What is the basis for reason? And mathematics?

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Skepdick
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### Re: P=P is a Contradiction

Immanuel Can wrote: Sat Sep 26, 2020 10:47 pm My sole point was that the law of identity is not optional.
Ironically, this is not a tautology which renders your point wrong.

The law of identity is optional in some interpretations of logic.

Rejection of the principle of identity
Schrödinger Logics
Immanuel Can
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### Re: P=P is a Contradiction

Skepdick wrote: Sat Sep 26, 2020 11:07 pm
Immanuel Can wrote: Sat Sep 26, 2020 10:47 pm My sole point was that the law of identity is not optional.
Ironically, this is not a tautology which renders your point wrong.
Actually, it is. Here's the definition you suggested: "In logic, a tautology (from Greek: ταυτολογία) is a formula or assertion that is true in every possible interpretation."

That would seem to fit P=P admirably.
Skepdick
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### Re: P=P is a Contradiction

Immanuel Can wrote: Sun Sep 27, 2020 12:45 am
Skepdick wrote: Sat Sep 26, 2020 11:07 pm
Immanuel Can wrote: Sat Sep 26, 2020 10:47 pm My sole point was that the law of identity is not optional.
Ironically, this is not a tautology which renders your point wrong.
Actually, it is. Here's the definition you suggested: "In logic, a tautology (from Greek: ταυτολογία) is a formula or assertion that is true in every possible interpretation."

That would seem to fit P=P admirably.
Clearly you are struggling with comprehension. Do you understand what the phrase "every possible interpretation" means?

P=P is not true in EVERY possible interpretation, because it's not true in Schrödinger's interpretation.

Therefore it doesn't satisfy the criterion of being a tautology.
Immanuel Can
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### Re: P=P is a Contradiction

Skepdick wrote: Sun Sep 27, 2020 1:17 am P=P is not true in EVERY possible interpretation, because it's not true in Schrödinger's interpretation.
No, it is only not KNOWN to be true in Shrodinger's thought experiment. Knowing and being are two different issues.

It is also not known to be untrue. It is not known at all. So whether or not it amounts to a tautology is a question only suspended...not ultimately unanswerable. What it IS and what it IS KNOWN AS are not the same problem.
Skepdick
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### Re: P=P is a Contradiction

Immanuel Can wrote: Sun Sep 27, 2020 1:29 am
Skepdick wrote: Sun Sep 27, 2020 1:17 am P=P is not true in EVERY possible interpretation, because it's not true in Schrödinger's interpretation.
No, it is only not KNOWN to be true in Shrodinger's thought experiment. Knowing and being are two different issues.

It is also not known to be untrue. It is not known at all. So whether or not it amounts to a tautology is a question only suspended...not ultimately unanswerable. What it IS and what it IS KNOWN AS are not the same problem.
Then how is it that you know the answer to a suspended question?

And if you don't know, then why do you assume identity as a "law" without evidence?

Surely you don't want us to think you are irrational?
Immanuel Can
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### Re: P=P is a Contradiction

Skepdick wrote: Sun Sep 27, 2020 1:30 am
Immanuel Can wrote: Sun Sep 27, 2020 1:29 am
Skepdick wrote: Sun Sep 27, 2020 1:17 am P=P is not true in EVERY possible interpretation, because it's not true in Schrödinger's interpretation.
No, it is only not KNOWN to be true in Shrodinger's thought experiment. Knowing and being are two different issues.

It is also not known to be untrue. It is not known at all. So whether or not it amounts to a tautology is a question only suspended...not ultimately unanswerable. What it IS and what it IS KNOWN AS are not the same problem.
Then how is it that you know the answer to a suspended question?
I didn't say I did. I said knowing and being are different.
And if you don't know, then why do you assume identity as a "law" without evidence?
If you ask that question, you don't know what the law of identity is.

It's a prerequisite and essential component of rationality itself. One can deny it, but only at the cost of becoming unable to use rationality afterward.
wtf
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### Re: P=P is a Contradiction

Immanuel Can wrote: Sat Sep 26, 2020 10:47 pm Well, I am content to mark our disagreement without impugning your character or intelligence.

