Nice article. It is nice to have axioms, and nice when they lead to a proof. The vast amount of human knowledge is not obtained this way. Even most theorem results start off as a pure guess. The proof proceeds as the very last thing, normally well after the mathematician was convinced the formal statement is provable, or meta-mathematically "true".
It is the same with both theoretical physics and engineering, which are the fields I work in. Probably also science and technology in general. Every good idea starts of as some wild hairbrained imagining.
Nice post, I agree that Gödel's theorem isn't formally about epistemology, however you can build up some claims about epistemology that are analogous to Gödel's theorem.
Let's first go through an informal sketch of Gödel's proof so that the analogy will be clear: he shows that there is one particular sentence, which is true in a certain axiomatic system, but not provable. This sentence states 'This statement is not provable'. Now think about it: the statement is either true or false. If this statement is false, then it is provable, but if it is provable it should be true (assuming that the system is consistent), which is a contradiction. If this statement is true, then it is not provable, but that doesn't contradict the fact that it is not provable. This implies the statement is true.
As an analogy you might picture an oracle that knows everything about the future, you could go to the oracle and ask: 'am I going to lift my left or my right hand in a minute?' Whatever the oracle answers whether 'left' or 'right' you can just do the opposite! Thereby anyone can disprove the oracle. So the oracle cannot know, which hands your are going to lift and neither can any one else you ask this question to.
The similarity is that in both cases the existence of the statement changes something about it's truth value / provability. You can imagine the statement: 'this statement is not provable' changing up it's truth value at will depending on whether we think it is provable, just as you can change the truth value of the statement 'you are going to lift your right hand' by raising your left.
In both examples we have a case of: giving justification to a claim undermines the very justification for that claim.
The problem with Gödel is not his conclusions, which don’t seem very useful IMHO, but his arguments, which depend on Cantor’s set-theoretical methods, e.g. self-membership, sets of sets, self-reference, etc. Any system that generates self-contradictions/inconsistencies is fundamentally false, even though it may demonstrate “Unreasonable Effectiveness”.
Yes, "axiomatization ≠ knowledge," and those who try to draw conclusions about all knowledge from the limitations of formal systems err as you say. But I do not believe you deal with the most important modern use of Gödel, which is Penrose's assertion, not that our knowledge is a formal system limited by Gödel's proof, but that any attempt to make mind / consciousness ("knowing?") reducible to computation is doomed. Tim Maudlin says he finds that argument flawed but I have not heard him state why and it is not obvious to me at all. Programmed algorithms are formal systems. Mind and all "knowledge" are not. No? I do not personally feel that the arguments you refute play a very strong role in modern thought. The assertion that mind is essentially computational is ubiquitous and very aggressive. Please comment on the Penrose Maudlin dispute. And thank you.
[edit added: The comment of Cathy Reason listed by RW Boyer below is very relevant. As she points out, your beef is not really with your Orthodox friend (in your last email) but with "a fundamental postulate of cognitive science that all mental processes can be represented in terms of recursive functions, and anything that can be represented using recursive functions can be represented as an axiomatizable formal system." I think that fundamental postulate is dead wrong and, if I understand him, so does Penrose. If this is what you are really about, too, I am with you 100% but let's get that out in the open. The fuzzy thinking of a few romantics and theists is not the real target. You are challenging much more powerful cultural assumptions.]
Thanks for the, I think generally valid, points on 'misinterpretations' of Godel’s Incompleteness Theorem.
Cathy Reason, a psychologist whom I respect for her knowledge of logic and mathematics, made this comment about your points that I thought you might want to ‘see:’
“He's technically right, but he misses the point. It's a fundamental postulate of cognitive science that all mental processes can be represented in terms of recursive functions, and anything that can be represented using recursive functions can be represented as an axiomatizable formal system. So while it's perfectly ok to postulate that human reasoning cannot be modeled with an axiomatic formal system, that is not a trivial postulate. It would be fundamentally incompatible with cognitive science as it is currently understood.”
