I was just digging this up on Metamath. Interestingly they don’t use x=x as an axiom since they can prove it with other axioms.
3.14159265358 - 3.14159265359 = -0000000 10 -11
Would you say that both are π? Or that they both represent π? Then π - π = -0000000 10 -11
Why isn’t that right? They both have the same symbol but are not equal. They have two different values for the same symbol.
I’m going to reflect upon that. I hope that you will expand upon this using an example.
I’m not sure I agree that axioms should be self evident. You give some good examples of things they should not be, but the axioms of non-Euclidean geometry are not self-evident, but are still useful. There are the set of theorms you can prove from axioms as one measure of utility, and the match of the system to the real world as another.
I’m not sure x = x is even an axiom. 1 = 1 is, and you can prove x = x from that, I think. (I just skimmed them a while ago, so I might be wrong.)
sqr(x) is not a function, in that it is not 1 - 1. If f(x) (say f(x0 = x+1) is a function, then you can prove that f(x) = f(x). If it is not, as in your example, you cannot. You have to be careful about what you are applying things to, or you get into trouble, as you have shown.
What exactly do you mean by this? I know lots of values you can’t express as digits in a finite way. Remember also that 1/3 is fundamentally different from e.
I don’t think self evidence has much to do with math, though it might have a lot to do with how well you do on a math test. I successfully resisted the urging of the chairwomen of my HS math department to major in math because while I found algebra to be wonderfully self evident, I found integral calculus anything but. I’ve know topologists, and what is obvious to them is way beyond my imaginings. And I think there are some things (like multi-dimensional geometry) that aren’t self evident to anyone. You manipulate the equations with direction from a fuzzy understanding of what you should be getting, and hope for the best. At least that’s what I read.
Of course not.
Because they’re not equal.
This is impossible.
(I apologize if the characters don’t display properly. The only non-numeral in the equations is pi.)
Actually, to me, equality has everything to do with the distinction between a symbol and the object it represents. To me, x = y means that they symbol “x” denotes the same object as the symbol “y”; this is true even if “x” and “y” are complicated expressions like a series or integral. I don’t know enough about logic or the philosophy of mathematics to know if this is the standard view.
(By the way, I believe that most mathematicians follow the convention that sqrt(x) always denotes the positive square root of x. The negative root is denoted by -sqrt(x). So “sqrt(x)” is not ambiguous.)
Because neither 3.14159265358 nor 3.14159265359 equal π.
Also, they are clearly DIFFERENT values.
Koz, what is the hangup with pi here? What you just said is EXACTLY like me saying:
2.001 is pretty close to 2 so I’ll call it 2. And 1.999 is pretty close to 2 so I’ll call that 2 as well. Since 2.001-1.999=0.002 that means that 2-2=0.002.
Pi has a very particular, precisely-defined value. It’s irrational, so you can’t write it down in decimal form on a piece of paper, but the same goes for “one third”. We use that funny little greek symbol because it saves pencil lead. You do not get to redefine it as the mood strikes.
A computer can solve “x - x” or “π - π” to get the result “0”, just like a human, if you program it with the right axioms. Even a decent graphing calculator can do that.
The trick is manipulating the expression as a set of symbols, rather than numeric values. The computer knows that anything minus itself is 0, just like we do, so it doesn’t have to worry about the value of x. Similarly, you can enter “π + π” and get “2π”, or “1/3 + 1/3” and get “2/3”, or “(√2)[sup]2[/sup]” and get “2”, even though the decimal values of pi, 1/3, and the square root of 2 can’t be written out precisely.
Okay, I’m going to try to throw my hat in here.
I believe that you are confusing the computation of a mathematical sentence with the meaning of the sentence itself. Allow me to provide an analogous (though not isomorphic) example.
