"Science is fundamentally flawed"
January 4, 2012 6:00 PM Subscribe
Do Godel's theorems refute all of science and logic?
I was speaking to a theist today, and he stated:
Kurt Godel proved nearly 100 years ago that there are some true statements that cannot be logically i.e. mathematically, scientifically true.
He also proved that there is no way to no how many true statements cannot be proved true.
This means science is fundamentally flawed and incapable of finding truth.
Basically I took it to mean that he was arguing that Kurt Godel's theorems refute the viability of science. This is not the first time that I've encountered this argument.
I've researched it a bit in the past, and not being an avid logician (though I am rather proud of my ability in terms of using logic) or a mathematician, I'm afraid that a lot of the explanation in my research materials were lost on me.
So basically, what I'm asking is, can someone put it into laymen's terms for me, what his 'Theorem of Incompleteness' really means (because I assume that if it really did refute the application of science, the majority of the modern world wouldn't be bothering itself so much with it daily), and if my opponent is really using it in the correct context? Can you suggest counterarguments to this?
Note, I understand that science does not endeavor to 'find truth' whatever that means.
As Dr. Jones so succinctly put it: " If it's truth you're looking for, Dr. Tyree's philosophy class is right down the hall."
I was speaking to a theist today, and he stated:
Kurt Godel proved nearly 100 years ago that there are some true statements that cannot be logically i.e. mathematically, scientifically true.
He also proved that there is no way to no how many true statements cannot be proved true.
This means science is fundamentally flawed and incapable of finding truth.
Basically I took it to mean that he was arguing that Kurt Godel's theorems refute the viability of science. This is not the first time that I've encountered this argument.
I've researched it a bit in the past, and not being an avid logician (though I am rather proud of my ability in terms of using logic) or a mathematician, I'm afraid that a lot of the explanation in my research materials were lost on me.
So basically, what I'm asking is, can someone put it into laymen's terms for me, what his 'Theorem of Incompleteness' really means (because I assume that if it really did refute the application of science, the majority of the modern world wouldn't be bothering itself so much with it daily), and if my opponent is really using it in the correct context? Can you suggest counterarguments to this?
Note, I understand that science does not endeavor to 'find truth' whatever that means.
As Dr. Jones so succinctly put it: " If it's truth you're looking for, Dr. Tyree's philosophy class is right down the hall."
Your friend's third statement doesn't follow from the first two statements. Just because you can't know everything doesn't mean you can't know anything.
posted by John Cohen at 6:06 PM on January 4, 2012 [12 favorites]
posted by John Cohen at 6:06 PM on January 4, 2012 [12 favorites]
Gödel's theorems can be used to prove that it is not possible to know everything about a system; for example, science.
But science does not make absolute statements of fact. Science is about finding the best theorem that fits the data. For example, the current theory of gravity fits the data better than the theorem that it is "turtles all the way down." But science is a process for closing in on the best fit; it doesn't ever assume that it has reached the best fit.
So science is not about absolute truth. It is about relative truth.
posted by musofire at 6:06 PM on January 4, 2012 [3 favorites]
But science does not make absolute statements of fact. Science is about finding the best theorem that fits the data. For example, the current theory of gravity fits the data better than the theorem that it is "turtles all the way down." But science is a process for closing in on the best fit; it doesn't ever assume that it has reached the best fit.
So science is not about absolute truth. It is about relative truth.
posted by musofire at 6:06 PM on January 4, 2012 [3 favorites]
Wikipedia actually has a fairly decent discussion of these theorems. Specifically, it states "The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, a corollary of the first, shows that such a system cannot demonstrate its own consistency."
You might get some better answers to your question if you tell us what it is about the above that you don't understand.
posted by dfriedman at 6:08 PM on January 4, 2012
You might get some better answers to your question if you tell us what it is about the above that you don't understand.
posted by dfriedman at 6:08 PM on January 4, 2012
I'm no mathematician and I don't really understand this theorem either. But the way I've - perhaps incorrectly - understood is that there are truths about a system that are impossible to prove using only the tools within that system.
For example, we may never really be able to understand everything about the human mind because we are trapped within the human mind.
If you are technically inclined, I found this blog post interesting:
http://www.bootstrappingindependence.com/technology/how-to-build-a-computer-model-of-god/
Basically, the author goes through a thought experiment showing it may be possible for a soul to exist without there being any physical proof of it. He analogizes to virtualized computers. In case you are unfamiliar, these days one computer can run many "virtual machines", where each instance thinks it's a physical individual computer. But it's all software. So one virtual PC that thinks it's a PC may have no way of ever knowing it's actually a vitualized PC, even though it is.
posted by User7 at 6:17 PM on January 4, 2012 [1 favorite]
For example, we may never really be able to understand everything about the human mind because we are trapped within the human mind.
