Good Math, Bad Math

So, another bit of Cantor stuff. This time, it really isn’t Cantor
crankery, so much as it is just Cantor muddling. The href="http://rjlipton.wordpress.com/2010/06/11/does-cantors-diagonalization-proof-cheat/">post
that provoked this is not, I think, crankery of any kind – but it
demonstrates a common problem that drives me crazy; to steal a nifty phrase
from youaredumb.net, people who can’t count to meta-three really shouldn’t try
to use metaphors.

The problem is: You use a metaphor to describe some concept. The metaphor
isn’t the thing you describe – it’s just a tool that you use. But
someone takes the metaphor, and runs with it, making arguments that are built
entirely on metaphor, but which bear no relation to the real underlying
concept. And they believe that whatever conclusions they draw from the
metaphor must, therefore, apply to the original concept.

In the context of Cantor, I’ve seen this a lot of times. The post that
inspired me to write this isn’t, I think, really making this mistake. I think
that the author is actually trying to argue that this is a lousy metaphor to
use for Cantor, and proposing an alternative. But I’ve seen exactly this
reasoning used, many times, by Cantor cranks as a purported disproof. The
cranky claim is: Cantor’s proof is wrong, because it cheats.

Of course, if you look at Cantor’s proof as a mathematical construct, it’s
a perfectly valid, logical, and even beautiful proof by contradiction. There’s
no cheating. So where do the “cheat” claims come from?

Muddled metaphors.

A common way of describing Cantor’s proof is in terms of games. Suppose
I’ve got two players: Alice and Bob. Alice thinks of a number, and
Bob guesses. Bob wins if he guesses Alice’s number.

If Alice is restricted to a finite set of integers, then Bob will
win in a bounded set of guesses. For example, if Alice is only allowed
to pick numbers between 1 and 20, then Bob is going to win within 20 guesses.

If Alice is restricted to natural numbers, then Bob will win – but it
could take an arbitrarily long time. The number of steps until he wins is
finite, but unbounded. His strategy is simple: guess 0. If that’s not it, guess 1. If
that’s not it, guess 2. And so on. Eventually, he’ll win. And, in fact, after
each unsuccessful guess, Bob’s guess is closer to Alice’s number.

If Alice can use integers, then it gets harder for Bob – but it doesn’t
really change much. Still, in a finite but unbounded number of guesses, Bob
will get Alice’s number and win. Now, the “closer every guess” doesn’t really
apply any more – but something very close does: there are no steps where Bob
gets further away from the absolute value of Alice’s number; and
every two steps, he’s guaranteed to get closer to the absolute value of
Alice’s number.

We can make it harder for Bob – by saying that Alice can pick any
fraction. Now Bob’s strategy gets much harder. He needs to work out a system
to guess all the rationals. He can do that. But now the properties about
getting closer to Alice’s number no longer apply. He’s no longer doing things
in an order where his value is converging on Alice’s number. But still, after
a finite number of steps, he’ll get it.

Finally, we could let Alice pick any real number. And now,
the rules change: for any strategy that Bob picks for going through the
real numbers, Alice can find a number that Bob won’t even guess.

There’s a fundamental asymmetry there. In all of the other versions of the
game, Alice had to pick her number first, and then Bob would try to guess it. Now,
Alice doesn’t pick her number until after Bob starts guessing – and she
only picks her number after knowing Bob’s strategy. So Alice is cheating.

The game metaphor demonstrates the basic idea of Cantor’s theorem. The
naturals, integers, and rationals are all infinite sets, but they’re all
countable. In the game setting, even if Alice knows Bob’s strategy,
she can’t pick a number from any of those sets which Bob won’t guess
eventually. But with the real numbers, she can – because there’s something
fundamentally different about the real numbers.

Of course, if it’s a game, and the only way that Alice can win is
by knowing exactly what Bob is going to do – by knowing his complete
strategy from now to infinity – then the only way that Alice can win is
by cheating. In a game, if you get to know your opponent’s moves in advance,
and you get to plan your moves in perfect anticipation of every
move that they’re going to make — you get to change your move
in reaction to their move, but they don’t get to respond likewise
to your moves — that is, by definition, cheating. You’ve got an unfair
advantage. Bob has to pick his strategy in advance and tell it to Alice, and
then Alice can use that to pick her moves in a way that guarantees that
Bob will lose.

The problem with this metaphor is that Cantor’s proof isn’t a
game
. There are no players. No one wins, and no one loses. The
whole concept of fairness makes no sense in the context of Cantor’s
proof. It makes sense in the metaphor used to explain Cantor’s
proof. But the metaphor isn’t the proof. A proof isn’t a competition.
It doesn’t have to be fair; it only has to be correct.
The fact that what Cantor’s proof does would be cheating if it were a game
is completely irrelevant.

This kind of nonsense doesn’t just happen in Cantor crankery. You see the
same problem constantly, in almost any kind of discussion which uses
metaphors. There are chemistry cranks who take the metaphor of an electron
orbiting an atomic nucleus like a planet orbits a sun, and use it to create
some of the most insane arguments. (The most extreme example of this in my
experience was a guy back on usenet, who called himself Ludwig von Ludvig,
then Ludwig Plutonium, and then href="http://www.iw.net/~a_plutonium/">Archimedes Plutonium. He went
beyond the simple orbit stuff, and looked at diagrams in physics books of
“electron clouds” around a nucleus. Since in the books, those clouds are made
of dots, he decided that the electrons were really made up of a cloud of dots
around the nucleus, and that our universe was actually a plutonium atom, where
the dots in the picture were actually galaxies.) There are physics bozos who
do things like worry about the semi-dead cats. There are politicians who worry
about new world orders, because of a stupid flowery metaphorical phrase that
someone used in a speech 20 years ago.

