# 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:** 2
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 13, 2005, 1:27pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/21 "2005-10-13T13:27:04Z")
It is \*not \* self-evident that 3 \* 3 = 9 It is self-evident that 9 / 3 = 3 since 3 \* 3 = 9 It is self evident that 9 \* 3 = 3 since 3 / 3 = 9 but it is FALSE I suppose that symmetry related to equality. x = x Here there is symmetry \*and \* equality; therefore, it is self-evident. I believe that is why this is the most fundamental of all axioms. But are there axioms that are asymmetrical but not equal _or_ axioms that are equal and asymmetrical? Can axioms be self-evident without both properties? Is is logical to speak of an equation being self-evident and TRUE _or_ is it illogical to speak of an equation being self-evident and FALSE? How to put up a good example… hmmm… Ms. Tillingham writes this on the chalk board: 3 \* 3 = Now some lucky student will have the chance to solve it. It is not self-evident, otherwise, it would not need to be solved… 3 \* 3 = This is WITHOUT PROOF AND WITHOUT KNOWLEDGE In other words, this is before Johnny takes the chalk and thinks about the problem. Good for Johnny, he solves the problem. 3 \* 3 = 9 Now you have KNOWLEDGE _WITHOUT_ PROOF. How can we go from KNOWLEDGE _WITHOUT_ PROOF to KNOWLEDGE WITH PROOF? Let’s see: The next day, Ms. Tillingham rolls in the chalkboard with the equation Johnny solved next to the chalkboard with the equation that Mary solved. (Suprise! They didn’t know about this experiement) Marry sees that 3 \* 3 = 9. Johnny sees that 9 / 3 = 3. Being bright students, they find it self-evident that 9 / 3 = 3 since 3 \* 3 = 9. Johnny knows that 3 \* 3 = 9 Mary knows that 9 / 3 = 3. See, Johnny knew the definition for multiplication but didn’t know the definition for division. On the other hand, Mary knew the definition for division but didn’t know the definition for multipication. Between Mary and Johnny it is self evident that 9 / 3 = 3 since 3 \* 3 = 9. On the third day, Rodger see this equation: 9 / 3 = 3 since 3 \* 3 = 9 and he writes 9 \* 3 = 3 since 3 / 3 = 9. See, Roger, knew the proof that 9 / 3 = 3 since 3 \* 3 = 9 and he inferred therefore that 9 \* 3 = 3 since 3 / 3 = 9. Proof without knowledge. How could Johnny and Mary who didn’t know the other equation, proof that 9 / 3 = 3 since 3 \* 3 = 9. And now for the kicker: how could Roger “know” that 9 / 3 = 3 since 3 \* 3 = 9 without knowing that 9 / 3 = 3 since 3 \* 3 = 9? He simply interchanged the mathematical signs. But what is really going on? This question is tricky but it is at the heart of the symmetry problem. 9 / 3 = 3 since 3 \* 3 = 9 There is numerial symmetry: 933339 Then there is equational symmetry: 3 = 3 _and_ 3 \* 3 Then sign symmetry: 9 / 3 \*and \* 3 = 9 Roger supposes that this is a axiom. For he also knows that 4 / 2 = 2 since 2 \* 2 = 4 Rodger see this equation 9 / 3 = 3 since 3 \* 3 = 9 and he writes 9 \* 3 = 3 since 3 / 3 = 9 He says, “My proof is TRUE, I haven’t destroyed symmetry.” Why is symmetry FALSE in certain equations? x = x x = x 3 - 3 = 0 since 0 + 3 = 0 3 + 3 = 0 since 0 - 3 = 0 9 / 3 = 3 since 3 \* 3 = 9 9 \* 3 = 3 since 3 / 3 = 9 ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 13, 2005, 1:54pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/22 "2005-10-13T13:54:39Z")
> [@Mangetout](#): > > I have the nagging suspicion that we’re only looking at the _preamble_ to some puzzle **Kozmik** intends to reveal with a flourish (and possibly an exclamation of “Aha! Then…”) when the requested proofs are given and accepted; Am I grasping completely the wrong end of the stick, **Kozmik** , or do you have something up your sleeve that will appear to prove that x = 0 for any value of x? Your previous intuition was correct. > [@Mangetout](#): > > If we agree that x = x, what is the specific problem with x - x = 0 ? It can never be solved. 3 – 3 = 3 / 3 – 1 Can easily be solved. PROOF WITH KNOWLEDGE x – x = x / x – 1 Can never be solved. KNOWLEDGE _WITHOUT_ PROOF ---
