Maybe you could find this article interesting:
https://en.wikipedia.org/wiki/Logicism

3rd cousins on the distaff side!

"Kurt Gödel's incompleteness theorem undermines logicism because it shows that no particular axiomatization of mathematics can decide all statements" Can anyone explain to me why the theorems undermine logicism?

Someone else can explain Godel's Incompleteness, but it breaks down to you need physical evidence to prove classical logic and mathematics are meaningful to begin with. Assuming they aren't then statistical physical evidence should support a systems logic being capable of describing both as merely pragmatic conventions only applicable in specific contexts.

"Kurt Gödel's incompleteness theorem undermines logicism because it shows that no particular axiomatization of mathematics can decide all statements" Can anyone explain to me why the theorems undermine logicism? — MonfortS26

It undermines the original project by Bertrand Russell to collect a set of rules and axioms that would allow to build all mathematical reasoning in a formal way. Godel proved that even the set of arithmetical truths cannot be reduced to a finite amount of information.

you need physical evidence to prove classical logic — wuliheron

No, you absolutely do not. Logicists hold that all truths within any system of logic can be deduced from logical propositions within it. Godel proved that this is fallacious. Neither appeals to external evidence physical or otherwise.

No, you absolutely do not. Logicists hold that all truths within any system of logic can be deduced from logical propositions within it. Godel proved that this is fallacious. Neither appeals to external evidence physical or otherwise. — Barry Etheridge

You are using classical logic to define the meaning of classical logic. Donald Hoffman is a game theorist who spent ten years researching the neurological evidence and running one computer simulation after another only to conclude that if the human mind and brain had ever resembled anything remotely like reality we would already be extinct as a species. Life and all of physical reality including our minds and brains obey an analog logic where humor and beauty are indivisible complimentary-opposites that Intuitionistic mathematics can handle. By merely comparing how logic and mathematics actually apply statistically to the physical world we can learn how applicable they are whether or not they fit the classical definition of being true.

All very fascinating but totally beside the point. You claimed that Godel's incompleteness theorem

breaks down to you need physical evidence to prove classical logic — wuliheron

.
That is incorrect!

All very fascinating but totally beside the point. You claimed that Godel's incompleteness theorem
breaks down to you need physical evidence to prove classical logic"
— wuliheron — Barry Etheridge

It requires physical evidence to prove that classical mathematics are a subtype of Intuitionistic mathematics that are more fully expressed using a metaphoric emotional-logic, hence, Godel's Theorem can merely be considered to be begging the question and demonstrating that classical mathematics are incomplete. That would make it official that classical logic describes about a quarter of everything observable really well and another quarter to a more limited extent.

I'll be quick, since I'm in a bit of a hurry.

No, you absolutely do not. Logicists hold that all truths within any system of logic can be deduced from logical propositions within it. Godel proved that this is fallacious. Neither appeals to external evidence physical or otherwise. — Barry Etheridge

Logicism is a very broad umbrella, so it wouldn't surprise me if some logicists did hold the views you're attributing to them. Nevertheless, that's not how we generally characterize, e.g., Frege's logicism or contemporary logicism (defended mainly by Crispin Wright and Bob Hale). The main point is not that one about truths, but about concepts, namely that every mathematical concept is reducible to a logical concept. For instance, Frege thought (correctly!) that the concept of "number" was suitably reducible to the concept of "class", and the latter was supposedly logical. Obviously, this implies that a good number of theorems that we consider as characteristic of numbers will need to follow from this new characterization, which is why Frege spent so much time trying to show that this indeed happens. As it turns out, he was also correct in this: Crispin Wright, George Boolos, Richard Heck, and others have shown that the axioms for second-order Peano Arithmetic are actually derivable from a single principle known as Hume's Principle (it should actually be called "Frege's Principle", but Frege modestly attributed this principle to Hume, and the name stuck), a principle that can be taken as an implicit definition of the concept of number, a remarkable fact that is known as Frege's Theorem in the literature. If this is enough to vindicate logicism depends on the logicality of Hume's Principle, and this is a highly disputed matter. Regardless, this has nothing to do with Gödel's theorems.