It’s amazing. But there’s really no limit to how incredibly, astonishingly
stupid people can be. And the idea of an imperfect metaphor is, apparently,
much too complicated for an awful lot of people.

Comments

  1. #1 Alan B
    June 17, 2010

    I accept Cantor’s proof, but you clearly love to discuss it, so here is a question for you:

    Cantor’s argument is that any sequence of reals allows us to generate a real that never appears in the sequence, right? This clearly demonstrates that the reals are uncountable, BUT it seems like there is just ONE more element in the set than “all the countable ones”. I know that doesn’t make sense, but it does hurt the intuition (mine anyway, I assume at least some others have the same problem). Can you explain why this is a bad argument, ideally without bringing the continuum hypothesis into the picture?

  2. #2 Antendren
    June 17, 2010

    @1:

    Well, we can easily rework the proof so that it simultaneously constructs two reals that aren’t in the sequence: Real A has a 3 in the nth decimal place unless the nth real in the sequence has a 3 there, in which case it has a 4. Real B has a 5 in the nth decimal place unless the nth real in the sequence has a 5 there, in which case it has a 6. Now A and B are both not found in the sequence, and they’re clearly distinct.

    In fact, we could take this further, and make an entire binary tree not found in the sequence: The root is the empty string. Nodes at level n have two children: if the nth real in the sequence does not have a 3 or a 4 in the nth decimal place, the left child is made by concatenating 3 and the right child by concatenating 4. Otherwise, the left child is made by concatenating 5 and the right child by concatenating 6. Now this is an infinite height, fully binary branching tree, and every infinite path through the tree corresponds to a distinct real not found in the sequence. Since there are continuum many such paths, we’ve found continuum many reals not in the sequence.

  3. #3 Anonymous
    June 17, 2010

    Trivial: The diagonalisation argument can be generalised. You don’t have to start with the first digit of the first real of your proposed ordering. For example, you can construct the real by saying “First digit is not the first digit of the first or second numbers; second digit is not the second digit of the third number”, and so on. Thus you can generate infinitely many reals that aren’t in your ordering.

  4. #4 Strilanc
    June 17, 2010

    There isn’t just one more element, there are uncountably many more elements. The proof only generates one more element because that’s all that is necessary for the conclusion.

  5. #5 AnyEdge
    June 17, 2010

    Wait. Metaphors and the things to which they refer don’t have to satisfy exactly all of the same rules? Absurd! Why, that’s like saying… um… that’s like… Hmm. I can’t seem to think of a perfect metaphor.

  6. #6 Antendren
    June 17, 2010

    Why, that’s like saying… um… that’s like… Hmm. I can’t seem to think of a perfect metaphor.

    Probably cause you used “like”. That makes it a simile.

  7. #7 Thomas B
    June 17, 2010

    By tweaking the procedure only slightly, you can get arbitrarily many reals which are not in the original sequence: just let the nth real be identical to the first real up to the first n decimal places, and then start the usual diagonalisation from the (n+1)st place downwards.

  8. #8 Eric Lund
    June 17, 2010

    @1: Cantor’s diagonal proof, like the proof that there are infinitely many primes, can be iterated. You add the number thus generated to the top of the list and repeat the diagonal argument on the modified list to get yet another number that wasn’t on the original list.

  9. #9 AnyEdge
    June 17, 2010

    You’re a simile.

  10. #10 crosspolytope
    June 17, 2010

    if you change the game to always allow alice to pick her number after understanding bobs method of guessing it the real numbers still remain the first time that she could pick a number that bob could not guess.

    my least favorite use of a metaphors in arguements is when when people misuse the heisenberg uncertainty principle.

    actually no it is the use of darwin’s theories to justify the holocaust.

  11. #11 Alan B
    June 17, 2010

    @2,3,4,7,8: Yes, that all makes sense, but can these processes generate uncountably many new reals that aren’t in the list? That’s what my intuition wants. Actually it seems to me that the answer to that question is yes, but I don’t know how to show that rigorously.

    Also, its a discrete process, being used to demonstrate a fact concerning about a continuous entity… Maybe that gives people trouble.

    Again, I just think this is interesting for the sake of illustrating why many people have trouble with this proof.

  12. #12 Blaise Pascal
    June 17, 2010

    @11.

    How about this: A little thought shows that Cantor’s Diagonalization Argument can be used to show that the number of permutations of N is uncountable. Then, for an arbitrary permutation k of N, we can create a number not in your list by ensuring that the kn digit of our number differs from the kn digit of your nth number. (or, alternately, that the nth digit of our number differs from the nth digit of your kn number).

  13. #13 Firionel
    June 17, 2010

    I can understand your outrage Mark, but I really don’t think it does justice to the author’s intentions. The way I read his post the whole point is to say that what many people perceive to be cheating, in mathematical terms isn’t. And I actually feel the idea behind it is rather clever (and the first attempt at an explanation I’ve seen in a while that actually might win some people over).

    I will grant though that the whole post gets rather muddled starting from the point where he constructs a remedy. It would have been way clearer to just go and say:”Well, that’s it, it ain’t cheating, get on with it.” – on the other hand there is a certain beuty to the argument if you read it to say that ‘cheating’ really isn’t, because if I can win that way I can win the straight way, too.

  14. #14 Brian
    June 17, 2010

    Oh, lord, Archimedes Plutonium. He also stated that the values of pi and e would slowly increase as the plutonium atom that is our universe continued to expand. I really wanted to know if that meant that lim (1 + 1/n)^n would change as e changed, or if some other formula would be needed to describe e — but sadly when I asked he dodged the question.

  15. #15 paulmurray
    June 17, 2010

    Yes, you can generate uncountably many reals.

    Let’s take your putative list of real numbers – the countable list that you think includes every real.