**Author:** ![SentientMeat](https://avatars.discourse-cdn.com/v4/letter/s/43a26b/32.png) [@SentientMeat](https://boards.straightdope.com/u/SentientMeat)\ **Post date:** [October 13, 2005, 1:59pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/23 "2005-10-13T13:59:15Z")
> [@](#): > > x – x = x / x – 1 > Can never be solved. It can be solved, indeed it is solved for all values of x. And it can be proved given the two axioms i) x-x=0 and ii) x/x=1, as I did before. I don’t think you’re using the words “solve”, “prove” or “symmetry” correctly. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 13, 2005, 2:08pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/24 "2005-10-13T14:08:23Z")
> [@Kozmik](#): > > Your previous intuition was correct. > It can never be solved. > 3 – 3 = 3 / 3 – 1 > Can easily be solved. > PROOF WITH KNOWLEDGE > > x – x = x / x – 1 > Can never be solved. > KNOWLEDGE _WITHOUT_ PROOF What, to you, is the essential difference between the two cases? What you seem to be missing is that 1, 3, -, / and = are not mysterious symbols that come down from heaven on stone tablets. They’re defined symbols whose meanings have been determined by human mathematicians, and as such we can prove many things about them. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [October 13, 2005, 3:02pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/25 "2005-10-13T15:02:31Z")
> [@Kozmik](#): > > It is \*not \* self-evident that 3 \* 3 = 9 > It is self-evident that 9 / 3 = 3 since 3 \* 3 = 9 > It is self evident that 9 \* 3 = 3 since 3 / 3 = 9 but it is FALSE > > I suppose that symmetry related to equality. > > x = x > > Here there is symmetry \*and \* equality; therefore, it is self-evident. I believe that is why this is the most fundamental of all axioms. > > But are there axioms that are asymmetrical but not equal _or_ axioms that are equal and asymmetrical? > Can axioms be self-evident without both properties? But axioms don’t have to be self evident. As **Driver8** said, they are merely the starting assumptions. The only requirements are that they not be contradictory and that they be independent of each other, _i.e_ not deriveable from each other. Like **Driver8** I think you are building a big structure on a faulty assumption {axiom) about axioms. And no, 3_3 isn’t self evident. However 3_3 means 3 added to itself 3 times. ||| and ||| and ||| If I put that string of |'s in one to one correspondence with the numbers starting at 1 1 2 3 4 5 6 7 8 9 10 11 … | | | | | | | | | I find that I have 9 things. What’s mysterious about that? ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [October 13, 2005, 3:13pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/26 "2005-10-13T15:13:12Z")
x-x=0 cannot be solved? What’s to solve? Any lone thing, when removed, leaves nothing. x-x=0 is the same as x=x+0, or for that matter, x+0=x+0 I’ll chime in with the others in saying that it sounds like your argument boils down to ‘axioms cannot be proven’. ---
**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 13, 2005, 4:17pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/27 "2005-10-13T16:17:41Z")
Try looking up Peano, [here](http://www.cut-the-knot.org/do_you_know/mul_num.shtml) seems to be an understandable discussion. Given 1 and the successor function, you can define addition and multiplication (and everything else.) So the knowledge in the OP is with proof. I think there can be knowledge without proof, by having enough evidence of something to be certain of it to a desired degree. Math does not have that issue. Axioms are axioms, and theorems are true given the axiom. I’d advise the OP to read up on this (in far greater depth than the link I gave) and be enlightened. ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 13, 2005, 5:43pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/28 "2005-10-13T17:43:01Z")