If you are technically inclined, I found this blog post interesting:
http://www.bootstrappingindependence.com/technology/how-to-build-a-computer-model-of-god/
Basically, the author goes through a thought experiment showing it may be possible for a soul to exist without there being any physical proof of it. He analogizes to virtualized computers. In case you are unfamiliar, these days one computer can run many "virtual machines", where each instance thinks it's a physical individual computer. But it's all software. So one virtual PC that thinks it's a PC may have no way of ever knowing it's actually a vitualized PC, even though it is.
posted by User7 at 6:17 PM on January 4, 2012 [1 favorite]
Response by poster: Hm. I don't really understand the second theorem it seems. Can you provide an example?
And what does "For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system" mean?
posted by Peregrin5 at 6:18 PM on January 4, 2012
And what does "For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system" mean?
posted by Peregrin5 at 6:18 PM on January 4, 2012
Godel demonstrated that no reasonably expressive system is capable of producing all truths without also generating contradicting results. It primarily discusses formal systems, but relaxing rigor doesn't "save" you anything, it just becomes sillier to talk about less formal systems that way. But it is fundamentally true about math, science, religion, whatever.
Does that render science worthless? Well, that's really a value question, but for most people, the fact that we can explain, predict, and control a tremendous amount of our world still does have a lot of value.
A good analogy is when you pan for gold, some gold will inevitably be washed out of your pan in the process, but you still can end up with more gold than you started with.
posted by aubilenon at 6:19 PM on January 4, 2012 [2 favorites]
Does that render science worthless? Well, that's really a value question, but for most people, the fact that we can explain, predict, and control a tremendous amount of our world still does have a lot of value.
A good analogy is when you pan for gold, some gold will inevitably be washed out of your pan in the process, but you still can end up with more gold than you started with.
posted by aubilenon at 6:19 PM on January 4, 2012 [2 favorites]
Math and physics, and indeed all of science, is not truth. It's an attempt at describing truth. Of course Godel's theorem doesn't in any way prove that science is incapable of discovering truth.
We can easily show that science can help us arrive at a truth. For example, we can use Newton's laws to calculate where a projectile will land every time. There is proof by contradiction that science is useful, and has done an excellent job at describing the truth about where the projectile will land. If he doubts that, go stand there and pray.
Godel say that a given system may not allow us to arrive at a given truth, but we are free to invent new systems. Science is a methodology for giving us new tools in the quest for truth, and for measuring whether or not a given proposal approximates truth or not.
Furthermore, it's just stupid to argue that math (and science) is useless and incapable of finding truth by citing what is, famously, a mathematical statement. The paradox there makes his whole argument useless.
posted by jeffamaphone at 6:22 PM on January 4, 2012
We can easily show that science can help us arrive at a truth. For example, we can use Newton's laws to calculate where a projectile will land every time. There is proof by contradiction that science is useful, and has done an excellent job at describing the truth about where the projectile will land. If he doubts that, go stand there and pray.
Godel say that a given system may not allow us to arrive at a given truth, but we are free to invent new systems. Science is a methodology for giving us new tools in the quest for truth, and for measuring whether or not a given proposal approximates truth or not.
Furthermore, it's just stupid to argue that math (and science) is useless and incapable of finding truth by citing what is, famously, a mathematical statement. The paradox there makes his whole argument useless.
posted by jeffamaphone at 6:22 PM on January 4, 2012
I just started re-reading "Godel, Escher, Bach" (Douglas Hofstadter) the other day, and the Introduction has a great and very accessible explanation of exactly what you're asking for. The rest of the book is worth it too.
posted by ella wren at 6:26 PM on January 4, 2012 [3 favorites]
posted by ella wren at 6:26 PM on January 4, 2012 [3 favorites]
Your friend is conflating mathematical truth and scientific truth. Gödel was talking about mathematical truth, which is grounded in axioms: If you can derive a statement by repeatedly applying axioms, then it is true and you have proved it.
Scientific truth, or "religious" truth, concerns the fundamental nature of the universe. Any scientist will happily admit that we can never know the absolute truth; just that we're confident with high probability because our models correctly predict real phenomena. So yeah, science doesn't know the absolute truth, nor does it claim to, and that in no way invalidates it.
For the future, here's a response to this silly theist reasoning:
"If you insist on dragging logic into this, let me point out that if your axioms lead to even one single contradiction, then your system of reasoning is logically meaningless. Apply that to the Bible (hint: full of contradictions)." (more specifically: if there exists any proposition P for which you can prove both P and ¬P , then your system is fundamentally flawed because you can prove every possible statement to be both true and false)
posted by qxntpqbbbqxl at 6:45 PM on January 4, 2012 [6 favorites]
Scientific truth, or "religious" truth, concerns the fundamental nature of the universe. Any scientist will happily admit that we can never know the absolute truth; just that we're confident with high probability because our models correctly predict real phenomena. So yeah, science doesn't know the absolute truth, nor does it claim to, and that in no way invalidates it.
