Maybe Europeans don't soon need to discuss SMS anymore; in some (or many?) European countries you can already send MMS messages (short for multimedia message service I think; next generation of SMS basically). This means you can for example take photos with your phone, edit the into MMS messages and send them to a friends phone.
So if in the States there still is problems delivering approx. 160 character long messages from phone to phone, MMS probably will not be around soon if your operators don't get their act together. (you have the technology so there's got to be something not so right with how the operators / phone market is arranged? technically messages a few bytes long cannot be that difficult to deliver)
If remember correctly in Europe SMS generates around 10% of revenues for the operators. So that is big business allready and you surely have plenty of users. This has not yet happened in the States because the networks have not been up to it. (and yes, e-mail is not a substitute for SMS).
Also, I have never noticed that a SMS would not have gotten through. Actually one time we tried to send around 300 sms messages during a time of few minutes (this was done with a crowd). All messages got through but some messages took a few hours to arrive. Apparently the local network cell had trouble relaying the messages but none were lost.
And third: I have never received any SPAM. I have heard a few times that somebody had got a few spam sms messages. Finland for example has got tough laws against spamming and advertising directly to customer. You can add your name to a centrally maintained list which is a list for people who do not wish to be telemarketed etc. If you then receive some kind of direct marketing, you just mention this to the authorities and the marketer automatically goes to court and will have to pay penalties (or may lose license to market if this happens regularly, so they need to be carefull). Therefore, this kind of spamming is not a large problem in Finland.
I agree that teens use a lot of text messages but I don't think they account for the majority of the messages sent. For example I usually send maybe 2-10 text messages during a work day. One actually a very good use is during business meetings: Let's say I don't remeber some figure or technical detail during the meeting, but I do not wan't to make the call publicly (during the meeting). So I send a text message to a co-worker, for example "how much did the license for software x cost". He/she will then send a message back and I get the information discreetly.
One other typical use might be that there's something a bit more urgent than an e-mail, but you still don't wan't to bother the other party by calling him (he might be in a meeting, on the phone etc.) A text message is something you don't have to answer right away but usually you do answer soon; an e-mail might lay around in my Inbox for days un-answered.
Yes, I agree that the sms messages are a very good business for the network providers. They are however quite usefull to me also.
I think it is a common opinion (in Europe?) that one of the more important reasons for the US to have been behind in cell phone usage is that the called party pays => this will incur costs on you if let's say you boss calls you on a work related issue and you really don't like to pay for that. If somebody calls you it's usually because they wan't to get in touch with you and not the other way around. Other reason for the US to be behind is surely the decision to not to adopt a common standard such as the gsm. Well, everybody knows this, do they...
In my experience people in Finland for example basically don't use landlines anymore, at least not for their private calls. Some percentage of work related calls are still to landlines, usually because you are calling a company number or a call center or similar. A person without a cell phone is an oddball (I think my grandfather doesn't own a cell phone, but everybody else I know does). People get landlines because they come as an add-on when you get an adsl line for example. So, if my mother or my sister calls me, she will use her cell phone. And she probably won't call from home, she might call on they way home from work for example. People just don't use landlines too much.
Also, in Finland typically your company pays your phone bill and gives you the phone (let's say if you work is it-related for example). Actually from my friends/relatives only the people who work for schools or universities etc. don't have a company paid phone. So giving the phone for you is good for the company, because at least during work hours you can allways be reached. This is good for you too, because the company pays also your private calls. And the network providers must be jumping in joy, I think.
But anyway, if the called party pays, then do people in the US pay for work related calls or does your boss pay the bill for those? Or does your company usually pay your bills alltogether? (if so this discussion wouldn't probably be on slashdot but between your company and the network provider?)
My friend works for a cell phone manufacturer. They are going to produce a car set based around bluetooth (might be in the stores allready). The nice point is that you do not have to throw any switches to activate the set when you go into the car; the car set activates if you have your phone with you. So actually, you can be on the phone, enter the car (the set activates) and pocket your phone and continue talking trough the car set (microphones, loudspeakers).
I think basically almost all (from mid-range upwards) cell phones do have a voice/speach recognition system for dialing. This would be usefull if you are using a mike or a car set. Anyway, typically the phone has memory reserved for 10-20 names (at least the old one I have).
Great lines! As the MS jokes do have some truth in them, also the above Linux jokes do.
Really, IMHO Linux and MS platforms are for different purposes - at least in practice. If I were to choose Linux over MS for some project the purpose would not be to save money on OS cost. In short: For MANY (of course, not all by far) sw projects the cost of OS and also the cost of sw tools or components does not matter (it is neglible). What would matter is how long time and how many people you will need to finish the job. I had linux first installed in 1994 (Windows really was crap then), but I still have done a lot of work on MS platform. I have the feeling that lately the MS platform is quicker to develop on (!!!when you don't have hordes of experienced coders available, which is often the situation) as compared to Linux or even Java world. You might not be able to get the same stability with MS but w2k is good enough for many real world solutions. Wouldn't use it for a space shuttle or similar purpose...
