# Knowledge without Proof **URL:** \ **Category:** Great Debates\ **Created:** [October 13, 2005, 8:15am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040 "2005-10-13T08:15:46Z")\ **Posts on this page:** 20\ **Page:** 5
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 18, 2005, 9:47pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/81 "2005-10-18T21:47:45Z")
> [@Orbifold](#): > > computers don’t have infinite precision and therefore cannot demonstrate that e-e=0 just by subtracting one from the other Yes, that is my point. ---
**Author:** ![Voyager](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/voyager/32/133_2.png) [@Voyager](https://boards.straightdope.com/u/Voyager)\ **Post date:** [October 18, 2005, 9:57pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/82 "2005-10-18T21:57:34Z")
> [@Kozmik](#): > > 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. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [October 18, 2005, 9:58pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/83 "2005-10-18T21:58:55Z")
> [@ultrafilter](#): > > …and the definition of =, and the definition of -, and the definition of 0… 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: 1. x = x (Identity Axiom) 2. e = e (Leibnitz Rule: If P = Q is a theorem, then so is U[x, P] = U[x, Q]) 3. e - x = e (Definition of Subtraction and Zero) Conclusion: e - e = x (Law of Contrapositive: P -\> Q \<-\> ~Q -\> ~P) QED ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [October 18, 2005, 10:01pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/84 "2005-10-18T22:01:12Z")
Actually, number 3 need define only subtraction. Zero is defined in the premise. ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 18, 2005, 10:24pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/85 "2005-10-18T22:24:24Z")
> [@Voyager](#): > > I didn’t get the impression that he understood about precision, only about that any representation of e as a number does not terninate. I think he’s talking about the computation, not the representation. 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. ---
**Author:** ![andros](https://avatars.discourse-cdn.com/v4/letter/a/e480ec/32.png) [@andros](https://boards.straightdope.com/u/andros)\ **Post date:** [October 18, 2005, 10:34pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/86 "2005-10-18T22:34:27Z")
Again, what’s your point? What’s the debate? ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 18, 2005, 10:47pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/87 "2005-10-18T22:47:19Z")
> [@andros](#): > > 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 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 18, 2005, 11:05pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/88 "2005-10-18T23:05:16Z")
> [@Kozmik](#): > > How can I know that π - π = 0, when I can never know π with precision? 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? ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 18, 2005, 11:31pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/89 "2005-10-18T23:31:34Z")
> [@ultrafilter](#): > > 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 ---
**Author:** ![Valgard](https://avatars.discourse-cdn.com/v4/letter/v/7feea3/32.png) [@Valgard](https://boards.straightdope.com/u/Valgard)\ **Post date:** [October 18, 2005, 11:45pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/90 "2005-10-18T23:45:30Z")
> [@Kozmik](#): > > 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? ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 19, 2005, 12:01am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/91 "2005-10-19T00:01:52Z")
> [@Voyager](#): > > no computer can even represent 1/3 with perfect accuracy 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. ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 19, 2005, 12:12am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/92 "2005-10-19T00:12:18Z")
> [@Valgard](#): > > there’s nothing magical about pi-pi=0 compared to 3-3=0 is there? 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 ? ---
**Author:** ![Orual](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orual](https://boards.straightdope.com/u/Orual)\ **Post date:** [October 19, 2005, 1:01am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/93 "2005-10-19T01:01:09Z")
> [@Kozmik](#): > > 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. ---
**Author:** ![Valgard](https://avatars.discourse-cdn.com/v4/letter/v/7feea3/32.png) [@Valgard](https://boards.straightdope.com/u/Valgard)\ **Post date:** [October 19, 2005, 1:48am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/94 "2005-10-19T01:48:12Z")
> [@Kozmik](#): > > For any real number x, x - x = 0, therefore, > 123456789 - 123456789 = 0 Correct. > [@Kozmik](#): > > How many digits are in x: 9 Correct but no more relevant to this discussion than the color of my socks. > [@Kozmik](#): > > For any real number x, x - x = 0, therefore, > π - π = 0 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. > [@Kozmik](#): > > How many digits are in x ? 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? ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 19, 2005, 2:13am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/95 "2005-10-19T02:13:18Z")
> [@Valgard](#): > > As many as you like and it makes absolutely no difference. If you really believe that then answer this equation: π + π = 🆒 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 19, 2005, 2:19am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/96 "2005-10-19T02:19:50Z")
> [@Kozmik](#): > > π + π = 2π. ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 19, 2005, 2:29am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/97 "2005-10-19T02:29:33Z")
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 = ---
**Author:** ![ShepherdWong](https://avatars.discourse-cdn.com/v4/letter/s/b38774/32.png) [@ShepherdWong](https://boards.straightdope.com/u/ShepherdWong)\ **Post date:** [October 19, 2005, 2:54am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/98 "2005-10-19T02:54:42Z")
π + π = π/3+5π/3=2π 🆒 ---
**Author:** ![Valgard](https://avatars.discourse-cdn.com/v4/letter/v/7feea3/32.png) [@Valgard](https://boards.straightdope.com/u/Valgard)\ **Post date:** [October 19, 2005, 3:08am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/99 "2005-10-19T03:08:24Z")
> [@Kozmik](#): > > If you really believe that then answer this equation: > π + π = > > 🆒 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? ---
**Author:** ![bitwise](https://avatars.discourse-cdn.com/v4/letter/b/ea5d25/32.png) [@bitwise](https://boards.straightdope.com/u/bitwise)\ **Post date:** [October 19, 2005, 3:30am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/100 "2005-10-19T03:30:23Z")
> [@Voyager](#): > > No computer ever can prove these things, even in theory. All computers have finite precision, though it can be very large if you try hard enough. Thus, no computer can even represent 1/3 with perfect accuracy. There are methods, like interval arithmetic, to understand this loss of precision, but you can never prevent it. So are friend **Kozmik** appears to be as ignorant about computers as he is about algebra. 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. [Previous page](https://boards.straightdope.com/t/knowledge-without-proof/326040.md?page=4) [Next page](https://boards.straightdope.com/t/knowledge-without-proof/326040.md?page=6)