This is why mathematics makes my head hurt... - by catbarf
Jeshibu on 29/9/2007 at 22:38
Quote Posted by catbarf
2. If the universe is infinitely large, there is an infinite number of planets.
No!
Quote:
3. The point is that if you're finding the number of intelligent beings per planet, you divide the number of intelligent beings by the number of planets. If we are to assume that the number of planets harboring intelligent life is finite, being smaller than the total number of planets, we are dividing a finite number by an infinite number. That is where the '0' comes from.
Why is the subset of what you just said was infinite necessarily finite now?
How about: there's a 5% chance of intelligent life per planet (ludicrously high, but simplified). That would mean there are infinity/20 planets with intelligent life, which is still infinity.
Gingerbread Man on 30/9/2007 at 00:35
Quote Posted by catbarf
2. If the universe is infinitely large, there is an infinite number of planets.
You can't
prove that.
Which is the problem. If you can prove that the universe is "infinitely large" -- well,
then you can go on with that premise.
In fact, you'd have to prove
both of those assertions if you're going to go all calculus in the next clause. The infinite size of the universe and the infinite number of planets it contains is pretty much the stark staring error of the whole thing. It's possible that what?
a) the universe is infinite and contains an infinite number of planets;
b) the universe is infinite and contains a certain number of planets (let's say a bazillion) scattered randomly around in an infinite universe;
c) the universe is finite in size and contains a certain, finite, number of planets scattered randomly around in a finite universe.
You obviously can't have a finite universe that contained an infinite number of planets, so there are only three possible outcomes. You rely on the assumption that the infinite universe / infinite planets possibility is either certain or extraordinarily more probable to be true, but you can't prove it. And it's not unreasonable to believe that the infinite universe / finite planets possibility is true, so it's different from arguing whether or not dogs can look up.
Chade on 30/9/2007 at 04:32
Even if there was a universe with an infinite number of planets, where only a finite number of planets contained life, this only means that the fraction of planets containing life is 0. This is a pretty meaningless statement when you have an infinite number of planets to start with.
It doesn't mean that there is no life in the universe. Zero times infinity can be anything. It just means if you pick a planet completely at random, there won't be any life on it.
Tocky on 30/9/2007 at 04:51
Indeed. I might also add I've yet to have a dog that looks down on me but it would be fun to ride one to work.
What if there is an infinite number of universes along different infinite timelines? Put that one in a fraction and mail it to the popes cat.
*Zaccheus* on 30/9/2007 at 09:19
Quote Posted by catbarf
The Mandelbrot set? We learned about this in Computer Programming, here's a link to one you can explore:
(
http://math.hws.edu/javanotes/c12/s5.html)
At the bottom of the page. Pretty cool.
That's the one. I've just finished writing the first version of (
http://www.codeguru.com/forum/showthread.php?t=435227) my own renderer in C#.
What I love about it is the simple formula leading to great complexity. As you can see in the source code, it is REALLY simple.
I've started looking into the maths behind it, they are quite fascinating and not so simple: Mandelbrot was investigating the Julia fractals, J(a,b), which can either be 'wholly connected' or 'wholly disconnected', depending on a and b. So Mandelbrot decided to plot a representation of each J(a,b) by interpreting a and b as a complex number (a + bi) near the origin of the complex plane. If J(a,b) is 'wholly connected' then that point is in the Mandelbrot set. That's what the Mandelbrot fractal is. :)
Briareos H on 30/9/2007 at 10:37
I'd be interested in what an astrophysicist has to say about that. The statement that an infinite set's subset is infinite is true if the subset is dense in the original set. The universe being dense in the mathematical sense ("it's possible to find a finite measure that will include at least one planet within its reach from any point of the universe") would be equivalent to say that the universe is homogeneous and isotropic in the physical sense.
And, oh, aren't we lucky, the Cosmological Principle on which most models are based states that the universe is just that, homogeneous and isotropic (more precisely, it states that there is a scale for which the universe is homogeneous and isotropic).
What it all means is that if we were to follow the Cosmological Principle for an infinite universe, then it would indeed have an infinite number of planets. And this is what I'd like to have more insight in, the applicability of the Cosmological Principle to an infinite universe and its validity as of today.
RocketMan on 30/9/2007 at 16:36
As an extension of what you're talking about, I've read of a theory that says that no particular point in space can be considered "special". This isn't a law, its more of a logical conjecture. So we can't for example say that earth is the center of the universe because of the exceedingly low probability of that being true and the homogeneity we observe. Anyway according to this logic, since we cannot say our local space is special, then every other space must look SIMILAR to our own in order for our space to be non-special. If that is true, then an infinite universe would have infinite planets or whatever else we have in our local space. Don't get me wrong though, you cannot prove that there are infinite planets by some sort of logical reduction alone.
Briareos H on 1/10/2007 at 00:36
on a different note, i call unfair preference on AR master and righteously demand a cock icon above each and every of my posts
you know what cock i am talking about sirs
Peanuckle on 1/10/2007 at 00:44
If you want something else that makes your head hurt, think of this:
According to prevailing theories of aerodynamics, it is physically impossible for a bumblebee to fly. Same for a helicopter. Should not work, but it does!
catbarf on 1/10/2007 at 01:34
Quote Posted by Peanuckle
If you want something else that makes your head hurt, think of this:
According to prevailing theories of aerodynamics, it is physically impossible for a bumblebee to fly. Same for a helicopter. Should not work, but it does!
But... a helicopter uses the equal and opposite reaction from the air pushed downwards to lift it off the ground, IIRC. How does it violate the theories of aerodynamics?