For the future, here's a response to this silly theist reasoning:
"If you insist on dragging logic into this, let me point out that if your axioms lead to even one single contradiction, then your system of reasoning is logically meaningless. Apply that to the Bible (hint: full of contradictions)." (more specifically: if there exists any proposition P for which you can prove both P and ¬P , then your system is fundamentally flawed because you can prove every possible statement to be both true and false)
posted by qxntpqbbbqxl at 6:45 PM on January 4, 2012 [6 favorites]
There is basically a disconnect between sentences 1 and 2, and sentence 3.
As far as I understand it, Gödel's theory s a very specific definition of 'true,' based on understanding the propositional and logical relationships between taken-for-granted and fundamental axioms.
If this is all that science does, then there are things that cannot be proved in science.
But if you think of science as a practice that continually refines best guesses through a process of experimental confirmation and refutation, carried out by communities of scientists who share some common beliefs, then science is not necessarily propositional in the first place, and so it is not Gödelian in terms of completeness. - Therefore the third sentence does not follow.
Science is very untidy and messy at the edges. I think a lot of scientists would like to be able to express their knowledge propositionally and unambiguously, but this is in reality very difficult, because the world is messy and descriptions of the world are also therefore messy.
The only example I was thinking about here was quantum mechanics, about which I keep hearing things like "none of the predictions of quantum mechanics have ever been proved wrong by experiment."
posted by carter at 6:46 PM on January 4, 2012
As far as I understand it, Gödel's theory s a very specific definition of 'true,' based on understanding the propositional and logical relationships between taken-for-granted and fundamental axioms.
If this is all that science does, then there are things that cannot be proved in science.
But if you think of science as a practice that continually refines best guesses through a process of experimental confirmation and refutation, carried out by communities of scientists who share some common beliefs, then science is not necessarily propositional in the first place, and so it is not Gödelian in terms of completeness. - Therefore the third sentence does not follow.
Science is very untidy and messy at the edges. I think a lot of scientists would like to be able to express their knowledge propositionally and unambiguously, but this is in reality very difficult, because the world is messy and descriptions of the world are also therefore messy.
The only example I was thinking about here was quantum mechanics, about which I keep hearing things like "none of the predictions of quantum mechanics have ever been proved wrong by experiment."
posted by carter at 6:46 PM on January 4, 2012
Based on my college education as a math major I would agree with others that your theist friend is warping Godel's work. It is more likely to be a reason for not wasting your time trying to create a single grand "explanation of everything" than it is a demonstration of what is and is not knowable.
At this point I should mention that I am a theist, and that even though I believe Godel's work, I do not think it relevant to theism and do not cite him for that reason.
Two other tangential points. #1 - people believe a great many things that are not true, and do not believe a great many things that are (try asking someone why a mirror reflects things left and right but not up and down). #2 - in addition to also recommending "Godel, Escher, Bach", I would recommend "The Mind's I", if you want to think about truth and illusion and what is knowable.
posted by forthright at 6:52 PM on January 4, 2012
At this point I should mention that I am a theist, and that even though I believe Godel's work, I do not think it relevant to theism and do not cite him for that reason.
Two other tangential points. #1 - people believe a great many things that are not true, and do not believe a great many things that are (try asking someone why a mirror reflects things left and right but not up and down). #2 - in addition to also recommending "Godel, Escher, Bach", I would recommend "The Mind's I", if you want to think about truth and illusion and what is knowable.
posted by forthright at 6:52 PM on January 4, 2012
We may never really be able to understand everything about the human mind because we are trapped within the human mind.
That's not exactly true -- the "because" part. It's NOT because we're in a human mind.
No mind -- human or Jovian or any other kind -- will ever understand everything about ANYTHING. No mind will ever understand everything about a flea, a dandelion, a lost scrabble tile, or pi. NOT EVER.
That's the Universe, buddy. We can accept it or not, but IMO -- By God, we'd better.
Apologies to RW Emerson.
posted by LonnieK at 7:10 PM on January 4, 2012
That's not exactly true -- the "because" part. It's NOT because we're in a human mind.
No mind -- human or Jovian or any other kind -- will ever understand everything about ANYTHING. No mind will ever understand everything about a flea, a dandelion, a lost scrabble tile, or pi. NOT EVER.
That's the Universe, buddy. We can accept it or not, but IMO -- By God, we'd better.