> [@David Simmons](#): > > 3\*3 isn’t self evident. I wrote that 3 \* 3 is not self evident but, and this is a crucial distinction, what is self-evident is that 9 / 3 = 3 since 3 \* 3 = 9. I will show you why. > [@David Simmons](#): > > ||| and ||| and ||| > > If I put that string of |'s in one to one correspondence with the numbers starting at 1 > > 1 2 3 4 5 6 7 8 9 10 11 … > | | | | | | | | | > > I find that I have 9 things. > > What’s mysterious about that? 3 \* 3 = 9. And we can see that from your example. Now I will show you my example so you can see. 3 \* 3 = 9 It can be shown that, ||| _and_ ||| _and_ ||| = ||||||||| 9 / 3 = 3 It can be shown that, ||||||||| _divided by |||_ = ||| _and_ ||| _and_ ||| (this is tricky to show graphically but the idea is that division groups 9 sticks into groups of three or think of it this way: you divide a whole into parts such as a whole pie into four pieces so that 1/4 + 1/4 + 1/4 + 1/4 = 1) 3 \*3 = 9 since 9 / 3 = 3 It can be shown that, ||| _and_ ||| _and_ ||| = ||||||||| since ||||||||| _divided by |||_ = ||| _and_ ||| _and_ ||| OR ||||||||| = ||||||||| since ||||||||| _divided by |||_ = ||||||||| THEREFORE, ||||||||| = ||||||||| Now let’s take division which created the problem. Remember, though, division is just as natural as addition or subtraction. Lets say you have: | You _divide by_ _||||_ and you get: - - - - 1 / 4 = .25 Suppose that there is only one pie that will ever exist and you divided it. There are only four people that exist. You divide the pie four ways. Joe get 1/4, Mark gets 1/4, Ryan gets 1/4 and David gets 1/4. 1 / 4 = .25 since 1 / 8 = 0.25 since 4 / 16 = .25 since 8 / 32 = .25 … Is it true that .25 = x/x where x/x can go on infinitely? If does go on infinitely we cannot continue that above but we know that that one one 1/4 is the “beginning”, if there was ever a vauge term. ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 13, 2005, 5:56pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/29 "2005-10-13T17:56:03Z")
3 - 3 = 0 This takes up the least bit of chalkboard. 154879874 - 154879874 = 0 This takes up much more chalkboard than the first equation. 4878751387876434544123 - 4878751387876434544123 = 0 Ah, who am I kidding! Here’s an equation that takes up even more chalkboard. I promise you all this - There is not enough chalkboard space in the entire world that could fill this equation: x - x = 0 ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 13, 2005, 5:58pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/30 "2005-10-13T17:58:35Z")
> [@Kozmik](#): > > I promise you all this - > > There is not enough chalkboard space in the entire world that could fill this equation: > > x - x = 0 You just wrote it yourself. Would you feel better if it was written as ([symbol]"[/symbol]x)(x - x = 0)? ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 13, 2005, 6:29pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/31 "2005-10-13T18:29:25Z")
> [@ultrafilter](#): > > You just wrote it yourself. Would you feel better if it was written as ([symbol]"[/symbol]x)(x - x = 0)? Ha Ha! That gave me a good laugh. Alright, I’ll give you a hint. x is not 2, 3, 4 , 5… I promise you it’s a real number but it might be irrational. ---
**Author:** ![SentientMeat](https://avatars.discourse-cdn.com/v4/letter/s/43a26b/32.png) [@SentientMeat](https://boards.straightdope.com/u/SentientMeat)\ **Post date:** [October 13, 2005, 6:37pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/32 "2005-10-13T18:37:14Z")
I give up. For which real value of x does x-x not equal zero? I prepare to be seriously underwhelmed. ---
**Author:** ![andros](https://avatars.discourse-cdn.com/v4/letter/a/e480ec/32.png) [@andros](https://boards.straightdope.com/u/andros)\ **Post date:** [October 13, 2005, 7:41pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/33 "2005-10-13T19:41:38Z")
Yup, I can remember back when _I_ took Freshman Algebra. ---
**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 13, 2005, 8:01pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/34 "2005-10-13T20:01:16Z")
> [@Kozmik](#): > > I promise you all this - > > There is not enough chalkboard space in the entire world that could fill this equation: > > x - x = 0 Does the term “proof by induction” mean anything to you at all? You don’t prove properties of variables by ennumerating the possible values of the variable. Here is a sketch of a proof, (where is **Lib** when we need him?) Basis: 1 - 1 = 0. You can show that on your chalkboard. Induction step: Given that x - x = 0, show succ(x) - succ(x) = 0. succ(x) = x + 1. So, we can rewrite as x +1 - (x + 1) = 0 x - x + (1 - 1) = 0 0 + (1 - 1) = 0 - since we know x -x = 0 0 = 0 - from the basis step. QED. Happy? ---
