h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Anthea
Belief is very important, you've listed yours.
Time spent philosophising, means less time to make war

i think what i was trying to say came across wrong
belife is fine its when you start killing/ torchering people in its name that is wrong

Don't worry - I'm going to answer the scary questions. Just thought I'd point out a weird paradox I discovered a few years ago. I know some things in mathematics take a stretch of the imagination to understand, but this one hits reality with a sledgehammer. I'm convinced it can't be true. What's more, it's quite simple, everyone here should understand what it means, though its deriviation is probably beyond me.

From 'The Free On-line Dictionary of Computing': -

Banach-Tarski paradox

It is possible to cut a solid ball into finitely
many pieces (actually about half a dozen), and then put the
pieces together again to get two solid balls, each the same
size as the original.

This paradox is a consequence of the Axiom of Choice.

The axiom of choice simply states that you can make a function that allows you to choose an element out of each set in a set of sets - for example: -

f(X) = the smallest element in each subset of X.

For example: -

S = {{1, 2, 3}, {4, 5, 6}, {1.2048, 5.5603, 0.3958}, {23, 1/2, 11}}

So f(S) = {1, 4, 0.3958, 1/2}

The axiom of choice says you can always make a function like f(x) (though it might be different).

Amazing how something so simple can bring about such an extraordinary result!

- Jordan

are the second set of balls hollow

No, they're both solid. That's the weird thing. It completely violates the law of conservation of mass. You have one ball, and you cut it up and put the bits together to make two more solid balls the same size and mass. It's impossible!

- Jordan

Well, Jordan, it just goes to show that you have to be careful about trusting mathematical models of the real world. I would be inclined to distrust the math personally, anyway.

I don't see where it says f(S) = S

Where does it say that the resulting items are all equal to the original?

Or that the axiom of choice is a bit screwy!

Also from the Foldoc: -

"Even if one accepts the axiom, it doesn't tell you how to construct a choice function, only that one exists. Most mathematicians are quite happy to use AC if they need it, but those who are careful will, at least, draw attention to the fact that they have used it. There is something a little odd about Choice, and it has some alarming consequences, so results which actually "need" it are somehow a bit suspicious, e.g. the Banach-Tarski paradox [the one I spoke of]. On the other side, consider Russell's Attic.
"AC is not a theorem of Zermelo-Frænkel set theory (ZF) [what us mathematicians use when we're dealing with sets]. Gödel and Paul Cohen proved that AC is independent of ZF, i.e. if ZF is consistent, then so are ZFC (ZF with AC) and ZF(~C) (ZF with the negation of AC). This means that we cannot use ZF to prove or disprove AC."


Russell's Attic:

An imaginary room containing countably many
pairs of shoes (i.e. a pair for each natural number), and
countably many pairs of socks. How many shoes are there?
Answer: countably many (map the left shoes to even numbers and
the right shoes to odd numbers, say). How many socks are
there? Also countably many, we want to say, but we can't
prove it without the Axiom of Choice, because in each pair,
the socks are indistinguishable (there's no such thing as a
left sock). Although for any single pair it is easy to select
one, we cannot specify a general method for doing this.


I'm shocked sometimes that mathematics will sink so low to demonstrate something so obvious. If you get rid of the axiom of choice, you can't prove it, but if you don't, you get the other nasty paradox.

- Jordan

It doesn't. I'm baffled.

I'm assuming you're talking about the Banach-Tarski problem. It's to do with some pretty involved maths, so far as I can see.

- Jordan

Neither - it's Georg. Just Georg. I knew that, it was a dense misspelling.

- Jordan

well, personally i would say fiction based in fact. hmmm...god exists as an idea and too many people have anthropormorphosised this idea. Like Death, it exists, so does an anthropomorphication of death (see Terry Pratchett!) but we don't believe he actually wanders around with a sythe! Doh!

Jordan. Here's one that has baffled me for a long time. Ever since I first heard of prior probability, I guess. Say we have a bag containing red and blue balls but the numbers of each are unknown. In fact it may even be zero in one case or the numbers may be precisely equal. What is the probability of blindly selecting a red ball? We are tempted to say that it's 1/2. But then someone notices that the blue balls come in two shades, dark and light. Do we now want to say that the probability of selecting red is now 1/3?

