LEM is a basic law of thought, not limited to mathemathics.godelian wrote: ↑Sat Mar 09, 2024 7:41 amI reject your question on formalist grounds:
Every possible answer to your question is simply irrelevant and in severe violation of the fundamental ontology of mathematics.https://en.wikipedia.org/wiki/Formalism ... thematics)
A central idea of formalism is that mathematics is not a body of propositions representing an abstract sector of reality.
According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.
The constructivist animosity against the law of the excluded middle
Re: The constructivist animosity against the law of the excluded middle
Re: The constructivist animosity against the law of the excluded middle
I reject the psychologist ontology of mathematics:
Computers can perfectly handle mathematics, while they are in no way or fashion psychological beings.https://en.wikipedia.org/wiki/Philosophy_of_mathematics
Psychologism in the philosophy of mathematics is the position that mathematical concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws).
Re: The constructivist animosity against the law of the excluded middle
The LEM is a basic law of thought. It has been one of the three fundamental laws of thought since Aristotle. I don't know why you bring up the philosophy of mathemathics now.godelian wrote: ↑Sat Mar 09, 2024 7:50 amI reject the psychologist ontology of mathematics:
Computers can perfectly handle mathematics, while they are in no way or fashion psychological beings.https://en.wikipedia.org/wiki/Philosophy_of_mathematics
Psychologism in the philosophy of mathematics is the position that mathematical concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws).
Re: The constructivist animosity against the law of the excluded middle
Of course, I do not deny Aristotle's merit or the merit of his work. However, it was merely the starting point of a long journey. It is not because Aristotle initially identified three candidate laws as fundamental that they eventually turned out to be fundamental. The same holds true for Isaac Newton. He originally produced fantastically good research. But then again, physicists no longer use his classical mechanics when dealing with very small or very big objects. In logic, we no longer use Aristotle's LEM when dealing with the inevitable class of undecidable problems.
Really?
The final answer to a non-trivial problem in mathematics is always a philosophical choice. That is simply the nature of the beast.
Re: The constructivist animosity against the law of the excluded middle
Well at this point we can only repeat ourselves. Logicians are inept for not seeing the layers of abstract thinking. Of course we can't use the LEM for issues we can't use it for, like undecidable problems. But that doesn't invalidate the LEM in any way as it still holds fundamentally.godelian wrote: ↑Sat Mar 09, 2024 8:03 amOf course, I do not deny Aristotle's merit or the merit of his work. However, it was merely the starting point of a long journey. It is not because Aristotle initially identified three candidate laws as fundamental that they eventually turned out to be fundamental. The same holds true for Isaac Newton. He originally produced fantastically good research. But then again, physicists no longer use his classical mechanics when dealing with very small or very big objects. In logic, we no longer use Aristotle's LEM when dealing with the inevitable class of undecidable problems.
Really?
The final answer to a non-trivial problem in mathematics is always a philosophical choice. That is simply the nature of the beast.
It's like saying, science can't tell what's outside the observable universe therefore science is wrong.
Re: The constructivist animosity against the law of the excluded middle
You got the gist of it. Now do a U-turn back to Rice's theorem....godelian wrote: ↑Sat Mar 09, 2024 3:05 amI think that I finally understand the gist of the issue now.Skepdick wrote: ↑Fri Mar 08, 2024 3:36 pm Let the predicate φ be LEM itself in predicate form. Which is exactly the EITHER monad.
If LEM is an axiom (and therefore always true) then the predicate is always-satisfiable.Code: Select all
(either R ∈ R or R !∈ R)
But neither disjunct is true! So the search will never terminate. Obviously. You can't find what doesn't exist...
Axiomatizing the truth of "P or not P", i.e. the LEM, assumes that all problems are decidable. This view is absolutely unrealistic:
The indiscriminate use of the LEM denies the historical fact that the answer to David Hilbert's Entscheidungsproblem is a resounding "no":https://en.m.wikipedia.org/wiki/Undecidable_problem
Since there are uncountably many undecidable problems, any list, even one of infinite length, is necessarily incomplete.
