Can we truly prove anything?

For the discussion of math. Duh.

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Re: Can we truly prove anything?

Postby gmalivuk » Mon May 10, 2010 5:42 pm UTC

rhetorical wrote:One of the underlying assumptions of mathematics is that the human mind is logically perfect.
How do you figure that?

One could say that if our definition of 'number' reflects the way the world actually functions
We're not talking about how the external physical world functions. We're talking about mathematics. The effectiveness of mathematics in telling us about the external world is a scientific question, as mathematics would exist as a logical structure even if the external world was completely random and arbitrary.

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.
What is your experience with pi? Does the fact that circles in the real world are never perfect mean the definition of pi is completely irrelevant?

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.
Yes, and no one says the mind is completely logical. But I doubt even you believe the human mind is completely *illogical*, either. And the very reason why we use formal logic is because we know that logical thinking doesn't come especially easily or naturally to us. So we formalized it in a way that other people can follow it and check it, and even a purely mechanistic process can run through certain kinds of proofs to check them.
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Re: Can we truly prove anything?

Postby Yakk » Mon May 10, 2010 7:19 pm UTC

Is it conventional to not implicitly include the axioms of logic when talking about ZF?

Anyhow, I was simply asserting that there are ways, without the axiom of choice, to assert that an object exists without uniquely specifying it.

On revue, I now see that you where talking more directly -- that AoC is an axiom that talks about existence of something without that existence being "close to unique" (ie, seemingly easy to munge into it talking about a unique object).

I now understand what you where saying, and my response was off topic. Sorry!
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Re: Can we truly prove anything?

Postby skeptical scientist » Mon May 10, 2010 10:43 pm UTC

Yakk wrote:Is it conventional to not implicitly include the axioms of logic when talking about ZF?

The standard in mathematical logic is to use a universal deduction system for all languages and theories. This deduction system will include inference rules like modus ponens, and logical axioms like De Morgan's laws—the details don't particularly matter, as long as your deduction system is consistent and complete (for classical logic). Then when you want to study different theories, like group theory, or the theory of algebraically closed fields, or ZF, you keep the logical structure fixed and just consider the particular axioms of the theory you're looking at. So the logical axioms are implicitly included, but not as axioms of ZF, and rather as part of the basic logical structure which is universal to all theories.

The exception to this is when people are interested in studying nonclassical logics, like intuitionistic logic.
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Re: Can we truly prove anything?

Postby rhetorical » Tue May 11, 2010 1:32 pm UTC

First, a definition: The way I am interpreting the question "Can we truly prove anything" involves "prove" to be proving something in the physical world around us.

Perhaps that is the origin of this disagreement, as you may be thinking of only theoretical mathematics. If not, then would you mind showing me where you disagree, and where you agree?

Do you agree that:

Mathematics is based on assumptions

If those assumptions are incorrect, so is mathematics

Mathematics can only function correctly if the mind (be it a human mind or a computer) is purely logical, or logical enough that the probability that a certain conclusion is true is close enough to 1 so as to make accepting it as true, reasonable



The human mind may/does have flaws that prevent it from reasoning logical or correctly all the time (i.e., it is logical or correct only some of the time.)

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Re: Can we truly prove anything?

Postby heyitsguay » Tue May 11, 2010 1:51 pm UTC

rhetorical wrote:First, a definition: The way I am interpreting the question "Can we truly prove anything" involves "prove" to be proving something in the physical world around us.

Perhaps that is the origin of this disagreement, as you may be thinking of only theoretical mathematics. If not, then would you mind showing me where you disagree, and where you agree?

Do you agree that:

Mathematics is based on assumptions

If those assumptions are incorrect, so is mathematics

You seem to be very fundamentally misunderstanding what function the "assumptions" of mathematics serve. Don't think of axioms as physical laws, but as rules of a game. We can define the rules for checkers, and there is no reasonable way to say whether those rules are right or wrong, only whether or not the resulting game is interesting and worth playing. If the rules you define end up being contradictory, you just change them in such a way that you still keep the fun/interesting parts of the game. In math, that amounts to saying that even if a contradiction were found in ZFC, it wouldn't invalidate the achievements mathematics has had, particularly in the realms of science and engineering, it would just mean that we'd have to find a new set of axioms that skirted the contradiction and still gave us a system with all the properties that make math so nice. Math's history has many examples where just this sort of thing has occurred, and it never ruined mathematics or invalidated (many) results, in fact it just led to a firmer understanding of just what math is.

rhetorical wrote:Mathematics can only function correctly if the mind (be it a human mind or a computer) is purely logical, or logical enough that the probability that a certain conclusion is true is close enough to 1 so as to make accepting it as true, reasonable

If you aren't certain your mind is functioning correctly enough for math, how would you define such a probability ? :P

rhetorical wrote:
The human mind may/does have flaws that prevent it from reasoning logical or correctly all the time (i.e., it is logical or correct only some of the time.)


