If intelligence were to evolve to be more efficient than emotions, would emotions still have any evolutionary purpose? — MonfortS26
I've seen things about the fractal being key to our concept of beauty, but I've always thought it was superstition. Do you have any type of source that you could include that would say the same? — MonfortS26
Beauty can't exist without an experiencer to experience it. Art, music and fractal dragon equations are all just data in the great objective whatever. — intrapersona
So are you saying that emotion responses are differential and logical responses are integral? — MonfortS26
So it sounds like the key to an accurate intuition is exposure? What do you mean by differentials? — MonfortS26
Is it the most accurate way of processing information though? — MonfortS26
I believe that the key to being genuine in life lies in your intuition. Intuition is the core of who we are as a person and everything else is just whatever our intuition chooses to perceive us to be. The way to live in the present is to be in touch with your intuition.
Anyone disagree? — MonfortS26
Are you putting forth a radically hard-determinist perspective? — Metaphysician Undercover
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. — Nagase
Unfortunately, I can't understand how your reply has any bearing on what I said... — Nagase
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
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
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
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
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
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
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
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
If beauty is inherent in nature, how do you account for individual taste?
You understand that symmetry is just a surrogate for genetic fitness. So what then is inherently special about symmetry? If there were a unsymmetrical being capable of appraising beauty, it would undoubtedly find it's own brand of asymmetry beautiful.
Are there supposed to be fractal dragon equations inherent in the shit-stained canvas? — hypericin
Consider the case of two paintings hanging in a gallery. One is the Mona Lisa; the other is simply a canvas randomly smeared with feces and vomit. A man walks into the gallery, finds the first pleasing, and recoils in disgust from the second. Then a dog walks into the gallery, sniffs them both, and finds the second to hold vastly more aesthetic interest.
Beauty is *both* in the object and in the eye of the observer. That is because it is a relation, between the properties of an object and the nature and tastes of the observer. — hypericin
Assuming they can create self-awareness to perceive the beauty. That still is an assumption.
5 hours ago ReplyShareFlag — intrapersona
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