Apologies to RW Emerson.
posted by LonnieK at 7:10 PM on January 4, 2012
what does "For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system" mean?First, note that 'system' roughly means 'a formal language'. A set of statements that are deterministically true or false.
Your quote means that if a formal language is rich enough, you will be able to use it to express something which is unprovable. Formal english lets you define one:
This sentence is false.That sentence is unprovable. Not even an omnipotent god can sort out whether it is true or false. (This is one of my favourite things to point out to theists)
Moving on:
I don't really understand the second theoremIt roughly says that if your pet system is shown to be consistent, then that is meaningless. Inconsistent theories can prove anything, including their own consistency. Hence, you can never know whether your consistency proof is true.
A formal language can not be used to show that it is itself consistent.
posted by clord at 7:22 PM on January 4, 2012 [2 favorites]
Here's an analogue to the Imcompleteness Theorem: Suppose that all particles in the universe are describable, that is, their location and momentum can be measured. You could then, in theory, construct a computer that, given the location and momentum of every particle in the universe at one point in time, could then predict the locations and momenta at a future moment in time.
Except to construct such a computer, you'd need more building blocks than there are particles.
I sincerely hope that your theist friend does not use any technology, since he seems to think that the science that underpins technology can't be trusted.
posted by notsnot at 7:28 PM on January 4, 2012
Except to construct such a computer, you'd need more building blocks than there are particles.
I sincerely hope that your theist friend does not use any technology, since he seems to think that the science that underpins technology can't be trusted.
posted by notsnot at 7:28 PM on January 4, 2012
It helps to keep in mind that Gödel was doing all of this in order to show that Principia Mathematica would be fruitless.
PM was trying to formally define an internally consistent basis for Mathematics. Gödel was saying two things about it:
posted by clord at 7:29 PM on January 4, 2012
PM was trying to formally define an internally consistent basis for Mathematics. Gödel was saying two things about it:
- You'll never capture everything,
- and even if you think you did, you'd be wrong.
posted by clord at 7:29 PM on January 4, 2012
Seconding the Hofstadter recommendation, first of all. "GEB" is a really beautiful book, although I haven't read it in a long time and IANAL(ogician). IAAG(eometer), though.
Your friend is probably confused about at least two things, regardless of the "answers" to his questions, which I don't think are stated precisely enough to be meaningful (in any sense of "meaningful").
First, he is probably considering a much broader range of "universes" than Goedel did. Second, he is understandably confusing the words "true" and "false" as they are used formally with the colloquial usages. These two ambiguities are the source of his apparent misconceptions.
One has to understand, first of all, the objects that Goedel's theorems are talking about. Goedel's theorems talk about formal theories that are effectively generated and have sufficient axiomatic juice to describe the natural numbers*. One therefore has to Wikipediate or Hofstadterize what each italicized thing means, and this will show how specific Goedel's theorems are, and the extent to which Goedel is simply not talking about any question remotely close to the way "truth" is treated in, say, natural science.
A formal theory can be viewed as a kind of "universe of discourse". Very roughly, a formal theory consists of an "initial" collection of statements, the "axioms", and a collection of rules for building new statements, given a statement in the theory. A "proof" of a statement S is constructed, beginning with the axioms, by applying the rules to get a new statement, applying the rules to the new statement to get a newer statement, apply the rules to the newer statement to get an even newer statement, and continue in this manner to eventually reach S. In this case, S is a "theorem" of the system.
One can construct formal theories until one is blue in the face, but when one ventures into a conceptual setting that doesn't resemble mathematics (broadly construed), it's kind of unlikely that one can describe everything one wants to describe using a formal theory. However, for example, chess can be interpreted as a formal theory -- the axiom is the initial board setup, the rules for putting the pieces in a new state are the rules of chess, and the theorems are the chessboard setups that can be reached in an actual game.
Our system is "consistent" if, for each theorem S, the negation of S is not a theorem. In other words, one cannot prove S from the axioms, and then go back to the axioms, crank through some other sequence of applications of the rules, and get "not S".
The system is "complete" if each "true" statement is a theorem. To say what I mean by "true" requires explaining a bit more about how the rules have to work, but if you're googling, I mean that S is a "tautology". So, the system is complete if each true statement can be proved, within the system, by applying the rules to the axioms.
Goedel's first theorem says that a sufficiently robust formal theory is either not consistent, or not complete. In other words, either there is a theorem S for which "not S" is also a theorem, or there is a true statement S that is not a theorem, i.e. that has no proof.
Goedel's second theorem says that, if a (sufficiently robust) formal system contains the statement "the system in which I live does not contain two contradictory statements", then the system does in fact contain two contradictory statements, more or less.