Ok maybe you will clarify this for me (remember stochastics and analysis not in-depth computational/CA/related theory ok)? If we add axioms to Peano axioms and use maybe a higher order logic the Gödel theorem will still hold, true? So if we are working just based on a set of axioms needed for Riemann Hypothesis we do not yet know that a proof exists, do we? I would say that it is likely that we will someday find a proof, but to my understanding as of yet there is not any proof that such a proof actually exists (we'll actually if we had that we would be done:> ). What would it mean to study Riemann hypothesis outside the system? To start from scratch? And could we still claim to be studying the Riemann hypothesis?
Ok, I read that his work should include some original work that would be important for CA field. Hopefully everybody will be able to make out what is to be credited to Mr. Wolfram.
The Rule of the Universe -part in the Wired article however is something I can only chuckle about. Hope he get's it done;)
Your points are absolutely true and I agree. However if all the mathematicians would go and claim that: "We don't care about your stupid applications; we're doing this for our own fun! Go stuff you apps!" the rather quick implication would be that the guys with the applications would take their money and and put it somewhere else (in computer science?).
The point is that let's say Wiles has just proven Fermat's last theorem and a guy comes up and says: 'that's neat, I think I can use this in my device'. Then Wiles should go: 'Very interesting. Can I help you with that?'. Actually, however unpropable the above would be I hope/think Wiles would respond just so. So I think we should continue developing math as we do but never forget that the final reason is not to do it just for the heck of it. Maybe we should even venture so far as to look for applications after creating something totally new? Anyway, I think you would be in your right to say that this it not a problem today.
As considering AoC I'm a believer. That's propably because I did some research on stochastics at the university (markov operators; the asymptotic properties thereof) and you will not get anywhere without Measure and Integration theory which is actually the basis for the whole thing. In my mind most of the stuff that follows from AoC seems to be natural (barring the unit ball problem but that can be explained too...).
I don't think I've ever met anybody claiming to study set theory per se (those guys are all dead by now?). Some guys however studied areas that to my understanding were closely related to set theory. Nothing I would understand however...
Ok, I'll try to give out a dummy proof for Gödel's Incompleteness theorem (the whole thing is apparently about 30 pages, I'll admit I haven't read the whole thing; I've read a partial proof in Russel's and Norwig's 'Artificial Intelligence'). This should clear things out a little bit and give insight to the discussion.
We'll start with the observation that in number theory we have names for all the natural numbers. This is seen as follows: let's say we have the successor function S and a single constant 0; then let S(0) denote 1, S(S(0)) denote 2 etc. By induction we have names for all the natural numbers.
Gödel also included the following function symbols: +, * and Exp and also the usual set of logical connectives and qualifiers in first-order logic. It is now obvious that that the set of sentences we can write in this language can be enumerated (order the symbols in alphabetical order, then do the same with sentences of lenght 1, then with 2 and so on). We can therefore number any sentence a with a unique natural number #a (the Gödel number). Therefore: Number theory contains a name for each of it's own sentences!!! In the same way we can number each possible proof P with a Gödel number G(P) because a proof is a finite sequence of sentences.
Then let us assume that we have an arbitrary set A of true statements about natural numbers. Because A can be named by a given set of integers we propose that it is possible to write the following sentence in our language: a(j,A) =
All i for which i is not the Gödel number of a proof of the sentence whose Gödel number is j, where the proof uses only premises in A.
Furthermore, let r be the sentence r(#r,A) i.e. a sentence that states its own unprovability from A. Can such a sentence exist for all A? Don't ask me, but apparently Gödel would have said that the answer is yes.
The rest is rather simple alltough rather ingenious. We need to prove that r is true. We'll go with reductio ad absurdum: Let's first suppose that r is provable from A (that r actually is false statement! remember that r was stating it's unprovability from A). But this would mean that we have a false statement provable from A. Therefore A cannot consist of only true sentences. This is a contradiction since according to our premises A consists of only true sentences! Therefore r must not be provable from A which is exactly what r claims.
So from the above (assuming that we believe the sentence r can be constructed) we have seen that for any set A of true sentences in number theory we have statements that cannot be proven from A. As a special case we can choose A = axioms of the number theory. Hence number theory containts statements that cannot be proven!
Feel free to complain about the inaccuracies in the above; all I can do is to suggest you get Gödel's proof into your hands. Anyway to my mind (if I do not miss any subtleties) the above goes on to establish that we can never prove all the theorems of mathematics within any given system of axioms (as the above problem appears allready with the natural numbers). This is apparently why Hilbert was pissed about Gödel's proof.