Cathy also commented:
“Godel's theorem does not only apply to Peano arithmetic, but also to any formal system which has at least the power of Peano arithmetic. That makes it much more general.”
Cathy Reason and Kushal Shah published a ‘proof’ of the incompleteness of any physical model with respect to addressing consciousness -- in Journal of Consciousness Studies – their “No Supervenience Theorem”, which I also think you would want to ‘see’:
Reason CM, Shah K. Conscious Macrostates Do Not Supervene on Physical Microstates. Journal of Consciousness Studies, 28, No. 5-6, 2021,102-20.
Reason CM. (2018). A Theoretical Solution of the Mind-Body Problem: An Operationalized Proof that no Purely Physical System Can Exhibit all the Properties of Human Consciousness. Preprint.
In case you might be interested, my paper “A Psychological Critique of Mathematicians” on ResearchGate.net discusses these theorems further in the context of the means of gaining human knowledge beyond the ordinary waking state of consciousness in which modern science has been conducted, and which is starting to become recognized as inadequate to address TOE.
Interesting papers by Catherine Reason. I read through them, but I don’t know that I’m convinced. It seems that the fundamental assumption is that humans can know with absolute non-contextual certainty that they are conscious. It then proves that machines cannot have this kind of certainty and deduce that the human mind cannot be modeled by a computer.
However, it’s impossible to be certain of anything without any assumption. I don’t see how humans can have that kind of absolute certainty without using the same kind of circular logic.
Great post! There is so much confusion about Gödel's theorems, which by definition apply to formal systems and by Turing's extension to machines/digital computers. But somehow people insist on misapplying these to humans while ignoring them where they apply exactly i.e. digital computers and artificial intelligence. In fact, Gödel himself drew very similar conclusions to yours: "My incompleteness theorem makes it likely that mind is not mechanical, or else mind cannot understand its own mechanism. If my result is taken together with the rationalistic attitude which Hilbert had and which was not refuted by my results, then [we can infer] the sharp result that mind is not mechanical. This is so, because, if the mind were a machine, there would, contrary to this rationalistic attitude, exist number-theoretic questions undecidable for the human mind."
Curt, thanks for the clarification: it is important our philosophers have a clear picture as to the true sources and limitations of human knowledge. As such, and to echo your analysis, it is important we are clear on definitions. And in that context, i.e., the context of human cognition, it is important to distinguish between our two forms of cognition: knowledge, which is most closely associated with discrete cognition and science; and the more complex form of cognition we call understanding that includes implicit elements of cognition that are commonly associated with philosophy. Epistemology, the awkward technical term that describes the philosophical study of how we come to understand the world, involves both basis of cognition. Most importantly, both forms of cognition involve concept-formation, the essential underlying cognitive process that defines humans.
In attempting to understand human cognition many philosophers (and scientists) tend to avoid the subject of concept-formation. This is unfortunate because it is this process that results in math, which is akin to what you refer to as “axiomatization,” our highest form of cognition that is scalable and explicit, making it our most reliable form of cognition (though admittedly not everything is discrete and explicit, and thus math has its limits, including capacity limits, as you note). The other key output of concept-formation is language, which is notable for its significant implicit considerations making it simultaneously subtle and flexible, but also subject to manipulation and deception (which is why definitions are so important). Earlier this year I completed a book for teenagers that describes how our conceptual processes work, including how they relate to Iain McGilchrist’s hemisphere (divided brain) hypothesis as well as Nick Lane’s bioelectric hypothesis. You and your readers can access this transcript for free at www.find-your-map.com. Respectfully, Brad.
Fun fact, Godel was a mathematical realist - he believed mathematics was what reality was made of on a fundamental level. It always gets me when people say mathematics can't be fundamental because Gödel proved it...