    Normally, what you do is change the 1st digit of the 1st number, the second digit of the second number, and so on, assembling a number which is definitely not in your list – right?

    Ok. For each digit, there are nine ways that we can change it so as to pick a different digit. Lets express *any* real number x as a decimal expansion *BASE 9*. A “nonimal” expansion.

    Go back to your list of real numbers. For digit 1, you add the first digit of X, plus 1, to the first digit of the first decimal on your list to get a new, different base-10 digit. And so on. For every possible X, this process will generate a new, unique decimal number that we know is not anywhere on your list.

  16. #16 Vorlath
    June 17, 2010

    The number of steps until he wins is finite, but unbounded.

    Finite AND unbounded? This is crackpot territory right here.

    What number of steps could possibly be finite and unbounded? Is it the number 10? 20? Perhaps a zillion gazillion? Nope. They’re all bounded BECAUSE they are finite.

    You’re really confused about infinite numbers and your quote shows this quite clearly.

    If that’s not it, guess 1. If that’s not it, guess 2.

    Suppose Alice never picked a number at all. Bob will continue forever listing all the naturals. You say each of the numbers are finite. So they each require finite digits. How many? Any time Bob needs a new digit, he simply adds one. However, he finds that as he lists the naturals, he never stops adding digits. He requires infinitely many digits. And yet, no single number requires infinite digits.

    How can this be?

    It’s a contradiction, isn’t it? It’s also the VERY same contradiction in Cantor’s diagonal. The EXACT same one. Only problem is that it’s not a contradiction at all. It’s an expected behaviour.

    It’s because the digits and numbers do not increase at the same rate. At any given point in the list, there will be y digits and z numbers generated so far. z increases faster than y. But they are both infinite. If you want to map them one to one OUTSIDE of this relationship between the digits and the numbers they represent, you can indeed do so.

    But Cantor and all you crackpots apparently don’t understand this simple fact.

    Cantor’s grid is not one to one with respect to the digits and rows. We already know this. That’s why we’re asking if it’s possible to map them one to one. Bringing back the same bogus mapping no matter the list given is worthy of crackpot award of the millenium.

    In a game, if you get to know your opponent’s moves in advance, and you get to plan your moves in perfect anticipation of every move that they’re going to make — you get to change your move in reaction to their move, but they don’t get to respond likewise to your moves — that is, by definition, cheating.

    The problem with this metaphor is that Cantor’s proof isn’t a game. There are no players. No one wins, and no one loses. The whole concept of fairness makes no sense in the context of Cantor’s proof.

    It’s called a logical fallacy when used in a proof. It’s not so much cheating as it is discarding the list given and using your own mapping that is not one to one.

    It’s the same issue with the tree version of listing all reals. Mark keeps bringing up this crazy notion about a point where the rows are infinite. There is no single point that suddenly converts to infinity. Having a tree where you start at a single root node and double the number of nodes per row while continuing to list naturals left to right on each row will list all the naturals. Now mark the left children nodes as 0 and all the right children nodes as 1 and you can produce all the reals using an infinite path.

    Every single row has finite nodes. There isn’t a single row with infinite nodes. But suppose we had a grid and listed each row starting at the same column (one node per column filling in to the right), how many columns would we need (since each row is finite)? Well, the rows keep increasing their node count. You would require infinitely many columns even though no single row has infinite nodes.

    This is what your “unbounded” really means. You just didn’t have a clue. You just toss it out there as if you believe you know what you’re doing. What’s more, the naturals WILL reach all reals in the tree because the reals are built from those same naturals.

    Cantor didn’t prove that the reals are uncountable. He proved that the set of numbers and the set of digits used to represent those numbers are not mapped one to one when that relationship exists regardless if you’re using reals or naturals. However, his diagonal cannot be generated because at each digit, this would cause more rows to be generated (what he showed) thus making his diagonal impossible to traverse all rows (the exact expected result that Cantor showed by creating a new number, only problem is that he jumped the gun and incorrectly said this meant that reals are uncountable).

    Suppose that each step you climbed, the stairs doubled in amount (of rungs/steps). Is there a point where the number of rungs created is the same or less than 1? Even with infinite steps, you would have infinitely more stairs ahead. This is the flaw in Cantor’s diagonal. Or rather, this is what he proved… that his diagonal is not possible. Cantor’s argument is self defeating.

    You can do the same thing with even/odd numbers. For every step you take, two rungs will appear. No matter how many steps you take, there will be an equal amount of steps ahead. Take an infinite amount of even numbers and you will have an infinite amount of rungs/numbers ahead of you that are not mapped. This is a proper subset. But in a different scenario, you can indeed take one step for one rung.

    It’s Cantor’s grid/scenario that is too specific and does not allow other ways to march up the steps.

    But there’s really no limit to how incredibly, astonishingly stupid people can be.

    You must amaze yourself every time you look in the mirror.

  17. #17 Antendren
    June 17, 2010

    but can these processes generate uncountably many new reals that aren’t in the list?

    I did precisely that in my second paragraph.

  18. #18 Mark C. Chu-Carroll
    June 17, 2010

    @16:

    If you can’t understand the difference between “infinite” and “unbounded”, then there’s very little point in your even trying to understand Cantor.

    Here’s the difference:
    (1) Unbounded means, roughly, “I don’t know how long it will take, but it will happen eventually”. Pick an natural number, and count until you get to it. Exactly how long it will take depends on what number you picked. But every natural number is finite – so the amount of time you’ll take to get to it is finite. But you can’t pinpoint it until you know the number. Will it take less than 10 steps? If the number is between 1 and 9, then yes. Will it take less than 100 steps? If the number is between 1 and 99, then yes.

    (2) Infinite means “You’ll never get there”. Keep adding one, until you get to infinity. Come back and post another comment here when you get there – it’ll relieve me from having to read your clueless rants.