Again, the checkers analogy. When playing you might slip up and move a piece the wrong way, perform a move incorrectly, but these have no bearings on the rules themselves. We're human, we make mistakes, and the mistakes are corrected in the long run. Math doesn't require that we are 100% logical all the time, only that we're logical enough to eventually catch the errors when they crop up. I'm no professional mathematician, but from what I've seen I'd say that that is done reasonably well. As to some deeper meaning of truth and whether we can ever really "know" that what we've proven is objectively "true", that is not a question of mathematics.
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Re: Can we truly prove anything?

Postby Talith » Tue May 11, 2010 1:52 pm UTC

This thread was posted in the Mathematics forum and the content of the majority of the thread, including the opening post, referred heavily to proof in mathematics which is a very different thing to proving theories in the real world (we call this science).

Mathematics is based on assumptions: True, this gives maths its richness and without any assumptions we wouldn't have an area of study called 'mathematics'.

If those assumptions are incorrect, so is mathematics: Mathematics is not the study of real world phenomena. Mathematics is the study of systems of formal logic which are built on the precise definitions of assumptions made in the system which we call axioms. In terms of mathematics, axioms can't 'be incorrect'. It's only when we try to apply our mathematics to the real world that we see if the axioms in our system are reasonable or not and once again this is science not mathematics.

I don't think your last two bullet points need comment given the reply to your first two. I will just say that talking about the human mind in this discussion really is leaving the OP at a tangent and is probably a better discussion for the science forum. It's already been said that we know we humans aren't brilliant at intuitive reasoning which is the precise reason we axiomatised our logic systems and applied rigorous logical reasoning to them, rather than relying on our intuition - this is mathematics.

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Re: Can we truly prove anything?

Postby rhetorical » Tue May 11, 2010 7:39 pm UTC

gmalivuk wrote:Yes, and no one says the mind is completely logical. But I doubt even you believe the human mind is completely *illogical*, either. And the very reason why we use formal logic is because we know that logical thinking doesn't come especially easily or naturally to us. So we formalized it in a way that other people can follow it and check it, and even a purely mechanistic process can run through certain kinds of proofs to check them.


I do not believe that the human mind is completely illogical. But it doesn't make a difference to the truth what I think. We, as humans, can never know whether or not humans are logical. If we could show that we are, it would be equivalent to a system proving itself consistent. The most we can say about mathematics is that it works only in our limited frame of mind. Therefore, we cannot "truly prove anything," as humans do not have the ability to establish absolute truth.

Also, there is no "purely mechanistic process" that is completely separate from the human mind. All the processes we have now that could do this are built by us, to operate by the rules we perceive. If those rules are wrong, then once again we are only 'proving' things within our own little bubble, and saying nothing about what is really true.


Edit:

Talith wrote:I will just say that talking about the human mind in this discussion really is leaving the OP at a tangent and is probably a better discussion for the science forum.


This is a discussion about proofs in mathematics, and mathematics as we know it now is based on the human mind and the way it functions. Understanding the way the mind works to understand mathematics is just as important as learning about computer science to understand why a piece of code runs the way it does. If this were purely a discussion on the limitations of the human mind without reference to mathematics, then yes, you are correct, this would better belong in a science forum, or better yet a religious forum, if we had one.

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Re: Can we truly prove anything?

Postby achan1058 » Tue May 11, 2010 8:12 pm UTC

@rhetorical: You are completely missing the point. Modern mathematics can be reduced to a set of statements which are derivable from certain given statements and rules. At it's very core, it is nothing more than a game of string manipulation, something that a computer can do. Whether computers can do it efficiently, and find results which humans find interesting, is another matter. "Truth" is only relevant when we attempt to apply any of this to the real world, and we use science to decide what to apply. (For example, you apply real numbers and not finite fields when you talk about things like temperature, since the former fits better.)

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Re: Can we truly prove anything?

Postby Yakk » Tue May 11, 2010 8:31 pm UTC

The game to play here is to imagine if one of the "obvious" rules of mathematical logic is actually a pile of garbage, but all humans are blind to the hole.

