IMO, the truths of mathematics are objectively true - and exist independently of human thought. You will nowhere find aliens who have proved that 1+1 = 3, though you may well find some who think the whole idea uninteresting or useless, and even some who cannot understand what you mean by "one". And this applies equally to excruciatingly difficult truths of mathematics. Fermat's Last Theorem is objectively true, and all that Wells did was find a way to show everyone that.
Nevertheless, mathematics is an intensely creative process, and the creativity is in what you think about. There are lots of mathematical objects out there, and some of them are more fruitful and more interesting to us humans than others. Oftentimes, the key to a hard proof is coming up with the right objects, the right definitions, to allow you to make the leap of logic to the larger goal. These defined objects have also always been out there in the timeless mathematical universe, but right along with them are an infinite number of useless objects and concepts, as well as some wondrous objects that no one has yet found the use of.
This is one reason that human mathematicians beat theorem-provers. The mechanical theorem provers have no concept of interestingness, and will prove an infinite number of theorems of no possible interest. Real mathematicians have the intuition, and the cleverness, to not waste their time in the chaff, and to seek out the beautiful and productive. It is extraordinarily creative. And the key act is often "Let's give this thing I have just come up with a name, so it is easier to work with". In other words, the act of definition.
I am not a mathematician, but I studied to be one, and this stuff was going to be my specialty, until I figured out there was more money in programming than in math.
It is not the case that the "continuum hypothesis is known to be true". Nor is it the case that it has been proven to be unprovable, though that is closer to being correct.
The continuum hypothesis is a statement about entities which do not exist in the universe. We know what the statement "2+2 = 4" is about; it's about integers, and since we can count, we're pretty sure that integers exist. The statement "the universe is expanding" is a statement about things we can observe. There can be quibbles about how much of the universe we can see, whether our understanding is really great enough to answer such questions, and so on, but in the end, practically everyone would say that the question has meaning and, therefore, has some kind of answer, even if the answer is no better than "the parts we can observe indeed appear to be expanding".
The continuum hypothesis is different. It is a statement about uncountable sets, which are creations of our mind. If we are right about the laws of physics, there are *no* uncountable sets existing as physical entities in our universe. What this means is that the continuum hypothesis is not a statement relevant to physical reality, and therefore is of quite different character than either "2+2 = 4" or "the universe is expanding". It is a completely reasonable belief system to hold that the continuum hypothesis, being entirely about non-existent mentally generated entities, has no meaning, and is therefore neither true nor false.
To believe that the continuum hypothesis has a definite truth value is a strong philosophical statement. The mathematical philosophy called Platonism holds that mathematical objects, such as uncountably infinite sets, actually exist, and therefore that statements about them such as the continuum hypothesis have meaning, and in fact that such statements are either true or false. Another philosophy of mathematics is formalism, which holds that mathematics is a game we play according to rules. If someone proves a complicated mathematical result about uncountable sets, we admire this as brilliant play of the game, but do we "believe" it? We believe it only if we believe those statements from which the reault was proved. To play and appreciate the game, we don't have to believe in the axioms, and in fact may find it entertaining to play the game starting from axioms we believe to be false. A formalist is unlikely to regard the continuum hypothesis as either true or false.
Another poster said that the continuum hypothesis has been proven to be unprovable. This is an oversimplification. What has been proven is that the continuum hypothesis is unprovable from the standard set theoretic axioms, using standard logic. A formalist admires this statement as itself brilliant game play, but understands that it is meaningful only for this game. Add another axiom, and suddenly you can prove CH. Unless you find the axioms compellingly true, you probably regard a claim of the truth (or falsity) of CH as dubious as a claim that one's goal in life should be to own Park Place. Truth is relative to where you started from.
A good Platonist on the other hand, will generally believe that the contiuum hypothesis is meaningful, and either true or false, if only we were clever enough to figure out which. Since we know we can't prove it from the standard axioms using the standard logic, a Platonist must hope for discovery of a new axiom or a new logic which is intuitively compelling, and which will also allow CH to be proved or disproved. So, to ask "Is CH true?" is assuming a Platonic view of the Universe, and can be answered only by mathematical creativity ("I propose Axiom X, which settles it"), not merely by a clever play of the game of mathematical deduction.
It is my understanding that most mathematicians who care about these issues are in fact Platonists.
Excellent article. Now will someone do the sequel, explaining to both sides the much greater rift between the UNIX culture and the (IBM) mainframe culture?
The world will be a better place when the UNIX partisans understand exactly why the "Those who do not understand UNIX are doomed to revinvent it, badly" quote makes IBM mainframe guys go ballistic. Compare mainframe security to UNIX security sometime for just a hint of what I'm going on about.
