Originally posted by royalchickenThat you would be thrilled if the Doctor would react to your first question you asked in this thread. The Doctor is, according to the information he gave, a professional mathematician .....
I was talking about his specific statements regarding vistesd's participation in this thread. What did you tell Dr. S?
Originally posted by DoctorScribblesI'm a programmer by trade, but I fail to understand your entire reasoning and so, this may be pretty lame:
So far so good. I'll look forward to the meat tomorrow.
"A final remark which will be useful later: an existence theorem with a constructive proof can be strengthened to a theorem which provides an example of the object whose existence it guarantees. Conversely, every theorem of the form '(specific) O has properties P' implies the existence t ...[text shortened]... g and one Design Pattern that mimics this observation. Responses are intended to be subjective.
Theorem is a class. Specific O is an instantiation of a class with the member P (property). The instantiation is the constructive proof of the existence of the class. ???
That's as far as OOP goes, but I fail to see how design patterns are related to your constructive/non-constructive proof. In fact, I fail to understand the concept completely, so I should probably just keep reading and stay out of this discussion.
😳
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Originally posted by DoctorScribblesWait a sec... I think I'm getting there. You're asking for specifically two concepts in OOP and one DP...
So far so good. I'll look forward to the meat tomorrow.
"A final remark which will be useful later: an existence theorem with a constructive proof can be strengthened to a theorem which provides an example of the object whose existence it guarantees. Conversely, every theorem of the form '(specific) O has properties P' implies the existence t ...[text shortened]... g and one Design Pattern that mimics this observation. Responses are intended to be subjective.
Classes and their instantiations [edit: that would be objects] ???
But the DP... hmmm... could it be Factory? The factory pattern creates instances and so it "guarantees" the existence of new instances... ???
Wheew, you're deep, Doctor Scribbles. I will not be ashamed if you correct me. 🙂
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Originally posted by stockenAs I said, I didn't have a correct answer in mind.
Wait a sec... I think I'm getting there. You're asking for specifically two concepts in OOP and one DP...
Classes and their instantiations [edit: that would be objects] ???
But the DP... hmmm... could it be Factory? The factory pattern creates instances and so it "guarantees" the existence of new instances... ???
Wheew, you're deep, Doctor Scribbles. I will not be ashamed if you correct me. 🙂
The Factory pattern was one of the two Design Patterns I had in mind. Singleton was the other.
Regarding OOP, the concepts I had in mind were constructors and reflection. It seems like we're on the same page.
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Originally posted by royalchickenRegarding: 2. PvQ becomes "We can construct a proof of either P or Q and specify which we have proved."
Chew on my sausage.
I demand to have (A OR NOT-A), and all of its classical close relatives, be a tautology, unless the issue below can be resolved.
In classical logic, a well-defined proposition must be something that is true or false, and the above tautology directly reflects this. Further, the proposition remains well-defined, and the tautology holds, even for propositions whose truth value is unknown, or even unknowable.
What would it mean for a proposition to be well-defined in a constructivist system? Would we have to supply a proof of P or NOT-P before claiming that P is well-defined? If not, that is, if P is well-defined prior to either proof, can we or can we not say that P must be either true or false? That is, can we or can we not assert (P OR NOT-P) for all P, without specifying which has a proof?
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Originally posted by DoctorScribblesI think I have confused myself while trying to resolve this. I need some clarification.
Regarding: [b]2. PvQ becomes "We can construct a proof of either P or Q and specify which we have proved."
I demand to have (A OR NOT-A), and all of its classical close relatives, be a tautology, unless the issue below can be resolved.
In classical logic, a well-defined proposition must be something that is true or false, and the ...[text shortened]... at is, can we or can we not assert (P OR NOT-P) for all P, without specifying which has a proof?[/b]
Consider these propositions.
p1: There exists a prime number less than 5.
p2: There exists a constructive proof of p1. (Read alternatively, One could know how to construct a prime number less than 5, if one is clever enough to discover the proof.)
p3: I already do know how to construct a prime number less than 5.
Which of these sorts of propositions do the constructivist logic operators operate on?
Or could they operate on all three, always with a final constructivist layer of semantics applied? If so, what exactly is that semantic transformation? Suppose a constructivist proof concludes C. Does it simply assert the truth of C, or the existence of a constructive proof of C, or the knowledge of a constructive proof of C, or what?
I guess what I'm asking is whether the assertions about the proofs (their existence, or the prover's knowledge of them) reside in the propositions or in the semantics of the operators. It seems to me that they must reside in the semantics of the constructivist operators, which operate on classical propositions like p1.
It might be time to wimp out and examine a concrete example.
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I think one source of confusion is the wording of (1) and (2). Should "we can construct a proof" be read "there exists a proof, which the clever among us could discover," or does it entail knowledge of that proof, like "there exists a proof, and we know what it is"?
If the latter, then I think this a major aesthetic flaw. Existential theorems should not acquire truth by virtue of mathematians discovering their constructive proofs. This is because one could imagine two universes, identical in all regards except the cleverness of the mathematicians. A proposition could be true in one, and false in the other, which means that all theorems implicitly make a claim about the intellgence of those discussing them. This kind of creeps me out.
Originally posted by DoctorScribblesI'm going to answer in more detail later today, but this is one of the things that is troubling me as well, because I think the second is the standard interpretation.
I think one source of confusion is the wording of (1) and (2). Should "we can construct a proof" be read "there exists a proof, which the clever among us could discover," or does it entail knowledge of that proof, like "there exists a proof, and we know what it is"?
If the latter, then I think this a major aesthetic flaw. Existential theor ...[text shortened]... icitly make a claim about the intellgence of those discussing them. This kind of creeps me out.