The boundaries of Gödel's Incompleteness theorems (particularly of the second one) are also quite interesting. If your axiomatization is not r.e. then you can reach a failure.
If your logic itself is too weak, it can fail (eg it does so in some substructural logics).
Finally there is the question of how weak is your theory allowed to be. Where is the boundary of the Incompleteness theorems, and are there properties invariant under some theory transformations that preserve the second Incompleteness Theorem between the theories?
PS: On things like the Paris Harrington theorem (and ordinal analysis), they essentially avoid the Incompleteness Theorem either by jumping into meta theory (and state for example: "if Con(T) is consisten with my theory, I can just add it" and repeat that) or by loosening yet another unspoken assumption that you have to use finite means. If instead you can use finitary means (like finitely presented ordinal notations), you can prove Con(T).
PSS: I have not delved into ultrafinitism to know what the situation could look like there. It is still a somewhat in development formally as it's tricky to have a finite set of numbers and induction work together.
Alright, now that I proofed you wrong, you got me busy with some new fundamentals. Thank you!
I don't know. Should I show you my proof? It would devastate your career, as I'm showing that everything you do is wrong. Does that do anything reasonable?
You should overlook your point of view. You should also consider if you are within Gödel's framework, or if you happened to leave it. You should also consider in how much such a choice could affect the outcome to which you came.
It's so easy to proof you wrong... You just shouldn't play with things you don't understand.
Here's your task: I'll give you a couple of initial numbers, and you predict the remainder of the chain. I'm pretty sure you can do it, but at least, you're busy with something that prevents you to write stupid articles that people even refer to. Oh, the numbers are: 111, and the chain has an infinite length. It's already growing in size as I write this, I've got it right here
Firstly, axiomatization is just fancy technical jargon for "making your underlying assumptions explicit." ;-)
The epistemic issue that seems to be poorly understood on many fronts is that knowledge, ie, whether something can be said to be true or false, can only be determined within the limits of a model, regardless of whether that model has been formalized or is informal. Newton's model of gravity renders stupidly precise "true" predictions within it's limits, but at more massive/faster scales it breaks down, and the Relativistic model picks up the slack. Despite what George Box would say, the category error is in saying that a model is itself true or false: a model can only be useful or not useful within a certain context, but it doesn't make sense to call it true or false, because the model is itself a representation of something else.
With all that said, I think most philosophers (myself included) don't have the maths to understand Gödel’s incompleteness theorem thoroughly, but the metaphor of incompleteness resonates with our intuitions about epistemological limitations, so for those who use the term non-metaphorically, yeah, category error.
I have a short question/polemic (but it's more of a question because I'm not an expert on this): If one is a Foundationalist, doesn't that imply that one axiomatizes everything? i.e. A Foundationalist cannot articulate any argument or statement without axioms (unless I've got it all wrong). Personally, I don't see how anyone can do it, not just Foundationalists. I ask because if we give credence to Foundationalism, then wouldn't your argument suggest that Godel's theorem in fact does generalize to any result of Foundationalism, and wouldn't that render it self-contradicting? What am I missing?
You are correct. Even physics relies on axioms to build foundational theories that describe the real world. However, those axioms are bound to find undecidable statements, because "truth" is not absolute but contextual. And yes, humans are able to "break" self referential loops through meta-thinking, but then, any additonal axioms added to solve the initially undecidable statement, will lead to contradictions at the meta level, unless you repeat the process, ad infinitum. Thus, self-reference is unescapable.
Thank you for your usual excellent exposition of what should have been easy to understand from the beginning. You are an excellent teacher.
*your high school math teacher after all these years*
I knew Curt reported me!
You deserved it!
Nice article. It is nice to have axioms, and nice when they lead to a proof. The vast amount of human knowledge is not obtained this way. Even most theorem results start off as a pure guess. The proof proceeds as the very last thing, normally well after the mathematician was convinced the formal statement is provable, or meta-mathematically "true".
