Gödel: The Continuation of Mathematics and Science
Some people wonder to themselves, why did mathematics and science continue despite the findings of Gödel's Incompleteness Theorems.
I have a logical and concise explanation for that, which, for some reason, goes unnoticed.
Suppose we have a theory that is incomplete with axioms that are consistent. Some of these axioms will remain unprovable as long as the theory is incomplete. The theory remains incomplete because there will always remain a possible world where some things, mathematical truths, physical laws, could have logically happened otherwise (invoking infinity here). Inconsistencies will remain in the theory, and there's no hope of ascertaining how much longer the journey will continue.
Now, the important part. As long as the theory is consistent, then, one can always add new axioms to the theory to expand its power and magnitude. Therefore, mathematics and science continue to this day...
Is that a coherent explanation for why we still go on or should we stop at some point and realize some deeper truth?
I have a logical and concise explanation for that, which, for some reason, goes unnoticed.
Suppose we have a theory that is incomplete with axioms that are consistent. Some of these axioms will remain unprovable as long as the theory is incomplete. The theory remains incomplete because there will always remain a possible world where some things, mathematical truths, physical laws, could have logically happened otherwise (invoking infinity here). Inconsistencies will remain in the theory, and there's no hope of ascertaining how much longer the journey will continue.
Now, the important part. As long as the theory is consistent, then, one can always add new axioms to the theory to expand its power and magnitude. Therefore, mathematics and science continue to this day...
Is that a coherent explanation for why we still go on or should we stop at some point and realize some deeper truth?
Comments (61)
So, it seems you don't understand Gödel either. That's alright.
This is the absurd thing with people.
Anything that shows a possible limit or is seen as a limit is thought to be wrong, fatal, an end. As an obstacle.
As if a mathematical/logical theorem would doom science and scientific research.
The same way we have approached paradoxes or antinomies, people have desperately looked for a way to brush them under the carpet and declare them solved.
This is truly the problem.
It's not an "as if" here. One can always expand Gödel's alphabet to account for more than previously hoped for.
And this process, could, in theory, go on forever.
Honestly, there is something really incredible in the negative self reference, which you find in all incompleteness results. Gödel's incompleteness theorems, Turing's answer to the Entscheidungsproblem, at the most simple version in Cantor's diagonal argument. We simply cannot make the connection to the larger picture, but there surely is one.
We simply don't understand what it actually means. I think this is the most important thing to be discovered (if you can use that word) in mathematics and logic.
In my view, negative self reference shows what limitations subjectivity gives.
The normal response is quietism. I mean with the above logical preponderance, then what's the point of continuing research? Does it all boil down to psychologism?
You know, it's been a burning thought of mine as to why Wittgenstein called Gödel's Incompleteness Theorems as "logical tricks", and I believe the above is the answer why.
Supposedly, Gödel hated Wittgenstein when seeing him at the Vienna Circle.
Please elaborate.
Ehh, that's an artifact of assuming that monads (logical atomism?) are logically the simplest possible things. But, that's off-topic, I think?
Ehh, like what? It seems to me that the vast majority of the world decided to ignore it despite it cropping up in Tarski's, Turing's, and other work (diagonal lemma).
Quoting Gregory
Meinong's jungle.
Plato's beard.
Because first, Gödelian incompleteness does not apply to physical theories. It applies (loosely speaking) to axiomatic systems of a particular logical structure, that support mathematical induction. No physical theory posits the existence of an infinite set of natural numbers. Incompleteness simply doesn't apply.
Secondly, incompleteness is not a statement about mathematical truth. It's a statement about axiomatic theories. Gödel himself was a Platonist. He was pointing out the limitations of axiomatic theories in discovering mathematical truth. But he did believe that "the truth is out there," as they used to say on the X-Files.
You almost caught me off guard there. I think, there's more to it than meets the eye... Ehem, Platonism? And if not, then why not?
Quoting fishfry
Yeah, that follows.
Quoting fishfry
Uhh, don't you mean non-denumerable? Cantor's program could have been completed, he just assumed that the program would account for everything, where Godel just kinda dashed those hopes. Just saying.
Quoting fishfry
Which is?
Quoting fishfry
Like I said, there's nothing wrong with axioms if they don't assume everything to be a certain case. I'm pretty rough with set theory; but, it just morphed into something else. Or is it really true that his Incompleteness Theorems only apply to set theory?