It requires physical evidence to prove that classical mathematics are a subtype of Intuitionistic mathematics that are more fully expressed using a metaphoric emotional-logic, hence, Godel's Theorem can merely be considered to be begging the question and demonstrating that classical mathematics are incomplete. That would make it official that classical logic describes about a quarter of everything observable really well and another quarter to a more limited extent. — wuliheron

I don't understand what you say when you say that classical mathematics is a subtype of intuitionistic mathematics. Given that there are theorems which can be proved in an intuitionistic setting, but not in a classical setting, and vice-versa, they should be disjunct "types", that is, there is no relation of inclusion among them. Can you clarify?

Also, you said that Gödel's theorem is "begging the question"; begging the question against what? What is it assuming that shouldn't be assumed?

"It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong." - Richard P. Feynman

Godel's Theorem assumes the classic logic positions of the laws of noncontradiction and the excluded middle, that is, it assumes that everything, including mathematics and logic, are always either true or false and never equally true and false which is considered merely nonsensical gibberish. The problem is its uncompromising position contradicts observation which is why upon discovering quantum mechanics Max Planck begged his colleges to please explain the joke complaining that a sense of humor was never amongst his list of job requirements. Hence, the theorem merely begs the question of whether classic logic and mathematics are true according to their own standards when the weight of all the physical evidence says they are not.

Subtypes are the Intuitionistic equivalent of subsets in classic mathematics and Intuitionistic mathematics are about four times as complex allowing a quarter of their mathematics, or subtype, to express all of classical mathematics.

I would say that that Feynman quotation is incredibly naive in our post-Kuhnian age, but no matter. Gödel didn't assume that classical mathematics was "true"; rather, his result is about classical mathematics. An analogy: Gödel's theorems suppose that the theory in question is recursively axiomatizable. That does not mean that it "begs the question" as to whether all mathematical theories are recursively axiomatizable, which would be plainly false. Rather, it is a theorem about such theories.

As for classical and intuitionistic mathematics, well, classical analysis proves the intermediate value theorem, which is not provable in intuitionistic mathematics. On the other hand, it seems that every total function from R to R in an intuitionistic setting is continuous, something that is clearly false in the classical setting. So one does not seem to be a subset of the other (unless they're inconsistent, in which case they're the same).

I would say that that Feynman quotation is incredibly naive in our post-Kuhnian age, but no matter. Gödel didn't assume that classical mathematics was "true"; rather, his result is about classical mathematics. An analogy: Gödel's theorems suppose that the theory in question is recursively axiomatizable. That does not mean that it "begs the question" as to whether all mathematical theories are recursively axiomatizable, which would be plainly false. Rather, it is a theorem about such theories.

As for classical and intuitionistic mathematics, well, classical analysis proves the intermediate value theorem, which is not provable in intuitionistic mathematics. On the other hand, it seems that every total function from R to R in an intuitionistic setting is continuous, something that is clearly false in the classical setting. So one does not seem to be a subset of the other (unless they're inconsistent, in which case they're the same). — Nagase

Kuhn is merely another historian giving his personal interpretation of history in the name of science and philosophy. I'll take experimental evidence over the word of a historian or even the consensus of the scientific community any day.

Godel used classical logic to formulate his theorem and, by the standards he used, if he was not asserting his theorem was true, than he was asserting it was false!

Mathematicians have already demonstrated that all of classical mathematics and causal physics can be fully represented using any number of simple metaphors or analogies such as asserting everything is merely composed of bouncing springs, balls of string, or vibrating rubber sheets for all I know. Another study similarly concluded they can be fully represented using only two dimensions. In other words, all of causality and causal mathematics are demonstrably based upon what I like to call "Cartoon Logic", that is, the logic of small children who will pick whatever explanation sounds good to them at the time or happens to contradict reality less. The implication is clear that mathematics and logic are merely pragmatic conventions just as quantum mechanics suggest our concepts of reality are.