It is the same with both theoretical physics and engineering, which are the fields I work in. Probably also science and technology in general. Every good idea starts of as some wild hairbrained imagining.
Curt,
Cathy Reason just clarified the 2018 paper I referenced is outdated -- so no need to consider it (just the 2021 JCS reference).
Thanks much for the excellent interviews/contributions on your site,
Bob (RW Boyer)
PS -- Years ago, due to incompleteness, I liked to refer to scientific theories of everything as
the "Big TOE."
Nice post, I agree that Gödel's theorem isn't formally about epistemology, however you can build up some claims about epistemology that are analogous to Gödel's theorem.
Let's first go through an informal sketch of Gödel's proof so that the analogy will be clear: he shows that there is one particular sentence, which is true in a certain axiomatic system, but not provable. This sentence states 'This statement is not provable'. Now think about it: the statement is either true or false. If this statement is false, then it is provable, but if it is provable it should be true (assuming that the system is consistent), which is a contradiction. If this statement is true, then it is not provable, but that doesn't contradict the fact that it is not provable. This implies the statement is true.
As an analogy you might picture an oracle that knows everything about the future, you could go to the oracle and ask: 'am I going to lift my left or my right hand in a minute?' Whatever the oracle answers whether 'left' or 'right' you can just do the opposite! Thereby anyone can disprove the oracle. So the oracle cannot know, which hands your are going to lift and neither can any one else you ask this question to.
The similarity is that in both cases the existence of the statement changes something about it's truth value / provability. You can imagine the statement: 'this statement is not provable' changing up it's truth value at will depending on whether we think it is provable, just as you can change the truth value of the statement 'you are going to lift your right hand' by raising your left.
In both examples we have a case of: giving justification to a claim undermines the very justification for that claim.
The problem with Gödel is not his conclusions, which don’t seem very useful IMHO, but his arguments, which depend on Cantor’s set-theoretical methods, e.g. self-membership, sets of sets, self-reference, etc. Any system that generates self-contradictions/inconsistencies is fundamentally false, even though it may demonstrate “Unreasonable Effectiveness”.
Yes, "axiomatization ≠ knowledge," and those who try to draw conclusions about all knowledge from the limitations of formal systems err as you say. But I do not believe you deal with the most important modern use of Gödel, which is Penrose's assertion, not that our knowledge is a formal system limited by Gödel's proof, but that any attempt to make mind / consciousness ("knowing?") reducible to computation is doomed. Tim Maudlin says he finds that argument flawed but I have not heard him state why and it is not obvious to me at all. Programmed algorithms are formal systems. Mind and all "knowledge" are not. No? I do not personally feel that the arguments you refute play a very strong role in modern thought. The assertion that mind is essentially computational is ubiquitous and very aggressive. Please comment on the Penrose Maudlin dispute. And thank you.
[edit added: The comment of Cathy Reason listed by RW Boyer below is very relevant. As she points out, your beef is not really with your Orthodox friend (in your last email) but with "a fundamental postulate of cognitive science that all mental processes can be represented in terms of recursive functions, and anything that can be represented using recursive functions can be represented as an axiomatizable formal system." I think that fundamental postulate is dead wrong and, if I understand him, so does Penrose. If this is what you are really about, too, I am with you 100% but let's get that out in the open. The fuzzy thinking of a few romantics and theists is not the real target. You are challenging much more powerful cultural assumptions.]
Curt,
Thanks for the, I think generally valid, points on 'misinterpretations' of Godel’s Incompleteness Theorem.
Cathy Reason, a psychologist whom I respect for her knowledge of logic and mathematics, made this comment about your points that I thought you might want to ‘see:’
“He's technically right, but he misses the point. It's a fundamental postulate of cognitive science that all mental processes can be represented in terms of recursive functions, and anything that can be represented using recursive functions can be represented as an axiomatizable formal system. So while it's perfectly ok to postulate that human reasoning cannot be modeled with an axiomatic formal system, that is not a trivial postulate. It would be fundamentally incompatible with cognitive science as it is currently understood.”