Ie, imagine if any logical system involving the "or" operator is actually capable of deriving any statement. We just cannot notice it, because it is a blind spot. We go along, using the system that we think makes sense, and never notice the huge gaping hole that is right there. Even more, the blind spot is such that when we build mechanical proof checking apparatus, the things we build also have that hole.

Now, at this point you should be going either "whoa" or "that's dumb", depending on your recent weed exposure.

Why? Because you are positing a connection between a mental blind spot in a human with regards to logic, and a blind spot in humans in regards to every device we build that "tests" that blind spot (accidentally or not).

If there was something ridiculously fundamental wrong with logic (at the level of or being garbage), and that thing was constructive, the tower of cards we built on top of it would just happen to happen to stand for reasons completely unrelated to the logic it is built on.

Now, if there was something fundamentally wrong with logic on the level of "something doesn't quite work, there is a teeny tiny problem that we are blind to", that won't show up in the towers of cards we build on top of logic. But the hole in our perceptions sort of has to be teeny tiny for that to be the case, and that sort of implies that even if logic isn't perfect, it is still close enough for skyscrapers.

On the other hand, things like the Axiom of Choice, the Axiom of Infinity, and a whole other pile of mathematical axioms do not describe anything that we can test by building skyscrapers. In a sense, if there was a gaping blind spot in our vision around that kind of mathematics, then we might not notice it. On the other hand, it would still have to be a really strange shaped blind spot: when people do mathematics, they don't just run out on a line. Mathematics is full of cross-hatches -- crazy ass things end up being warped around to proving results regarding simple number theory. And number theory, for one, is full of recent explosive "checking of results" far beyond what anyone doing the math originally thought of (computers sort of do that).

That gives us a "hand grenade radius" asto how big our blind spot can be.

There is, of course, the most ridiculous option: we are brains in a vat, and a superior intelligence is twerking our perception and our world to make us think that we are building skyscrapers from our logic and math, but our logic and math is actually utterly insane, and they just make the skyscrapers work by fiat. They also make other skyscrapers fall down, also by fiat, because it amuses them to watch us with our blind spot... Even worse, the logic we all think we know and applied is just implanted memories.

However, when it reaches the point where your argument requires that you posit that you are a brain in the vat being manipulated by an infinite intelligence mad scientist, it is time to give up and take a high school Philosophy course or something.
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Re: Can we truly prove anything?

Postby rhetorical » Tue May 11, 2010 9:28 pm UTC

Yakk wrote:The game to play here is to imagine if one of the "obvious" rules of mathematical logic is actually a pile of garbage, but all humans are blind to the hole.

Ie, imagine if any logical system involving the "or" operator is actually capable of deriving any statement. We just cannot notice it, because it is a blind spot. We go along, using the system that we think makes sense, and never notice the huge gaping hole that is right there. Even more, the blind spot is such that when we build mechanical proof checking apparatus, the things we build also have that hole.

Now, at this point you should be going either "whoa" or "that's dumb", depending on your recent weed exposure.

Why? Because you are positing a connection between a mental blind spot in a human with regards to logic, and a blind spot in humans in regards to every device we build that "tests" that blind spot (accidentally or not).

If there was something ridiculously fundamental wrong with logic (at the level of or being garbage), and that thing was constructive, the tower of cards we built on top of it would just happen to happen to stand for reasons completely unrelated to the logic it is built on.

Now, if there was something fundamentally wrong with logic on the level of "something doesn't quite work, there is a teeny tiny problem that we are blind to", that won't show up in the towers of cards we build on top of logic. But the hole in our perceptions sort of has to be teeny tiny for that to be the case, and that sort of implies that even if logic isn't perfect, it is still close enough for skyscrapers.

On the other hand, things like the Axiom of Choice, the Axiom of Infinity, and a whole other pile of mathematical axioms do not describe anything that we can test by building skyscrapers. In a sense, if there was a gaping blind spot in our vision around that kind of mathematics, then we might not notice it. On the other hand, it would still have to be a really strange shaped blind spot: when people do mathematics, they don't just run out on a line. Mathematics is full of cross-hatches -- crazy ass things end up being warped around to proving results regarding simple number theory. And number theory, for one, is full of recent explosive "checking of results" far beyond what anyone doing the math originally thought of (computers sort of do that).

That gives us a "hand grenade radius" asto how big our blind spot can be.