**Author:** ![Lemur866](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lemur866/32/434_2.png) [@Lemur866](https://boards.straightdope.com/u/Lemur866)\ **Post date:** [October 13, 2005, 8:14pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/35 "2005-10-13T20:14:45Z")
Heh. You mathmeticians, with yer fancy “logic” and book-lernin’. You think you’ve got everthin’ figgured out, eh? Well, there be some things ya’ll will never figgur out. Like, why durn’t a duck’s quack echo, and how can a bumblebee fly when yer “math” sez it cain’t? Answer me that, smart guys. ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 13, 2005, 9:19pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/36 "2005-10-13T21:19:28Z")
> [@SentientMeat](#): > > I give up. For which real value of x does x-x not equal zero? > > I prepare to be seriously underwhelmed. I would be too if x - x _did not_ equal zero. Let me sum up from here: First axiom: x = x Second axiom: x - x = 0 Third axiom: x / x = 1 First principle: Symmetry is integral, 3 \* 3 = 9 since 9 / 3 = 3 Second principle: a prori knowledge is that 3 - 3 = 0 Third principle: knowledge is based on things Forth principle: things are represented by ideas Fifth principle: ideas can exist without things Sixth principle: we can have direct knowledge of things and indirect knowledge of ideas Seventh principle: there are two classes of ideas; abstract and representational, representational are defined such as 3 - 3 = 0, abstract ideas are undefined. Eighth principle: Abstract ideas are either imaginary or real Ninth principle: A thing can never be an idea; the thing we call a sphere can never be a real sphere 10th principle: binary code is the basis of reality 11th principle: omniscience is having direct knowledge of abstract ideas 12th principle: since we cannot be omniscient, it would be impossible to know x – x = x / x – 1 but because this equation can be expressed in absolute symmetry we can prove that whether or not there is a God, we can have proof that we have the potential to reach direct knowledge of abstract ideas > [@ultrafilter](#): > > What you seem to be missing is that 1, 3, -, / and = are not mysterious symbols that come down from heaven on stone tablets. They’re defined symbols whose meanings have been determined by human mathematicians, and as such we can prove many things about them. We can only have knowledge and proof that 3 – 3 = 3 / 3 – 1 We cannot have knowledge and proof that God exists or that “I think, therefore I am” Yet we can have knowledge without proof that x – x = x / x – 1 ---
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 13, 2005, 10:48pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/37 "2005-10-13T22:48:46Z")
I would like you all to consider these three dialouges carefully: I know God exists. _Prove it._ Well, let’s see, there the ontolog- _No, no, no, no! You’ve got to prove God exists using an equation._ I can’t. _Then you cannot prove God exists._ “I think therefore I am.” _Prove it._ Hmm. I have knowledge of my senses which allow me to know myself. _Hold on a minute. You’re not proving anything unless you take out pencil and paper!_ _We can have knowledge without proof_. Prove it. _If we know that x – x = x / x – 1 where x - x = 0 and where x / x - 1 = 0 Therefore, 0 = 0 Thus, it is a self-evident truth that x – x = x / x – 1 and we conclude that 0 = 0 Since we conclude symmetrically in a self evident truth that x – x = x / x – 1 then we can have knowledge without proof. If x was 3 then we can prove the solution; we can write it out on the chalkboard. We know that x – x = x / x – 1 concluded with 0 therefore we know x without proof_ Hey! What do you mean we can’t prove x – x = x / x – 1? _Young man! There isn’t enough chalk in the universe to write out that equation. An infinite number of supercomputers with infinite computing power calculating for an infinite number of years would never prove x – x = x / x – 1_ What are you talking about? I can solve it. \*In hell you will! Because after an infinite number of years of calculating out the first side of the equation, you would still have an infinite number of years to calculate the second half. And even if you only had one side to solve, you’d still never ever solve it! Hahaha!! For all these reasons, we can have knowledge without proof. Since we conclude symmetrically in a self evident truth that x – x = x / x – 1 then we can have knowledge without proof.\* ---