Kuhn is merely another historian giving his personal interpretation of history in the name of science and philosophy. I'll take experimental evidence over the word of a historian or even the consensus of the scientific community any day. — wuliheron

Actually he was a physicist by formation. In any case, you may do whatever you like, but the point is that scientists don't often proceed in the way Feynman describes, and that's not how science generally progresses.

Godel used classical logic to formulate his theorem and, by the standards he used, if he was not asserting his theorem was true, than he was asserting it was false! — wuliheron

Again, you're misunderstanding the theorems. The theorems are conditional in nature, i.e. they say that "under this and that circumstances, this result follows". In Gödel's case, the circumstances are (i) classical logic, (ii) recursively axiomatized theories which (iii) contain a modicum of arithmetic and (iv) are consistent. So the theorems are, if (i), (ii), (iii), (iv) hold for a given theory, then the theory is incomplete and can't prove its own consistency. There are many theories for which (i)-(iv) don't hold, and the theorem is silent about those (for instance, (ii) fails for the theory of the natural numbers, (iii) fails for Presburger arithmetic, (iv) fails for the inconsistent theory; these theories are all complete, trivially so in the last case). Given that the intuitionists also accept conditional reasoning, it follows that the theorem is valid also in an intuitionist setting.

Mathematicians have already demonstrated that all of classical mathematics and causal physics can be represented using any number of simple metaphors or analogies such as asserting everything is merely composed of bouncing springs, balls of string, or vibrating rubber sheets for all I know. Another study similar concluded they can be fully represented using only two dimensions. In other words, all of causality and causal mathematics are demonstrably based upon what I like to call "Cartoon Logic", that is, the logic of small children who will pick whatever explanation sounds good to them at the time or happens to contradict reality less. The implication is clear that mathematics and logic are merely pragmatic conventions just as quantum mechanics suggest our concepts of reality are. — wuliheron

I don't understand the relevance of the above, since nothing I said contradicts or is even remotely connected to that.

Regardless, I'm still curious about your notion of "subtypes". You said that classical mathematics is a subtype of intuitionistic mathematics. I took that to mean that every theorem of classical mathematics is a theorem of intuitionistic mathematics, i.e. classical mathematics is a (proper?) subset of intuitionistic mathematics. But then that doesn't seem to follow, since, e.g., the intermediate value theorem is a theorem of classical, but not of intuitionistic mathematics. So, is there any other way of understanding this subtype relation?

Actually he was a physicist by formation. In any case, you may do whatever you like, but the point is that scientists don't often proceed in the way Feynman describes, and that's not how science generally progresses.

Again, you're misunderstanding the theorems. The theorems are conditional in nature, i.e. they say that "under this and that circumstances, this result follows". In Gödel's case, the circumstances are (i) classical logic, (ii) recursively axiomatized theories which (iii) contain a modicum of arithmetic and (iv) are consistent. So the theorems are, if (i), (ii), (iii), (iv) hold for a given theory, then the theory is incomplete and can't prove its own consistency. There are many theories for which (i)-(iv) don't hold, and the theorem is silent about those (for instance, (ii) fails for the theory of the natural numbers, (iii) fails for Presburger arithmetic, (iv) fails for the inconsistent theory; these theories are all complete, trivially so in the last case). Given that the intuitionists also accept conditional reasoning, it follows that the theorem is valid also in an intuitionist setting.

I don't understand the relevance of the above, since nothing I said contradicts or is even remotely connected to that.

