# 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:** 4
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 17, 2005, 6:36pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/61 "2005-10-17T18:36:32Z")
It’s still not “infinity”, whatever that is. ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 17, 2005, 6:42pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/62 "2005-10-17T18:42:41Z")
> [@ultrafilter](#): > > It’s still not “infinity”, whatever that is. So you say it’s not “infinity”. What is it? It never ends and it can never be completely written out. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 17, 2005, 6:45pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/63 "2005-10-17T18:45:22Z")
> [@Kozmik](#): > > So you say it’s not “infinity”. What is it? A real number, unlike ∞. > [@](#): > > It never ends and it can never be completely written out. No real number has a finite decimal representation. As a matter of convenience, we usually write .5000… as .5, but that doesn’t mean that those trailing zeros aren’t there. ---
**Author:** ![andros](https://avatars.discourse-cdn.com/v4/letter/a/e480ec/32.png) [@andros](https://boards.straightdope.com/u/andros)\ **Post date:** [October 17, 2005, 6:50pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/64 "2005-10-17T18:50:53Z")
> [@Kozmik](#): > > This is an infinite number. Defintion: it never ends; it can never be written out. > BUT if I subtract it by itself, I know the answer is 0 Ah, you mean it’s an _irrational_ number. Yes, they never terminate and never repeat, and cannot be expressed as _p/q_. It is not an infinite number, but that’s ok. What you pasted in above was _not_ a number, however–it’s an _approximation_ of a number. _e_, _pi_, root2, root3 are all irrational numbers. When decimalized, they do not terminate–they extend for an infinite number of decimal places. Decimalized, however, they are only approximations, and cannot really be manipulated unless they are rounded to a finite number of decimal places. Therefore we can say that 2.71828 - 2.71828 = 0 without problems–an approximation of _e_ minus itself. We can also say that _e_ - _e_ = 0 without problems. However, when we say that 2.71828… - 2.71828… = 0 we recognize that the terms are not themselves numbers, but an approximate symbolic representation of the number _e_. ---
**Author:** ![andros](https://avatars.discourse-cdn.com/v4/letter/a/e480ec/32.png) [@andros](https://boards.straightdope.com/u/andros)\ **Post date:** [October 17, 2005, 6:59pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/65 "2005-10-17T18:59:17Z")
And hey, if you really want to see blown minds, check out the application of the Euler identity in which _x_=_π_: _e[sup]iπ[/sup]_ + 1 = 0 Put _that_ in your pipe and smoke it. ---
**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 17, 2005, 7:28pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/66 "2005-10-17T19:28:43Z")
I think we’ve established that our friend understands neither infinity or what a variable is. **Kozmik** , my advice to you is to get either a good math book or a very patient math tutor. ---
**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, 2:51pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/67 "2005-10-18T14:51:34Z")
> [@andros](#): > > Ah, you mean it’s an _irrational_ number. Yes, they never terminate and never repeat, and cannot be expressed as _p/q_. It is not an infinite number, but that’s ok. > > What you pasted in above was _not_ a number, however–it’s an _approximation_ of a number. > > _e_, _pi_, root2, root3 are all irrational numbers. When decimalized, they do not terminate–they extend for an infinite number of decimal places. Decimalized, however, they are only approximations, and cannot really be manipulated unless they are rounded to a finite number of decimal places. > > Therefore we can say that > > 2.71828 - 2.71828 = 0 > > without problems–an approximation of _e_ minus itself. We can also say that > > _e_ - _e_ = 0 > > without problems. However, when we say that > > 2.71828… - 2.71828… = 0 > > we recognize that the terms are not themselves numbers, but an approximate symbolic representation of the number _e_. **Andros** , you got it! 🙂 ---
**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, 4:46pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/68 "2005-10-18T16:46:12Z")
> [@Kozmik](#): > > **Andros** , you got it! 🙂 Well, yippy skippy for me, I guess. If you had simply wanted to lay out some of the most basic ideas in mathematics, as I did, you could have done so at any time. What’s the debate again? I still have no idea what it was you wanted to discuss in the first place. ---
**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, 4:47pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/69 "2005-10-18T16:47:17Z")
New topic for debate: **Resolved: Have you ever looked at your hands, man? I mean really _looked_ at them? Whoa.** ---
**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, 8:51pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/70 "2005-10-18T20:51:34Z")
\*e \* - _e_ = 0 You know that; I know that. But just to make sure 😉 we have our super computer solve _e_ - _e_ The super computer must complete two calculations; first, the super computer must find out the exact value of _e_ and then, the supercomputer must subtract the exact value of _e_ from the exact value of _e_. KNOWLEDGE: e - e = 0 PROOF: when the supercomputer completes those two calculations Since the supercomputer can never complete those two calculations, we have knowledge without proof. ---
**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, 9:08pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/71 "2005-10-18T21:08:39Z")
> [@Kozmik](#): > > \*e \* - _e_ = 0 > > You know that; I know that. > > But just to make sure 😉 we have our super computer solve _e_ - _e_ > The super computer must complete two calculations; first, the super computer must find out the exact value of _e_ and then, the supercomputer must subtract the exact value of _e_ from the exact value of _e_. No - if you find anyone, programmer or not, who attempts to compute the exact value of an irrational number (and Euler proved e irrational) you fire them. Or worse. BTW, are you scared to respond to me? ---