Cantor's work is not a physical theory.
Quoting Wallows
That no formal axiomatic theory (that satisfies some key technical assumptions) can express all mathematical truth. Which is the answer to your original question. That's why we keep looking for mathematical truth. Because no formalism can capture it all. So we're never done just because we have a formalism.
But, think of it this way. If there exists, a non-denumerably infinite alphabet, then we can enjoy everything there is to say about Cantor's work and program, ya?
In a physical theory?
Quoting Wallows
I enjoy Cantor's beautiful work and everything there is to say about it even though it doesn't exist!
Math [math]\neq[/math] Physics.
I don't know where the certainty in that negation is stemming from. Care to clarify?
The discovery of non-Euclidean geometry in the 1840's by Bernhard Riemann and others.
The discovery of individually consistent but mutually inconsistent geometries was the moment we understood that Math [math]\neq[/math] Physics. Math alone can never tell us what's true.
Russell put it well.
https://www.goodreads.com/quotes/577891-pure-mathematics-consists-entirely-of-assertions-to-the-effect-that
When we don't have the answers, it can be so.
Quoting Wallows
Sometimes brilliant minds don't get the point of the other. I don't remember where I read it, but I remember Wittgenstein accusing Gödel of simply finding again the paradox. Yet Russell's paradox is different and Gödel doesn't fall into it. Perhaps someone could view them as "logical tricks". And as one teacher in the university said to me over twenty years ago "from time to time someone attacks Gödel on the basis that it has 'circular reasoning'.
Quoting Wallows
We make models of reality, for example mathematical formulas that portray some aspect of the complex reality around us. Fine, but the problem of subjectivity comes with when that model itself has an impact on what it's modelling. Then it has to model itself into the model. Now you might argue that this can be still modeled and in many cases it surely can be, but not when the 'correct' answer is something that the model doesn't give.
This isn't a small difficulty. In the social sciences it has been understood for a long time that when studying economics or sociology, the theories themselves have an effect how we picture the World and thus effects our actions, hence theories are also subjective, not only objective. An needless to say how things change in physics when the measurement has an effect on what is measured.
Quoting Gregory
Negative self reference is different from ordinary self reference.
Example:
Try writing a response that you never will write.
Are there such responses? Yes. As we have a finite life we surely cannot write all responses, hence there exists those responses. I or Wallows of Fishfry can write responses that you don't write. But for you to write something that you don't write is impossible. Hope you notice the negative self reference.
Do you have evidence that the universe is eternal? My understanding is that this is an open question in physics, the solution to which would make someone famous. Did Aristotle hold that time is eternal? How would he have known one way or the other?
The referent of the axiom of infinity is the abstract idea of the endless sequence of counting numbers 1, 2, 3, 4, ... but NOT any claim that such a thing has physical existence.
I knew what you meant lol
What's the different between logical atomism and Hilbert's program? I thought they were the same. It seems to me Godel threw the same monkey wrench into the latter the Russell through into the former, although Godel's theorem threw Russell while his own did not. I think it was just more techniquely drawn out, which is why Wittgenstein said it was the same
Better to have stayed w/Cantor. Hilbert's program got demolished by Gödel.
Even though I formally agree with your views on the matter, I still want to point out what Stephen Hawking wrote on the subject:
Quoting Stephen Hawking, Gödel and the end of physics
Hawking may indeed be overstepping boundaries there. What he says, is also not genuine syntactic entailment, but rather some kind of wholesale "intuition". Furthermore, Hawking is also not respectful of the plethora of fine print surrounding Gödel's incompleteness theorems.
But then again, I was just pointing out that more or less "serious" opposite views also exist!
Quoting fishfry
Yes, that is indeed some of the fine print. Gödel's incompleteness is a statement that is even only about Peano-Arithmetic-like (PA) first-order axiomatic theories, and not even about all first-order theories.
There are weaker first-order theories that are complete (and consistent):
Quoting Wikipedia on the completeness of arithmetic
Within the confines created by a lot of fine print, stronger, second-order theories can also be complete (and consistent):
Quoting Wikipedia on the possibility of completeness in the context of second-order logic
So, yes, agreed, it is quite easy to overstep the boundaries of Gödel's incompleteness theorems by incorrectly applying them where they do not apply. One really has to read the fine print!
