Guys, guys, we’re all forgetting two very important facts here: One, this is a different universe, with different laws, different physics, different mathematics. Pi does not have to be infinite here, as Equestria is fundamentally different from our world, to the point that it is safe to assume that they are operating under different set of universal constants, physics, and mathmatics.
… second, Celestia is basically Horse God, what she says go.
@LuminoZero
Yeah, I could tell that by the way you were trying to argue with mathematical certainties. Saying that pi can be rational is the equivalent of saying that we can’t be positive 1 = 1.
Really? That’s the part of the quote you’re going to ding on historical accuracy? Not the line about aliens existing on the planet? BTW, the point still stands.
@LuminoZero
Uh, but that’s wrong. Everyone already knew the world was round by the time Columbus sailed. Its circumference had been charted with relatively high accuracy, even…using math. Even though they had no way to prove it.
I’ll leave you with a pair of quotes. I know a losing battle when I see one.
“The only fool bigger than the person who knows it all is the person who argues with him.”
-Stanislaw Jerzy Lec
“Fifteen hundred years ago everybody knew the Earth was the center of the universe. Five hundred years ago, everybody knew the Earth was flat, and fifteen minutes ago, you knew that humans were alone on this planet. Imagine what you’ll know tomorrow.”
-Men in Black, 1997
The greatest part of knowledge is accepting the fact that everything you know could be wrong, and maintaining an open mind as you move into the future. If you get locked into the past, you’ll never grow.
I see no evidence that you have understood a single thing we’ve said, or even tried.
Einstein has nothing to do with this. The proof of pi’s irrationality does not rely on authority or trust. Math is not engineering.
The degree to which the mathematical claim “pi is irrational” is contingent knowledge is roughly the degree to which the claim “I am body in a house, not a brain in a vat” is contingent knowledge. It might be false, but if it is, everything we know about the world is so false that the entire corpus of human knowledge is pointless, so entertaining the possibility is a trivial exercise in imagination. Any attempt to make practical use of the possibility founders on the simple fact that situations in which it has any utility are situations in which utility itself becomes an absurd concept. Try engineering a test for the brain-in-a-vat hypothesis, bearing in mind that the supposed simulator can already run physics simulations to the molecule.
Shorter:
Pi is irrational. It has no last digit. Any argument that the contrary is even possible is best suited for pseudo-deep conversations in a college dorm.
@mathprofbrony
I just want to reinforce this, just in case people don’t understand the point here.
Pi never ends.
Pi is a never-ending number
Pi has no last digit
Pi is the ratio of a perfect circle’s circumference to its diameter. The diameter of a perfect circle goes into its circumference a very specific number of times that, in base 10, leaves an infinite, non-repeating string of decimal places. No matter how many digits of pi you calculate, you will never reach the end. It has no end.
Twilight is perfectly capable of disproving her for all the reasons stated multiple times in this thread, which you apparently did not read. Pi has no last digit.
If she decreed by law that pi had a last digit of 2, she would have decreed something incontrovertibly wrong, and it would damage her authority and lead to questions regarding her sanity. Serious efforts to organize a sun-moving unicorn contingent would probably arise.
Equestrian science is probably different from ours. Equestrian math is almost certainly not.
I think the point people are missing here (eggheads, the lot of you) is that Celestia has, in fact, had centuries to compute Pi. At her own leisure, I might add. She has theoretically FORGOTTEN more than anyone else will ever know. She ruled Equestria alone for a thousand years, and I imagine boredom set in after maybe…a year.
She has, in this comic, victimized Twilight with a prank a thousand years in the making. Twilight is incapable of disproving her, and Celestia could never rattle off the entirety of her implied calculation for the same reason.
Furthermore, as ruler of Equestria, it is WELL WITHIN her capabilities to simply DECREE that the final digit of Pi is 2. Once again, NOBODY can disprove her because she has more knowledge in her cranium than anyone can fathom. Not that anyone would dare to challenge her Highness’ Royal Decree.
Remember, the calculations of modern science and mathematics don’t necessarily apply to Equestria or it’s government. All you eggheads have done is prove the math doesn’t work in real life. In Equestria, Celestia could well have WRITTEN THE BOOK HERSELF. Apples to oranges, mates.
Your claims go to the heart of a longstanding philosophical debate over whether math is discovered or invented. Googling that phrase will get you some interesting papers on both sides of the subject. You have taken the side of “invention,” holding that mathematical truths are not objectively pre-existent.
For what it’s worth, I am firmly on the side of “yeah sure invented, but for all practical purposes, discovered.” It’s true, as you say, that humans make up the axioms and theorems necessary to model the physical world. (I will note that much supposedly “pure” mathematics is done with little connection to physics these days, and we are often surprised to find that physics is the more valuable later for the abstractions thus created – this is the problem of “the unreasonable effectiveness of mathematics,” which is an interesting problem on its own.) However, once the axioms are accepted, the theorems must follow, and the axioms are statements like “zero exists” – so fundamental to our understanding of the world that a world in which they weren’t true would be unrecognizeable, literally a drink-the-glass, shatter-the-chocolate-milk chaos.
This is why mathematics isn’t contingent knowledge in the same way Einstein’s theory of relativity is. Look at that proof of the irrationality of sqrt(2). A world in which any possible step in that proof was wrong – maybe one in which integers have multiple possible prime factorizations? – would be a world in which physics stopped working as we understood it. In short, either math works, or magic does. Since we’ve never seen magic working, for all practical purposes, I don’t find that to be a plausible choice.
