Yes, that is my point.
But that is not a reasonable way to prove anything. Why do you not consider the other proofs given valid? Do you understand them? And computer precision really has nothing to do with it - you couldn’t do the subtraction even on a Turing machine with an infinite tape.
But be careful. Strictly speaking, we can’t let definitions make logical implications. Otherwise, we could prove that pigs fly by defining “fly” to mean “wallow in mud”. Besides, equality is defined within the indentity axiom. And the definitions of zero and subtraction can be formed into a premises as a tautology. I would do the proof this way:
Premise:
Let x = 0
Proof:
-
x = x (Identity Axiom)
-
e = e (Leibnitz Rule: If P = Q is a theorem, then so is U[x, P] = U[x, Q])
-
e - x = e (Definition of Subtraction and Zero)
Conclusion:
e - e = x (Law of Contrapositive: P -> Q <-> ~Q -> ~P)
QED
Actually, number 3 need define only subtraction. Zero is defined in the premise.
π has thus far been calculated to 1.24 trillion places.
We know more digits of π than Leibniz or Newton. Their knowledge of Pi was not dependent on knowing Pi with precision.
My knowlege that π - π = 0 does not depend on knowing π with precision.
Again, what’s your point? What’s the debate?
How can I know that π - π = 0, when I can never know π with precision?
Is is it knowledge based on inferrence such that:
I know that 4 - 4 = 0,
therefore, I know that π - π = 0
Becaues there’s a proof that for any real number x, x - x = 0. This has been pointed out to you numerous times; why haven’t you even acknowledged it?
I do acknowledge it.
For any real number x, x - x = 0, therefore,
3 - 3 = 0
BUT
For any real number π, π - π = 0, therefore,
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706… - 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706… = 0
What do you mean by “But”? But what? pi-pi = 0 is not an exception or an unusual case or whatever; there’s nothing magical about pi-pi=0 compared to 3-3=0 is there?
What is the point, or rather what is the debate?
No computer can represent π with perfect accuracy either.
3.14159265358979… / 3.14159265358979… = 1
Imperfect accuracy divided by itself yields perfect accuracy.
3 / 3 = 1
Perfect accuracy divided by itself yields perfect accuracy.
For any real number x, x - x = 0, therefore,
123456789 - 123456789 = 0
How many digits are in x: 9
For any real number x, x - x = 0, therefore,
π - π = 0
How many digits are in x ?
It doesn’t matter! Something minus itself equals nothing. It doesn’t matter how big, how small or to how many decimal places ‘something’ is.
I’ll just add my voice to the growing chorus of people desiring to know what your point is.
Though don’t I expect that we’re going to get an answer.
Correct.
Correct but no more relevant to this discussion than the color of my socks.
Incorrect. There is no “therefore”; x-x=0 isn’t a step in a proof showing that n-n=0, they are the Exact Same Equation.
As many as you like and it makes absolutely no difference. So long as x (or n or whatever you want to call the variable) is a real number it’s still true. What color are my socks?
If you really believe that then answer this equation:
π + π =
2π.
Solve the last equation so that x + x = y + z
π - π = π / π - 1
3 - 3 = 3 / 3 - 1
3 + 3 = 2 + 4
π + π =
π + π = π/3+5π/3=2π
May we take that to indicate that you don’t believe it to be true? Do you have an actual counterexample (not another question on an unrelated tangent)?
FWIW, yes I do think it’s true and luckily Ultrafilter did the complicated graduate-level work to solve that mathematical stumper. Hopefully a second thread devoted to why 1+1=2 will not be forthcoming.
So returning to pi-pi=0…what about this gives you pause, makes you think it hasn’t been proven true, etc? Is it the fact that pi is irrational?
This is not quite correct; automated theorem proving is a field of study. It is also possible to represent 1/3 with perfect accuracy on a computer. It really depends on your representation. I can represent 1/3 as a pair of integers; I could also use some kind of base-3 representation.