However:
Even the most traditionally "rigorous" natural science -- physics -- is very far from being "axiomatized", as far as I know. To formalize physics in this way was a pipe dream of David Hilbert in a much more naive era. The point is that the type of thing I think you're discussing with your friend (given that it matters that he is a theist) is not really even in the same world as Goedel's theorems. In natural science, one doesn't really care about "truth". Although it's a matter of some debate (you might want to look up Karl Popper), "true" in science more or less means "not demonstrably false, with bonus points for evidentiary support, depending on your philosophical orientation". This is a very different notion of truth than "is a tautology in a formal system", because one isn't even working with statements in a formal system; one is working in a much more nebulous context.
I don't really even see mathematical truth and scientific truth as related: when I say something is "true" in mathematics, I mean that, given some initial assumptions, I can produce a proof of that statement. "Truth" in any other context is variably defined and contingent on frequently-changing information. "Truth" in the setting that Goedel is talking about is very different.
TL;DR and not very well-explained, but maybe points you in the right direction.
*This is because of the extremely clever trick Goedel uses in his proof. Statements in the formal theory -- you'll find out what these are in the next paragraph of the main text of this post -- are numbered in a specific way. Goedel shows that manipulating statements in the theory -- i.e. deciding whether they have various properties -- corresponds to deciding whether the corresponding numbers have various properties. Roughly speaking, in order to Goedel's trick, while remaining "inside" the theory, the theory is required to be able to describe the properties of numbers being used.
posted by kengraham at 7:32 PM on January 4, 2012 [4 favorites]
Your friend is probably confused about at least two things, regardless of the "answers" to his questions, which I don't think are stated precisely enough to be meaningful (in any sense of "meaningful").
First, he is probably considering a much broader range of "universes" than Goedel did. Second, he is understandably confusing the words "true" and "false" as they are used formally with the colloquial usages. These two ambiguities are the source of his apparent misconceptions.
One has to understand, first of all, the objects that Goedel's theorems are talking about. Goedel's theorems talk about formal theories that are effectively generated and have sufficient axiomatic juice to describe the natural numbers*. One therefore has to Wikipediate or Hofstadterize what each italicized thing means, and this will show how specific Goedel's theorems are, and the extent to which Goedel is simply not talking about any question remotely close to the way "truth" is treated in, say, natural science.
A formal theory can be viewed as a kind of "universe of discourse". Very roughly, a formal theory consists of an "initial" collection of statements, the "axioms", and a collection of rules for building new statements, given a statement in the theory. A "proof" of a statement S is constructed, beginning with the axioms, by applying the rules to get a new statement, applying the rules to the new statement to get a newer statement, apply the rules to the newer statement to get an even newer statement, and continue in this manner to eventually reach S. In this case, S is a "theorem" of the system.
One can construct formal theories until one is blue in the face, but when one ventures into a conceptual setting that doesn't resemble mathematics (broadly construed), it's kind of unlikely that one can describe everything one wants to describe using a formal theory. However, for example, chess can be interpreted as a formal theory -- the axiom is the initial board setup, the rules for putting the pieces in a new state are the rules of chess, and the theorems are the chessboard setups that can be reached in an actual game.
Our system is "consistent" if, for each theorem S, the negation of S is not a theorem. In other words, one cannot prove S from the axioms, and then go back to the axioms, crank through some other sequence of applications of the rules, and get "not S".
The system is "complete" if each "true" statement is a theorem. To say what I mean by "true" requires explaining a bit more about how the rules have to work, but if you're googling, I mean that S is a "tautology". So, the system is complete if each true statement can be proved, within the system, by applying the rules to the axioms.
Goedel's first theorem says that a sufficiently robust formal theory is either not consistent, or not complete. In other words, either there is a theorem S for which "not S" is also a theorem, or there is a true statement S that is not a theorem, i.e. that has no proof.
Goedel's second theorem says that, if a (sufficiently robust) formal system contains the statement "the system in which I live does not contain two contradictory statements", then the system does in fact contain two contradictory statements, more or less.
However:
Even the most traditionally "rigorous" natural science -- physics -- is very far from being "axiomatized", as far as I know. To formalize physics in this way was a pipe dream of David Hilbert in a much more naive era. The point is that the type of thing I think you're discussing with your friend (given that it matters that he is a theist) is not really even in the same world as Goedel's theorems. In natural science, one doesn't really care about "truth". Although it's a matter of some debate (you might want to look up Karl Popper), "true" in science more or less means "not demonstrably false, with bonus points for evidentiary support, depending on your philosophical orientation". This is a very different notion of truth than "is a tautology in a formal system", because one isn't even working with statements in a formal system; one is working in a much more nebulous context.