Hasslefree
Hi, one of the problems for me in your initial reply was the certitude of your comments on miracles (post 4999) "miracles happen every day around us" and the, what I took to be, patronising tone of your final paras 'we'd all become blase and say 'whatever'...' or something like that. <> one of the main problems with this form of communication for me is the lack of real time connection and also the mechanical difficulty in going back mid-reply to check what somebody has actually said, which wipes out what you have typed.<>
I do not say 'whatever' as a blase response, and I still don't know the combination of numbers that gets me an accent on the 'e'. I feel that the word has been co-opted by people who use it as an answer for everything, that is why I made the point-that for once she had got it right. I do not think that we are going to get an answer that satisfies us I simply think that the explanation 'there'll be an omnipotent, omniscient being behind this hiding from showing himself because, of course, that would deny us the chance to have faith in them, stands to reason dunnit?' seems a little daft. I am a rationalist in the sense that I believe what I feel compelled to believe in an emotional and totally irrational way, and stick with that but try to form these irrational and more than likely totally wrong ideas based on something slightly more substantial than fairy tales and a form of social conscription. One of the main problems I have about belief in a God is that it is a largely taught phenomenon. Children will think that there is stuff going on they cannot fathom but rarely develop a belief in a deity. Or they will base a belief on observed, rational principles, such as 'they feed me-I look up to them'. Even being born Catholic means that should you decide that you do not wish to play on the team you remain for life a 'lapsed Catholic', same applies to Jewish people. Also the act of Christening children, whilst largely social still carries with it the middle ages type guff of 'getting them done because if anything happens to them and they're not done they go to hell' So much of the God stuff seems like a wonderful and elegant hoax, with everything 'interpreted' by us to prove that the entirely missing central figure is there but resting, or hidden , or 'moving in one of his mysterious ways, possibly the butterfly stroke. I know this will upset some of the christians but that is why most organised Christian faiths, at least in my opinion, push Easter as the 'He so loved the world He gave us his only son'season. Basically, in the generally taught version of the story He did a long weekend up there and then came back to life again-not something generally available to your average punter even with BUPA platinum membership. How many Christians do you think celebrate Ascension Day?In the same way your Witnesses rarely call on a house with children and start the conversation with 'how'd you like a Christmas with no presents?', or,for that matter 'join us and bleed to death if you are ever unfortunate enough to need a transfusion' work like a charm that one would. These are my personal moans (and there are many, many more) on the organised stuff but I think we have differentiated (enough) the God existing/organised religion being a touch suspect bit.

Incidentally the full quote you begin with starts "I cannot prove that..." which does kind of change the thrust of the sentence. Given that you don't attempt to change the views of people who believe in something that works for them how on Earth did we get here? Looking forward to more of the same,

Rain.

Sorry, I keep misspelling your name- Hasselfree.

Hi raindog
Being absolutly right does not make you absolutly believeable. Being absolutly believeable does not make you absolutly right.

Both of these are true and verifiable statements, through experience.

Explaining these in the most flowering speech does not assure that others will understand.

If I have carried anything forward from this thread it is that each of us have an opinion, seek to explain it to others, seek to understand others opinion to the best of our ability (apologies to the math guys), and discuss the merit and downfall of each.

And then, to quote a small child, mumble "whatever" into the top of our tea cup, shut off the computer, and go to bed.

Diversity

Jordan. You are still being a little vague, but I guess you are talking about an actual infinity when you say you can add to it and it still remains (only!!!) infinity. OR are you saying that you cannot add to it? Your post seems to assume that these are two ways of putting the same assertion. I'm not a mathematician, but it seems a tad woolly to me.

I guess it depends on the definition of 'add'. Does it mean to make something larger, or does it mean to perform a certain mathematical operation involving at least two mathematical entities? Depending on which of these we accept, the answers come out opposite!

Rain
So I think your tone was cynical and you think my tone is patronising The joy of words !
sometimes it's not what the words say but how they are said.
The current teenage way of saying 'whatever' is done with an attitude and that's what I hear in my head.
With the written word and no body language or tone to our words we can only put our own interpretation on what we read and this can add to the confusion. Unless we
It could equally be said in an agreeable tone that 'whatever 'means 'I agree with you.'
I agree with everything you say about religion, but we are discussing the possibilty of the creation of all things and if there is an instigator or not.
Can the something come from the nothing question?
We all have our own truths, they are the only ones we can reasonably rely upon.
So Rain tell me is there such a thing in the world as love?
Can you give me evidence of it's existance- non existance

Jordan. Is there any reason why we can't subtract from an actual infinity, leaving infinity of course? If so why can't we add to it with a similar result, or can we?

Damn that b*****d God for His book debases me!

Jordan...are both those balls stationary? For, we all know (at least i hope we all know!) that velocity increases mass...

??? ..
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