There is, of course, the most ridiculous option: we are brains in a vat, and a superior intelligence is twerking our perception and our world to make us think that we are building skyscrapers from our logic and math, but our logic and math is actually utterly insane, and they just make the skyscrapers work by fiat. They also make other skyscrapers fall down, also by fiat, because it amuses them to watch us with our blind spot... Even worse, the logic we all think we know and applied is just implanted memories.

However, when it reaches the point where your argument requires that you posit that you are a brain in the vat being manipulated by an infinite intelligence mad scientist, it is time to give up and take a high school Philosophy course or something.



So the most ridiculous option is that there is a supernatural power, and you make a straw man argument, creating a stupid scenario that virtually no one believes, of brains in a vat and an infinitely intelligent mad scientist? Since our argument is about logic, I'd expect a little more seriousness regarding other possibilities.

That point aside, you seem to expect that eventually humans will discover all the holes in our logic system. We do not know that this will ever happen.

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Re: Can we truly prove anything?

Postby jestingrabbit » Tue May 11, 2010 11:00 pm UTC

rhetorical wrote:So the most ridiculous option is that there is a supernatural power, and you make a straw man argument, creating a stupid scenario that virtually no one believes, of brains in a vat and an infinitely intelligent mad scientist? Since our argument is about logic, I'd expect a little more seriousness regarding other possibilities.


It was a pretty good approximation of Descartes argument. He imagined an evil demon who was tricking his mind to perceive his reality around him. He concluded that god would not allow that to occur. In the next paragraph (or perhaps a few after that) he concluded "cogito ergo sum" ie I think therefore I am. This was the statement that Descartes decided he could not reasonably doubt, and so he put it as the foundation of his logical system. Descartes might propose that your own existence is something that is, in some sense, truly proven by the evidence of your own perception every instant of every day.

So it comes across as silly, perhaps, but it has excellent pedigree.
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Re: Can we truly prove anything?

Postby heyitsguay » Wed May 12, 2010 12:00 am UTC

rhetorical wrote:
So the most ridiculous option is that there is a supernatural power, and you make a straw man argument, creating a stupid scenario that virtually no one believes, of brains in a vat and an infinitely intelligent mad scientist? Since our argument is about logic, I'd expect a little more seriousness regarding other possibilities.

That point aside, you seem to expect that eventually humans will discover all the holes in our logic system. We do not know that this will ever happen.


So, you are worried that somewhere in the process of human logical thought, there is a huge hole, but it's a hole we can't notice, and maybe can never notice. But then by that criteria, what effect could this hole have on the consistency of the logical system? Surely if this hole in our logic produced a contradiction or some other inconsistency, we'd be able to see this, and understand where it came from, or at least see that a flaw existed. So if the hole were to not be noticeable, it could not produce any contradiction or inconsistency in the manner in which we used the logical system, so where then is the contradiction?

You seem to enjoy the philosophical side of logic, and that is great, it's definitely very interesting, and I'm sure that many people here, myself included, have pondered these same sorts of questions. However, lots of us have also taken the time to learn the mathematical side of all this as well, and it's from this education that we gain our viewpoint which seems to be so against yours. It is apparent that you have not had this mathematical education, which is fine, but if you really want to get another perspective on the questions you are having, consider taking a course in formal mathematical logic. Actual work in the area will be far more eye-opening than any hand-wavy argument we can give you here.

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Re: Can we truly prove anything?

Postby Tirian » Wed May 12, 2010 12:52 am UTC

Talith wrote:Mathematics is the study of systems of formal logic which are built on the precise definitions of assumptions made in the system which we call axioms. In terms of mathematics, axioms can't 'be incorrect'.


Erm. They can certainly be inconsistent. An example is naive set theory, which put forward that there is an inconceivably large universe of mathematical objects and a set can be defined as all of those objects that unambiguously satisfy any property you can name. I don't know whether you'd call that axiom "incorrect" just because it leaves you in the spot of working with the set of all objects that have the property of not including themselves as members. But it does show that the system is so broken that you can't depend on its results to model real-world behavior.

I don't know what would happen if someone came across a similar problem for ZF and/or type theory. It would probably be the most profound mathematical result of the year, but I wouldn't be surprised if we just threw a patch on the system and mostly plowed forward. Heck, we still use naive set theory 99% of time even though we know that it is thin ice, on the assumption that someone has (or could) translate our work into a more sound framework.

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Re: Can we truly prove anything?