I don't really even see mathematical truth and scientific truth as related: when I say something is "true" in mathematics, I mean that, given some initial assumptions, I can produce a proof of that statement. "Truth" in any other context is variably defined and contingent on frequently-changing information. "Truth" in the setting that Goedel is talking about is very different.
TL;DR and not very well-explained, but maybe points you in the right direction.
*This is because of the extremely clever trick Goedel uses in his proof. Statements in the formal theory -- you'll find out what these are in the next paragraph of the main text of this post -- are numbered in a specific way. Goedel shows that manipulating statements in the theory -- i.e. deciding whether they have various properties -- corresponds to deciding whether the corresponding numbers have various properties. Roughly speaking, in order to Goedel's trick, while remaining "inside" the theory, the theory is required to be able to describe the properties of numbers being used.
posted by kengraham at 7:32 PM on January 4, 2012 [4 favorites]
@notsnot: I like that metaphor; that seems closer to some of the newer, more computer-sciencey proofs of Goedel's theorem, and reminds me to tell the asker to look at this.
posted by kengraham at 7:36 PM on January 4, 2012
posted by kengraham at 7:36 PM on January 4, 2012
As usual, remind your friend that one had best understand the shit out of something before claiming it to be "fundamentally flawed".
posted by kengraham at 7:37 PM on January 4, 2012 [2 favorites]
posted by kengraham at 7:37 PM on January 4, 2012 [2 favorites]
Different epistemic categories. Logic, science, faith and truth are seldom more than occasional drinking buddies. Your friend is comparing apples, oranges, toyotas and charging rhinos.
The Gödel result your friend is referring to was a refutation of an epistemically optimistic hope -- popular at the turn of the last century -- that we'd be able to mechanize the problem of finding true things and their proofs. Or, well, "mechanize" if need be. Possibly provided a really solid foundational work like the Principia Mathematica, we'd just be able to turn a crank and enumerate the facts (with proofs).
Turns out if the theory is juicy enough, we can't do that. There are all sorts of tangles that any machine set on the task can get stuck in, including various self-representations and contradictions. Sad times. Mathematicians can't just retire yet.
posted by ead at 7:43 PM on January 4, 2012 [1 favorite]
The Gödel result your friend is referring to was a refutation of an epistemically optimistic hope -- popular at the turn of the last century -- that we'd be able to mechanize the problem of finding true things and their proofs. Or, well, "mechanize" if need be. Possibly provided a really solid foundational work like the Principia Mathematica, we'd just be able to turn a crank and enumerate the facts (with proofs).
Turns out if the theory is juicy enough, we can't do that. There are all sorts of tangles that any machine set on the task can get stuck in, including various self-representations and contradictions. Sad times. Mathematicians can't just retire yet.
posted by ead at 7:43 PM on January 4, 2012 [1 favorite]
Perhaps one way to think about the proof is that it demonstrates that science and math will never be 'perfect'. But still very very good.
posted by sammyo at 7:52 PM on January 4, 2012
posted by sammyo at 7:52 PM on January 4, 2012
As qxntpqbbbqxl said upthread, if your friend believes that Gödel's theorems invalidate science, he's really not going to like what they do to his religion.
posted by General Tonic at 8:00 PM on January 4, 2012 [2 favorites]
posted by General Tonic at 8:00 PM on January 4, 2012 [2 favorites]
Let me recommend a good book: Godel's Proof. The book goes through the incompleteness theorems along with their historical background in a very accessible way.
Your statement that Godel showed that "there are some true statements that cannot be logically i.e. mathematically, scientifically true" is not quite correct. What Godel showed was that using the rules of classical logic, any consistent system of axioms (basically a list of sentences) powerful enough to represent ordinary arithmetic on the natural numbers will be such that there will be some sentence that is true yet cannot be proved from those axioms.
Lots of smart mathematicians thought that all of mathematics could be proved to be consistent. Many (but not all) people working on the foundations of mathematics today think that Godel smashed that idea. One thing that Godel did not thereby smash was logic itself. In fact, Godel also proved that unlike ordinary arithmetic, first-order logic is complete. So, there are no truths in first-order logic that cannot be proved from the axioms of first-order logic. One reason this is important is that many scientific theories turn out to be first-orderizable: that is, they can be formulated in first-order logic. For example, special relativity is first-orderizable. So, yeah, science (as a whole, anyway) is not really threatened by Godel's theorems.
Also, you should note the following. Suppose you have an axiom system S. And suppose that it is true that p, but you cannot prove it in S. You might formulate a new axiom system S* by adding p to your old system S. In this way, you will be able to prove p in your new system S*. The problem is that you will either create new truths that cannot be proved in S* or you will make the system inconsistent.