Yes, I totally agree. Should you read my comment and the thread carefully (yes, a lot of text...) you'll see that the AoC was presented just as an example for an axiom and how the axioms affect mathematics. The discussion was about whether math can be done without basing it on some formalized assumptions or axioms. Obviosly you can do something just based on your gut feeling, but you will not know whether your work is worth anything, agree?
Also you were pointing out that if ZF is ok, then it is so still if AoC is included. Still the beginning of the previous centure saw a lot of discussion whether AoC should be included as one get's some apparently unreasonable results. The point being two-fold: 1) do we get contradictions because of AoC 2) do we get just an abstract mathematical construct or then again something that really can be used (analysis being closer to applications at that time; now of course almost all math is applied). To my mind mathematics as a whole should always bear in mind that it's primarily purpose is to aid other sciences, not to do art for art's sake.
Yes! The first part in your response about the axioms is what I meant. Choosing different axioms yeilds different theory (and possibly rubbish); for example the Axiom of Choice is necessary for basically all modern analysis, but you can have a lot of classical analysis without it. It is a requirement for measure theory (a measure cannot be constructed without it). Still the Axiom of Choice allows for some very non-intuitive results: for example you can break the unit sphere (3d) into finitely many (however immeasuralbe) pieces and then proceed to construct two unit spheres out of those (Banach-Tarski Decomposition). The axiom itself is however very intuitive and is part of established mathematics (from 1920s on I think). One can only wonder... Anyway excellent page here. Includes comment by Jerry Bona: The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma? (the three are equivalent). Luckily the mathematics we now have seems to portray nature rather well, so I think we can rest assured.
You still fail to understand the meaning of axioms. Let's forget the name 'axiom' and talk about just assumptions. For example you might implicitly use some basic assumption when calculating 2+2=4 (at least, you apparently are calculating in Z, not in Z_2 for example). You see, every time you try to set up some proposal or theorem you need to assume something. Without assuming some underlaying construct what is there to deduce (based on nothing)? The reason we talk about 'axioms' is because we wish to emphasize the importance of these basic assumptions. You should go to some mathematician you respect and discuss the matter with him, if cannot convey it over here.
The point however is indisputable: all mathematics is logically based on some set of axioms (or assumptions, if you will). These assumptions need to be correct for the mathematics to be correct. The actual process of axiomatization has got nothing to do with this; here you are mixing history with mathematical constructs to prove something. In mathematics, the most important thing is to completely understand what you are doing. It may be your intuition that is guiding you: intuition is necessary but can just as easily lead you to wrong theorems. Only by complete understanding and carefull verifiying should you be confident on your results. This is however very difficult; recently a friend of mine had to 'cancel' several of his published articles, because he was using an established ten year old result that was proven to be wrong. So mathematics (remember the 'empirical' point) is not unerring.
With your deduction about Gödel Incompleteness theorem you are also mixing things; namely mathematical constructs and mathematicians themselves. As with mathematics (and with all kinds of logic), if you choose a set of assumptions which is allready conflicting within itself you can prove anything. This will have nothing to do with nature however. So, the Gödel Incompleteness (GI) result applies because of the following first-order logic: GI applies to all mathematical constructs which include at least the Peano axioms AND Number Theory as a mathematical construct includes Peano axioms => GI applies to number theory. It couldn't get more simple! (hope I got my assumptions right...)
As to Number Theory being the 'Queen of Mathematics'; this is the general opinion. I myself do think that Complex Analysis is the most beautiful part of mathematics (eloquent proofs, non-intuitive results (at first), all accessible to a first or second year student). Anyway I've allways disliked purely discrete things (such as integers). I don't study complex analysis by the way; I've done research on Markov operators (stochastics stuff) and now I'm back to basic applied stuff (cutting and packing; you even get to see actual results!).
It is as you said that there are claims i) known to be true ii) known to be false iii) known to be not provable true or false (altough these might be true or false) and also finally claims iv) for which we do not know yet to which of the first three categories the claim would belong to. However to my understanding Gödel's theorem is not at least directly connected algorithmic theory (abstract machines etc).
Saddly I'm not an expert on Turing machines or AI theory (as considering whether a machine can be built to mimick human mind; I can only believe that this might be possible, but however not in the foreseeable future). BTW: a mathematician called Yiannis Moschovakis has proposed an alternative for the abstract machine approach (Mathematics Unlimited 2001).
I do actually practice mathematics (research and also application to practice) so maybe I'm a step ahead here. I would say that your view of mathematics (as a partially non-axiomatic science) is flawed in a quite a common way (for example for people in engineering, physics and computer science). I hope you will read my lengthy comments below.