Postby Talith » Wed May 12, 2010 1:08 am UTC

Sure the axioms we choose can be inconsistent, but we can still do mathematics with the axioms even if our reasoning then leads to a contradiction; we find the contradiction, we clearly state that we made a poor choice, and we move on to the next system, having gained the knowledge we did about the previous system. I didn't mean for a set of axioms not being correct to be read as not being consistent, in the same sense that in a system where 1=2 is an axiom, saying '1=2' is correct. However, we will, at some point down the line, realise that we made a poor choice of our system and it's through the power of mathematics that we can show this.

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Re: Can we truly prove anything?

Postby Twistar » Wed May 12, 2010 2:14 am UTC

Talith wrote:Sure the axioms we choose can be inconsistent, but we can still do mathematics with the axioms even if our reasoning then leads to a contradiction; we find the contradiction, we clearly state that we made a poor choice, and we move on to the next system, having gained the knowledge we did about the previous system. I didn't mean for a set of axioms not being correct to be read as not being consistent, in the same sense that in a system where 1=2 is an axiom, saying '1=2' is correct. However, we will, at some point down the line, realise that we made a poor choice of our system and it's through the power of mathematics that we can show this.

So what are the types of things that clue us off that there is a problem with the axioms we've chosen? Is it just that we start to notice they produce boring results? The reason I ask this question is is you choose axioms, and always apply them correctly you can never find anything "wrong" because you've applied the axioms correctly.
I've never taken a mathematical logic course (I plan to though) so I'm not familiar with ZF or naive set theory. From Tirian's post it looks like the problem with set theory is that there was an assumption that seemed intuitively impossible. What types of not favorable results did this assumption lead to?
I guess my question ultimately boils down to: since every system based on a set of axioms is correct with regards to those axioms we can't use the systems or axioms to compare different sets of axioms so what extra-axiomatic thing do we compare our structures/axioms to?

Also, I wasn't quite clear on what consistency meant so I looked it up: a consistent system is one that does not contain a contradiction. I have taken a course on symbolic logic (sentential, predicate...) so I know what that means. What are some inconsistent sets of axioms we've had in the past and what contradictions did they contain?

edit: one more question, by what set of rules is 1=2 a contradiction? Clearly if our axioms were (1) A (2) ~A, our system would contain a contradiction. what if like Talith suggests, our axiom was (1) 1=2

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Re: Can we truly prove anything?

Postby PM 2Ring » Wed May 12, 2010 2:57 am UTC

What types of not favorable results did this assumption lead to?


Russell's paradox wrote:In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard Dedekind and Frege leads to a contradiction. The very same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen. Therefore priority goes to Russell and the paradox is known by his name.

It might be assumed that, for any formal criterion, a set exists whose members are those objects (and only those objects) that satisfy the criterion; but this assumption is disproved by a set containing exactly the sets that are not members of themselves. If such a set qualifies as a member of itself, it would contradict its own definition as a set containing sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox.

In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZF).

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Re: Can we truly prove anything?

Postby WarDaft » Wed May 12, 2010 4:00 am UTC

If we suppose that we are completely insane, we cannot be sure that we are not merely subconsciously choosing to view our little world as consistent no matter what evidence to the contrary... but if we are really that crazy, it's hard to imagine how assuming we aren't could possibly do any more harm.
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Re: Can we truly prove anything?

Postby gmalivuk » Wed May 12, 2010 11:46 am UTC

Yeah, there are plenty of extreme hypotheticals you can posit that would mean all of our math and science are utterly worthless. But the thing is, in those magical universes every other possible belief would be equally worthless. So we may as well stick with systems that would at least work for not-completely-crazy people in a not-completely-random universe.
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Re: Can we truly prove anything?

Postby Yakk » Wed May 12, 2010 1:15 pm UTC

No, I didn't presume we'd spot all of the blind spots in our logic system. I just described ways in which it seems, reasonably, that the certain kinds of blind spots are either ridiculously large (ie, to the level of the mad scientist twerking our brains and our universe), or are bounded in size.

The flaws in naive set theory do make the theory inconsistent. However, most arguments made using naive set theory are not garbage, and where ported over to non-naive set theory pretty much intact. Mathematics intertwined with naive set theory ties back to real life, constructive, implementations of crazy-ass shit that works in a number ways: either our universe is wonkey (see mad scientist), or that provides an external-to-brain method to test the solidity of our logic. So that explains why, despite the fact that naive set theory is inconsistent, it isn't far away from something that is consistent.

Ie, while in formal logic a single contradiction means that all is lost (or all is gained), practically it doesn't seem to work that way (barring the mad scientist).

