. (Exceptions arguably include the axiom of infinity and the axiom of collection, but both of those can be made specific by requiring a minimal set satisfying the existential, which results in equivalent statement, over the remaining axioms.) So in that sense, the axiom of choice is less constructive than the other axioms, so proofs which use it are less constructive than proofs which do not.
[math]\exists x \epsilon X\ P(x)\ \Leftrightarrow\ !\forall x \epsilon X\ !P(x)[/math]
I believe, even in ZF, that you can make the above statement. Does it rely on the axiom of infinity or the axiom of collection?
I suppose without the axiom of infinity, you cannot find a set large enough that the above isn't equivalent to "make a statement about each element, and or them together"?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: Can we truly prove anything?
Yakk wrote:[math]\exists x \epsilon X\ P(x)\ \Leftrightarrow\ !\forall x \epsilon X\ !P(x)[/math]
I believe, even in ZF, that you can make the above statement. Does it rely on the axiom of infinity or the axiom of collection?
Well, \( ((\exists x)(\phi \land \psi) \Leftrightarrow \lnot(\forall x)(\phi \Rightarrow \lnot\psi))\) will be a theorem of your first order logic for any formulae \(\phi\) and \(\psi\), so no, what you've got there doesn't depend on any axiom of ZF. That said, what you *haven't* got there is a sentence of the form \((\exists x) P(x)\). You could use the above to derive one, but then you'd need to prove something of the form \(\lnot(\forall x) \lnot P(x)\), which is going to have to come from somewhere...
Last edited by Token on Mon May 10, 2010 4:15 pm UTC, edited 1 time in total.
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Re: Can we truly prove anything?
We cannot completely prove anything. Every proof ever put forward is based on certain assumptions. If any of those assumptions are wrong, we have not proven anything. For example, almost every proof, if not all proofs, is based on the assumption that numbers do not randomly change in value throughout the proof. If they do, then the proof collapses. To our minds, numbers may not change value. However, the numbers may be changing value constantly in a way we cannot perceive.
Also, if there is an error in the logical functioning of the human mind, nothing the human mind can come up with, let alone prove, can be correct. In addition, our understanding of mathematics depends on the fact that there are such things as absolute truths. There are those who do not believe in absolute truths (such as the New Age movement.) If they are correct, then our study of mathematics is useless.
Therefore, if the assumptions that form the the basis of our understanding of all of mathematics are correct, then the proofs of mathematical concepts that we devise are most likely correct. One of the most important of these assumptions, not just in mathematics but in all of life, is that the human mind is without error. If it is, mathematics will lose most of our trust, because although the errors may be reduced by collecting and comparing the conclusions of many people, there still exists some error.
Just something to ponder, as its conclusion affects all of life, not just the area of mathematics.
Also, if there is an error in the logical functioning of the human mind, nothing the human mind can come up with, let alone prove, can be correct. In addition, our understanding of mathematics depends on the fact that there are such things as absolute truths. There are those who do not believe in absolute truths (such as the New Age movement.) If they are correct, then our study of mathematics is useless.
Therefore, if the assumptions that form the the basis of our understanding of all of mathematics are correct, then the proofs of mathematical concepts that we devise are most likely correct. One of the most important of these assumptions, not just in mathematics but in all of life, is that the human mind is without error. If it is, mathematics will lose most of our trust, because although the errors may be reduced by collecting and comparing the conclusions of many people, there still exists some error.
Just something to ponder, as its conclusion affects all of life, not just the area of mathematics.
 Talith
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Re: Can we truly prove anything?
rhetorical wrote:We cannot completely prove anything. Every proof ever put forward is based on certain assumptions. If any of those assumptions are wrong, we have not proven anything. For example, almost every proof, if not all proofs, is based on the assumption that numbers do not randomly change in value throughout the proof. If they do, then the proof collapses. To our minds, numbers may not change value. However, the numbers may be changing value constantly in a way we cannot perceive.
