Immanuel Can wrote: ↑Sun Sep 20, 2020 4:43 am
But my real interest in doing so was to speak about the law of identity, which is a truth without which no mathematics would be possible. So I'm sure you can get behind the law of identity.
Fair enough?
Yes and no but mostly no. First, I don't actually have much interest in the subject. I agree with you that the law of identity is a law of logic that is assumed throughout math; and so that like a fish never thinking about water, mathematicians never think about the law of identity.
So in this sense I agree with you.
But I found your response somewhat disingenuous; so much so, in fact, that although my natural inclination is to just let this whole subject go; I find that I am compelled to respond.
You never took responsibility for, or addressed my questioning, of your use of the word "tautology." In so doing you are interpreting my objections in a completly wrong manner and totally misunderstanding (accidentally or deliberately) what I'm saying.
Like I say it's not a big deal either way; but you continue to avoid taking responsibility for what you plainly said.
I'll respond to the individual points now.
Immanuel Can wrote: ↑Sun Sep 20, 2020 4:43 am
Oh, I see where you got confused. You thought the word "trivial" had to have meant "useless." It did not: by it, I meant "obvious" or "circular," not "useless."
No, I understood you to say that tautologies are obvious and thereby useless. In which case you are wrong, since Wiles's proof of Fermat's last theorem is a tautology. It's a theorem that holds in every model of the axioms from which it's derived. And since FLT is most definitely NOT trival or obvious in any sense of those words, it follows that you are wrong about tautologies.
But instead of simply acknowledging this totally simple point and moving on, you just change the subject. You IGNORE the fact that I've written you now two lengthy posts challenging your use of the word "tautology." You simply ignore it.
I find this baffling.
Immanuel Can wrote: ↑Sun Sep 20, 2020 4:43 am
If you check, you'll find that there is more than one definition for "trivial" in the dictionary. And the term "trivially true" is an idiom, meaning, "true, but in a way that adds no new information." So when we say that something is "trivially true," we mean that it is true, but not in any way that is surprising.
In your original statement that I objected to, you said that P = P is trivial
because it is a tautology. But that's wrong. Fermat's last theorem is a tautology but is highly nontrivial. So now you're changing the subject.
Immanuel Can wrote: ↑Sun Sep 20, 2020 4:43 am
Now, P=P is as
obvious a truth as one can possibly get. But since the second P adds no new information to the first P, we can say it's "trivially true." Or to put it another way, it's a "truism."
I will say here that I am not a logician, so I take no position on whether the law of identity is true or not, obvious or not, trivial or not. As I say as a math-oriented person I just take the law of identity for granted. But if I put on my logician hat, I don't think it's obvious at all. After all you are literally not the same person you were a microsecond ago. Your cells all regenerate after a few days. And so forth. Or we can say, "I'm not myself today." That is a perfectly correct English usage; yet it
violates the law of identity.
So I don't think the law of identity is particularly true; and even if it is true in some particular domains, it's far from clear to me that it's "obvious." It's something we ACCEPT WITHOUT PROOF in order to get logic and math and science off the ground. I think the law of identity is a rather a profound statement because (1) it is essential to rationality, science, and civilization; yet (2) it is manifestly false!
So there. That's my two cents on the law of identity. But even so ... I am not talking about the law of identity. I don't CARE about the law of identity. I am asking you to defend or explain or correct your statement that the law of identity is trivial by virtue of its being a tautology. That claim is as false as false can be.
Immanuel Can wrote: ↑Sun Sep 20, 2020 4:43 am
I now see that you've been getting excited over a mere misunderstanding of usage.
I'm simply disappointed to watch you change the subject and either deliberately or inadvertently pretend I'm "excited" over something you just made up as you wave your hands and ignore, YET AGAIN, what I am saying to you.
Immanuel Can wrote: ↑Sun Sep 20, 2020 4:43 am
And maybe that's my fault for assuming people would understand the implications of my use of the idiom "trivially true." But I trust this clears all that up.
You mean you thought I would be fooled by your entirely changing the subject and ignoring what I've been saying? Well I guess that didn't happen.
Immanuel Can wrote: ↑Sun Sep 20, 2020 4:43 am
Now, look back at the top of this thread. What is the claim? That P=P is not a truth at all; that it is, in fact, a "contradiction." That is clearly false. It was to that claim that I was referring.
Well that's why I wasn't following the thread. I don't tend to follow some of these, let's call them off-beat threads that pop up on this forum. Which is exactly WHY I responded to what you wrote without bothering to read what came before.
But why should that matter? The paragraph you wrote, where you said P = P is a tautology hence trivial, is false on its face. I somehow I have failed to convey that this is my concern; then the fault is on me for poor communications. So let me just say this again.
When you wrote that P = P is trivial because it's a tautology, did you mean that or did I misunderstand that or are you simply perhaps mistaken about what a tautology is?
Here is the original quote.
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.
You're wrong. You're flat out wrong. Replace the string "P = P" with the string "Wiles' proof of FLT" and you will see why.
Peace brother. I'm sure this is an honest misunderstanding.