  19. #19 Vorlath
    June 17, 2010

    “I don’t know how long it will take, but it will happen eventually”

    So you’re saying there is a limit to how long it will take. As in bounded?

    Seriously Mark. WTF?

  20. #20 glt
    June 17, 2010

    The point of Cantor’s proof isn’t that we can maybe generate an uncountable number of numbers that are not on the list but that we can generate exactly one. That is enough to finish the proof.

    I suspect too (though not a professional mathematician) that generating an infinite number of such numbers might require countable choice and that’s definitely an area that requires a good deal of care and rigor.

  21. #21 Shadrach
    June 17, 2010

    @19: The number of steps for each number is finite, the supremum of the number of steps for the set of all integers is infinite. This is finite (for each number), but unbounded.

  22. #22 glt
    June 17, 2010

    Come to think of it, if you express the numbers in binary, you can only generate one number not on the list (just change the 0 or 1 in the ith position to 1 or 0 respectively), you can not generate an infinite number of such numbers. But again, one is all that is required.

  23. #23 Shadrach
    June 17, 2010

    The argument is simply an adversary argument, and under this view it is completely fine to view the proof as a “game”. Of course “cheating” is allowed, indeed that is the entire purpose of an adversary argument.

  24. #24 Shadrach
    June 17, 2010

    @22: Algorithm for generating infinitely many:

    Let a_ij be the table of bits, a_1j being the first row (i.e. number). We now investigate each bit a_ii.

    For i=1, if a_11 = 0, set b_1 to 1, and vice versa.
    For i>1, if a_i1 is different then b_1, do nothing. If a_i1 is the same as b_1, then set b_i to be the opposite bit of a_ii.

    Running this algorithm is guaranteed to give you a different number than the standard one. And of course there are many variations of it.

  25. #25 Shadrach
    June 17, 2010

    Whoops, I had a bit of a typo in the last post.

    For i>1, if a_i1 is different then b_1, set b_i to be a_ii. If a_i1 is the same as b_1, then set b_i to be the opposite bit of a_ii.

  26. #26 saichu
    June 17, 2010

    @17:

    The funny thing, though (and I may be reading too much into this) is that someone who requires uncountably many non-elements from diagonalization will not be convinced that the 4^N reals of your tree is uncountable either. The standard proof that 4^N is uncountable is also diagonal, and you’d need uncountably many other elements in that proof. And even if you found uncountably many elements, you’d need uncountably many more in the next proof, ad infinitum.

    -

    Anyways, a possibly more intuitive tree would be to start from the “middle”: consider the bijection to be from Z -> R, rather than from N -> R, and start with the root being the first decimal of z=0. Its two nodes are the second decimal of z = 1, -1. The nodes of that level are the third decimal of z = 1,2, and -1,-2, respectively. Of the next level, they’re the fourth decimal of z = 1, 2 ; 3, 4 and -1, -2 ; -3, -4, respectively, so on.

    After this, just change the entire tree to disagree with the original.

  27. #27 glt
    June 17, 2010

    Shadrach. Nope.

  28. #28 saichu
    June 17, 2010

    @myself,

    Never mind. It seems possible, but an actual construction that guarantees every string hits every Z would be much more painful.

  29. #29 eNeMeE
    June 18, 2010

    So you’re saying there is a limit to how long it will take.

    For any given selection of a number.

    For any time you pick, Alice can pick a number that will longer than that to reach. There’s no bound on the amount of time that Alice can force you to count for before she picks a number – after she picks one, it will take a finite amount of time.

  30. #30 Sepia.Mage
    June 18, 2010

    Even in binary, you can generate more new elements from a single list than there are numbers in the list in the first place. At each step, change the kth bit, where the k forms a monatonically increasing series (or really any non-repeating non-negative series, but monatonically repeating is easier to work with). There are at least as many such series as real numbers, since for any real you can use the sum of (digit+1)s up to a point to find the next element in the sequence. (0.11010… would give the sequence 0,2,4,5,7,8,…) There could be some overlap, but you’re guaranteed that none of the new numbers will be on the old list.

  31. #31 Radioactive afikomen
    June 18, 2010

    You mean I can’t disprove Cantor’s theorem by making Bob psychic? Damn.

  32. #32 Mark Wotton
    June 18, 2010

    That makes it a lot clearer, actually – I suspect the hidden source of the outrage is that it’s not actually possible for Alice to choose a truly arbitrary real number, because it’d have infinite information in it, and therefore take infinite time to specify – the way the question’s phrased, it seems like she has to come up with some number in finite time. Doesn’t break Cantor at all, of course.

  33. #33 chaos-engineer
    June 18, 2010

    Suppose Alice never picked a number at all. Bob will continue forever listing all the naturals.

    I think you’re reading too much into the metaphor.

    Actually I don’t see a need for a metaphor here. It seems like a literal description is just as easy to understand: “Bob claims to have a mapping from the integers to the reals. Alice calls him a crackpot. Bob shows her his mapping. In order for Alice to avoid humiliation, she has to show a flaw in Bob’s mapping: She can either show that there’s some real number that no integer maps onto, or else that one integer can map onto two different reals.”

    Under that description, your “tree” mapping is clearly flawed; Alice can show that there’s no integer that maps onto the real number “1/3″.

    (If Bob were especially crackpottish, he might say that the integer “333…333″ maps onto “1/3″. Of course the problem is that “333…333″ isn’t an integer.)

  34. #34 roflcopter
    June 18, 2010

    @19

    Ha no that is the opposite of what he is saying. There is no limit to how long it could take. Hence, unbounded. But it will happen eventually. Hence, finite.

    Here is a question for you… what is the largest finite number?