WHen you were young, and first learning your mathematical facts, you likely performed the computation of 8 + 5 by counting on your fingers, as I did. If someone asked your 8-year-old (or whatever age is appropriate) self, “Hey, Kozmik, how do you know that 8 + 5 = 13?” a possible, accurate reply might be, “I started with eight fingers sticking up. I knew it was eight fingers because I remember what eight fingers looks like. I began to call out numbers in order from one to five, and each time I called out a number, I would stick up another finger. After I stuck up two fingers, I had no more fingers left to stick up, so I put all of my fingers down and then continued my counting. When I reached five, I had three fingers sticking up, and I remembered the ten fingers I had passed by in the meantime, so I knew that the answer was thirteen.”
In other words, the eight-year-old conception of math involves the use of physical objects. In a hypothetical existence where the child is without the use of any of his senses (or, if you prefer, an existence where “things” and “other things” are indistinguishable) and only the memory of 8 + 5 = ?, this computation is impossible for him to perform.
This is the key, however: that the answer to the expression 8 + 5 is independent of the methodology. The child counting on his fingers is not in actuality performing the computation 8 + 5. In reality, he is performing this operation: 8(fingers) + 5(fingers) = 13(fingers) and making the inference from that 8(somethings) + 5(somethings) = 13(somethings), though it is likely that the child does not realize this inference, and quite possible that the child would not understand the difference even if explained to him. His conceptual framework of mathematics depends on the presence of objects; without that, he has nothing.
Mathematics, however, as we understand it, does not depend on the presence of distinct things (though it is likely that our understanding of it does). When I am asked for the answer to 8 + 5 (happens more often than you’d think, I’m a math tutor), I don’t go to my fingers. I don’t even go to the image of fingers in my head. I don’t think about the concepts of “eightness” or “fiveness”, or what it means to have eight things and be given five more things. I know the answer purely by rote. Does that make my answer less correct than the boy who knows it through his concept of fingers? No, we get the same answer every time.
How do I know that we will get the same answer every time, though? By what right do I claim that eight and five always make thirteen? By what right does the boy claim that if eight fingers and five fingers make thirteen fingers, that eight pencils and five pencils will make thirteen pencils? What about diapers? DVDs? Cents? Wives? Lawsuits? Governments? Bottles of cleaning fluid? Some of these I have personal experience with adding eight and five of. Others, I do not. So, by what right can I say that eight and five make thirteen? Actually, I’m not sure. I’d like to claim empiricism, but I remain unconvinced currently.
Thus, at least from my point of view, neither 8 + 5 = 13 nor 8x + 5x = 13x nor 8e + 5e = 13e necessarily come from perfect knowledge. None of them is known any better than another. But, if you accept certain statements either as axiomatic or as provable from other axioms, then there’s no reason to view 1 - 1 = 0 any differently from x - x = 0, as they’re both essentially the same thing.
Here are the statements (some of which may be axioms, and some of which may be provable from other axioms, so I will stick with calling them statements) you need to accept for the following proof:
(S1) ax - bx = (a-b)x
(S2) 0x = 0
(S3) 1 - 1 = 0 (you have stated your acceptance of this one, that’s why I’m making use of it)
(S4) 1x = x
If you have problems with these statements, please state which ones are acceptable and which ones are not. I’ve probably written more statements than I need to for an orderly proof, but I’d rather not spend too much longer on this.
Proof of x - x = 0
x - x
= 1x - 1x (by S4)
= (1 - 1)x (by S1)
= 0x (by S3)
= 0 (by S2)
Algebra is played by certain rules. Accepting these rules is a key of playing the game, otherwise we can’t even get to 8 + 5 = 13. (Eight whats? Have you proven this empirically? etc…) So, if you’re playing by the rules, x - x = 0. If we’re not playing by the same rules, please state the rules that you are playing by, or else this discussion is pointless.
