"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
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. — Barry Etheridge
.breaks down to you need physical evidence to prove classical logic — wuliheron
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
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
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 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. — wuliheron
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
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
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. — wuliheron
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
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
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
Quantum mechanics are noncommutative and you are merely arguing that classical logic and mathematics must be commutative and Godel's theorem is classical. — wuliheron
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
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. — wuliheron
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
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
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
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
"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
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
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
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
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
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
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
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
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
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
Generally, "provable" means roughly follows from the axioms by acceptable rules of inference. — Nagase
Unfortunately, I can't understand how your reply has any bearing on what I said... — Nagase
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