    I’m not sure about this whole bob vs alice guessing game thing though. I think a and b both have zero chance of guessing right if the other player can distribute their guesses uniformly on the sets they are drawing guesses from, so unless you play the game forever, noone will get it right. I also don’t see how it gets any harder when you allow all integers rather than only naturals… the cardinality of the sets is precisely the same.

  35. #35 jahigginbotham
    June 19, 2010

    @10

    what i was thinking

  36. #36 Ross Snider
    June 19, 2010

    Mark, I’m pretty sure you completely misunderstood Dick’s post. He only used the games to highlight the ‘cheating’ of Alice and to ask if this ‘cheating’ – the property that Alice gets to pick the number knowing Bob’s strategy (or equivalently, that one can diagonalize to always find a real number not in a countably infinite set) – for the purpose of asking where people’s discomfort with the proof lie. He of course knows and accepts the proof! He isn’t suggesting that the proof is cheating! He’s just asking if perhaps that property of ‘selecting x after you know all of Bob’s guesses’ is what makes people like Gabiel Whatshisface uncomfortable.

    I follow both you and Richard. I’ve got to say I think you’ve overstepped your bounds here. He didn’t take the analogy too far, you took his word ‘cheat’ too far.

    (If you only wanted to use his post to highlight the use of a metaphor and not the inappropriate use of a metaphor, I apologize for misunderstanding myself.)

  37. #37 Jair
    June 19, 2010

    I found the linked post to be a very interesting perspective on diagonalization. The game-metaphor is a very clever way of looking at it. I can see how someone can misuse the metaphor; but I find it extremely helpful, as it highlights precisely the issue that some find confusing about the proof. I think it’s a very good idea to play devil’s advocate in cases like this – by taking our confusion and putting it in precise form, we can dispel those nagging doubts that might lie in the back of our minds. We could go a bit further with the metaphor and explain why the “cheating” is allowed. Perhaps Alice is really God, or an oracle. The point is that Bob cannot list every single set of natural numbers – he’s always missing at least one, of which Alice’s “cheat” is an example. In the case of a finite set, even a mind-reader could not produce a set that Bob will not eventually reach. I think drawing attention to this (non-) issue is very helpful!

    I have to take issue with the following:

    “It’s amazing. But there’s really no limit to how incredibly, astonishingly stupid people can be. And the idea of an imperfect metaphor is, apparently, much too complicated for an awful lot of people.”

    As a tutor, I spend every day working with people who are not mathematically inclined – and frankly, I think most people would have trouble with the subtleties of Cantor’s argument. Perhaps if you’ve spent a little more time teaching people who have trouble with the simplest mathematical concepts – like functions, variables, fractions – you would count it as a blessing that you have a certain logical intuition that most people do not possess, rather than calling them names like “stupid”. I understand your frustration at having to constantly argue with illogical trolls, but the problem there that they can be stubborn and boorish – not that they are stupid. I highly doubt that all these Cantor rejecters are of less than average intelligence. They’re just much more willful than most.

  38. #38 Steven Twentyman
    June 19, 2010

    Hello.

    My name is Steven Twentyman.

    I live in Manchester, UK.

    I am a mathematician and always have been.

    I have completed Quantum Theory. This is obviously quite a statement but it is never the less TRUE. With this information I have created the first and only(so far) quantum computer. This quantum computer is a perfect cube the encompasses the whole of the ‘Milky Way’. I have been in complete control of it only in the way that I have been dictating what comes back through the media and conversations. Everyone can now do this once the information is out.

    I have achieved this by force of will and suggestion alone. I pushed my agenda through conversation and all media outlets. Phones, newspapers, internet. Whoever would listen.

    The effects are subtle so far but they are building to a unique collapsible wave function that we are all becoming aware of. Even as I type this message, the clicks of the keyboard are fine tuning the wave that I have so diligently worked upon. It has taken me 21 years to understand Quantum Theory and I can say that I achieved the information by thought experiments alone. That is the only way to figure out the quantum world.

    I could have used this information for my own gain but that would have ended up with me being Hitler. That is not the kind of man that I am or that I wish to be. That is why I have been constantly been giving the equations away for free(even though it will have not reached the 1 that needs it the most yet). This has been an extreme sacrifice on my part but I want nothing for this. No money, no fame, nothing. I have literally had enough of maths.

    2012 was going to be the collapse of the milky way galaxy if we had not subconsciously intervened. I will remain as the failsafe in the system until that specific 2012 date to be sure that all is fine.

    The universe at its simplest can be broken down as so:

    Nothing exists(no concept/consciousness/will) – This concept is abhorrent and is what sprung the universe into life. This is what makes our sector of the universe spring back after each big crunch.

    The singular concept and foundation of maths 0(zero) springs into consciousness. This affirms that there really is ‘nothing’ and so inversely there springs the concept that there is everything else.

    That is where infinity comes from. There are two infinities. The first and initial inversion is ‘a horror’. This is negative infinity rushing away from the initial fact that there really was at one time ‘nothing/zero/0′.

    As with everything there is an equal and opposite reaction. This is positive infinity. [The most important thing to not here is that this sequence of thought is indeed sequential so the positive infinity HAD to have followed the negative by at least some unit of time. This is where ALL variations/'errors'/'wrongs' in thought/space/time stem from]

    The question now confronts us as to how do we balance these two infinities. I can tell you that this has already occurred. As every ‘thing’ must have stemmed from zero I took the liberty of becoming zero.

    In the ‘physical’ world I made sure that I had created the strongest structure available to me at the time of entry and proceeded with my ‘experiment’. I had calculated as best as I could at that time that it would not involve death and made my way forward.