Regardless, I'm still curious about your notion of "subtypes". You said that classical mathematics is a subtype of intuitionistic mathematics. I took that to mean that every theorem of classical mathematics is a theorem of intuitionistic mathematics, i.e. classical mathematics is a (proper?) subset of intuitionistic mathematics. But then that doesn't seem to follow, since, e.g., the intermediate value theorem is a theorem of classical, but not of intuitionistic mathematics. So, is there any other way of understanding this subtype relation? — Nagase

The idea that any theory is demonstrably incomplete is the heart of the matter. For me, a context without significant content or any content without a significantly greater context is an oxymoron along the lines of a statistic of one. What is incomplete defines what is complete just as you cannot have an up without a down, a back without a front. What Godel showed is that it is incomplete by the standards of classical logic and the principles of the excluded middle and noncontradiction. What he did not do is take it that next step further and show how logic itself is context dependent as quantum mechanics suggests. What is a joke and what makes sense is merely a question of the context.

Intuitionistic subtypes are metaphors meaning the subsets of classical logic must also be treated as metaphors if they are to be compatible with the physical evidence and statistically demonstrated to be valid.

The idea that any theory is demonstrably incomplete is the heart of the matter. For me, a context without significant content or any content without a significantly greater context is an oxymoron along the lines of a statistic of one. What is incomplete defines what is complete just as you cannot have an up without a down, a back without a front. What Godel showed is that it is incomplete by the standards of classical logic and the principles of the excluded middle and noncontradiction. What he did not do is take it that next step further and show how logic itself is context dependent as quantum mechanics suggests. What is a joke and what makes sense is merely a question of the context. — wuliheron

Emphasis mine. If that is the heart of the matter, then it can be quickly be made to rest, since that particular claim is not what Gödel's theorems are about, but only a popular misconception. In fact, Hilbert, Ackermann, Presburger, Tarski and others had shown that many mathematical theories are complete before Gödel proved his theorems, so obviously the latter can't apply to the theories proven complete by those gentlemen (e.g. various weak forms of arithmetic, the theory of real closed fields, the theory of algebraic fields of a given characteristic, etc.). As I mentioned in my last post, Gödel's theorems apply only to recursively axiomatized theory which contain enough arithmetic. By recursively axiomatized, I mean that the set of axioms of the theory should be decidable by an algorithm. By "contain enough arithmetic", it means that the theory should have enough arithmetic to capture the primitive recursive functions (or, as we know nowadays, the theory should contain Robinson's minimal arithmetic). Any theory that fails these two requirements will not be subjected to Gödel's theorems, and thus may be complete (though it's not automatically complete! The theory of groups clearly fails them, but it's incomplete, since it doesn't decide whether a group is abelian or not).

Intuitionistic subtypes are metaphors meaning the subsets of classical logic must also be treated as metaphors if they are to be compatible with the physical evidence and statistically demonstrated to be valid. — wuliheron

Maybe I'm just being dense, but I don't understand what that means or how it answers my question. What you appear to be saying is that a classical theorem should be "compatible with the physical evidence and statistically demonstrated to be valid" before it is accepted as true. But this has nothing to do with relations of inclusion between intuitionistic and classical mathematics. Suppose, for the sake of the argument, that the intermediate value theorem was shown to be "compatible with the physical evidence and statistically demonstrated to be valid". Then we would have to accept a theorem of classical mathematics which is not a theorem of intuitionistic mathematics. On the other hand, suppose that we could somehow show that it is "compatible with the physical evidence and statistically demonstrate to be valid" that every total function from R to R is continuous. Then we would have to accept a theorem from intuitionism that is false in classical mathematics. Either way, though, there wouldn't be any inclusion relation between them, so that none would be a "subtype" of the other.