It is true that a lot of early mathematics was not first developed based on a set of axioms. You refer to axiomatization that was started by Hilbert; what you in some way fail to see is reason for the the need of axiomatization (alltough you point out contradictions etc.). Mathematics not based on axioms (or this actually means: based on wrong axioms; axiom = some basic assumption) is just speculation and can lend to wrong results (some examples below). Number theory however can be and fortunately is based on axioms; they are not the tools but instead the basis on which number theory is built (logically, not in terms of development time-line). Alltough a lot of number theory was developed before complete axiomatization (actually these basic axioms are really intuitive so it's not easy to go wrong), number theory is still an axiomatic science and the Gödel result applies. If it were not so (that number theory can be based on axioms), number theory would not be the 'queen of mathematics' but just a lot of worthless speculation. E.g. (as you might know) propability theory was considered just as play before it was axiomatized (your typical text book in propability theory usually forgets a rigid axiomatization and just goes on to display things in an intuitive manner; ok for text book but not for research work).
Also, obviously you can use basic axioms such as Peano's without your knowledge of doing so. But if you were to choose a wrong set of axioms you might end up with wrong results. Let us propose for example an axiom: there exists a largest natural number. We can then easily go on to prove that '1' is this number (work it out, if you will).
There's actually a quite famous case in the 19th century where a famed mathematician called Frege started his work on set theory. He chose a set of axioms which is now known as 'naive set theory' as it is flawed. Frege however did not understand this and proceeded to work on the naive set theory for some dozen years. When his work was finally published an other famous mathematician (philosopher, logician) Bernard Russell wrote to Frege and pointed out that his set of axioms is flawed. Frege was despaired as now the major body of his life's work might be worhtless. (It turned out that many parts of his work were usable but some major conclusions were wrong; As I remember Frege stopped publishing for several years because of this).
So the point that I'm trying to carry out here is that if a theory is based on wrong assumptions it might be worthless and therefore axiomatization is integral for mathematics. The problem is to choose the right axioms, however even now we sometimes do not know if we have chosen correctly (e.g. Axiom of Choice). Somebody pointed out that mathematics could be seen as an empirical science; it is a science where one empirically tries out different thoughts and chooses the ones that are correct.
Yep, apparently he's proposing Dynamical systems theory ('Chaos theory') and CA as his own ideas which is obviously not true. E.g. Poincare discovered 'complexity' in similar simple dynamical systems some 100 years ago. Somebody collected a set of ANKOS reviews here. Incidentally the idea that world would consist also out of CA's is actually also not originally Wolfram's (I think I have even read a scifi story based on this idea).
I am sorry to downplay you but apparently you do not have a strong background in math and maybe some knowledge in algorithm theory? In short: Gödel's theorem applies to all axiomatic systems containing Peano axioms. The system that number theoreticians study contains the Peano axioms (obviously as it is a theory of numbers). Therefore the Gödel theorem applies as considering the Riemann hypothesis.
You'll find the
Peano axioms here. The most important thing about the Peano axioms is that they state the existence of natural numbers {1,2,3,...} (note the axioms give out an infinite number of such objects). So no Peano axioms => no theory of numbers...
See also Some Theorems Derivable from Peano's Axioms. It should help to understand what signigicance these axioms have. Also, all mathematics is axiomatic. For mathematicians if a claim is not based a axiomatic system then it is just speculation...
I read about Wolfram's book in Wired. I got the impression he's gone a bit over the top however. The article talked about a lot of things that supposedly were "new inventions" but mostly seemed to be old news (e.g. complex things arise from simple dynamical systems etc.) and then proceeded to Wolfram admitting that he could deduce the "rule of the universe" in the near future...
So has he done anything that could be used in mathematics or is the book just full of such speculations? It would be rather suprising to see CA used to solve the Riemann Hypothesis...
If you look at the page you specified:
"Number theory is a theory about the integers, a set which we call Z where..." Obviosly to represent integers you first need the Peano axioms to get the natural numbers. Therefore Gödel's Incompleteness Theorem applies (you only need the Peano axioms).
To my understanding Gödel proved by example that any system containing at least the Peano axioms (natural numbers) is incomplete. As this is true for basically all mathematics, the previous comment obviosly applies here. We know that we do have mathematical theorems that cannot be proven to be true or false. We do not know which theorems would be such beforehand; to my understanding there is no reason for Riemann's hypothesis to be provable. Anybody know of any categarization system that would help to deduce whether a theorem is provable or not?
BTW: There was (is?) actually a group called constructivists that tried to do without Peano axioms (they disargeed with complete induction). To my understanding they did not get much done...
Slashdot Mirror
Check out:
= /m ms/
http://www.nokia.com/nokia/0,1522,,00.html?orig
(SMS with pictures etc.)