**Author:** ![Lemur866](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lemur866/32/434_2.png) [@Lemur866](https://boards.straightdope.com/u/Lemur866)\ **Post date:** [October 13, 2005, 11:05pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/38 "2005-10-13T23:05:26Z")
Hmmm. I believe I can prove that x-x=(x/x)-1, given your axioms. First axiom: x = x Second axiom: x - x = 0 Third axiom: x / x = 1 x-x=(x/x)-1 x-x=0 so 0=(x/x)-1 x/x=1, if x!=0 so 0=1-1 1-1=0 so 0=0 I proved it! And it didn’t take infiniity time! What do I win? Ah yes, a Godless meaningless existance! I’m soaking in 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 13, 2005, 11:37pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/39 "2005-10-13T23:37:21Z")
I’m pretty confused about what you are saying, but I hope to hell you aren’t on my side. > [@Kozmik](#): > > Let me sum up from here: > > First axiom: x = x > Second axiom: x - x = 0 > Third axiom: x / x = 1 Only if x != 0, remember. > [@](#): > > First principle: Symmetry is integral, 3 \* 3 = 9 since 9 / 3 = 3 > Second principle: a prori knowledge is that 3 - 3 = 0 > Third principle: knowledge is based on things > Forth principle: things are represented by ideas > Fifth principle: ideas can exist without things Buzz! Sorry, you lose. Ideas exist only in brains or on paper (maybe) - in all cases they are associated with material objects. Show me an idea without a physical base. The ideas of love, truth, beauty and justice all exist in our brains. > [@](#): > > Sixth principle: we can have direct knowledge of things and indirect knowledge of ideas Some would dispute we have direct knowledge of things. > [@](#): > > Seventh principle: there are two classes of ideas; abstract and representational, representational are defined such as 3 - 3 = 0, abstract ideas are undefined. > Eighth principle: Abstract ideas are either imaginary or real > Ninth principle: A thing can never be an idea; the thing we call a sphere can never be a real sphere > 10th principle: binary code is the basis of reality Nope. If you knew anything about electronics, everything is analog. Even binary. Some wires inside a computer can be floating, at neither a 1 or 0 state. > [@](#): > > 11th principle: omniscience is having direct knowledge of abstract ideas Sez you. > [@](#): > > 12th principle: since we cannot be omniscient, it would be impossible to know x – x = x / x – 1 but because this equation can be expressed in absolute symmetry we can prove that whether or not there is a God, we can have proof that we have the potential to reach direct knowledge of abstract ideas But we do know it (for x != 0, at least.) > [@](#): > > We can only have knowledge and proof that 3 – 3 = 3 / 3 – 1 > > We cannot have knowledge and proof that God exists or that “I think, therefore I am” > > Yet we can have knowledge without proof that x – x = x / x – 1 I’ve had math profs that would have beaten you about the head and shoulders with a pointer. Want to address the fact that I proved what you claimed could not be proven in finite time? Do you know anything about math beyond the 10th grade level? It is not apparent. And, by the way - are you saying god exists or does not exist? I haven’t a clue. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [October 14, 2005, 12:14am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/40 "2005-10-14T00:14:45Z")
Just out of interest, **Kozmik** , what’s your take on mechanical public suicide machines? [Previous page](https://boards.straightdope.com/t/knowledge-without-proof/326040.md?page=1) [Next page](https://boards.straightdope.com/t/knowledge-without-proof/326040.md?page=3)