Well, no, disagreed. Hilbert merely received a negative answer to one of his many questions ...
Oh boy, here we go again. Might want to reconsider; it's theorems not axioms that could possibly be true, but not provable. :roll:
Adding axioms does not necessarily increase the power of a theory, but it pretty much always increases the amount of trust that the theory requires.
For example, number theory can "see" all the Gödelian numbers representing the theorems and their proofs in set theory. So, number theory "knows" all theorems in set theory, but there are quite a few of these theorems that it does not trust. So, set theory is not necessarily more powerful than number theory. It could just be more gullible!
If you make a model of reality, then you are engaged in an empirical discipline (such as science). Such model cannot possibly be an exercise in mathematics any more, because the model-theoretic model for a theory in mathematics is NEVER the real world. Such mathematical real-world theory would be the elusive theory of everything (ToE) to which we do not have access, and do not even expect to ever have access.
The problem is truly about crossing boundaries between mathematics and downstream user disciplines of mathematics (such as science), which merely use mathematics to maintain consistency in their use of language. These downstream disciplines talk about something else. They do not talk about mathematics. They merely use mathematics.
Furthermore, such empirical discipline always requires its own regulatory framework that duly constrains what exactly it is willing to talk about. They do not use just mathematics either. There is always also a completely native bureaucracy of rules.
Well, I tend to think that axioms can be theoretical in nature? Have you ever encountered such a sentiment in your line of work?
Hence inter-subjectivity? The observer effect seems to play a role here.
But, that aside, I can see the point of utilizing some ideas from Godel to justify the need for us, as a species, to slow down, as there doesn't seem to be a light at the end of the tunnel. You can thank Godel for that.
Or social sciences, like economics.
Quoting alcontali
Uh, the math used in the models have to be correct. Yes, math used as a tool (as you point yourself also), but just as all tools, you have to use it correctly.
Quoting alcontali
Say that to an economist.
No really, math is a very useful tool. It really is. We have to be logical in every field of study. Math has a huge advantage in being logical. Something like statistics, which one can argue isn't pure math, is truly an inherent part of many fields of study. Think about how much we use a thing called a mean average.
I think that something as profound like the incompleteness results of Gödel (and others) has also huge implications to math as a tool and our use of mathematical models in picturing reality. Put it another way, if you have problems to mathematically model some phenomenon, some event, and cannot make a mathematical function y=f(x) out of it, perhaps the problem lies is that the correct model would be something that falls into the category of the incompleteness results, in a way is a Gödel number the Gödels incompleteness theorem (is it the first theorem?) talks about.
This might be confusing to understand, but what I mean that there would be something that we have extreme difficulties in modelling with a mathematical formula. Then what would we do?
We do have an unmathematical method. And what is that?
We use narrative: "First happened this, which then lead to that". How we got from this to that and onward cannot be explained by a variable and made into a function (used in it's broadest definition possible). We tell a story. Best example of this is the field of History. There hasn't been much use of math, except statistics, in history and the field of cliometrics hasn't given much to the field. And many don't even consider history as a science. Yet history is one of the most important things for social sciences and our understanding of ourselves and our societies is firmly based on history.
So I personally believe that the incompleteness results have far more to give to our understanding of the World than now is thought.
Yes, the observer plays the dominant role.
The observer cannot simply observe without interaction in this case. Hence the inescapable subjectivity. When the correct answer is what the observer doesn't give, how could the observer give then the correct answer? In the end we are a part of the universe, hence the idea of a Laplace's Demon is utterly wrong: not everything can be extrapolated from all knowledge and understanding of the present, if the observer is part of the universe. And the remarkable thing here is that this isn't in conflict with determinism. There is a correct extrapolation of the future (that Laplace's demon ought to know). The basic line, which is not understood now, is that this is because of mathematics, the incompleteness results.
In 2008 a guy at NASA (now at Santa Fe Institute) called David H Wolpert showed that the "problem" (it isn't a problem, you know) was incompleteness results, or in the most simple way, Cantor's diagonalization. Wolpert had written about it even far earlier. Once the observer is part of the universe, it cannot avoid this problem. The problem is that this is not understood or it simply hasn't broken ground in the wider scientific and mathematical community.