As I mentioned in my last post, Gödel's theorems apply only to recursively axiomatized theory which contain enough arithmetic. By recursively axiomatized, I mean that the set of axioms of the theory should be decidable by an algorithm. By "contain enough arithmetic", it means that the theory should have enough arithmetic to capture the primitive recursive functions (or, as we know nowadays, the theory should contain Robinson's minimal arithmetic). Any theory that fails these two requirements will not be subjected to Gödel's theorems, and thus may be complete (though it's not automatically complete! The theory of groups clearly fails them, but it's incomplete, since it doesn't decide whether a group is abelian or not). — Nagase

Quantum mechanics are noncommutative and you are merely arguing that classical logic and mathematics must be commutative and Godel's theorem is classical.

Maybe I'm just being dense, but I don't understand what that means or how it answers my question. What you appear to be saying is that a classical theorem should be "compatible with the physical evidence and statistically demonstrated to be valid" before it is accepted as true. But this has nothing to do with relations of inclusion between intuitionistic and classical mathematics. Suppose, for the sake of the argument, that the intermediate value theorem was shown to be "compatible with the physical evidence and statistically demonstrated to be valid". Then we would have to accept a theorem of classical mathematics which is not a theorem of intuitionistic mathematics. On the other hand, suppose that we could somehow show that it is "compatible with the physical evidence and statistically demonstrate to be valid" that every total function from R to R is continuous. Then we would have to accept a theorem from intuitionism that is false in classical mathematics. Either way, though, there wouldn't be any inclusion relation between them, so that none would be a "subtype" of the other. — Nagase

As best I can tell you are confused over the central issue. Classical logic proving internally consistent, yet, contradicting the physical evidence means all classical truths are context dependent and become a jokes in other contexts. The law of identity itself is going down the nearest convenient rabbit hole or toilet of your personal preference and what is classical mathematics or Intuitionistic mathematics also becomes context dependent.

Photons provide a similar example because what appears to be a shadow in a well lit room can become a faint blob of light in a dark one even though it is identical in every other respect other than the changing context.

Quantum mechanics are noncommutative and you are merely arguing that classical logic and mathematics must be commutative and Godel's theorem is classical. — wuliheron

I quite frankly don't see how you could give this reading to what I said. What does it mean to say that classical mathematics is "commutative"? Some classical theories (Peano Arithmetic) have an axiom stating the commutative of certain operations, others do not (non-abelian groups). So what?

In any case, I repeat: if your problem with Gödel's theorem is that it allegedly claims that every mathematical theory is incomplete, then you have no problem with Gödel's theorem at all, since it does not claim that every mathematical theory is incomplete.

As best I can tell you are confused over the central issue. Classical logic proving internally consistent, yet, contradicting the physical evidence means all classical truths are context dependent and become a jokes in other contexts. The law of identity itself is going down the nearest convenient rabbit hole or toilet of your personal preference and what is classical mathematics or Intuitionistic mathematics also becomes context dependent.

Photons provide a similar example because what appears to be a shadow in a well lit room can become a faint blob of light in a dark one even though it is identical in every other respect other than the changing context. — wuliheron

But how does this answer my question about the inclusion relationship between classical and intuitionist mathematics? Is there any such relationship? If yes, how should we characterize it?

I quite frankly don't see how you could give this reading to what I said. What does it mean to say that classical mathematics is "commutative"? Some classical theories (Peano Arithmetic) have an axiom stating the commutative of certain operations, others do not (non-abelian groups). So what?

In any case, I repeat: if your problem with Gödel's theorem is that it allegedly claims that every mathematical theory is incomplete, then you have no problem with Gödel's theorem at all, since it does not claim that every mathematical theory is incomplete.

But how does this answer my question about the inclusion relationship between classical and intuitionist mathematics? Is there any such relationship? If yes, how should we characterize it? — Nagase

My assertion is that Godel's theorem begs the question and is demonstrably useless outside of classical mathematics and limited physical applications.

Categorization is part of the confusion because there is no way to characterize or categorize Indeterminacy. Calling something like quanta random or a joke meaningless or insisting a shadow has no properties is merely another way of saying we can't define them as anything other than false or context dependent. Clearly shadows, for example, exist and calling them false can only have limited usefulness when they can be more broadly defined as context dependent and sharing their identity with photons.

