One of the great mysteries of the universe for me is how on earth do such threads get to three pages long, when more interesting ones die out before reaching the second page.
If anyone can solve this mystery, I think they will be able to start the next world religion.
But remember, .999 repeating = 1, folks. Right, Kosmik?
Hey, that was my joke, not andros’ (see #99).
His math skills are probably better than mine of course 
Shit, that ought to be good for another three pages.
Weird. I swear I just hit the “reply to post” button. Database error?
Anyway, sorry.
YES!! Now were on to something! How can we know π - π = 0 unless we calculate π out to infinite decimal places? Well, my friend, it’s through the thing that seperates us from the computers; inferrence with a little bit of common sense.
I know because 4 - 4 = 0 ergo π - π = 0. In other words, I know through inferrence.
We can just cite the Perfect Master on that one and call it good =)
No. Inference is not an allowed epistemological technique in mathematics. Either you know something because you have a proof, or you don’t know it at all.
Not at all. It’s a matter of definition. Think of it as n-point-zero. It is exactly n. It is defined that way. If you have a 1 out in the zillionth decimal place, then the number is no longer n. n refers to that number which is exactly n, and no other. How do we know there’s not a wee little 1 waaaaaaaaaaaaay out there in the numerical string? Because the number we’ve chosen to work with has no 1 out there by definition. n-point-zero minus n-point-zero = 0, the same way we know 4 - 4 = 0.
There is absolutely no inference involved. π - π = 0, because we’re working exactly with π, not π + .000000000000000000001.
So, um . . . your point is that humans are smarter than computers?
In other news, new research suggests water might be wet. Stay tuned!
And if I might add, you’re really hung up on numerals. Numbers are more than numerals.
But how do you know he’s perfect unless you take it out to infinite decimal places? 
No computer can define Cecil, because he is infinite.
Cecil’s perfection is one of those most glorious and enigmatic of mathematical concepts: A Given.
Only because you’ve chosen a number base in which that is true. Try doing this math in base-e and none of your examples will terminate. Do it in base-π, and you’ll see that integers don’t terminate, but all the “impossible” examples are nice and clean. The accuracy and validity of simple arithmetic does not rely on our having ten fingers.
Ten fingers? Pshaw. I’m personally able to count up to 21. Explain that with your “science”.
I may be foolish to add my thoughts to this thread, but I believe I see the point Kosmik is trying to make, and even after 130-odd posts think there may be something worth discussing here.
Sticking with the x-x=0 example, many here have correctly observed that this is derivable since we take x=x to be an axiom for all real numbers x. The question, I think, is how did we choose this to be an axiom? Obviously, our motivation for choosing axioms is irrelevant, as long as the axioms are not self-contradictory, derivable from one another, or pointless (e.g. using “all White Sox fans are idiots” as an axiom of mathematics–sorry I’m also a jealous Cubs fan); the only thing that matters is that things can be proved from these axioms, and in fact a set of axioms in one system may become theorems if a different set of axioms were the starting point.
But we are motivated (to some extent) to make the axioms of a system self-evident. I suppose a theory of the natural numbers could still be derived if, say we replaced one of the Peano axioms with a lengthy axiom defining (a+b)^2, but that set of axioms would be criticized because a more self-evident set exists that could give us the same derived theorems (let’s leave Godel’s incompleteness theorem aside for now).
But on what basis do we decide the self-evident staus of an axiom? The self-evidence of x=x, I think, is based on the very definition of equality, the fact that a number has one and only one value always and forever. This ultimately derives from the idea that an object is in fact identical with itself (e.g. something cannot both exist and not exist).
Now we do have examples where two items may have the same symbol but not be equal. An old paradox: “Paris” represents both a city in France and one in Texas. From math, if “Sqr(4)” is the symbol for the square root for 4 (I don’t know how to write the usual radical sign here), it is not always true that Sqr(4)=Sqr(4), since this symbol can represent both 2 and -2.
We get around this paradox by saying we have to be more specific: “Paris” and “sqr(4)” are ambiguous, and we reject their use because it conflicts with our self-evident axiom x=x. By doing this, we have added a (slight) complication to the idea of equality–the distinction between symbols and the things they represent. No biggie, but it could be argued this makes the original axiom a little less “self-evident”.
Turning to the use of irrational numbers in the axiom x=x, I think we see another way the self-evidence of the axiom could be problematic. This is only because of the ambiguity in common notions of “value” (e.g. all values I know about can be written using digits in a finite way). As someone pointed out, the value of pi and e are arrived at via an infinite process, which is unlike the finite processes used to evaluate rational numbers (and geometric irrationals like sqrt(2), though this is obviously done using a geometric interpretation of “value”).
To summarize a lengthier-than-expected post, can the idea of equality be extended to items derived from an infinite process, given that it’s self-evidence is based on our experience with items derived from a finite process? I would answer yes; the self-evidence of x=x is actually based on personal familiarity with a finite number of items derived from a finite process, yet we’re applying it to an infinite number of such objects. I see no difference in accepting the same limitation when there’s an infinite process operating on a single item (e.g. at some point you stop the process, spend a finite time to check they are equal, and move to accepting the axiom for these processes). I guess the notion becomes self-evident after a little exploration, which I would encourage Kosmik to reflect upon.
The axiom this boils down to is one of identity; x-x=0 works because x=x, which is another way of saying:
“Whatever something is, that’s what it is”
How could it be any different for real values?
“Sometimes, a thing is not the same as it is” is self-contradictory.
I don’t personally see how you can get any more basic that this.
Heya, CJJ*.
Except pi, e, and root2 are not derived from infinite processes. Their numerical, decimal values are nonterminating, and can only be “precisely” determined by an infinite process, true. But what they are, what they represent, is a product of a finite and clear process.
What did I miss?
My first hypothesis is that you’re male.