Even extremely abstract things (like AoC) can be tied back, under certain restrictions, to world-testable results. AoC's consistency with ZF means, practically, it describes things that you cannot disprove constructively, so it describes the inverse image of things that cannot be ruled out. And when we use AoC to prove that something doesn't exist, that ends up working as well as using completely constructive tools to do the same.
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Re: Can we truly prove anything?

Postby rhetorical » Wed May 12, 2010 11:13 pm UTC

All I am saying is that we cannot prove absolute truths. This is how I interpret the OP's question: Can mathematics prove anything to be an absolute truth. If fact, you are very correct in most everything you have said so far, in terms of our perception of the world. I only disagree with you when you try to extrapolate that out to absolute truths of the universe.

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Re: Can we truly prove anything?

Postby Black » Thu May 13, 2010 12:36 am UTC

How can you doubt everything about reasoning except the ability to define what truth is?

If you have the faculties to decide what 'absolute truth' is you have everything you need to accept constructive mathematics.

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Re: Can we truly prove anything?

Postby rhetorical » Thu May 13, 2010 1:09 pm UTC

Black wrote:How can you doubt everything about reasoning except the ability to define what truth is?

If you have the faculties to decide what 'absolute truth' is you have everything you need to accept constructive mathematics.



That's what I'm saying: We do not have the faculties to decide absolute truth.

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Re: Can we truly prove anything?

Postby Yakk » Thu May 13, 2010 2:20 pm UTC

He said define, not decide. They are different.
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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Re: Can we truly prove anything?

Postby gmalivuk » Thu May 13, 2010 2:39 pm UTC

Right. And the point is that if rhetorical is making claims about "absolute truth", then rhetorical must have some definition in mind for "absolute truth". But if that definition is okay, then why do you have a problem with every *other* definition?
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Re: Can we truly prove anything?

Postby rhetorical » Thu May 13, 2010 8:56 pm UTC

Yakk wrote:He said define, not decide. They are different.



I meant define.

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Re: Can we truly prove anything?

Postby gmalivuk » Thu May 13, 2010 9:16 pm UTC

But if you say we can't define absolute truth, then why are you going on to make claims about absolute truth, if we can't even define it?
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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Re: Can we truly prove anything?

Postby Yakk » Thu May 13, 2010 9:23 pm UTC

I hereby define "absolute truth" to be sentences with an even number of vowels in English.

You may disagree with my definition, but how is it not a definition, again?
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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Re: Can we truly prove anything?

Postby rhetorical » Thu May 13, 2010 9:32 pm UTC

gmalivuk wrote:But if you say we can't define absolute truth, then why are you going on to make claims about absolute truth, if we can't even define it?


Nice equivocation. I'm using define in the same way one defines a variable. You are using define to mean "give another wording."

In the end, this is what my statement is:

Humans have no say in what is or is not absolutely true, i.e., what is absolute truth.

Basically, humans are limited in their ability.

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Re: Can we truly prove anything?

Postby Yakk » Thu May 13, 2010 9:40 pm UTC

When I define a variable, I express what I want that variable to mean. Prior to the definition, the variable didn't have that meaning.

So, by "absolute truth", what are you talking about?

I assume you aren't using my definition (which was given as an example of how it is seemingly possible for a human to define "absolute truth"). What definition are you using for "absolute truth"?

You are aware that you can type words that represent nonsense, right? I'm trying to find out if whatever you mean when you say "absolute truth" is meaningless nonsense or not. For me to have a chance at determining this, I need you to explain what you mean by it...

There are ways to approach this that aren't "complete"... Is there some kind of test you can do to determine if something is absolute truth? Some test that shows if something is not absolute truth? (these are different kinds of test, in a very neat way)

Your claim about properties of "absolute truth" is seemingly coming from a human. And the claim is that humans can not ... what can they not do again? They can have "no say" in matters pertaining to what is and what isn't "absolute truth"?

I'm not doing this in order to tear apart your argument, as much as to express that the kind of argument you seem to be making is an extremely problematic one to actually express in a way that doesn't reduce the phrase "absolute truth is something humans cannot express, and has no other interesting properties, including any relation to 'truth'".
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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Re: Can we truly prove anything?

Postby Daggoth » Wed Apr 27, 2011 5:03 pm UTC

I think that if the OP was asking something along the line of "Can proof of something be offered which does not require any assumption?"

State the following:
"There exists an absolute truth"
Since you have to assume that the statement is true, it isnt an absolute truth, so there you go.