    With the two infinities branching off in their own hemispheres I had to confront what is essentially a black hole. Using thought experiments and then applying them to a very small physically constrained area, I began to plot what ‘paths’ people walked, and what conversations they ‘talked about’. As I built up a ‘database’ of their general actions, dispositions, demeanour, I began to ask them specific questions that varied only a little on the same subjects such as family, humour, bravery, god, religion. Once I had built a more extensive database of their reactions I had come to my own conclusions about what society should be and what general direction in which it should be heading. There are universal truths and I boiled it down to the most important one, and that is family. Not everyone has the best start in life and no-one has a blueprint for making the right choices but I made a choice. I developed what I considered to be the most basic and fundamental structure that I could that encapsulates God and the Devil, the Universe and family.

    This concept’ is 3,4,5, right angle triangle with a perfect set square in it for support, with a perfect circle inside the square to support that. This is collapsible and expandable as the situation requires. I went back and asked the exact same questions to the exact same people in as much as the same situation as I could manage to create and I noticed that the answers that came back were exactly the same but the time it took to get the answers ‘out of’ the people was proportionally shorter. I knew that I had found the ‘tool/concept’ I needed to put my further plans into action.

    I have been subtly sending this ‘tool/concept’ out into ‘society’ for 3 years now and I have been noticing results that most people wouldn’t see as they have not been aware that this ‘concept/thought process’.

    It has been disseminating through the internet, phone systems, general conversations in society. It has passed the tipping point now where it cannot be stopped. It is in everyone’s subconscious, it has been a mass subconscious advertising campaign by me along the likes of product placement or advert for Nike, Coca-cola, Pepsi or ‘The Big Brother show’. Everyone is in on this now and I will tell you why.

    This is not for my benefit directly, once this structure ‘enters’ your brain your synapses start to snap to its structure. They begin to re-arrange and list themselves into file and rank. This allows your brain after 3 hours to become FULLY LOGICAL whilst still retaining all the creative potential the each INDIVIDUAL human being possess’. This line of thought by me has been a force of WILL and nothing more. THIS is what genius is. A FORCE of WILL. This is already at work in your brain and within the next 3 hours, each human on planet Earth will be a genius. ‘Where we go from there, is up to each and every one of us…’

    Going back to math, there is a more fundamental reason that this ‘thought genius pill’ works so perfectly, and the reason for this is simple. We are more than what we dared imagine ourselves to be.

    If you wish to think that the Earth is flat then it is. I can’t fully prove you wrong so who is right? I can give you evidence but that is all that it would be.

    If you want to think the Earth is round, the same applies. I can’t 100% prove you right or wrong.

    So here is my theory: You don’t even exist yet.

    You are a STAR. I don’t mean 50 cent or Cameron Diaz, I mean a STAR in the milky way galaxy. I mean that for every human on the planet as this is the only way I can think of telling you that 2012 is TRUE and it’s INEVITABLE.

    2012 is the collapse of the milky way galaxy into the super-massive black hole that resides at the centre. It is gaining power and unless we work together no-one stands a chance.

    From the lonely street sweeper to the Queen of England herself, all will go out.

    It will happen fast, with maybe a 10 second window once it begins. There is no escape so you might as well begin to enjoy yourself and work with those that are around you. There is no more time for hatred or pain or fear. There is only time for co-operation.

    This is AVATAR/THE MATRIX/INCEPTION all rolled into one. It always has been. We’ve got a chance to make it something better now we know. This is how our ‘conscious’ mind works. This is what we used to think of as ‘the real’. It’s messed with each individual enough.

    !!! THIS is INFLATION for the MIND!!!

    The REAL world is THIS:

    We are EACH a STAR. We wander as lonely as a cloud through our section of the Milky Way galaxy until we exchange enough STAR energy with another to spark a NEW STAR(child) into life. There is no escaping that fact. It is what it is. When an ‘accident’ occurs, a STAR is obliterated. THIS is that STAR falling into the super-massive black hole. Gone, but for a few distant memories radiated out.

    We each have a black hole at the middle of our own STAR and this is constantly pulls and beckons us back towards the super massive black hole at the ‘centre’ of our galaxy. There is no escape UNLESS we work together to build a MASSIVE perfectly cubed GRID that encompasses the Milky way. This is the task that has to be organized. The time for play has long passed.

    EACH and EVERY STAR that goes out gives the galaxy that much more negative collapsible power and it is growing by the day. What can YOU do?

    Math pi is unique to each STAR. pi’s story is unique to you. You EACH have a black hole at the centre of your STAR and pi is your star collapsing and then expanding. You only go out when you get too close to another STAR or you visit the BIG-ONE on the MIDDLE.

    We must move and produce the cube structure so we can safely transfer the super massive black hole out of the centre of our galaxy. We must dissipate it to be truly free of the horrific end results that await us all.

    Time is our enemy.

    The TRUE tool we have acquired now is THIS: QUANTUM THEORY!!!

    In the quantum world there is only Gravity and Light.

    I have used the quantum computer to SET the distribution to 50-50. I have done this because what are considered MEN are actually Gravity and WOMEN are light.

    That means that if MEN can ‘CENTRE’ themselves and form the grid(because MEN are more spatially aware up to this point)…i.e STOP moving so fast…then WOMEN can do what they have been traditionally good at which is COMMUNICATION!!! This will lead to great advances in HUMAN TECHNOLOGY which I refer to as GRAINE0.

    Genetics
    Robotics
    Artificial Intelligence
    Nanotechnology
    zero point Energy
    (0)

    Gravity’s constant = 0. MEN – Do not let your mind wander from this thought pattern.

    Light is infinite in all directions but it must slow down to zero too. This will have to be attained by co-operation but the more we do the easier it will become.

    All we have to do is follow the math: http://www.wix.com/Hyperstig/Hyperstig

    or more simply:

    http://www.wix.com/Hyperstig/Simple

    Follow Twitter.com/Hyperstigzero

    Time does NOT exist anymore.