Cathy also commented:
“Godel's theorem does not only apply to Peano arithmetic, but also to any formal system which has at least the power of Peano arithmetic. That makes it much more general.”
Cathy Reason and Kushal Shah published a ‘proof’ of the incompleteness of any physical model with respect to addressing consciousness -- in Journal of Consciousness Studies – their “No Supervenience Theorem”, which I also think you would want to ‘see’:
Reason CM, Shah K. Conscious Macrostates Do Not Supervene on Physical Microstates. Journal of Consciousness Studies, 28, No. 5-6, 2021,102-20.
Reason CM. (2018). A Theoretical Solution of the Mind-Body Problem: An Operationalized Proof that no Purely Physical System Can Exhibit all the Properties of Human Consciousness. Preprint.
http://arxxiv.org/abs/1706.04192.
In case you might be interested, my paper “A Psychological Critique of Mathematicians” on ResearchGate.net discusses these theorems further in the context of the means of gaining human knowledge beyond the ordinary waking state of consciousness in which modern science has been conducted, and which is starting to become recognized as inadequate to address TOE.
Thanks for considering it,
Bob (RW Boyer)
Interesting papers by Catherine Reason. I read through them, but I don’t know that I’m convinced. It seems that the fundamental assumption is that humans can know with absolute non-contextual certainty that they are conscious. It then proves that machines cannot have this kind of certainty and deduce that the human mind cannot be modeled by a computer.
However, it’s impossible to be certain of anything without any assumption. I don’t see how humans can have that kind of absolute certainty without using the same kind of circular logic.
Great post! There is so much confusion about Gödel's theorems, which by definition apply to formal systems and by Turing's extension to machines/digital computers. But somehow people insist on misapplying these to humans while ignoring them where they apply exactly i.e. digital computers and artificial intelligence. In fact, Gödel himself drew very similar conclusions to yours: "My incompleteness theorem makes it likely that mind is not mechanical, or else mind cannot understand its own mechanism. If my result is taken together with the rationalistic attitude which Hilbert had and which was not refuted by my results, then [we can infer] the sharp result that mind is not mechanical. This is so, because, if the mind were a machine, there would, contrary to this rationalistic attitude, exist number-theoretic questions undecidable for the human mind."
Curt, thanks for the clarification: it is important our philosophers have a clear picture as to the true sources and limitations of human knowledge. As such, and to echo your analysis, it is important we are clear on definitions. And in that context, i.e., the context of human cognition, it is important to distinguish between our two forms of cognition: knowledge, which is most closely associated with discrete cognition and science; and the more complex form of cognition we call understanding that includes implicit elements of cognition that are commonly associated with philosophy. Epistemology, the awkward technical term that describes the philosophical study of how we come to understand the world, involves both basis of cognition. Most importantly, both forms of cognition involve concept-formation, the essential underlying cognitive process that defines humans.
In attempting to understand human cognition many philosophers (and scientists) tend to avoid the subject of concept-formation. This is unfortunate because it is this process that results in math, which is akin to what you refer to as “axiomatization,” our highest form of cognition that is scalable and explicit, making it our most reliable form of cognition (though admittedly not everything is discrete and explicit, and thus math has its limits, including capacity limits, as you note). The other key output of concept-formation is language, which is notable for its significant implicit considerations making it simultaneously subtle and flexible, but also subject to manipulation and deception (which is why definitions are so important). Earlier this year I completed a book for teenagers that describes how our conceptual processes work, including how they relate to Iain McGilchrist’s hemisphere (divided brain) hypothesis as well as Nick Lane’s bioelectric hypothesis. You and your readers can access this transcript for free at www.find-your-map.com. Respectfully, Brad.