I dont think that logic or math has to be considered "fundamentally flawed" just because certain assumptions are needed as stepping stones.I believe that imperfect things have the capacity to be functional, and they can even be improved to reduce the degree of their imperfection.

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Re: Can we truly prove anything?

Postby Yakk » Wed Apr 27, 2011 5:10 pm UTC

Kurushimi wrote:IIf they turned out to be inconsistent, wouldn't that make any proof we've ever made completely useless?

Nope. For an example, take a look at some of the reverse-proof work that has been done: determining not what you can prove from axioms, but rather what axioms are needed to prove a statement.

If our system is inconsistent, quite possibly there is a similar system in which most of our proofs can be picked up and dropped in. Maybe there will be extra qualifications, or some interesting corner cases.

We have some evidence that our proofs are not completely unrelated to truth by the fact that they work when applied to the world. Maybe their connection to the world isn't the connection we think -- but that's ok as well!
And wouldn't this mean that believing in a magical unicorn is every bit of justified as believing in gravity?

Believing in gravity seems to work better than believing in a magical unicorn. And the people who believe in magical unicorns, when push comes to shove, fall down like gravity says: while the gravity believers don't fall down like the magical unicorn says.

It could be the case that the magical unicorn is just biding its time. In which case, I, for one, welcome our new magical unicorn overlords.
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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Re: Can we truly prove anything?

Postby Mindworm » Wed Apr 27, 2011 6:01 pm UTC

I always thought we could prove without doubt that a great bunch of theorems follow from ZFC (most of what we call math). If ZFC turns out to be inconsistent our proofs would still be true, but not terribly relevant, since everything else including their negations also follows from ZFC.

It could be the case that the magical unicorn is just biding its time. In which case, I, for one, welcome our new magical unicorn overlords.

And if they turn out to be just regular unicorns filmed in an awkward angle which makes them appear magical?
The cake is a pie.

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Re: Can we truly prove anything?

Postby Ankit1010 » Thu Apr 28, 2011 6:42 am UTC

Instead of saying that "we cant really prove anything because weve made some assumptions we cant prove", I like to think of it as that we can pick the assumptions we want as properties of the system were dealing with, so its just kind of a choice on which system you want to prove things for. The axioms of PA and most common systems use simple properties that just make sense to us because of the way we interpret our world, stuff like a=a.

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Re: Can we truly prove anything?

Postby dissonant » Thu Apr 28, 2011 10:12 am UTC

"Paradox"

Not truth, nor certainty. These I forswore
In my novitiate, as young men called
To holy orders must abjure the world.

"If ..., then," this only I assert;
And my successes are but pretty chains
Linking twin doubts, for it is vain to ask
If what I postulate be justified,
Or what I prove possess the stamp of fact.

Yet bridges stand, and men no longer crawl
In two dimensions. And such triumphs stem
In no small measure from the power this game,
Played with the thrice attenuated shades
Of things, has over their originals.

How frail the wand, but how profound the spell


- Clarence R. Wylie Jr.

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Re: Can we truly prove anything?

Postby antonfire » Thu Apr 28, 2011 10:46 am UTC

Wow, thank you for that.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Can we truly prove anything?

Postby 314man » Thu Apr 28, 2011 4:52 pm UTC

What makes a truth absolute?
I take it as something that is impossible for it to be false. So then we're applying skeptical thinking:

I think X is true
If y, then X is false
I can't doubt y
Therefore X isn't absolutely true
(really it's I know X, If Y, then I don't know X, etc. but this is essentially the same).

Which then you can't be sure of anything except that you exist, since you know you are thinking which requires for you to exist (this is all from Descartes Meditations btw).
Now since we know for sure that we exist, we can extend our knowledge a bit. Basically we can define things, and those definitions have to be true because we created it. Eg. Bachelors are unmarried men, a triangle has 3 sides, etc.

Now, math breaks down to a set of axioms. These are just a set of definitions that we deem important. The rest of math comes from deductive thinking with the axioms as the main premises. So as long as the axioms hold, everything we discover in math is true.

We really just have to watch out that we do not contradict ourselves, saying something like "a square is a triangle with 4 sides". If we come up with a contradiction, then we twerk our definitions to get rid of them. However it's never a question if the axioms are true or not, because we defined them to be true. If we did define a square to be a triangle with 4 sides, it would still be true that a square is a triangle with 4 sides. We just happened to not define a square to be that definition (because that would mean squares do not exist, and why define that?)

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Re: Can we truly prove anything?

Postby Talith » Thu Apr 28, 2011 5:11 pm UTC

Just a quick note on your last point: we define things which don't exist all the time, normally so that we can prove they don't exist.

