## P=P is a Contradiction

What is the basis for reason? And mathematics?

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

raw_thought wrote: Fri Sep 18, 2020 4:18 pm https://en.wikipedia.org/wiki/Taxicab_g ... stance.svg
So you are saying that the green line is not shorter then the red, blue or yellow lines. READ YOUR OWN ARTICLE!
Idiot. I can read it for you. I can't understand it for you.

The green line represents Euclidean distance.
The blue line represents Taxicab distance.

If you are in an Euclidian geometry, then you measure with Euclidian distance.
If you are in a Taxicab geometry, then you measure with a Taxicab distance.

If you are in an Euclidian geometry, but measuring with Taxicab distance then you are measuring wrong.
If you are in a Taxicab geometry, but measuring with Euclidian distance then you are measuring wrong.

HOW you measure "distance" is determined by the geometry you happen to be in.

The Universe we live in is NOT Euclidian, so which geometry are you defaulting to?

Taxicab geometry - Wikipedia 2020-09-18 17-22-44.png (25.47 KiB) Viewed 286 times
Impenitent
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### Re: P=P is a Contradiction

non euclidean space is fun

I love triangles with 3 right angles

-Imp
wtf
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### Re: P=P is a Contradiction

Immanuel Can wrote: Fri Sep 18, 2020 1:22 am I just live for your good opinion.
I do wonder why you can't answer a simple question. You seemed to imply that tautologies add no new information. If I misunderstood your intent, just tell me that. Why are you playing games instead?

If you meant to say what you seemed to, you're wrong; because tautologies are important and often highly nontrivial. Wiles's proof of Fermat's Last Theorem is a tautology; a statement derivable (in principle) from axioms that therefore by Gödel's completeness theorem (not his more famous incompleteness theorems) is true in any model of those axioms.

On the other hand, even if you are saying that P = P is trivial and adds no information, even that is wrong. For example in set theory we can define the empty set as ∅ = {x : x ≠ x}, which implicitly relies on the law of identity. And likewise we can define the universe as U = {x : x = x} and then show that U can not be a well-founded set; so that in the presence of the axiom of foundation, U is a proper class. I just showed two nontrivial mathematical applications of the law of identity. [The axiom of foundation says a set can't be a member of itself].

So just wrap this up. Did I or didn't I misunderstand your quote? What is your thing that you can't answer that? After all if I misunderstood you, you should just set me straight. And if I understood you correctly and showed your argument unsound by virtue of one of its premises being false, you should just thank me for the correction and move on.

And now I've even falsified your MAIN argument that P = P is trivial. It's not trivial at all. It's arguably not even true. "I'm not feeling myself today." "I'm not the man I used to be." Or the fact that every few days all our cells get regenerated into new ones. The law of identity by itself is highly nontrivial; and tautologies are highly nontrivial. Literally everything about your argument is wrong.

But it's ok. Just toss out another insult or two. I'm sure readers will be convinced of your superior intelligence.
Immanuel Can
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### Re: P=P is a Contradiction

wtf wrote: Sun Sep 20, 2020 3:12 am ...tautologies are important and often highly nontrivial.
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."

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.

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 now see that you've been getting excited over a mere misunderstanding of usage. 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.

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.

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?
raw_thought
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### Re: P=P is a Contradiction

Skepdick
https://en.wikipedia.org/wiki/Taxicab_g ... stance.svg
Suppose you are a taxi driver and your boss tells you that since there isn't any street that goes straight from point A to point B it will take longer than a direct route. However, he has devised a mathematical technique that will tell you how long your actual journey will take. Would you be shocked and and say, " this violates common sense geometry!!" The only thing your article is saying is that if you cannot take a direct route ( diagonally ) it will take you more time. It then measures the MORE DISTANT length.
raw_thought
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### Re: P=P is a Contradiction

Actually taxi cab geometry IS EUCLIDIAN geometry! Its just saying that if you cannot go the shortest route because you live on a grid, this is how you can calculate the distance.
raw_thought
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### Re: P=P is a Contradiction