Maybe Europeans don't soon need to discuss SMS anymore; in some (or many?) European countries you can already send MMS messages (short for multimedia message service I think; next generation of SMS basically). This means you can for example take photos with your phone, edit the into MMS messages and send them to a friends phone.
So if in the States there still is problems delivering approx. 160 character long messages from phone to phone, MMS probably will not be around soon if your operators don't get their act together. (you have the technology so there's got to be something not so right with how the operators / phone market is arranged? technically messages a few bytes long cannot be that difficult to deliver)
If remember correctly in Europe SMS generates around 10% of revenues for the operators. So that is big business allready and you surely have plenty of users. This has not yet happened in the States because the networks have not been up to it. (and yes, e-mail is not a substitute for SMS).
Also, I have never noticed that a SMS would not have gotten through. Actually one time we tried to send around 300 sms messages during a time of few minutes (this was done with a crowd). All messages got through but some messages took a few hours to arrive. Apparently the local network cell had trouble relaying the messages but none were lost.
And third: I have never received any SPAM. I have heard a few times that somebody had got a few spam sms messages. Finland for example has got tough laws against spamming and advertising directly to customer. You can add your name to a centrally maintained list which is a list for people who do not wish to be telemarketed etc. If you then receive some kind of direct marketing, you just mention this to the authorities and the marketer automatically goes to court and will have to pay penalties (or may lose license to market if this happens regularly, so they need to be carefull). Therefore, this kind of spamming is not a large problem in Finland.
I agree that teens use a lot of text messages but I don't think they account for the majority of the messages sent. For example I usually send maybe 2-10 text messages during a work day. One actually a very good use is during business meetings: Let's say I don't remeber some figure or technical detail during the meeting, but I do not wan't to make the call publicly (during the meeting). So I send a text message to a co-worker, for example "how much did the license for software x cost". He/she will then send a message back and I get the information discreetly.
One other typical use might be that there's something a bit more urgent than an e-mail, but you still don't wan't to bother the other party by calling him (he might be in a meeting, on the phone etc.) A text message is something you don't have to answer right away but usually you do answer soon; an e-mail might lay around in my Inbox for days un-answered.
Yes, I agree that the sms messages are a very good business for the network providers. They are however quite usefull to me also.
I think it is a common opinion (in Europe?) that one of the more important reasons for the US to have been behind in cell phone usage is that the called party pays => this will incur costs on you if let's say you boss calls you on a work related issue and you really don't like to pay for that. If somebody calls you it's usually because they wan't to get in touch with you and not the other way around. Other reason for the US to be behind is surely the decision to not to adopt a common standard such as the gsm. Well, everybody knows this, do they...
In my experience people in Finland for example basically don't use landlines anymore, at least not for their private calls. Some percentage of work related calls are still to landlines, usually because you are calling a company number or a call center or similar. A person without a cell phone is an oddball (I think my grandfather doesn't own a cell phone, but everybody else I know does). People get landlines because they come as an add-on when you get an adsl line for example. So, if my mother or my sister calls me, she will use her cell phone. And she probably won't call from home, she might call on they way home from work for example. People just don't use landlines too much.
Also, in Finland typically your company pays your phone bill and gives you the phone (let's say if you work is it-related for example). Actually from my friends/relatives only the people who work for schools or universities etc. don't have a company paid phone. So giving the phone for you is good for the company, because at least during work hours you can allways be reached. This is good for you too, because the company pays also your private calls. And the network providers must be jumping in joy, I think.
But anyway, if the called party pays, then do people in the US pay for work related calls or does your boss pay the bill for those? Or does your company usually pay your bills alltogether? (if so this discussion wouldn't probably be on slashdot but between your company and the network provider?)
My friend works for a cell phone manufacturer. They are going to produce a car set based around bluetooth (might be in the stores allready). The nice point is that you do not have to throw any switches to activate the set when you go into the car; the car set activates if you have your phone with you. So actually, you can be on the phone, enter the car (the set activates) and pocket your phone and continue talking trough the car set (microphones, loudspeakers).
I think basically almost all (from mid-range upwards) cell phones do have a voice/speach recognition system for dialing. This would be usefull if you are using a mike or a car set. Anyway, typically the phone has memory reserved for 10-20 names (at least the old one I have).
Great lines! As the MS jokes do have some truth in them, also the above Linux jokes do.
Really, IMHO Linux and MS platforms are for different purposes - at least in practice. If I were to choose Linux over MS for some project the purpose would not be to save money on OS cost. In short: For MANY (of course, not all by far) sw projects the cost of OS and also the cost of sw tools or components does not matter (it is neglible). What would matter is how long time and how many people you will need to finish the job. I had linux first installed in 1994 (Windows really was crap then), but I still have done a lot of work on MS platform. I have the feeling that lately the MS platform is quicker to develop on (!!!when you don't have hordes of experienced coders available, which is often the situation) as compared to Linux or even Java world. You might not be able to get the same stability with MS but w2k is good enough for many real world solutions. Wouldn't use it for a space shuttle or similar purpose...