I apologize for my tone. We exchanged a couple of civilities and I should continue to assume this is an honest misunderstanding. I commend you for taking the high road.

To keep this simple, please simply read this quote, which consistes of your own words; and either explain, clarify, retract, defend, or place into context your use of the word tautology in that sentence.

As you do so, bear in mind that Wiles's proof of FLT is a tautology.

Immanuel Can wrote: Thu Sep 17, 2020 2:11 am P=P is also a tautology. The fault is not that it is wrong, or tells a lie; it's that even if true, it's utterly uninformative of anything new. It adds no value to our thinking at all.
I am challenging your use of the word tautology and I'd like you to explain or retract or place in context etc.

Addendum: It occurs to me that some readers may not know what I mean by saying a proof is a tautology.

Wiles's proof of FLT is a written proof. In principle it could be drilled down to a formal step-by-step proof in set theory starting with the axioms and proceeding step-by-step till FLT was proven.

[In practice this would be very difficult and I don't think it's been done or that anyone's trying.]

As such, it is true in every model of set theory. This is in fact the content of Gödel's completeness theorem, not to be confused with his more famous incompleteness theorems.

Gödel's completeness theorem says that a statement has a proof from a given set of axioms if and only if it's true in every model of those axioms. This is the case with FLT and in fact with every formal proof in mathematics.

That is, the very fact that there is a proof of FLT from the axioms of set theory; is equivalent to the statement that it's true in every model of those axioms.

The Wiki definition of a tautology, as I mentioned earlier, is "In logic, a tautology (from Greek: ταυτολογία) is a formula or assertion that is true in every possible interpretation." By this definition. all proofs in mathematics are tautologies.

https://en.wikipedia.org/wiki/Tautology_(logic)

This sounds counterintuitive, till you realize that tautologies are not trivial. They're not obvious, they're not simple, they're not intuitive. Just like a novel is made up of the same old 26 English letters and some punctuation; but the great writers are really good at arranging strings of those letters. In theory a computer (or infinitely many monkeys) could type out the works of Shakespeare; but in fact it took Shakespeare to do it.

It's the same in math. It's hard to find the proofs, even if they follow logically from axioms and could in theory have been cranked out by a computer. But computers don't know which proofs are important. That's what humans are for. And finding interesting tautologies is hard work.
Skepdick
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### Re: P=P is a Contradiction

Immanuel Can wrote: Sun Sep 27, 2020 4:02 am I didn't say I did. I said knowing and being are different.
Immanuel Can wrote: Sat Sep 26, 2020 10:47 pm My sole point was that the law of identity is not optional.
Knowing and being are different.

So how is it that you came to know the law of identity is "not optional", if the question of its being is undecided?
Immanuel Can wrote: Sun Sep 27, 2020 4:02 am It's a prerequisite and essential component of rationality itself. One can deny it, but only at the cost of becoming unable to use rationality afterward.
It's a prerequisite of "rationality" to blindly accept unjustified axioms?

I think I am much better off being irrational then...
Immanuel Can
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### Re: P=P is a Contradiction

I do prefer the kind of conversation you suggest. Thank you. I shall reply.
wtf wrote: Sun Sep 27, 2020 7:24 am These are your words.
Immanuel Can wrote: Thu Sep 17, 2020 2:11 am P=P is also a tautology. The fault is not that it is wrong, or tells a lie; it's that even if true, it's utterly uninformative of anything new. It adds no value to our thinking at all.
I am challenging your use of the word tautology and I'd like you to explain or retract or place in context etc.
Fine, we can do that.

First, notice that my statement pertains to only one item: "P=P." That is because it was Eodnhoj's equation, meant to disprove the law of identity, which was the question in hand. It is not a statement about any other mathematical equation, nor is it a statement about whether another tautology can have any use, as I said earlier.

Now: what is wrong with the claim, "P=P is a tautology," or "P=P tells us nothing new," or "P=P is uninformative of anything new"? All seem perfectly true to me.

P.S. -- It occurs to me, after the fact, that I may have to point out (or perhaps explain for the first time, if you didn't happen to know already) that "P" in logic stands for either "premise" or "predication." Thus it is not at all the same as a mathematical placeholder like "X" or "Y." It refers to a specific linguistic utterance, rather than to mathematical formulation or quantity.