Anyway, this may be beside the point, but I wouldn't give up on truth so quickly. Truth is important!
posted by Jonathan Livengood at 8:05 PM on January 4, 2012 [5 favorites]
Your statement that Godel showed that "there are some true statements that cannot be logically i.e. mathematically, scientifically true" is not quite correct. What Godel showed was that using the rules of classical logic, any consistent system of axioms (basically a list of sentences) powerful enough to represent ordinary arithmetic on the natural numbers will be such that there will be some sentence that is true yet cannot be proved from those axioms.
Lots of smart mathematicians thought that all of mathematics could be proved to be consistent. Many (but not all) people working on the foundations of mathematics today think that Godel smashed that idea. One thing that Godel did not thereby smash was logic itself. In fact, Godel also proved that unlike ordinary arithmetic, first-order logic is complete. So, there are no truths in first-order logic that cannot be proved from the axioms of first-order logic. One reason this is important is that many scientific theories turn out to be first-orderizable: that is, they can be formulated in first-order logic. For example, special relativity is first-orderizable. So, yeah, science (as a whole, anyway) is not really threatened by Godel's theorems.
Also, you should note the following. Suppose you have an axiom system S. And suppose that it is true that p, but you cannot prove it in S. You might formulate a new axiom system S* by adding p to your old system S. In this way, you will be able to prove p in your new system S*. The problem is that you will either create new truths that cannot be proved in S* or you will make the system inconsistent.
Anyway, this may be beside the point, but I wouldn't give up on truth so quickly. Truth is important!
posted by Jonathan Livengood at 8:05 PM on January 4, 2012 [5 favorites]
Also, see George Boolos' explanation of Godel's second incompleteness theorem using only one-syllable words.
posted by Jonathan Livengood at 8:49 PM on January 4, 2012 [1 favorite]
posted by Jonathan Livengood at 8:49 PM on January 4, 2012 [1 favorite]
And what does "For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system" mean?Well, let's take an example.
An example of a "statement about the natural numbers" would be the claim that "every natural number that is divisible by ten is also divisible by five".
That claim may or may not be true. We could check for a whole bunch of numbers: For example, 10 / 10 = 1, so 10 is divisible by 10, and 10 / 5 = 2, so 10 is divisible by 5. So it's good at least for 10 itself.
And 20 / 10 = 2, and 20 / 5 = 4, so it's good for 20, too.
We could check for 30, and 40, and 50, and 1,000,000. We would find that the claim is good for all of them, too.
But no matter how many numbers we check, there's a possibility that there's some number we didn't check, which would (if we checked it) show that the claim is false.
So that's where a proof comes in. Can we actually prove that the statement is true? For every number? Keeping in mind that we cannot explicitly check every number?
In this case, yes, we can:
Let X be a natural number that is divisible by 10. Then there's some natural number Y such that X = 10 * Y.
And since 10 = 5 * 2, that means X = 5 * 2 * Y.
Now consider the number 2 * Y. Let's call it Z. It's a natural number, since both 2 and Y are natural numbers. So now we have that X = 5 * Z, where Z is a natural number. Which means that X is divisible by 5.
So we just proved that any natural number that is divisible by 10 is also divisible by 5.
What Gödel says (among other things) is that there are statements about the natural numbers that have no possible proof, but which are nonetheless actually true.
If we happened to stumble upon such a statement, we could check all the live long day to see if it's true for 10, or true for 20, or true for 30, or true for 40, or true for 1,000,000, and we would find that it's true at least for every number that we check.
But there would be no actual way to prove that it's true for every number.
It's not terribly uncommon for theists to hear this and say "therefore God!", but generally speaking theists have a lot more opinions about God than Gödel's theorems do.
posted by Flunkie at 8:49 PM on January 4, 2012 [3 favorites]
Response by poster: Amazing answers everyone! Thank you so very much for your input! I will keep this question and its answers for posterity on my website.
Best answers soon to follow when I'm able to get a few minutes to really read in depth all of the answers given.
posted by Peregrin5 at 11:32 PM on January 4, 2012
Best answers soon to follow when I'm able to get a few minutes to really read in depth all of the answers given.
posted by Peregrin5 at 11:32 PM on January 4, 2012
A good rule of thumb is that anyone drawing big philosophical conclusions from either Gödel or from the Heisenberg Uncertainty Principle has no idea what they are talking about.
posted by thelonius at 12:08 AM on January 5, 2012 [4 favorites]
posted by thelonius at 12:08 AM on January 5, 2012 [4 favorites]
I think there's a pretty simple explanation that a lot of people are missing here.
Mathematical proofs are (largely) deductive. They take general statements, chain them together, and infer particular conclusions. In Flunkie's excellent example, one general statement is "If X is a natural number divisible by then, there is some other natural number Y such that X = 10 * Y". The particular conclusion is "Every natural number that is divisible by 10 is also divisible by 5".