Purely anecdotal and unscientific, but perhaps better than nothing.
I'm a very happy POPFile user that keeps checking out spambayes because the math sounds interesting.
spambayes has become quite good, but POPFile is phenomenal. Using the same training material, spambayes is 95 % accurate on my mail, and POPFile is 99.5 % accurate. Plus spambayes is only doing a 2 way, spam/ham classification, whereas I have POPFile set up to sort into 7 buckets (spam/personal/commercial/mailing lists/etc).
Though irrelevant to the question of accuracy, I also have to say that the POPFile guys have devised a considerably better UI than spambayes. (A friend with the spambayes Outlook plugin sings its praises highly. I don't use Outlook, so it does me no good...)
I run POPFile (and am extremely happy with it by the way). In my experience, this feature would be completely useless with POPFile, because when it misclassifies an email, there's no correlation with its probability estimate. When it mistakenly says a spam is not spam (or vice versa), it's just as likely to have come up with a spam probability of 1E-100 as.01 or.2. And an item with a.4 spam probability is at best only marginally more likely to be an actual spam than one with an infinitesimal spam probability estimate.
Perhaps some of the other filters behave differently.
Well, I am paying for XM, and am extremely happy with it. I'd pay more. My local stations either disagree with me musically or have gone "shock jock". (Sorry, I don't think that immigrant-bashing and rock-and-roll go all that well together.) I never got into music downloading, so I was mostly dependent on radio for keeping me connected with new music. I've become pretty disconnected by now. XM has me discovering new bands again (not to mention the old bands I missed out on because apparently the mass market *likes* immigrant-bashing).
I recommend it for anyone who can afford the subscription, and who misses what FM radio sometimes was before it became a wasteland. No doubt entropy will have its way with XM too, but why not enjoy it before that happens? (And the more money it makes before it "sells out", the longer said event will be delayed.)
Also, anything by Banco de Gaia. Last Train to Lhasa, Live at Glastonbury, Magical Sounds, Igizeh, it's hard to go wrong.
Plus, the soundtrack to Pi is a good compilation, for a little bit of a number of great artists, including Banco de Gaia, as well as the incessantly cited Aphex Twin, Autechre, Space-Time Continuum, etc. etc. It's a great place to start.
Slashdot Mirror
This is actually a smart answer.
IMO, the truths of mathematics are objectively true - and exist independently of human thought. You will nowhere find aliens who have proved that 1+1 = 3, though you may well find some who think the whole idea uninteresting or useless, and even some who cannot understand what you mean by "one". And this applies equally to excruciatingly difficult truths of mathematics. Fermat's Last Theorem is objectively true, and all that Wells did was find a way to show everyone that.
Nevertheless, mathematics is an intensely creative process, and the creativity is in what you think about. There are lots of mathematical objects out there, and some of them are more fruitful and more interesting to us humans than others. Oftentimes, the key to a hard proof is coming up with the right objects, the right definitions, to allow you to make the leap of logic to the larger goal. These defined objects have also always been out there in the timeless mathematical universe, but right along with them are an infinite number of useless objects and concepts, as well as some wondrous objects that no one has yet found the use of.
This is one reason that human mathematicians beat theorem-provers. The mechanical theorem provers have no concept of interestingness, and will prove an infinite number of theorems of no possible interest. Real mathematicians have the intuition, and the cleverness, to not waste their time in the chaff, and to seek out the beautiful and productive. It is extraordinarily creative. And the key act is often "Let's give this thing I have just come up with a name, so it is easier to work with". In other words, the act of definition.
It is not the case that the "continuum hypothesis is known to be true". Nor is it the case that it has been proven to be unprovable, though that is closer to being correct.
The continuum hypothesis is a statement about entities which do not exist in the universe. We know what the statement "2+2 = 4" is about; it's about integers, and since we can count, we're pretty sure that integers exist. The statement "the universe is expanding" is a statement about things we can observe. There can be quibbles about how much of the universe we can see, whether our understanding is really great enough to answer such questions, and so on, but in the end, practically everyone would say that the question has meaning and, therefore, has some kind of answer, even if the answer is no better than "the parts we can observe indeed appear to be expanding".
The continuum hypothesis is different. It is a statement about uncountable sets, which are creations of our mind. If we are right about the laws of physics, there are *no* uncountable sets existing as physical entities in our universe. What this means is that the continuum hypothesis is not a statement relevant to physical reality, and therefore is of quite different character than either "2+2 = 4" or "the universe is expanding". It is a completely reasonable belief system to hold that the continuum hypothesis, being entirely about non-existent mentally generated entities, has no meaning, and is therefore neither true nor false.
