Line is a Circle

SOL

Okay a line is a circle of infinite diameter,

So it follows that the line between two points is actually and arc not a line

Eh?

Thus ruining maths

Sev

Okay a line is a circle of infinite diameter

Well, the only reason that you would have made such an assumption is because such a circle would have no apparent curvature. You're clearly contradicting yourself.

Anyhow this is just stupid, Cormac, I hope you get lambasted by stiff arses with no concept of intuitive mathematics but with an obsessive and needless necessity for "rigorous" mathematical convention.

Just being bitter at this hour of the morning. Oh and the "QED" in your signature isn't very appropriate.

SOL

no ste it has infinitely small curvature and so it brings us back to the fact that maths is just a ****ty approximation and nothing more

Sev

You state that a circle of infinite radius is a line. Then you go on to deny the reason that this assumption is true.

And tell me cormac, what is the difference between infinitely small.. and zero?

ecksor

Okay a line is a circle of infinite diameter

How do you work that one out? Where is the center of this circle?

Originally posted by Sev
Anyhow this is just stupid, Cormac, I hope you get lambasted by stiff arses with no concept of intuitive mathematics but with an obsessive and needless necessity for "rigorous" mathematical convention.

You're sore because of that thread last week where your logic was picked apart? Rigour and intuition aren't exclusive. Intuition may be useful to find a solution or a direction of investigation, but it must stand up to rigour at the end of the day or it is at best unproven or at worst worthless.

scipio_major

I see someone has been introduced to the Riemann Sphere for the first time.

Fade to credits
Scipio_major

Sev

Stop confusing logic with convention.

ecksor

Right. I think you should stop trolling on this subject until you have something intelligent to say. Otherwise, we've already established your views on things here

Sev

Fair enough... I take it there's no way I can defend my position anyway without being debased with derision.

SOL

the centre is infinitely far away so it's position is irelevant
The difference between infinitley small and 0 is the smallest think we can think of cause maths is just an approximation,

bonkey

Originally posted by SOL
cause maths is just an approximation,

And what, praytell, is maths just an approximation of?

I mean - you say that a straight line has an infinitely small curvature. How can you prove this? The only way to do so would be to actually prove that a straight line, if extended to infinity in both directions would eventually meet itself.

While this can be proven, it relies on a curved space for it to exist within - such as our universe. However, in a flat (or uncurved) space, then a straight line extended to infinity will not meet itself (as that would be a contradiction of either the definition of "straight" , or of "uncurved space").

Where our conventional mathematics break down is when we start dealing with infinities. Because infinity is not an actual number, but rather a concept, we can only quantify what an infinity is by approximating a value for that.

However, to say that a numerical value for infinity is an approximation is a long way from saying that our entire mathematical system is an approximation.

And as one further question...if a line between two points is actually an arc, why does this ruin maths?

Surely all you have done is substituted "line" with "arc of circle with infinite radius". Given that one is - according to you - the definition of the other - not only is it a perfectly valid substitution (if you're right), but it doesnt change anything. Thus, it most certainly doesnt ruin maths, nor more than the knowledge that mathematics can formulate statements that can neither be proven nor disproven ruins maths.

At most, they mean that the simplified view of mathematics which was taught to you in school is just that - a simplified view of mathematics.

jc

Gordon

You should really have written "straight line" in your original post methinks.

Isn't it completely obvious that a circle of infinite radius would still be curved so would therefore not be straight hence nullifying the hypothesis that curvy=straight?

Or that could be what bonkey is saying . God I used to be great at maths until I moved to a country that allows underage drinking!

If you want to "ruin" maths look for the omega number (apparently). Like the comma of life, the omega number is something that doesn't add up, it is something that cannot be calculated but exists. Ah heck I don't know really, I just like reading the New Scientist.

ecksor

Originally posted by Sev
I take it there's no way I can defend my position anyway without being debased with derision.

By your own admission you're not even explaining your position correctly. If you wish to try to explain why exactly mathematics is so unsatisfactory to you or whatever its shortcomings are in your eyes, then I think an interesting discussion could follow. The suspicion that an interesting discussion was lurking was the only reason I didn't ignore your previous posts to begin with. All I'm trying to say is that you should stop waving around accusations of confusing logic with convention or phrases like 'intuitive mathematics' if you're not willing to explain your points. Up to now it has seemed that you've just got some misconceptions, but the person with the misconceptions isn't supposed to go around lecturing everyone else on how things work, you get me?