I think its philosophical value is as an analog to the recursive shape of epistemological limitation. Not as a proof, but as a metaphor.
Fun fact, Godel was a mathematical realist - he believed mathematics was what reality was made of on a fundamental level. It always gets me when people say mathematics can't be fundamental because Gödel proved it...
The boundaries of Gödel's Incompleteness theorems (particularly of the second one) are also quite interesting. If your axiomatization is not r.e. then you can reach a failure.
If your logic itself is too weak, it can fail (eg it does so in some substructural logics).
Finally there is the question of how weak is your theory allowed to be. Where is the boundary of the Incompleteness theorems, and are there properties invariant under some theory transformations that preserve the second Incompleteness Theorem between the theories?
PS: On things like the Paris Harrington theorem (and ordinal analysis), they essentially avoid the Incompleteness Theorem either by jumping into meta theory (and state for example: "if Con(T) is consisten with my theory, I can just add it" and repeat that) or by loosening yet another unspoken assumption that you have to use finite means. If instead you can use finitary means (like finitely presented ordinal notations), you can prove Con(T).
PSS: I have not delved into ultrafinitism to know what the situation could look like there. It is still a somewhat in development formally as it's tricky to have a finite set of numbers and induction work together.
Alright, now that I proofed you wrong, you got me busy with some new fundamentals. Thank you!
I don't know. Should I show you my proof? It would devastate your career, as I'm showing that everything you do is wrong. Does that do anything reasonable?
You should overlook your point of view. You should also consider if you are within Gödel's framework, or if you happened to leave it. You should also consider in how much such a choice could affect the outcome to which you came.
Hey Mr, I understand everything.
It's so easy to proof you wrong... You just shouldn't play with things you don't understand.
Here's your task: I'll give you a couple of initial numbers, and you predict the remainder of the chain. I'm pretty sure you can do it, but at least, you're busy with something that prevents you to write stupid articles that people even refer to. Oh, the numbers are: 111, and the chain has an infinite length. It's already growing in size as I write this, I've got it right here
Firstly, axiomatization is just fancy technical jargon for "making your underlying assumptions explicit." ;-)
The epistemic issue that seems to be poorly understood on many fronts is that knowledge, ie, whether something can be said to be true or false, can only be determined within the limits of a model, regardless of whether that model has been formalized or is informal. Newton's model of gravity renders stupidly precise "true" predictions within it's limits, but at more massive/faster scales it breaks down, and the Relativistic model picks up the slack. Despite what George Box would say, the category error is in saying that a model is itself true or false: a model can only be useful or not useful within a certain context, but it doesn't make sense to call it true or false, because the model is itself a representation of something else.
With all that said, I think most philosophers (myself included) don't have the maths to understand Gödel’s incompleteness theorem thoroughly, but the metaphor of incompleteness resonates with our intuitions about epistemological limitations, so for those who use the term non-metaphorically, yeah, category error.
I have a short question/polemic (but it's more of a question because I'm not an expert on this): If one is a Foundationalist, doesn't that imply that one axiomatizes everything? i.e. A Foundationalist cannot articulate any argument or statement without axioms (unless I've got it all wrong). Personally, I don't see how anyone can do it, not just Foundationalists. I ask because if we give credence to Foundationalism, then wouldn't your argument suggest that Godel's theorem in fact does generalize to any result of Foundationalism, and wouldn't that render it self-contradicting? What am I missing?
You are correct. Even physics relies on axioms to build foundational theories that describe the real world. However, those axioms are bound to find undecidable statements, because "truth" is not absolute but contextual. And yes, humans are able to "break" self referential loops through meta-thinking, but then, any additonal axioms added to solve the initially undecidable statement, will lead to contradictions at the meta level, unless you repeat the process, ad infinitum. Thus, self-reference is unescapable.
“However, those axioms are bound to find undecidable statements, because "truth" is not absolute but contextual.”
Can’t be said often enough