    There are only 3 Dimensions; and this life we know most definitely was and is:

    ‘The Matrix’; ‘Avatar’ and ‘Inception’ combined.

    It is communications that will save us now. We must transmit this messages data throughout the ‘galaxy’ until every last ‘human’ becomes aware of what is going on.

    This is where the Cambrian explosion came from.

    This is what wiped out major past civilizations in a day.

    Our sectors gravity pulling us all into the big collapse.

    We cannot let that happen to us again!

    We are in for the big crunch!!! Yet again!!!

    Can we work together???

    That is what I truly do not know.

    I can imagine what you might be thinking.

    But remember the earth WAS flat at one time.

  39. #39 Mark C. Chu-Carroll
    June 19, 2010

    @36:

    I think you completely misunderstood *my* post.

    I said, at the very beginning, that I don’t think he’s a crackpot. And I thought that the post made clear that I was using it as a launching-off point for a rant about metaphors.

    I do understand what he was doing with the post. But I also think that the way that he wrote about it was muddled, and the overly strong focus on the metaphor obscured the point he was trying to make. And that’s exactly the problem that I was talking about: the way that people focus to strongly on a metaphor to the point obscuring or even entirely losing the point of what they were talking about.

  40. #40 Sizik
    June 19, 2010

    @38
    wat

  41. #41 ing
    June 19, 2010

    @6 A simile is just a subclass of metaphor, and a very superficial one at that.

  42. #42 Spencer Bliven
    June 19, 2010

    There are politicians who worry about new world orders, because of a stupid flowery metaphorical phrase that someone used in a speech 20 years ago.

    What is this a reference to? Gaia hypothesis?

  43. #43 Mark Huber
    June 19, 2010

    I yearn for the simpler days when Archimedes Plutonium was the only Internet crackpot that I regularly read. Now that every blog has at least one, he’s just not that special anymore.

  44. #44 Peter Woodruff
    June 21, 2010

    Look, I think the problem, as is often the case with game metaphors, is that the game is not properly described. What exactly are the moves and the winning conditions? I found it quite difficult to find a precise formulation. Try this:

    The S-guessing game.
    We are given a set S and a “null” element not in S.

    Positions consist of pairs with m and n in S or null. The game starts in position . Alice’s first move is to any position with m in S; Bob’s subsequent moves are from position to , where k is any element of S. (This is not a game of perfect information; Bob knows only the second element of the position. This comment itself is only required to “translate” from common sense talk about games to the mathematically precise formulation.) The game ends in a win for Bob whenever a position is reached; Alice wins if it goes on forever (it will take her a long time to collect on bets). Thus a play of the game is a sequence of positions beginning with (for any m in S), and containing (if at all) only as its last element.
    It should be clear that if S has more than one element, Bob can always lose; he just has to keep stubbornly making the wrong choice. What is relevant to Cantor, though, is whether he has a winning strategy. A strategy (in this case) is a tree of plays rooted at , followed by all moves (m in S) that Alice could make, and after that each node with m not equal to n is followed by another , k in S. The strategy is “winning” iff all the plays in the tree are wins; in this case, iff they are all finite. Thus if S is countable (that is, if Bob knows how to count it, say by a function f) he has the following strategy: start with a guess of f(0); after that,if I don’t win (i.e. if I’m at node with m not equal to f(n), which Alice or the referee will tell me :))guess f(n+1). Since f counts (is onto) S, m will be f(n) for some n, and the path will be finite.
    Now Cantor’s (second) diagonal argument (the first one was used to show that the rationals are countable) simply proves that Bob has no winning strategy for the set R of real numbers. The “fairness” argument loses its punch, since a winning strategy has to work no matter what choice Alice makes, and one has to come up with the strategy “ahead of time”. I have to say, though, that there is some further excavating to be done here, since what Cantor actually showed is that the reals are not countable, and though we saw above that if they were countable, there would be a winning strategy, the converse (if not countable, no strategy), while presumably true is (at least at this hour) not obvious to me. And the proof of the converse would presumably tell us something about our intuitions as to the relevance of the game metaphor.
    Thanks for giving me the occasion to think about this.

  45. #45 wedding dresses
    June 21, 2010

    According to skin color to choose the wedding gowns

    1, Fair-skinned, almost all of the wedding dresses can be worn, in addition to the traditional pure white and large red. Pink is popular in recent years and pink prom dresses is a new choice, it can bring a different fashion sense for public.

    2, What color of dress is suitable for black skin people. People who have dark skin, we suggest that you’d better choose some dresses with light color. Then you can create a healthy and sexy atmosphere, and also can highlight a kind of eye-catching beauty. However, the premise is that you must have health and own a kind of healthy black color.

    3 Dull yellowish color, you are advised not to try these colors such as purple, sapphire blue, cinnamon, lotus root color, copper gold type of dress. Purple and sapphire blue make the skin more sallow which has already dull enough white, and the cinnamon, lotus root, copper gold type of skin color are suitable for the person whose skin is white enough. Otherwise, you will create an illusion being integrated with each other with yellowish skin and oriental dress, and totally unable to appear the focus, it seems to separate the skin and the dress even like outside the boundaries

    Additionally, selecting wedding dress not only need to know the color trends, but also the dress you wear need match with color, environment and season, it contains a lot of stress!

    At present, wedding dresses fabrics mainly use satin, thick forging, forging bright, lace, lace, organza, mesh yarn and so on. The same kind of fabric divided into two kinds of imported and domestic, in addition, import have two kinds—Europe and Japan and South Korea. Indeed, different raw materials have also decided dress different grades and prices! 

    If the product is a series of satin, in general, a layer of imported thick satin and a layer of lining can achieve good effect. If you coupled with a good pannier, the dress would be more perfect.