Also, if there is an error in the logical functioning of the human mind, nothing the human mind can come up with, let alone prove, can be correct. In addition, our understanding of mathematics depends on the fact that there are such things as absolute truths. There are those who do not believe in absolute truths (such as the New Age movement.) If they are correct, then our study of mathematics is useless.
Therefore, if the assumptions that form the the basis of our understanding of all of mathematics are correct, then the proofs of mathematical concepts that we devise are most likely correct. One of the most important of these assumptions, not just in mathematics but in all of life, is that the human mind is without error. If it is, mathematics will lose most of our trust, because although the errors may be reduced by collecting and comparing the conclusions of many people, there still exists some error.
Sorry but that just sounds like a load of crackpot speak to me. The way we define numbers and the axioms we use mean that "numbers changing value during a proof" can't happen from definition. Similarly the way we build logic from sets of definitions and the axioms that they're based on means that human intuition doesn't come into it. It is true that we can't build absolute truths from mathematics but we don't need to. We make reasonable choices of our logical system to work in and then the process of doing mathematics gives us facts that are true under the system we define.
To make it clear that mathematics is independent of the people doing it. An alien civilisation, even one based on a computer, would come to the same logical conclusions that human mathematicians come to give the formal system that they work in.
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Re: Can we truly prove anything?
Yakk wrote:skeptical scientist wrote:Yes, and I explained why that is, and the reason is not because "no one can decide whether it makes more sense to accept or reject it." The axiom of choice is the only axiom of ZFC which says that an object exists, when that object is not uniquely specified by the axiom. (Exceptions arguably include the axiom of infinity and the axiom of collection, but both of those can be made specific by requiring a minimal set satisfying the existential, which results in equivalent statement, over the remaining axioms.) So in that sense, the axiom of choice is less constructive than the other axioms, so proofs which use it are less constructive than proofs which do not.
[math]\exists x \epsilon X\ P(x)\ \Leftrightarrow\ !\forall x \epsilon X\ !P(x)[/math]
I believe, even in ZF, that you can make the above statement. Does it rely on the axiom of infinity or the axiom of collection?
I suppose without the axiom of infinity, you cannot find a set large enough that the above isn't equivalent to "make a statement about each element, and or them together"?
Actually, depending on your convention, "\(\exists x\)" is merely shorthand for "\(\lnot \forall x \lnot\)". However, there is no meaningful difference between the two, since a negated universal quantifier can be thought of as an existential quantifier (there exists a counterexample). If you converted the axiom into an equivalent axiom which contained only universal quantifiers (and not negated universal quantifiers), then you would have done something interesting.
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"With math, all things are possible." —Rebecca Watson

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Re: Can we truly prove anything?
Talith wrote:To make it clear that mathematics is independent of the people doing it. An alien civilisation, even one based on a computer, would come to the same logical conclusions that human mathematicians come to give the formal system that they work in.
Would they? You seem to be ignoring one of the points I mentioned in my post. One of the underlying assumptions of mathematics is that the human mind is logically perfect.
Also, you mentioned that numbers do not change value by definition. One could say that if our definition of 'number' reflects the way the world actually functions, then mathematics can prove things. Otherwise, mathematics cannot prove anything worthwhile to the world. Perhaps, in my original post, I should have said, "We cannot completely completely prove anything relevant to the world around us. We can easily prove things within the realms of definitions, but in order to prove something relevant to our world, the definitions must be consistent with our experiences, and our experience of the world must be an accurate picture of the way the world actually functions.
Talith wrote:Similarly the way we build logic from sets of definitions and the axioms that they're based on means that human intuition doesn't come into it.
I said nothing about human intuition, I said that the human mind must be assumed to be logically perfect. Intuition is different from logical reasoning, is it not? These "sets of definitions and the axioms that they're based on" came about by human reasoning, did they not? Therefore, if they are to be true, the mind must be able to reason perfectly and the mind must be able to perceive the world correctly.
Talith wrote:We make reasonable choices of our logical system to work in and then the process of doing mathematics gives us facts that are true under the system we define.
What if the human mind is incapable of making reasonable choices? It is obvious that the human mind is not completely logical just by looking at the way people live every day.
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