Hence, axiomatizing the LEM amounts to ignoring the answers by Alonzo Church and Alan Turing to David Hilbert's Entscheidungsproblem and actively denying the existence of fundamentally unsolvable problems, even though there is an infinitely long list of them.https://en.m.wikipedia.org/wiki/Entscheidungsproblem
The problem asks for an algorithm that considers, as input, a statement and answers "yes" or "no" according to whether the statement is universally valid, i.e., valid in every structure.
In 1936, Alonzo Church and Alan Turing published independent papers[2] showing that a general solution to the Entscheidungsproblem is impossible.
Syntactically speaking virtually nothing is decidable unless it's trivially true.
Given a particular syntax does the the symbol X represent a set?
That's a type-checking question. How do you type-check symbols without explicit type declarations? You can't - the required information is not encoded in your syntax. So you are back at the untyped lambda calculus.
Is 0 a number? Undecidable.
Is the successor function applicable to 0? Undecidable.
Can you ALWAYS call the successor function to get a larger number? Undecidable.
You can deconstruct the whole house of cards.
Re: The constructivist animosity against the law of the excluded middle
Oooo...kaaayyy... yes of course it gives you omniscience... You hit the nail on the head! Now I'll back away reaal slowly... okay? ... don't mind me...
Re: The constructivist animosity against the law of the excluded middle
Shame, you seem to have confused yourself on your (non?)dualism.
Dualism: X is either true or false.
Non-dualism: X is neither true nor false.
Either you are a dualist or you are a non-dualist
Re: The constructivist animosity against the law of the excluded middle
Here you go, ignoramus.
https://ncatlab.org/nlab/show/principle+of+omniscience
The belief in LEM causes the belief in omniscience. I can only explain it to you - I can't understand it for you.In logic and foundations, a principle of omniscience is any principle of classical mathematics that is not valid in constructive mathematics. The idea behind the name (which is due to Bishop (1967)) is that, if we attempt to extend the computational interpretation of constructive mathematics to incorporate one of these principles, we would have to know something that we cannot compute. The main example is the law of excluded middle (EM); to apply P ∨ ¬P computationally, we must know which of these disjuncts hold; to apply this in all situations, we would have to know everything (hence ‘omniscience’).
Re: The constructivist animosity against the law of the excluded middle
So it's a principle (still pretty dumb naming) but it doesn't give me actual omniscience? DaamnSkepdick wrote: ↑Sat Mar 09, 2024 9:50 pmHere you go, ignoramus.
https://ncatlab.org/nlab/show/principle+of+omniscience
The belief in LEM causes the belief in omniscience. I can only explain it to you - I can't understand it for you.In logic and foundations, a principle of omniscience is any principle of classical mathematics that is not valid in constructive mathematics. The idea behind the name (which is due to Bishop (1967)) is that, if we attempt to extend the computational interpretation of constructive mathematics to incorporate one of these principles, we would have to know something that we cannot compute. The main example is the law of excluded middle (EM); to apply P ∨ ¬P computationally, we must know which of these disjuncts hold; to apply this in all situations, we would have to know everything (hence ‘omniscience’).
Re: The constructivist animosity against the law of the excluded middle
word saladSkepdick wrote: ↑Sat Mar 09, 2024 2:13 pmShame, you seem to have confused yourself on your (non?)dualism.
Dualism: X is either true or false.
Non-dualism: X is neither true nor false.
Either you are a dualist or you are a non-dualist
Re: The constructivist animosity against the law of the excluded middle
Salad brain. It's an axiom. Which implies it's always true.
Go ahead and decide whether the number of stars in the universe is odd or even.
"I don't know" or "I can't count them" are NOT acceptable answers. You MUST know. Axiomatically.
Re: The constructivist animosity against the law of the excluded middle
You've been smoking too much again I think. There's certainly no such axiom or even such a principle about the LEM generally. Okay there is this principle in some obscure part of maths apparently.