Ok maybe you will clarify this for me (remember stochastics and analysis not in-depth computational/CA/related theory ok)? If we add axioms to Peano axioms and use maybe a higher order logic the Gödel theorem will still hold, true? So if we are working just based on a set of axioms needed for Riemann Hypothesis we do not yet know that a proof exists, do we? I would say that it is likely that we will someday find a proof, but to my understanding as of yet there is not any proof that such a proof actually exists (we'll actually if we had that we would be done :> ). What would it mean to study Riemann hypothesis outside the system? To start from scratch? And could we still claim to be studying the Riemann hypothesis?
Ok, I read that his work should include some original work that would be important for CA field. Hopefully everybody will be able to make out what is to be credited to Mr. Wolfram. The Rule of the Universe -part in the Wired article however is something I can only chuckle about. Hope he get's it done ;)
Your points are absolutely true and I agree. However if all the mathematicians would go and claim that: "We don't care about your stupid applications; we're doing this for our own fun! Go stuff you apps!" the rather quick implication would be that the guys with the applications would take their money and and put it somewhere else (in computer science?).
The point is that let's say Wiles has just proven Fermat's last theorem and a guy comes up and says: 'that's neat, I think I can use this in my device'. Then Wiles should go: 'Very interesting. Can I help you with that?'. Actually, however unpropable the above would be I hope/think Wiles would respond just so. So I think we should continue developing math as we do but never forget that the final reason is not to do it just for the heck of it. Maybe we should even venture so far as to look for applications after creating something totally new? Anyway, I think you would be in your right to say that this it not a problem today.
As considering AoC I'm a believer. That's propably because I did some research on stochastics at the university (markov operators; the asymptotic properties thereof) and you will not get anywhere without Measure and Integration theory which is actually the basis for the whole thing. In my mind most of the stuff that follows from AoC seems to be natural (barring the unit ball problem but that can be explained too...).
I don't think I've ever met anybody claiming to study set theory per se (those guys are all dead by now?). Some guys however studied areas that to my understanding were closely related to set theory. Nothing I would understand however...
Ok, I'll try to give out a dummy proof for Gödel's Incompleteness theorem (the whole thing is apparently about 30 pages, I'll admit I haven't read the whole thing; I've read a partial proof in Russel's and Norwig's 'Artificial Intelligence'). This should clear things out a little bit and give insight to the discussion.
We'll start with the observation that in number theory we have names for all the natural numbers. This is seen as follows: let's say we have the successor function S and a single constant 0; then let S(0) denote 1, S(S(0)) denote 2 etc. By induction we have names for all the natural numbers.
Gödel also included the following function symbols: +, * and Exp and also the usual set of logical connectives and qualifiers in first-order logic. It is now obvious that that the set of sentences we can write in this language can be enumerated (order the symbols in alphabetical order, then do the same with sentences of lenght 1, then with 2 and so on). We can therefore number any sentence a with a unique natural number #a (the Gödel number). Therefore: Number theory contains a name for each of it's own sentences!!! In the same way we can number each possible proof P with a Gödel number G(P) because a proof is a finite sequence of sentences.
Then let us assume that we have an arbitrary set A of true statements about natural numbers. Because A can be named by a given set of integers we propose that it is possible to write the following sentence in our language: a(j,A) =
All i for which i is not the Gödel number of a proof of the sentence whose Gödel number is j, where the proof uses only premises in A.
Furthermore, let r be the sentence r(#r,A) i.e. a sentence that states its own unprovability from A. Can such a sentence exist for all A? Don't ask me, but apparently Gödel would have said that the answer is yes.
The rest is rather simple alltough rather ingenious. We need to prove that r is true. We'll go with reductio ad absurdum: Let's first suppose that r is provable from A (that r actually is false statement! remember that r was stating it's unprovability from A). But this would mean that we have a false statement provable from A. Therefore A cannot consist of only true sentences. This is a contradiction since according to our premises A consists of only true sentences! Therefore r must not be provable from A which is exactly what r claims.
So from the above (assuming that we believe the sentence r can be constructed) we have seen that for any set A of true sentences in number theory we have statements that cannot be proven from A. As a special case we can choose A = axioms of the number theory. Hence number theory containts statements that cannot be proven!
Feel free to complain about the inaccuracies in the above; all I can do is to suggest you get Gödel's proof into your hands. Anyway to my mind (if I do not miss any subtleties) the above goes on to establish that we can never prove all the theorems of mathematics within any given system of axioms (as the above problem appears allready with the natural numbers). This is apparently why Hilbert was pissed about Gödel's proof.