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Re: Can we truly prove anything?

Postby cphite » Fri Apr 29, 2011 6:34 pm UTC

jestingrabbit wrote:
rhetorical wrote:So the most ridiculous option is that there is a supernatural power, and you make a straw man argument, creating a stupid scenario that virtually no one believes, of brains in a vat and an infinitely intelligent mad scientist? Since our argument is about logic, I'd expect a little more seriousness regarding other possibilities.


It was a pretty good approximation of Descartes argument. He imagined an evil demon who was tricking his mind to perceive his reality around him. He concluded that god would not allow that to occur. In the next paragraph (or perhaps a few after that) he concluded "cogito ergo sum" ie I think therefore I am. This was the statement that Descartes decided he could not reasonably doubt, and so he put it as the foundation of his logical system. Descartes might propose that your own existence is something that is, in some sense, truly proven by the evidence of your own perception every instant of every day.

So it comes across as silly, perhaps, but it has excellent pedigree.


I guess this Descartes guy just wasn't very serious about this stuff :roll:

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Re: Can we truly prove anything?

Postby Daggoth » Sat Apr 30, 2011 3:28 pm UTC

314man wrote:What makes a truth absolute?
I take it as something that is impossible for it to be false. So then we're applying skeptical thinking:

I think X is true
If y, then X is false
I can't doubt y
Therefore X isn't absolutely true
(really it's I know X, If Y, then I don't know X, etc. but this is essentially the same).

Which then you can't be sure of anything except that you exist, since you know you are thinking which requires for you to exist (this is all from Descartes Meditations btw).
Now since we know for sure that we exist, we can extend our knowledge a bit. Basically we can define things, and those definitions have to be true because we created it. Eg. Bachelors are unmarried men, a triangle has 3 sides, etc.

Now, math breaks down to a set of axioms. These are just a set of definitions that we deem important. The rest of math comes from deductive thinking with the axioms as the main premises. So as long as the axioms hold, everything we discover in math is true.

We really just have to watch out that we do not contradict ourselves, saying something like "a square is a triangle with 4 sides". If we come up with a contradiction, then we twerk our definitions to get rid of them. However it's never a question if the axioms are true or not, because we defined them to be true. If we did define a square to be a triangle with 4 sides, it would still be true that a square is a triangle with 4 sides. We just happened to not define a square to be that definition (because that would mean squares do not exist, and why define that?)


that definition of absolute truth works for a cave dwelling hermit but probably not for a conciousness that exists as a part of a society whereas you might be sure that you know something to be absolutely true while someone else considers it just a part of your imagination. so i guess an applicable extension would be "an absolute truth is one that is beyond all reasonable doubt". To "prove" such a truth would require a tautological method ( IF P then P, or a = a) to which a defiant wise turtle could say " Yeah thats what you're saying, but you can't force me to accept it as truth".

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Re: Can we truly prove anything?

Postby 314man » Sun May 01, 2011 6:18 am UTC

Daggoth wrote: that definition of absolute truth works for a cave dwelling hermit but probably not for a conciousness that exists as a part of a society whereas you might be sure that you know something to be absolutely true while someone else considers it just a part of your imagination. so i guess an applicable extension would be "an absolute truth is one that is beyond all reasonable doubt". To "prove" such a truth would require a tautological method ( IF P then P, or a = a) to which a defiant wise turtle could say " Yeah thats what you're saying, but you can't force me to accept it as truth".


But when you create your own definition, it does not matter if it is a part of your imagination because it only exists in your imagination. If I define a bachelor to be "a purple radio", that's what I have it defined as. If someone comes to you and says "A bachelor is an unmarried man, not a purple radio", he is really saying "I've defined a bachelor to be an unmarried man, not a purple radio". They are just opposing definitions, and both of them are true. Essentially what makes a definition true is if the person accepts the definition.

So in a sense, you are arguing with yourself if a definition is true, or "correct". You can't play the role of the wise turtle when arguing with yourself though. Or else it's like saying "I define a rectangle to be a four sided shape. However I don't accept a rectangle to be a four sided shape". If that were to happen, you would consider your definition not true, and thus it's not a definition.

Definitions that are common to a society (eg all of math) are just definitions that are common to everyone. But they are still individually defined by the person. Someone didn't go up to me and said "1+1=2" and I believed it to be true. I heard it many times and I decided to defined 1+1=2 to be true. However, I could just as easily define it to be 1+1=3 if I chose to


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