The way around the issue is to use a systems logic where even its own axioms and identity go down the proverbial rabbit hole into Indeterminacy, thus, displaying context dependence in everything which can be established statistically as factual in some contexts and metaphorical or a personal truth in others. Which, is something only Intuitionistic mathematics can do as far as I know, not being a mathematician myself.

My assertion is that Godel's theorem begs the question and is demonstrably useless outside of classical mathematics and limited physical applications. — wuliheron

And my assertion is that the theorem does not beg the question you're saying it begs, namely that classical mathematics is true, because it does not assume classical mathematics; rather, it is about classical mathematics. To put it more forcefully, it's possible to prove the theorem using as a background logic intuitionism, so it obviously doesn't assume any classical theorem. As for being useless outside of classical mathematics and with limited physical applications, yes, obviously, nobody (except maybe Penrose and Hawking) said anything to the contrary.

Categorization is part of the confusion because there is no way to characterize or categorize Indeterminacy. Calling something like quanta random or a joke meaningless or insisting a shadow has no properties is merely another way of saying we can't define them as anything other than false or context dependent. Clearly shadows, for example, exist and calling them false can only have limited usefulness when they can be more broadly defined as context dependent and sharing their identity with photons.

The way around the issue is to use a systems logic where even its own axioms and identity go down the proverbial rabbit hole into Indeterminacy, thus, displaying context dependence in everything which can be established statistically as factual in some contexts and metaphorical or a personal truth in others. — wuliheron

That's nice, but I still don't see how that answers my question. Is classical mathematics a subtype of intuitionist mathematics? Yes or no? If yes, what is the meaning of "subtype", here? Clearly it's not the subset relation, because we know that classical mathematics is not a subset of intuitionist mathematics. So what is it?

And my assertion is that the theorem does not beg the question you're saying it begs, namely that classical mathematics is true, because it does not assume classical mathematics; rather, it is about classical mathematics. To put it more forcefully, it's possible to prove the theorem using as a background logic intuitionism, so it obviously doesn't assume any classical theorem. As for being useless outside of classical mathematics and with limited physical applications, yes, obviously, nobody (except maybe Penrose and Hawking) said anything to the contrary. — Nagase

The foundations of Intuitionistic mathematics have yet to be fully developed and, as far as I can tell, they first need to be expressed as a systems logic along the lines of what I've described. That mathematicians are beginning to express things like Godel's theorem in Intuitionistic terms merely means they are working on the problem and not that they have left classical logic and mathematics behind as of this date.

That's nice, but I still don't see how that answers my question. Is classical mathematics a subtype of intuitionist mathematics? Yes or no? If yes, what is the meaning of "subtype", here? Clearly it's not the subset relation, because we know that classical mathematics is not a subset of intuitionist mathematics. So what is it? — Nagase

"Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds."

http://plato.stanford.edu/entries/intuitionism/

Hence, most certainly classical mathematics can be considered a subtype of Intuitionistic mathematics. My own belief is that everything is context dependent making even what is mental or physical a matter of the situation and, for example, the mind and brain have already been demonstrated to substitute for each other at the most fundamental level of their organization for increased efficiency and error correction. They express the particle-wave duality of quantum mechanics which, for me, is simply another way of saying the display extreme context dependence or are "yin and yang".

The foundations of Intuitionistic mathematics have yet to be fully developed and, as far as I can tell, they first need to be expressed as a systems logic along the lines of what I've described. That mathematicians are beginning to express things like Godel's theorem in Intuitionistic terms merely means they are working on the problem and not that they have left classical logic and mathematics behind as of this date. — wuliheron

Look, here's the fact of the matter: Gödel's theorems do not assume classical logic is true. They are about classical logic. If your logic contains conditional reasoning, then Gödel's theorems will be provable within it.

"Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds."

http://plato.stanford.edu/entries/intuitionism/

Hence, most certainly classical mathematics can be considered a subtype of Intuitionistic mathematics. My own belief is that everything is context dependent making even what is mental or physical a matter of the situation and, for example, the mind and brain have already been demonstrated to substitute for each other at the most fundamental level of their organization for increased efficiency and error correction. They express the particle-wave duality of quantum mechanics which, for me, is simply another way of saying the display extreme context dependence or are "yin and yang". — wuliheron

Question: what is the subtype relation? More to the point, if type A is a subtype of type B, does it follow that every theorem provable in type A is also provable in type B?

Look, here's the fact of the matter: Gödel's theorems do not assume classical logic is true. They are about classical logic. If your logic contains conditional reasoning, then Gödel's theorems will be provable within it. — Nagase

You cannot prove something is true without somehow demonstrating it is true! Conditional reasoning or otherwise, you must assume if nothing else that we can make clear distinctions between true and false! Godel's theorem is based upon the rules of classical logic in that, at the very least, the law of identity and noncontradiction must apply to any proof. You can play around with variations on the excluded middle all you want, but the essential nature of the logic remains the same.

Question: what is the subtype relation? More to the point, if type A is a subtype of type B, does it follow that every theorem provable in type A is also provable in type B? — Nagase

I'm not a mathematician and those that I've read about claimed the foundations are incomplete. That said, subtypes of the overall symmetry will always express a four fold symmetry or supersymmetry that can be expressed as root metaphors or axioms. In physics, a four fold supersymmetry should be expressed in everything observable and can be thought of metaphorically as infinite dimensions or universes all converging and diverging within the singular void and making it impossible for us to perceive anything less than a four fold symmetry in anything clearly discernible. Such a scenario could only be proven statistically by classical standards, but even if it can never be disproved it would mean everything must express four fold symmetry and so you can use eight dimensions and a singularity or 16 or 32 and so on depending on how much accuracy is desired.

You cannot prove something is true without somehow demonstrating it is true! Conditional reasoning or otherwise, you must assume if nothing else that we can make clear distinctions between true and false! Godel's theorem is based upon the rules of classical logic in that, at the very least, the law of identity and noncontradiction must apply to any proof. You can play around with variations on the excluded middle all you want, but the essential nature of the logic remains the same. — wuliheron

But Gödel's theorems do not state "classical logic is true". They state "if we assume classical logic and some other conditions, then there are some mathematical theories which are incomplete and can't prove their own consistency". In other words, they are of the form "if A, then B". Clearly I don't need to establish "A" in order to prove "If A, then B"; I can show that, if John is decapitated, then he will die, without thereby showing that John was decapitated!

I'm not a mathematician and those that I've read about claimed the foundations are incomplete. That said, subtypes of the overall symmetry will always express a four fold symmetry or supersymmetry that can be expressed as root metaphors or axioms. In physics, a four fold supersymmetry should be expressed in everything observable and can be thought of metaphorically as infinite dimensions or universes all converging and diverging within the singular void and making it impossible for us to perceive anything less than a four fold symmetry in anything clearly discernible. Such a scenario could only be proven statistically by classical standards, but even if it can never be disproved it would mean everything must express four fold symmetry and so you can use eight dimensions and a singularity or 16 or 32 and so on depending on how much accuracy is desired. — wuliheron

That doesn't answer my second question, which I repeat here for the sake of completeness: if A is a subtype of B, does it mean that every theorem provable in A is also provable in B?

But Gödel's theorems do not state "classical logic is true". They state "if we assume classical logic and some other conditions, then there are some mathematical theories which are incomplete and can't prove their own consistency". In other words, they are of the form "if A, then B". Clearly I don't need to establish "A" in order to prove "If A, then B"; I can show that, if John is decapitated, then he will die, without thereby showing that John was decapitated! — Nagase

I never said it proves classical logic true, merely, that it begs the question of whether it is true or not by assuming the position that it is true.