An interesting discussion, held many times over many coffees.
>>731069t (merged)
I hate to break it to you, Celestia, but that’s impossible. There are multiple conclusive mathematical proofs (dating back to the 18th century) that show pi is an irrational number (that is, a number that cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer). That means that it has an infinitely long decimal form, and that it never falls into a any sort of repeating pattern. So the final digit can’t be two, because it doesn’t have a final number.
I’m not going to dispute your math and calculations, since they are completely accurate. What I will dispute is the idea that something, ANYTHING, is 100% correct and true. There’s a saying I like about Engineering.
“An Engineer can make a productive and successful career by agreeing with Einstein. If you prove Einstein wrong, people will make productive and successful careers out of agreeing with you.”
Science and, due to its direct connection to the branch of physics, math change and evolve over time. The greatest enemy of knowledge is the assumption that what we know will never change. I will concede that math has proven much more stable over time than any science has, but there is also the fact that math is merely the imaginary rule set made up to work with physics.
Understanding math is kind of a cart before the horse problem. Reality doesn’t work because the numbers say so, the numbers work because that is what happens in reality. Much like all physics was birthed from observation, all math was born from the need to quantify what we observed.
When we observe things that are beyond the scope of our mathematical understanding, what then? It’s a big universe out there, and we may have to come to terms with the fact that our system of mathematics is a made-up system created by observing a highly localized set of rules.
If one is trying to prove that a number is irrational by looking at all of its digits and saying “they never repeat,” then the task is futile, as you say. I would certainly agree with that.
However, the proofs that π is irrational are not of that type. They are more like the proof that the square root of 2 is irrational, which we can easily write here:
Suppose sqrt(2) = a/b for some fraction of integers a and b in lowest terms.
Then square both sides of that equality. You get that 2 = a2 / b2.
Now multiply through by b2. You get 2b2 = a2.
Since we said a/b is in lowest terms, a and b can’t both be even integers. That is, the prime 2 can appear in one of their prime factorizations, or neither, but not both.
If 2 appears in the prime factorization of some integer x, than it appears an even number of times in the prime factorization of x2.
But then 2b2 is an integer with an odd number of 2s in its prime factorization, and a2 is an integer with an even number of 2s in its prime factorization. But they’re the same integer. So that can’t happen, because prime factorizations are unique. So 2 can’t be a/b, regardless of the integers a and b.
So while we can only write the square root of 2 to a finite number of digits, sqrt(2) = 1.414213562…, we don’t need to “check an infinite number of digits” to show that it is irrational.
The fact that mathematics can make and prove statements about infinite things even without writing them out explicitly is one of the neatest things about it.
Wait wait what? Statements which have been mathematically proved remain true forever, at least if you accept the underlying axiom systems – and the underlying axioms required to prove that pi is irrational are nothing more than statements like “The empty set exists” and “For every number, there is a number one greater.” It has been proved that for any right triangle with legs a and b, and hypotenuse c, you will always have a2+b2=c^2.
Take a look at https://en.wikipedia.org/wiki/Proof_that_CF80_is_irrational . I would particularly suggest Cartwright’s proof as the easiest proof on that page. The fact that pi is irrational follows from some basic calculus. All you need is integration by parts.
Mathematical statements are not contingent. If proved, then they are true now and will remain true forever. The only things that could go wrong with the theorems are:
(a) there have been logical errors overlooked by every reader in every proof of the irrationality of π for hundreds of years, or
(b) there is a fundamental inconsistency in the system of axioms we use to define mathematical logic, which would be exceedingly surprising since, again, the axioms say things like “you are allowed to take unions of sets” and “every non-empty set of positive integers contains a least element.”
Human knowledge may be incomplete, but logic is, within its sphere, consistent, and the theorem that π is irrational resides within that sphere.
@LuminoZero
Well, see the thing is, I get that people are defending pi as irrational, because you really cant say it’s not. Any idiot can look at a string of numders and say, ‘there’s no pattern here’, but the real argument isn’t in the physical maths, but the theoretical probability. See, we’re talking about an infinity here, and infinity is a sort of schrodingers cat of probability, but wrapped in a psuedo catch 22. The longer a string of numbers, the more possible combinations, but, the more combinations used, the more possible a pattern emerges. The problem is, the only way to prove it non irrational, is to see the pattern, or find the end, and, if it is infinate, then that’s impossible, meaning that, if pi were irrational, it’d be impossible to prove, but (and here’s the catch;) it also litterally CAN’T be at the same time, because, acording to the laws probability, when working with infinity, then the probability of any specfic outcome must be %100, because litterally everything must occur at some point within infinity.
Therefore, the cat can’t be alive, or dead, and, if he is dead, then he has to be alive too.
So, if the proof that Pi will always be irrational, regardless of how many digits it has, is a series of mathematical proofs designed to determine an irrational number.
And the only reason we know it is irrational is because of these proofs. (Should I have written proves there? I’m not sure on the proper plaurization of ‘proof’)
Are you seeing the issue I have with this?
Now, I am not claiming Pi is not irrational or all physics and math are wrong or any such silliness. I am merely reminding people that there was a time when the highest point of science was knowing that fire resided in the heavens, and the proof of this was that the embers from a fire drifted up because they were being called home.
Human knowledge has always been imperfect, and we correct ourselves when we discover our mistakes, eventually. So today we claim that Pi is irrational, and that is fine. Tomorrow, who knows? I mean, the impossible is just physics we don’t understand yet.
