Wikipedia:Reference desk/Archives/Mathematics/2012 March 22

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March 22

Sleeping Beauty problem

Sleeping Beauty problem is described as a puzzle in probability theory and formal epistemology. All the people mentioned in the references seem to be philosophers, there's not a single mathematician. Given the nature of the arguments, may I conclude that formal epistemology is a synonym for "bad at maths"? 84.197.178.75 (talk) 11:26, 22 March 2012 (UTC)

We have a list of philosophers and a list of mathematicians. Except for Bertrand Russell I know of nobody who is both, and so I tend to agree with you. But I am jumping to conclusions, because I have not actually merged the two lists to find the intersection (set theory). Bo Jacoby (talk) 14:03, 22 March 2012 (UTC).

Bo Jacoby: Raymond Smullyan is definitely both a mathematician and a philosopher, and I think he might be doing epistemology as well. – b_jonas 10:31, 23 March 2012 (UTC)

Ha, good one. More seriously, the problem is not really amenable to classical mathematical probability theory. The "paradox" is in determining what SB "ought" to believe. As such, it is a question of how we interpret rationality and belief. This is the domain of philosophy. Consider the basic differences between frequentist and bayesian probability. Mathematics cannot tell us which one is more "real", and we use ideas from outside of math to determine which set of assumptions we may use for a certain problem. This is also relevant for the unexpected hanging paradox discussed above. Philosophers may debate it, but the resolution is not a matter of mathematics. On the other hand, there are problems like the Monty hall problem or the pillow problem (redlink, see here [1]) that are amenable to careful classical probabilistic analysis, and have interesting, counter-intuitive answers. Maybe reading those pages will help you see the difference. Post WP:EC: Bo, it is true that in the modern day, there are few mathematicians that are also philosophers. But that doesn't mean all philosophers are bad at math. I've seen philosophical logicians run (rigorous, formal) circles around mathematicians. SemanticMantis (talk) 14:14, 22 March 2012 (UTC)

I've had philosophy lectures at university, but those were about Turing machines and Gödel. So yes, I can accept that some philosophers can be good at maths, but are those well represented in this field? I don't know what the current trends in philosophy are. It seems that Nick Bostrom is very succesful, selling books in 20 languages, but so do the rapture people and creationists. Is he famous because he's from some fringe movement in philosophy? 84.197.178.75 (talk) 16:00, 22 March 2012 (UTC)

And we also have an article on formal epistemology, though it is very stubby. SemanticMantis (talk) 14:17, 22 March 2012 (UTC)

I think that would be WP:SYNTH ;-) Whereas I do sometimes wonder why am I here now at the dawn of the human race rather than half a billion years in the future which should be much more probable it isn't something I would sit down and write a paper about. For philosophers it is. It doesn't mean they are necessarily bad at maths. Dmcq (talk) 14:23, 22 March 2012 (UTC)

This may be a problem of framing more than anything. If I were the Sleeping Beauty in question and I were given 1:1 odds each time I woke up (and I got to keep the accumulated winnings/losses), I know which case I'd be betting on... — Quondum^☏✎ 14:31, 22 March 2012 (UTC)

I'm having trouble interpreting Self-Sampling Assumption. Let's say you receive a phone call; the person says he flipped a coin and if it was heads, he'd call a random person from the phone book, if it was tails he would call all of them. Then you have two "worlds", and you have a 1/2 chance of being the only observer in world one, and a 1/2N chance of being the observer receiving the k-th call, with N being the total number of observers. So it's 50% chance of heads again? What is the use/significance of it? It's like new-agers or whatever you call them taking a statement from quantum mechanics out of context to support some esotheric theory. 84.197.178.75 (talk) 15:05, 22 March 2012 (UTC)

I think in this case, if you want something that makes sense, you need to assume that there are N observers in each world, except most of them don't receive a call in the first world. So you have a 1/2N chance of being any particular observer. Given the information that you've received a call, there's an N/(N+1) chance you're an observer in the second world. You'd need to reformulate the question so that the people in the first world who didn't receive a call had never been born, or something like that. For example, our guy flips a coin to decide whether to have one child or ten children (with different women, so the children won't know anything about each other). You're told this was your father and what he did, but not what the outcome of the coin flip was. How likely are you to have siblings? 64.140.121.1 (talk) 04:03, 23 March 2012 (UTC)