For example, economist and game theorist like Oscar Morgenstern understood the problem in economic forecasting and gave his example of Holmes and Moriarty (in 1928), where he showed that there's no possible way for Holmes to reason what Moriarty will do. A counterargument given by another future Nobel-laureate to Morgenstern noted, quite correctly, that there had to be a correct forecast, but then didn't grasp that here was the real issue, that there is a correct answer, but it's uncomputable.
Another ignorant answer is to avoid the problem by assuming premises that give you a min-max game theoretic answer.
In all, this is really a field which should be studied and understood. Perhaps even this meager forum can do something about it.
Quoting Wallows
Or perhaps to speed up, improve our understanding of reality.
You see, talking about this subject gets people simply to the defensive. If you go and say: "Hey, these incompleteness result have a HUGE ROLE in things!" it comes of as anti-science or something. And the first idea is that this person is showing a problem. As if to attack against science. And science has to be defended. Hence the emphasis goes to avoid it, go past. Try to show that the "problems" can be overcome. And then what is left out is that here is an extremely profound mathematical insight that we ought to understand.
It would mean that all sentences that are provable in the theory can also be confirmed to be true in its standard model-universe-world (=semantic completeness) but that there are also facts in that standard model-universe-world that are true but that are not provable from the theory (=syntactic incompleteness).
So, the situation with natural-number theory (PA) is:
The main problem will be that an empirical theory will not even be semantically complete. Concerning physics, Hawking said the following in that regard:
Quoting Hawking in Gödel and the end of physics
In my opinion, just like Hawking said, even semantic completeness is already an unattainable ideal for an empirical discipline.
By the way, the proof for Carnap's diagonal lemma does indeed use a self-referencing expression:
Quoting Wikipedia, start of the proof for Carnap's diagonal lemma
However, Carnap's diagonal lemma is itself not self-referencing, and Gödel's first incompleteness theorem is neither. The entire self-referencing thing is just a hack to get the proof going. I don't understand Hawking's obsession with the "self-referencing" thing. Alan Turing uses a similar hack to get the proof for the impossibility to solve his Halting Problem going. It is not that the Halting Problem itself would be self-referencing.
It's more like: You have a function that computes something about any function. Ok, let it now compute this about itself. Look, it explodes when it does that. So, the overly generalized claim that says x,y,z ... is wrong about what it computes.
That strategy is much more of a hack than anything else ...
As I earlier said, many don't see the subtle difference between Russell's paradox and Gödels (or Turings) finding. Wittgenstein accused Gödel of finding the paradox again and didn't hit it so well with Turing either. It's basically indirect reference.
Just as the real number that Cantor shows cannot be in any list. Of course, we do see the relation between the real number and the list of real numbers. Same thing. Not a paradox.
Hence the issue is about the limitations of computing, modelling, giving proofs. Not that there would be a correct model or proof. And the observation that non-computable mathematical objects can be very important to us. Even if we cannot compute them, they still are mathematical.
So math rules, I guess.
Yes, agreed. Russell's sentence actually is self-referencing.
Quoting ssu
Yes, agreed. As far as I am concerned, there is indeed nothing paradoxical about Cantor's theorem (countable versus uncountable infinite cardinalities).
His proof strategy, diagonalization, is certainly an interesting approach. I usually like it. For example, the diagonalization of the term "heterological" is really intriguing. The argument in the video below (just 3 minutes) is incredibly simple, pure math, and leads to a rather unexpected (surprising) result:
Diagonalization does not have to be hard (but, of course, sometimes it is ...)
EXACTLY!!!
You see, going through Gödel's Incompleteness Theorems are hard. Turings proof with a Turing Machine isn't also the most easiest thing. Even Gödel had to really think about it before he agreed that Turing's findings were equivalent with his theorems. But luckily in math things can be simplified.
But you are totally right, one should talk about in general about diagonalization. Referring to Gödel's incompleteness Theorem is confusing, because the theorems aren't easy to understand.
And of course, from Cantor's diagonal argument we next get to the Continuum Hypothesis, which is, well, a kind of Holy Grail in Math.
To be more specific, it's certain mathematical statements that could be undecidable, including theorems. There are a few examples in combinatorics and number theory, but I'm not aware of such things occurring in complex analysis, for example. If you come across something let me know. As for axioms themselves, I don't think Godel applies. Again, let me know if you come across a contrary opinion.