Suppose you are a taxi driver and after your boss explains what your traveling time will be and that a straight line from point A to point B cannot be taken. Would you then say, " WOW! Squares are circles and circles are squares because common sense geometry is out the window!"
Skepdick
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### Re: P=P is a Contradiction

raw_thought wrote: Wed Sep 23, 2020 1:03 pm Suppose you are a taxi driver and your boss tells you that since there isn't any street that goes straight from point A to point B it will take longer than a direct route. However, he has devised a mathematical technique that will tell you how long your actual journey will take.
Would you be shocked and and say, " this violates common sense geometry!!"
I am not "shocked" about anything here.

You keep referring to "common sense geometry" without being explicit.

Which geometry is that?

raw_thought wrote: Wed Sep 23, 2020 1:03 pm The only thing your article is saying is that if you cannot take a direct route ( diagonally ) it will take you more time. It then measures the MORE DISTANT length.
I don't understand why you are using the word "only" here. That is precisely what the article is saying. The particular geometry determines HOW you measure distance.

The geometry determines the paths available to you for getting from A to B.

The notion of "more" distant lengths is non-sensical. You are comparing apples (Euclidian distances) to oranges (Taxicab distances).
Last edited by Skepdick on Wed Sep 23, 2020 3:30 pm, edited 1 time in total.
Skepdick
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### Re: P=P is a Contradiction

raw_thought wrote: Wed Sep 23, 2020 1:11 pm Actually taxi cab geometry IS EUCLIDIAN geometry!
It's not!

In an Euclidian geometry there is only one "shortest line" between two points.
In Taxicab geometry there are multiple "shortest lines" between two points.
raw_thought wrote: Wed Sep 23, 2020 1:11 pm Its just saying that if you cannot go the shortest route because you live on a grid, this is how you can calculate the distance.
You CAN go the shortest route!

What "shortest" means is determined by the geometry!
Skepdick
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### Re: P=P is a Contradiction

raw_thought wrote: Wed Sep 23, 2020 1:53 pm Suppose you are a taxi driver and after your boss explains what your traveling time will be and that a straight line from point A to point B cannot be taken. Would you then say, " WOW! Squares are circles and circles are squares because common sense geometry is out the window!"
I already asked you this question and you ignored it.

Which geometry are you calling "common sense geometry"?

Euclidian geometry is NOT "common sense". It's at least two dimensions short of "common sense".
raw_thought
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### Re: P=P is a Contradiction

https://en.wikipedia.org/wiki/Taxicab_g ... stance.svg
Look at the example from your own link! It shows that the shortest distance from point A to point B is still a single straight line.
Of course if you arbitrarily say that one can only move back and forth and never in a diagonal straight line of course the distance will be longer. I can even claim that I live 4,000 miles from New York, 2,000 miles from New York, 1,000 miles from New York or any distance ( except shorter than a single straight line from my house to New York ) But I hope you see that that is silly.
I can even say that if one can only move up and down and right and left some of the distances ( NOT ALL THE DISTANCES LIKE IN A CIRCLE ) from the center are equal. Anyway, it is obvious that you have not shown me a square circle.
I cannot believe that we are debating if the shortest distance between 2 points is a single straight line!
raw_thought
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### Re: P=P is a Contradiction

https://en.wikipedia.org/wiki/Taxicab_g ... stance.svg
Your example is EUCLIDIAN GEOMETRY! In no place does it deny that the shortest distance between 2 points is a single straight line!
raw_thought
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### Re: P=P is a Contradiction

And since , the shortest distance between 2 points is always a single straight line. Your link does not show a square circle!
raw_thought
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### Re: P=P is a Contradiction

And the authors of that article never meant it to!
raw_thought
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### Re: P=P is a Contradiction

Sure in a city where diagonal lines are arbitrarily forbidden you cannot drive from some point A to point B in the shortest distance. But that doesn't mean that there is no shortest distance. It means that to take the shortest possible distance between 2 points you will have to trespass people's property. That is a question of law, not geometry.