Yes, I totally agree. Should you read my comment and the thread carefully (yes, a lot of text...) you'll see that the AoC was presented just as an example for an axiom and how the axioms affect mathematics. The discussion was about whether math can be done without basing it on some formalized assumptions or axioms. Obviosly you can do something just based on your gut feeling, but you will not know whether your work is worth anything, agree?
Also you were pointing out that if ZF is ok, then it is so still if AoC is included. Still the beginning of the previous centure saw a lot of discussion whether AoC should be included as one get's some apparently unreasonable results. The point being two-fold: 1) do we get contradictions because of AoC 2) do we get just an abstract mathematical construct or then again something that really can be used (analysis being closer to applications at that time; now of course almost all math is applied). To my mind mathematics as a whole should always bear in mind that it's primarily purpose is to aid other sciences, not to do art for art's sake.Yes! The first part in your response about the axioms is what I meant. Choosing different axioms yeilds different theory (and possibly rubbish); for example the Axiom of Choice is necessary for basically all modern analysis, but you can have a lot of classical analysis without it. It is a requirement for measure theory (a measure cannot be constructed without it). Still the Axiom of Choice allows for some very non-intuitive results: for example you can break the unit sphere (3d) into finitely many (however immeasuralbe) pieces and then proceed to construct two unit spheres out of those (Banach-Tarski Decomposition). The axiom itself is however very intuitive and is part of established mathematics (from 1920s on I think). One can only wonder... Anyway excellent page here. Includes comment by Jerry Bona: The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma? (the three are equivalent). Luckily the mathematics we now have seems to portray nature rather well, so I think we can rest assured.
You still fail to understand the meaning of axioms. Let's forget the name 'axiom' and talk about just assumptions. For example you might implicitly use some basic assumption when calculating 2+2=4 (at least, you apparently are calculating in Z, not in Z_2 for example). You see, every time you try to set up some proposal or theorem you need to assume something. Without assuming some underlaying construct what is there to deduce (based on nothing)? The reason we talk about 'axioms' is because we wish to emphasize the importance of these basic assumptions. You should go to some mathematician you respect and discuss the matter with him, if cannot convey it over here.
The point however is indisputable: all mathematics is logically based on some set of axioms (or assumptions, if you will). These assumptions need to be correct for the mathematics to be correct. The actual process of axiomatization has got nothing to do with this; here you are mixing history with mathematical constructs to prove something. In mathematics, the most important thing is to completely understand what you are doing. It may be your intuition that is guiding you: intuition is necessary but can just as easily lead you to wrong theorems. Only by complete understanding and carefull verifiying should you be confident on your results. This is however very difficult; recently a friend of mine had to 'cancel' several of his published articles, because he was using an established ten year old result that was proven to be wrong. So mathematics (remember the 'empirical' point) is not unerring.
With your deduction about Gödel Incompleteness theorem you are also mixing things; namely mathematical constructs and mathematicians themselves. As with mathematics (and with all kinds of logic), if you choose a set of assumptions which is allready conflicting within itself you can prove anything. This will have nothing to do with nature however. So, the Gödel Incompleteness (GI) result applies because of the following first-order logic: GI applies to all mathematical constructs which include at least the Peano axioms AND Number Theory as a mathematical construct includes Peano axioms => GI applies to number theory. It couldn't get more simple! (hope I got my assumptions right...)
As to Number Theory being the 'Queen of Mathematics'; this is the general opinion. I myself do think that Complex Analysis is the most beautiful part of mathematics (eloquent proofs, non-intuitive results (at first), all accessible to a first or second year student). Anyway I've allways disliked purely discrete things (such as integers). I don't study complex analysis by the way; I've done research on Markov operators (stochastics stuff) and now I'm back to basic applied stuff (cutting and packing; you even get to see actual results!).
It is as you said that there are claims i) known to be true ii) known to be false iii) known to be not provable true or false (altough these might be true or false) and also finally claims iv) for which we do not know yet to which of the first three categories the claim would belong to. However to my understanding Gödel's theorem is not at least directly connected algorithmic theory (abstract machines etc).
Saddly I'm not an expert on Turing machines or AI theory (as considering whether a machine can be built to mimick human mind; I can only believe that this might be possible, but however not in the foreseeable future). BTW: a mathematician called Yiannis Moschovakis has proposed an alternative for the abstract machine approach (Mathematics Unlimited 2001).
I do actually practice mathematics (research and also application to practice) so maybe I'm a step ahead here. I would say that your view of mathematics (as a partially non-axiomatic science) is flawed in a quite a common way (for example for people in engineering, physics and computer science). I hope you will read my lengthy comments below.