So Eodnoj's "P=P" would be a statement like, "A dog is a dog." It would not imply any mathematical utility, as would "X=X," say.
Last edited by Immanuel Can on Sun Sep 27, 2020 2:17 pm, edited 1 time in total.
Immanuel Can
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### Re: P=P is a Contradiction

Skepdick wrote: Sun Sep 27, 2020 10:54 am
Immanuel Can wrote: Sun Sep 27, 2020 4:02 am It's a prerequisite and essential component of rationality itself. One can deny it, but only at the cost of becoming unable to use rationality afterward.
It's a prerequisite of "rationality" to blindly accept unjustified axioms?
It's the prerequisite of all knowledge to have to accept some axiom or axiom before beginning. It is literally impossible to do otherwise, in fact.
I think I am much better off being irrational then...
Then you have already taken for granted the axiom "It's wrong to accept axioms before beginning," which is itself an axiom you have to accept before beginning, so you can say that.
henry quirk
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### Re: P=P is a Contradiction

Eodnoj's "P=P" would be a statement like, "A dog is a dog."

I think it's more along the lines of: this dog is this dog and no other dog
Skepdick
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### Re: P=P is a Contradiction

Immanuel Can wrote: Sun Sep 27, 2020 2:01 pm It's the prerequisite of all knowledge to have to accept some axiom or axiom before beginning. It is literally impossible to do otherwise, in fact.
You must have a proof of impossibility hiding up your sleeve, otherwise you should probably concede that you are wrong. Again.

Reverse mathematics does exactly what you claim is "impossible".

Immanuel Can wrote: Sun Sep 27, 2020 2:01 pm Then you have already taken for granted the axiom "It's wrong to accept axioms before beginning," which is itself an axiom you have to accept before beginning, so you can say that.
I've done no such thing. Before beginning I rejected all axioms.

Of course, to save yourself further embarrassment, I am going to venture a wild guess you'll (mis)interpret my rejection of axioms as an axiom.

Axiomatics is the poor man’s logic --Jean-Yves Girard
Immanuel Can
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### Re: P=P is a Contradiction

henry quirk wrote: Sun Sep 27, 2020 2:45 pm Eodnoj's "P=P" would be a statement like, "A dog is a dog."

I think it's more along the lines of: this dog is this dog and no other dog
Precisely. You have it.
Immanuel Can
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### Re: P=P is a Contradiction

Skepdick wrote: Sun Sep 27, 2020 3:25 pm
Immanuel Can wrote: Sun Sep 27, 2020 2:01 pm It's the prerequisite of all knowledge to have to accept some axiom or axiom before beginning. It is literally impossible to do otherwise, in fact.
You must have a proof of impossibility hiding up your sleeve...
Not "up my sleeve." In plain sight.

Try it. Even you can't do it.
Immanuel Can wrote: Sun Sep 27, 2020 2:01 pm Then you have already taken for granted the axiom "It's wrong to accept axioms before beginning," which is itself an axiom you have to accept before beginning, so you can say that.
I've done no such thing. Before beginning I rejected all axioms.
Then you started with the axiom," Before you begin, it is best to reject all axioms."

But you saw the flaw in your logic coming, I see. So I'll save myself further concern.
Skepdick
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### Re: P=P is a Contradiction

Immanuel Can wrote: Sun Sep 27, 2020 4:37 pm Not "up my sleeve." In plain sight.

Try it. Even you can't do it.
You are doing it right now!

You are practicing deliberate misinterpretation.

It requires no axioms.

Immanuel Can wrote: Sun Sep 27, 2020 4:37 pm Then you started with the axiom," Before you begin, it is best to reject all axioms."
Exactly! You mis-interpret the rejection of axioms as an axiom.
Immanuel Can wrote: Sun Sep 27, 2020 4:37 pm But you saw the flaw in your logic coming, I see. So I'll save myself further concern.
I sure pre-empted your sophistry. Science, eh?
Last edited by Skepdick on Tue Sep 29, 2020 9:12 am, edited 1 time in total.