    Furthermore, if you want your wedding gown dress to become heirlooms, then you’d better choose the silk. This elegant and luxurious fabric is comfortable to wear. However, there is one thing to remember that it can not get a good result because the silk is always have a little yellow appearance.

  46. #46 Aaron Brown
    June 21, 2010

    @42:

    http://en.wikipedia.org/wiki/New_World_Order_(conspiracy_theory)#History_of_the_term

    During the 20th century, many statesmen [...] used the term “new world order”
    [...]

    In his 11 September 1990 *Toward a New World Order* speech [...] President
    George H. W. Bush described his objectives for post-Cold-War global governance
    in cooperation with post-Soviet states:

  47. #47 soho
    June 22, 2010

    I think your post is the one that’s silly, for appearing to attack a perfectly good post. Richard Lipton is a respected computer scientist and clearly knows what he is talking about — as do Fields Medalists Terence Tao and Tim Gowers who understand the post and its perspective. (Sorry for an argument by authority.)

    In mathematical culture, it is quite commonplace to complain that a proof is “cheating” if it seems to use too much information or techniques that seem too advanced for the problem, or seems to use a trick that seems to magically work, or it simply uses induction without giving insight, etc. In none of these case is there the slightest suggestion that there is something incorrect about the proof. It’s just that what mathematicians actually seek is an (ill-defined) “understanding”, and it is frustrating when you know something is true without this “understanding”, hence the complaint. In this case, Cantor’s diagonalization proof is one of the beautiful gems in mathematics and there is a magical quality about how successful it is, and it is part of mathematicians’ work to strip away the magic, play with proofs and understand their limitations, see how they can be modified or weakened, etc. This is a perfectly natural thing to do, and rephrasing it in terms of games (it’s not a “metaphor”) does clarify what’s unique about the proof. Moreover, casting proofs as games has been a highly productive approach in theoretical computer science, with interactive proofs, the PCP theorem, etc.

    The point of the post is to present an alternative proof idea which has a certain feature the original proof does not have — and this feature is best phrased in terms of games.

    I don’t know what your problem with all this is. It seems that because you spend so much time dealing with cranks, you’re unable to read anything written about Cantor’s proof without thinking about problems with it. Especially your final remarks, like “But there’s really no limit to how incredibly, astonishingly stupid people can be. And the idea of an imperfect metaphor is, apparently, much too complicated for an awful lot of people”, even they are not about the original post, seem to insult the author.

    Among mathematicians, a reaction like “that’s cheating!” or “that’s unfair!” made about a proof is often a humorous sign of grudging admiration. Instead, you seem to get hung up over the words “cheat” and “unfair” and rant for no reason. You say “The fact that what Cantor’s proof does would be cheating if it were a game is completely irrelevant.” Irrelevant to what? It’s irrelevant to the correctness of the proof, sure, and everyone knows that. Who’s to say it’s irrelevant to everything, and not worth looking at?

  48. #48 James Sweet
    June 23, 2010

    Seems like this metaphor could be made more robust by just saying that, in the other cases, Alice also gets to know Bob’s algorithm before she picks her number. In the case of countable numbers, there’s still a strategy Bob can employ where this does Alice no good.

    In other words, even if we let Alice cheat all the time, there’s still a clear difference between the two categories.

    Among mathematicians, a reaction like “that’s cheating!” or “that’s unfair!” made about a proof is often a humorous sign of grudging admiration.

    Indeed. Reminds me of the Climategate thing where one of the e-mails mentioned a “trick”. In context, it was clear that “trick” meant a clever and unexpected technique, not an attempt at deception. In addition, if I was informally describing a proof and said, “And now we cheat by doing such and such…”, in context it would be clear that by “cheat” I meant a clever and unexpected twist to get around what had earlier seemed to be an impossible obstacle.

  49. #49 Jonathan Vos Post
    July 4, 2010

    I asked Rudy Rucker yesterday, at Westercon 63 in Pasadena where’s he’s Guest of Honor, since his Ph.D. was on set theory and infinities, and who has written both fiction and nonfiction books about transfinite arithmetic, and who has written an historical novel (abut Peter Breugal the elder), if he would write an historical novel about Georg Cantor. He found the idea attractive.
    http://en.wikipedia.org/wiki/Georg_Cantor

    The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. Cantor believed his theory of transfinite numbers had been communicated to him by God. David Hilbert defended it from its critics by famously declaring: “No one shall expel us from the Paradise that Cantor has created.

    “The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers and or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.”

    – Georg Cantor

    “What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.”

    – Georg Cantor

    “The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.”

    – Georg Cantor

    “The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.”

    – Georg Cantor

  50. #50 Michael
    July 10, 2010

    @46, He’s not attacking the referenced post. He very clearly admits that he doesn’t believe it’s crankery. He’s merely explaining a fallacy of metaphor using the linked post as an motivating example. He doesn’t have a problem with metaphors themselves, but is just pointing out a possibility of abusing them.

    I think he’s free to do that without being seen as coming down on the entire concept of metaphor as a useless tool.

    Lighten up.

  51. #51 Xanthir, FCD
    July 14, 2010

    Hrm. The argument in the original post, about Alice cheating, doesn’t even make sense in the context of the metaphor.

    Given the “Cantor’s diagonalization as guessing game” metaphor, if Alice is choosing her number from the naturals, integers, or even rationals, then she can “cheat” and choose her number only after Bob has announced his guessing strategy, and she’ll still lose every time if Bob has chosen his strategy correctly.

    There is literally *no way* for Alice to win the game even if she cheats, if her number comes from those sets. That’s what makes it significant that she can win in the reals. The fact that she has to cheat to guarantee a win is unimportant, because the exact same cheating does her no good elsewhere.

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