Okay, I think I see where this is going. One of the earlier posters suggested I “put it in my pipe and smoke it” which was what led to my breakthrough. I printed this post on a section of the People’s Free And Liberated Daily from three years ago – an article about a train wreck, with the SuDoKu puzzle on the reverse. The PFALD is printed on hemp paper with soy-based inks but I assure you this did not alter my calculations. Having filled in all of the answers to the SuDoKu with troubling subtracted integers and irrationals, I deemed the puzzle insoluble, and disconsolate, almost discarded it. However, I had need of it for use as a drop-cloth while synthesizing some of Owlsley’s Old Original, and some of the fruits of my labor may have spilled. My toner cartridges were refilled in Berkeley CA by a trusted holistic mathematician and herbalist who assured me that these would stabilize the chi (Chinese, not Greek) of any notes printed out by my printer; again, I hope that the importance of his participation in the process is not lost on you all. Having placed the above admixture in my pipe and smoked it, I have discovered the Straight Dope, and mastered Kozmik’s point, which I summarize hereafter in limerick form:
[sup]3[/sup]√3
⌠
( ⌡*z*[sup]2[/sup] d*z* ) * cos (3π / 9 ) = log [sup]3[/sup]√*e*
1
Which is to say:
The integral zee squared, dee zee
from one to the cube-root of three
times the cosine
of three pi over nine
equals log of the cube-root of e.
Look, the only way x - x != 0 is if x != x. Yes, you can reject the axiom that 1=1 and therefore x=x if you like. But you won’t have a very interesting mathematics if you do so.
Philosophers have long been aware of the quandry the OP finds himself in. So much so they have developed terms to describe them.
Ladies and Gentlemen, I give you A Priori and A Posteriori.
I realize this is a philosophical treatise I’m linking to Kozmik, but it would be less than nine pages if it were printed. By means of reference, this thread, if it were printed, would be ~fifty-nine pages. If you can make it through this thread, you can make it through the essay I linked to.
Enjoy,
Steven
No, because as Metamath points out, it is possible to prove that x=x from the other axiom’s, thus relinquishing the need for it to be an axiom.
It’s quite clear that Kozmik doesn’t understand that we use base 10 numbering in our daily lives because we have 10 digits.
He also does not understand that pi does not have to be written in decimal notation in order to be a number.
Kozmik; PI is the number…that’s it, period, the number is PI. We can approximate it as 3.141592654…on and on in decimal form, but that will always be an approximation. The number pi is PI. That’s it. If we represented PI in base 16, it would be:
0x3.243F6A8885A308D313198A2E0370734… Blowing your mind yet… OH MY GOD THERE ARE LETTERS IN THIS NUMBER!!!
No matter how you choose to approximate it in your chosen number base, the fact is, the number is PI, and by identity, pi-pi=0. Period, done, proven, accepted.
If you don’t understand anything beyond 3rd grade arithmetic, then don’t get involved in these sorts of discussions.
Speaking of number bases, in base PI, would you accept that 1 - 1 =0?
Because that’s the exact (not approximate) same thing as: PI-PI=0.
Guess what? in base 16: 10-8=8. (Boy, I’m really breaking out the advanced math now, aren’t I? :rolleyes: )
If we’re going to expand this discussion (and why not?) to general knowledge theory, it’s a pretty broad topic. Add to what you’ve mentioned “analytic” and “synthetic”, and we have Kant. Plato described knowledge as justified true belief — you have knowledge if there is justification for believing something is true. It was the bedrock theory for a long long time. But lately, other schools have emerged. Here’s some interesting reading:
The cool thing about that limerick is that it gives rise to yet another formula for [symbol]p[/symbol]:
[symbol]p[/symbol] = 6*sum( a[sub]k[/sub] * (-1/3)[sup]2k + 1[/sup], 0 < k ) for an appropriate sequence <a[sub]k[/sub]>. I’ll figure that out later.
To nitpick: 1 is conventionally still one, in pi or any other base. Pi expressed in base pi would be 10.
True, and general knowledge theory is a good topic, but I narrowed it specifically to the difference between a priori and a posteriori because that is exactly what the OP is having trouble with. We are asked to believe π - π = 0 is true even though we can’t work it all the way out and prove it in the real world. If we could work it out on our fingers it would be a posteriori because we would know from experience it was true. Things which can be worked out in the real world with real objects don’t seem to bother our OP. Mathematics when it can’t be clearly mapped to real objects is pure a priori and this is what seems to cause the OP’s frustration.
Enjoy,
Steven
That isn’t a nitpick - it is pretty fundamental. It’s true for every base except base one (my favorite.)
Base one computers are very energy efficient.