**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, 9:09pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/72 "2005-10-18T21:09:02Z")
Nope. We don’t need a supercomputer to “prove” anything. I can _prove_ that the square root of five is an irrational number. Similarly, I can _prove_ that the square root of two is irrational as well. In neither case do I need to calculate an infinite number of digits. Without a computer I can prove that _e_ - _e_ = 0: Additive inverses always exist for real numbers (proof available upon request) \*The additive inverse of a real number _n_ equals (-1)\*n _e_ is a real number Therefore, the additive inverse of _e_ is (-1)\*_e_ or -_e_ QED I guess I’m still missing your point. This is all really basic algebra. I promise, no computers are needed to “prove” these things. Would you like the proof that _e[sup]iπ[/sup]_ + 1 = 0? No computer can ever demonstrate that either, but it’s a fact. ---
**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, 9:12pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/73 "2005-10-18T21:12:00Z")
It doesn’t matter what the value of e is: e - e = 0 follows from the axiom of identity. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [October 18, 2005, 9:15pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/74 "2005-10-18T21:15:07Z")
> [@Kozmik](#): > > \*e \* - _e_ = 0 > > You know that; I know that. > > But just to make sure 😉 we have our super computer solve _e_ - _e_ > The super computer must complete two calculations; first, the super computer must find out the exact value of _e_ and then, the supercomputer must subtract the exact value of _e_ from the exact value of _e_. > > KNOWLEDGE: e - e = 0 > PROOF: when the supercomputer completes those two calculations > > Since the supercomputer can never complete those two calculations, we have knowledge without proof. Bull. As has been pointed out to you several times already, statements like “x-x=0 for all real numbers x” can be proven from the axioms of mathematics. It seems your claim is that “e-e=0” can’t be _demonstrated_. But of course it _can_. The demonstration just isn’t as simple as moving pebbles around on a counting board. Instead the demonstration consists of laying out the axioms of mathematics and the rules of logical deduction, and then showing step-by-step how “x-x=0 for all x” follows from those axioms and rules. It’s as much a demonstration of fact as taking three pebbles away from three pebbles to leave no pebbles. The only difference is that you have the option to personally reject the axioms or rules of deduction if you wish (good luck with that). ---
**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, 9:15pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/75 "2005-10-18T21:15:53Z")
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 lack of precision can have big consequences. When I was in grad school there were a few students working on interval arithmetic. They had a grant from the Army Corps of Engineers to evaluate the precision of their computations about how much dirt to move around the Mississippi to keep things from flooding. The analysis showed that they were off by millions of tons - that the number they came up with was practically random. This was almost 30 years ago, and had no part in the recent unpleasantness. ---
**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, 9:16pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/76 "2005-10-18T21:16:57Z")
> [@Liberal](#): > > It doesn’t matter what the value of e is: e - e = 0 follows from the axiom of identity. Don’t look upthread. You’ll get an ulcer. 😃 ---
**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, 9:20pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/77 "2005-10-18T21:20:46Z")
> [@Voyager](#): > > Don’t look upthread. You’ll get an ulcer. 😃 Thanks for the heads up! 🙂 ---
**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, 9:22pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/78 "2005-10-18T21:22:04Z")
> [@Liberal](#): > > It doesn’t matter what the value of e is: e - e = 0 follows from the axiom of identity. …and the definition of =, and the definition of -, and the definition of 0… ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [October 18, 2005, 9:27pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/79 "2005-10-18T21:27:59Z")
> [@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. Actually, I think that is in fact **Kozmik** ’s point (such as it is). Namely: no, computers don’t have infinite precision and therefore cannot demonstrate that e-e=0 just by subtracting one from the other. Which by itself is correct. However, **Kozmik** seems to be leaping from there to the idea that e-e=0 can not be proved or demonstrated by _any_ means, which is preposterous. ---
**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, 9:38pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/80 "2005-10-18T21:38:30Z")
> [@Orbifold](#): > > Actually, I think that is in fact **Kozmik** ’s point (such as it is). Namely: no, computers don’t have infinite precision and therefore cannot demonstrate that e-e=0 just by subtracting one from the other. Which by itself is correct. However, **Kozmik** seems to be leaping from there to the idea that e-e=0 can not be proved or demonstrated by _any_ means, which is preposterous. 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. Actually, since e is (1 + 1/n)\*\*n as n goes to infinity, you could probably prove e - e = 0 by induction on this, but that boils down to something just as trivial as what we’re talking about. [Previous page](https://boards.straightdope.com/t/knowledge-without-proof/326040.md?page=3) [Next page](https://boards.straightdope.com/t/knowledge-without-proof/326040.md?page=5)