In the actual work of mathematical research outside of set theory it seems very rare that one might encounter incompleteness. But I am familiar only with my own interests. :cool:
There is no repeating pattern, yet we can instantly notice it in it's decimal form 3,14159 26535 89793.... And you have even ways to calculate it.
If it would be repeating, then it would be a rational number and then you could square the circle (if I'm correct). Transcendental numbers are interesting. In the real numbers they are the numbers that with you get an infinity that cannot be mapped 1-to-1 to the natural numbers. Algebraic numbers you still can.
Yes, that seems to me to be a fair characterisation of why Godel's incompleteness theorem is no barrier to progress in mathematics in most areas of interest and use. The only maths I know of it having killed was Hilbert's project to try to prove maths to be complete and consistent.
There's a book devoted to exactly that. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzen. I have a copy, I should leaf through it.
Quoting alcontali
Adding any axiom that is independent of the other ones always increases the power of a theory. By definition that means it can prove more theorems. It can prove all the original theorems plus itself, as proof.
Quoting alcontali
Are you saying that number theory "knows" what theorems are true; but it does not necessarily have proofs for all of them? If so then I agree.
There's a system that contains exactly all the true theorems of number theory. It's called true arithmetic. Tarski's theorem says that this class is not arithmetically definable, which is why it can sneak under Gödel. But conceptually the class exists and is a consistent and complete account of arithmetic. As I understand it, anyway ... I don't know any more about it than the Wiki page.
Quoting alcontali
@andrewk says killed and that's good enough for me! :-)
Quoting andrewk
Thanks! I always confuse Hilbert's program with Hilbert's 23 questions anyway.
Quoting alcontali
I am constantly struck by the fact that physicists in general but particularly many celebrity physicists with TED talks and Youtube channels, don't know the first effing thing about math. Hawking, rest his soul and give him all the credit for all his amazing accomplishments, doesn't know the first thing about this subject.
First, he says: "whether we can formulate the theory of the universe in terms of a finite number of principles?" So he does not know that ZFC has infinitely many axioms, and that axiomatic systems with only finitely many axioms are barely of interest as far as I know. He's mixing up physical theories and mathematical logic into a mishmash to suit his confused purposes.
Secondly, when is says, that "... a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted," he is leaping from:
* A physical theory is a mathematical model;
* Therefore since SOME mathematical models are subject to incompleteness -- [and surely not physical models!!] -- therefore a physical theory must be subject to incompleteness.
This is a leap of bad logic and bad thinking. Most likely any physical axiomatic theory would not be subject to incompleteness. You would not accept this fallacious syllogism from an undergrad in logic 101.
If Hawking thinks there's a physical theory subject to incompleteness, he must think that this theory can represent the infinite set of natural numbers. Hawking is claiming that an actual infinity exists in the physical world. Else incompleteness doesn't apply.
To say this another way: In any finite domain, to determine whether a given proposition is true we just go enumerate every possibility and look. All finite systems are complete. We have a proof by examining cases for any question we could ask.
The only way to have incompleteness is if your domain is infinite. And if Hawking is making that claim, he ought to give evidence and consider the implications; not just write a bad syllogism to sell books.
If there is an actual infinity in the world: Why aren't physics postdocs applying for grants to study the physical truth or falsity of the Continuum hypothesis? Until that happens I know that physicists are talking through their hats about infinity, as Hawking is doing here.
What strikes me is that an amateur forum poster like me who studied math back in the day, knows far more about this subject than the great Stephen Hawking. And that this happens to me all the time when I watch celebrity physicist videos. I've seen many physicists make egregious math errors especially as regards set theoretic matters and the nature of the real numbers. Physicists I respect. Sean Carroll did it the other day.
So, uh, sorry for the rant there but that's what popped into my head when I read that quote.
Quoting Gregory
So then you agree that the physical world does NOT instantiate an actual infinity. Didn't you imply otherwise earlier, or did I misunderstand?
2) contingent truths (univetse)
A) infinite truths
B) finite truths.
1 is infinite because it includes mathematics. Leibniz thought the world infinite, so contingent truths are infinite in his system. Fish fry is right that you can't ad hoc apply Godel to the world though