To believe that the continuum hypothesis has a definite truth value is a strong philosophical statement. The mathematical philosophy called Platonism holds that mathematical objects, such as uncountably infinite sets, actually exist, and therefore that statements about them such as the continuum hypothesis have meaning, and in fact that such statements are either true or false. Another philosophy of mathematics is formalism, which holds that mathematics is a game we play according to rules. If someone proves a complicated mathematical result about uncountable sets, we admire this as brilliant play of the game, but do we "believe" it? We believe it only if we believe those statements from which the reault was proved. To play and appreciate the game, we don't have to believe in the axioms, and in fact may find it entertaining to play the game starting from axioms we believe to be false. A formalist is unlikely to regard the continuum hypothesis as either true or false.
Another poster said that the continuum hypothesis has been proven to be unprovable. This is an oversimplification. What has been proven is that the continuum hypothesis is unprovable from the standard set theoretic axioms, using standard logic. A formalist admires this statement as itself brilliant game play, but understands that it is meaningful only for this game. Add another axiom, and suddenly you can prove CH. Unless you find the axioms compellingly true, you probably regard a claim of the truth (or falsity) of CH as dubious as a claim that one's goal in life should be to own Park Place. Truth is relative to where you started from.
A good Platonist on the other hand, will generally believe that the contiuum hypothesis is meaningful, and either true or false, if only we were clever enough to figure out which. Since we know we can't prove it from the standard axioms using the standard logic, a Platonist must hope for discovery of a new axiom or a new logic which is intuitively compelling, and which will also allow CH to be proved or disproved. So, to ask "Is CH true?" is assuming a Platonic view of the Universe, and can be answered only by mathematical creativity ("I propose Axiom X, which settles it"), not merely by a clever play of the game of mathematical deduction.
It is my understanding that most mathematicians who care about these issues are in fact Platonists.
> I'm sorry, we had to outsource the sanskrit grammar nazis to India.
...
> oh, wait
Only makes sense. That is, after all, the home of the original Aryans.
Excellent article. Now will someone do the sequel, explaining to both sides the much greater rift between the UNIX culture and the (IBM) mainframe culture?
The world will be a better place when the UNIX partisans understand exactly why the "Those who do not understand UNIX are doomed to revinvent it, badly" quote makes IBM mainframe guys go ballistic. Compare mainframe security to UNIX security sometime for just a hint of what I'm going on about.
Sounds to me like the Orange is a better Apple!
I've been getting spams telling me to try out Napster 2 for weeks now. 'Nuff said.
Purely anecdotal and unscientific, but perhaps better than nothing.
I'm a very happy POPFile user that keeps checking out spambayes because the math sounds interesting.
spambayes has become quite good, but POPFile is phenomenal. Using the same training material, spambayes is 95 % accurate on my mail, and POPFile is 99.5 % accurate. Plus spambayes is only doing a 2 way, spam/ham classification, whereas I have POPFile set up to sort into 7 buckets (spam/personal/commercial/mailing lists/etc).
Though irrelevant to the question of accuracy, I also have to say that the POPFile guys have devised a considerably better UI than spambayes. (A friend with the spambayes Outlook plugin sings its praises highly. I don't use Outlook, so it does me no good...)
I run POPFile (and am extremely happy with it by the way). In my experience, this feature would be completely useless with POPFile, because when it misclassifies an email, there's no correlation with its probability estimate. When it mistakenly says a spam is not spam (or vice versa), it's just as likely to have come up with a spam probability of 1E-100 as .01 or .2. And an item with a .4 spam probability is at best only marginally more likely to be an actual spam than one with an infinitesimal spam probability estimate.
Perhaps some of the other filters behave differently.
Well, I am paying for XM, and am extremely happy with it. I'd pay more. My local stations either disagree with me musically or have gone "shock jock". (Sorry, I don't think that immigrant-bashing and rock-and-roll go all that well together.) I never got into music downloading, so I was mostly dependent on radio for keeping me connected with new music. I've become pretty disconnected by now. XM has me discovering new bands again (not to mention the old bands I missed out on because apparently the mass market *likes* immigrant-bashing).
I recommend it for anyone who can afford the subscription, and who misses what FM radio sometimes was before it became a wasteland. No doubt entropy will have its way with XM too, but why not enjoy it before that happens? (And the more money it makes before it "sells out", the longer said event will be delayed.)
Also, anything by Banco de Gaia. Last Train to Lhasa, Live at Glastonbury, Magical Sounds, Igizeh, it's hard to go wrong.
Plus, the soundtrack to Pi is a good compilation, for a little bit of a number of great artists, including Banco de Gaia, as well as the incessantly cited Aphex Twin, Autechre, Space-Time Continuum, etc. etc. It's a great place to start.