That doesn't answer my second question, which I repeat here for the sake of completeness: if A is a subtype of B, does it mean that every theorem provable in A is also provable in B? — Nagase

That's a tricky question and, as I keep saying, I'm not a mathematician and even they don't have the foundations of the mathematics complete as of yet. My own view is with everything being context dependent it depends upon what you mean by provable in any given situation.

I never said it proves classical logic true, merely, that it begs the question of whether it is true or not by assuming the position that it is true. — wuliheron

And I'm saying that no question is begged. If I say "If John is decapitated, then he will die", I'm not "begging the question" as to whether John was decapitated or not!

That's a tricky question and, as I keep saying, I'm not a mathematician and even they don't have the foundations of the mathematics complete as of yet. My own view is with everything being context dependent it depends upon what you mean by provable in any given situation. — wuliheron

Generally, "provable" means roughly follows from the axioms by acceptable rules of inference.

And I'm saying that no question is begged. If I say "If John is decapitated, then he will die", I'm not "begging the question" as to whether John was decapitated or not! — Nagase

A doctor has already successfully transplanted the head of one monkey onto another. The quality of life wasn't great, but it lived. For me, everything is literally and figuratively context dependent. If I say, "She's hot!" I could be talking about anything from a good looking woman to an overheating car engine and not only words, but mathematical axioms only have demonstrable meaning in specific contexts.

Generally, "provable" means roughly follows from the axioms by acceptable rules of inference. — Nagase

Yes, but my own view of everything being context dependent means the axioms can also be treated as root metaphors and how you interpret them simply depends upon the context. That's the only way the law of identity can consistently go down the rabbit hole and would mean you can interpret the mathematics either metaphorically or axiomatically with which one is more useful or appealing simply depending upon the context. The mathematics would still have to be self-consistent and prove to be at least statistically nontrivial, but proof takes on an entirely different meaning when it is context dependent.

Unfortunately, I can't understand how your reply has any bearing on what I said...

Unfortunately, I can't understand how your reply has any bearing on what I said... — Nagase

Quantum Cognition might be a good example. When sociologists applied quantum mechanics to some of their studies they discovered it could answer some of their most puzzling results and created the field of Quantum Cognition. A popular example is what they call the "Sure Thing" experiment where they offer people a 50-50 chance to either win $200.oo or lose $100.oo and such simple odds heavily stacked in their favor are something anyone can understand. Even if they lose a few rounds, they'll usually continue to play knowing the odds are in their favor. However, the minute they were not told the results of the last round they tend to stop.

According to classic logic it makes no sense because the odds are so heavily stacked in their favor they should keep playing. However, according to quantum mechanics without information on the last round they cannot predict the next. The context is determining everything with Monty Hall of Let's Make a Deal being another good example of related fuzzy logic.

After contestants choose from among three doors he often shows them a booby prize behind one of the two remaining doors and then offers them a chance to trade the original door they choose for the one they haven't seen yet. According to classical logic it makes no difference because the odds are merely 50-50, however, fuzzy logic says otherwise. According to fuzzy logic your first choice was between three doors and, therefore, more likely wrong than trading between the two remaining ones which is something contestants today are well aware of. The context is determining the law of identity including which type of logic is more useful.

An example such as someone getting a head transplant might be extreme, but that's the whole point of my insisting everything is context dependent. If we didn't have extremes like quantum mechanics nobody would be debating these issues.

You seem to be on some kind of tirade against classical logic. But this has nothing to do with what I asserted, namely that Gödel's theorems don't assume classical logic and that classical mathematics is not a subset of intuitionistic mathematics (where mathematics A is a subset of mathematics B iff all theorems provable in A are also provable in B). I have no intention of defending classical logic here, nor, for the matter, did Gödel in his incompleteness paper.