You might be interested in Anthropic principle. Some of the straightforward probability arguments break down if you get too many budding universes so we just can't make a reasonable argument one way or the other. Dmcq (talk) 11:18, 23 March 2012 (UTC)

See here. I'm mentioned in his article on page 12, footnote 2 :) . Count Iblis (talk) 00:32, 24 March 2012 (UTC)

That's one of the oddest footnotes I've ever seen in a scientific paper. 64.140.121.1 (talk) 23:00, 24 March 2012 (UTC)

I will have a good look at that, thanks. I find it quite fascinating seeing the extraordinary scenarios and theories people come up with! Dmcq (talk) 01:26, 25 March 2012 (UTC)

84, thanks for bringing this problem up. It's been a lot of fun thinking about it. My first instinct was to be a "halfer," but after further thought I've become a "thirder." What was most interesting was realizing that there's no unambiguously correct answer, and that it's necessary to formalize certain assumptions about what we're allowed to infer from the fact we exist. 64.140.121.160 (talk) 04:57, 26 March 2012 (UTC)

Ability to write complex numbers as real + imaginary (x +iy)

Hello, I was thinking about complex numbers and whether they are always of the form (or can be written in the form) $z=x+iy$ .

From wikipedia's complex_number it's almost as if complex numbers are defined to be of that form, there doesn't seem to be a discussion/proof that for arbitrary z one can write it as x + iy.

Does anyone know of a 'pathological' expression containing i that cannot be written as $z=x+iy$? Or perhaps a nasty function of z that cannot be written as: $f(z)=u(x,y)+iv(x,y)$ .

I suppose I'm asking given an arbitrary real function $g(\alpha )$ if we set $\alpha =i$ then can we always write $g(i)$ as $x+iy$ ?

Thanks, Christopherlumb (talk) 20:34, 22 March 2012 (UTC)

This is essentially a definitional statement. We can always write an arbitrary complex number z in the unique form x + iy because the basis { 1 , i } is both orthonormal and spanning over the complex plane. Nimur (talk) 21:52, 22 March 2012 (UTC)

(edit conflict) Moreover, the complex numbers form an algebraically closed field. — Fly by Night (talk) 21:59, 22 March 2012 (UTC)

And, you can continue working towards even greater generality by progressing to quaternions, where the basis { 1, i, j, k } is defined (essentially, these elements are arbitrarily chosen so that they satisfy a particular set of simple relations). I prefer to think of this in terms of a set of just a few simple commuting axioms, explained here. You can continue up the ladder of increasing generality for any arbitrary group of basis elements, and define an arbitrary n-dimensional complex plane... but this starts getting pretty silly and impractical. Nimur (talk) 22:01, 22 March 2012 (UTC)

(edit conflict) No, not always. You can if you work over the Riemann sphere. The function

$f(\alpha )={\frac {1}{1+\alpha ^{2}}}\,,$

causes problems when $\alpha =i$. — Fly by Night (talk) 22:03, 22 March 2012 (UTC)

Not a problem! Simply switch to homogeneous coordinates, which provide a finite value for all coordinates mapping to the projective plane, including ( ± z ∞) for arbitrary complex value z.

Like quaternions, these mathematical techniques have a practical application: they allow engineers and computer programmers to work math in the projective space, which simplifies certain common 3-D graphics calculations. Nimur (talk) 22:28, 22 March 2012 (UTC)

Could you please explain your construction? Simply linking to the article on homogeneous coordinates is of no use. Do you mean identifying the complex number $x+iy$ with the ratio $(x:y)$? If so, then that does not deal with infinity. You end up with a smaller space than you started with, namely the real projective line RP¹ instead of the complex plane C. If you mean the of ratios $(z:w)$, where z and w are both complex then you have the complex projective line, which is exactly the space I mentioned: the Riemann sphere! You need to be very careful; the real and complex projective spaces are quite different. — Fly by Night (talk) 22:50, 23 March 2012 (UTC)

Thank you to all for your replies, I'll have a look into some of the links. This is the peril of coming into maths from a physics/engineering background - not so hot on rings, fields etc.Christopherlumb (talk) 16:39, 23 March 2012 (UTC)