It is true that a lot of early mathematics was not first developed based on a set of axioms. You refer to axiomatization that was started by Hilbert; what you in some way fail to see is reason for the the need of axiomatization (alltough you point out contradictions etc.). Mathematics not based on axioms (or this actually means: based on wrong axioms; axiom = some basic assumption) is just speculation and can lend to wrong results (some examples below). Number theory however can be and fortunately is based on axioms; they are not the tools but instead the basis on which number theory is built (logically, not in terms of development time-line). Alltough a lot of number theory was developed before complete axiomatization (actually these basic axioms are really intuitive so it's not easy to go wrong), number theory is still an axiomatic science and the Gödel result applies. If it were not so (that number theory can be based on axioms), number theory would not be the 'queen of mathematics' but just a lot of worthless speculation. E.g. (as you might know) propability theory was considered just as play before it was axiomatized (your typical text book in propability theory usually forgets a rigid axiomatization and just goes on to display things in an intuitive manner; ok for text book but not for research work).
Also, obviously you can use basic axioms such as Peano's without your knowledge of doing so. But if you were to choose a wrong set of axioms you might end up with wrong results. Let us propose for example an axiom: there exists a largest natural number. We can then easily go on to prove that '1' is this number (work it out, if you will).
There's actually a quite famous case in the 19th century where a famed mathematician called Frege started his work on set theory. He chose a set of axioms which is now known as 'naive set theory' as it is flawed. Frege however did not understand this and proceeded to work on the naive set theory for some dozen years. When his work was finally published an other famous mathematician (philosopher, logician) Bernard Russell wrote to Frege and pointed out that his set of axioms is flawed. Frege was despaired as now the major body of his life's work might be worhtless. (It turned out that many parts of his work were usable but some major conclusions were wrong; As I remember Frege stopped publishing for several years because of this).
So the point that I'm trying to carry out here is that if a theory is based on wrong assumptions it might be worthless and therefore axiomatization is integral for mathematics. The problem is to choose the right axioms, however even now we sometimes do not know if we have chosen correctly (e.g. Axiom of Choice). Somebody pointed out that mathematics could be seen as an empirical science; it is a science where one empirically tries out different thoughts and chooses the ones that are correct.
Yep, apparently he's proposing Dynamical systems theory ('Chaos theory') and CA as his own ideas which is obviously not true. E.g. Poincare discovered 'complexity' in similar simple dynamical systems some 100 years ago. Somebody collected a set of ANKOS reviews here. Incidentally the idea that world would consist also out of CA's is actually also not originally Wolfram's (I think I have even read a scifi story based on this idea).
Better place to look for Peano axioms is here. It should give the Axiom of Induction in a more sensible manner.
I am sorry to downplay you but apparently you do not have a strong background in math and maybe some knowledge in algorithm theory? In short: Gödel's theorem applies to all axiomatic systems containing Peano axioms. The system that number theoreticians study contains the Peano axioms (obviously as it is a theory of numbers). Therefore the Gödel theorem applies as considering the Riemann hypothesis.
You'll find the Peano axioms here. The most important thing about the Peano axioms is that they state the existence of natural numbers {1,2,3,...} (note the axioms give out an infinite number of such objects). So no Peano axioms => no theory of numbers...
See also Some Theorems Derivable from Peano's Axioms. It should help to understand what signigicance these axioms have. Also, all mathematics is axiomatic. For mathematicians if a claim is not based a axiomatic system then it is just speculation...I read about Wolfram's book in Wired. I got the impression he's gone a bit over the top however. The article talked about a lot of things that supposedly were "new inventions" but mostly seemed to be old news (e.g. complex things arise from simple dynamical systems etc.) and then proceeded to Wolfram admitting that he could deduce the "rule of the universe" in the near future...
So has he done anything that could be used in mathematics or is the book just full of such speculations? It would be rather suprising to see CA used to solve the Riemann Hypothesis...If you look at the page you specified: "Number theory is a theory about the integers, a set which we call Z where..." Obviosly to represent integers you first need the Peano axioms to get the natural numbers. Therefore Gödel's Incompleteness Theorem applies (you only need the Peano axioms).
To my understanding Gödel proved by example that any system containing at least the Peano axioms (natural numbers) is incomplete. As this is true for basically all mathematics, the previous comment obviosly applies here. We know that we do have mathematical theorems that cannot be proven to be true or false. We do not know which theorems would be such beforehand; to my understanding there is no reason for Riemann's hypothesis to be provable. Anybody know of any categarization system that would help to deduce whether a theorem is provable or not? BTW: There was (is?) actually a group called constructivists that tried to do without Peano axioms (they disargeed with complete induction). To my understanding they did not get much done...