Scientific proofs are (always) inductive. They take particular statements, chain them together, and infer general conclusions. These "particular statements" are also known as "observations", and hence science is properly known as empiricism. The most accessible and famous discourse on inductive / empirical / scientific proofs is Chapter 6 of Bertrand Russell's "The Problems of Philosophy".
Gödel's theorems specifically refer to the consistency of systems of axioms. Axioms are general statements, i.e. the basis of mathematical proofs. Hence Gödel's theorems do not apply to empiricism. QED.
Here's another funamental difference for you. Mathematics is concerned with ontology, i.e. what is, whereas science is concerned with epistemology, i.e. what and how do we know?. Science couldn't care less about ontology; here I'm just restated what qxntpqbbbqxl said.
Religion, by definition, is an ontology. They couldn't care less how or what they know; they simply take their holy scripture to be axiomatic and deductively move on from there.
(I think, IMO).
posted by asymptotic at 4:54 AM on January 5, 2012 [1 favorite]
Mathematical proofs are (largely) deductive. They take general statements, chain them together, and infer particular conclusions. In Flunkie's excellent example, one general statement is "If X is a natural number divisible by then, there is some other natural number Y such that X = 10 * Y". The particular conclusion is "Every natural number that is divisible by 10 is also divisible by 5".
Scientific proofs are (always) inductive. They take particular statements, chain them together, and infer general conclusions. These "particular statements" are also known as "observations", and hence science is properly known as empiricism. The most accessible and famous discourse on inductive / empirical / scientific proofs is Chapter 6 of Bertrand Russell's "The Problems of Philosophy".
Gödel's theorems specifically refer to the consistency of systems of axioms. Axioms are general statements, i.e. the basis of mathematical proofs. Hence Gödel's theorems do not apply to empiricism. QED.
Here's another funamental difference for you. Mathematics is concerned with ontology, i.e. what is, whereas science is concerned with epistemology, i.e. what and how do we know?. Science couldn't care less about ontology; here I'm just restated what qxntpqbbbqxl said.
Religion, by definition, is an ontology. They couldn't care less how or what they know; they simply take their holy scripture to be axiomatic and deductively move on from there.
(I think, IMO).
posted by asymptotic at 4:54 AM on January 5, 2012 [1 favorite]
An example of a "statement about the natural numbers" would be the claim that "every natural number that is divisible by ten is also divisible by five".
That claim may or may not be true. We could check for a whole bunch of numbers: For example, 10 / 10 = 1, so 10 is divisible by 10, and 10 / 5 = 2, so 10 is divisible by 5. So it's good at least for 10 itself.
...
But no matter how many numbers we check, there's a possibility that there's some number we didn't check, which would (if we checked it) show that the claim is false.
As a counterpoint to this, consider the following statement: "Every even number other than 2 is the sum of two prime numbers." This statement has been checked for every even number up to about 1,609,000,000,000,000,000 (or so). But despite years and years of effort, it's never been proven to be true in the way that Flunkie proved above that all numbers divisible by ten are also divisible by five. Gödel's first theorem says that statements in mathematics exist that are true, but that cannot be proven using the logical structures of mathematics. Maybe it's the case that every even number is the sum of two primes, but that mathematical logic is incapable of proving it. Or maybe it's the case that nobody's been quite smart enough yet, and tomorrow some wunderkind will find a new proof of this statement. We simply can't distinguish between these two possibilities.
posted by Johnny Assay at 6:13 AM on January 5, 2012 [1 favorite]
That claim may or may not be true. We could check for a whole bunch of numbers: For example, 10 / 10 = 1, so 10 is divisible by 10, and 10 / 5 = 2, so 10 is divisible by 5. So it's good at least for 10 itself.
...
But no matter how many numbers we check, there's a possibility that there's some number we didn't check, which would (if we checked it) show that the claim is false.
As a counterpoint to this, consider the following statement: "Every even number other than 2 is the sum of two prime numbers." This statement has been checked for every even number up to about 1,609,000,000,000,000,000 (or so). But despite years and years of effort, it's never been proven to be true in the way that Flunkie proved above that all numbers divisible by ten are also divisible by five. Gödel's first theorem says that statements in mathematics exist that are true, but that cannot be proven using the logical structures of mathematics. Maybe it's the case that every even number is the sum of two primes, but that mathematical logic is incapable of proving it. Or maybe it's the case that nobody's been quite smart enough yet, and tomorrow some wunderkind will find a new proof of this statement. We simply can't distinguish between these two possibilities.
posted by Johnny Assay at 6:13 AM on January 5, 2012 [1 favorite]
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