Your intuition has its place, but it will mislead you if you're not critical of it.

Sev

Well, this all began with this post.

Originally posted by Sev
Hehe, me and Davey (PHB) have this running joke now about the value of 0/0

Some say its 0 as its 0 over a number. Some say its infinity because its a number over 0, and some say its 1. Its essentially infinity/infinity, which is the same as 6/6 which is the same as 5/5 which equals 1, so as far as im concerned the answer is 1.

But the question is so utterly stupid and trivial, I like to think of it as a question of faith. What do you believe?

I knew making that post, that there would be some who would take exceptions to what I had said, and not understood the underlying gest of the post. Which is why, I tried (for a moment) to dismiss the explicit technicalities of the mathematics and relate to it on a sheerly practical level to provoke a sense of humour I guess.

But the biggest mistake I ever made, was thinking that people would automatically understand me. For example, naturally I don't mean "intuition" in the most literal sense. I dont believe 6+4 is 10, because I 'feel' it is, but rather because I know it is. Because I instinctively grasp mathematics in a way that many others can't.

The fact that there would need to be a proof for why 1+1=2, simply annoys me. The concept is so unbelieveably strikingly obvious, and the proof is nothing but an attempt to try to put into mathematical logic what any normal brain would automatically assess without second thought. Seriously, it makes me want to lash out and hurt people. I had hoped the majority would automatically agree, but it appears now, to my surprise, that there are many, (yourself included I think), who feel that such a proof (and such trivial mathematics that relates) has importance and is worth the effort to even explain in its own merit.

I make my arguments (if you can say I ever had any), in the same sense that underlines the core of the subject of my running debate. That such trivialities are so benal that there is no need to even explain thoroughly my reasoning, but that most would take for granted the fundamental obviousness of what I'm saying, and just accept and understand based on experience. Clearly I was wrong. People do NOT think like I do, and some just take things quite literally.

You seem quite fond of highlighting what I find to be strikingly obvious, to present your case, in such away that it appears quite condescending, as if I hadn't already realised. But I guess thats a personal trait, and reflects your mindset with which you approach a logical argument, wether it be verbal or mathematical. Although I accept that nothing you have ever said has actually been flawed or incorrect, but in my opinion , it has neither affected, discounted or disproven my beliefs too. We just have rudimentary differences in our thought process. I aplogise for the cheap stab I made earlier in this thread.

With that in mind, I don't believe there is anything I have ever said on these boards that has been logically 'flawed', if only for the reason that it doesnt comply with standardised mathematical convention and procedure. Ive always held the view that some people can be taught maths, whereas some don't need to be.

I would appreciate if you would not try to break down the points of what I have said now to refute and negate in some kind of strictly logical, exclusive, procedural way. That will just show you have completely missed or misunderstood the fundamental message of this argument. If people cannot still come to terms with what I'm saying, then I have failed, I give up.

ecksor

Because I instinctively grasp mathematics in a way that many others can't.

My point is that the maths is entirely subjective.

Those two statements contradict each other. You may have a talent for mathematics, but you haven't quite grasped what it is all about as far as I can see. I suppose I sound condescending again, but bear in mind that you sound equally condescending when you tell us what to think about things you don't appear to understand.

At the very least, if you do have interesting ideas going about your head, you can't or won't put them into a context where the rest of us can think about them. Most of getting into mathematics is learning the language that lets one person express their ideas in a formal and unambiguous way so that others can think about and use the ideas too.

Some more of my nitpicking:
I accept that it is obvious that 6 + 4 = 10, but I'm not quite sure where that obviousness came from. I probably learned it or calculated it at some point and when I did that, it made sense and so was 'obvious', I don't know.

But, again, I come back to my usual questions "what set is this" and "how do you define +". Supposing that the set is all the integers from 0 to 7 inclusive and that + is the operation of modulo arithmetic, then 10 doesn't even exist in that set, and most certainly isn't the correct answer. You might feel that this is irrelevant, but perhaps the triviality that such a proof is investigating can illuminate something about the nature of sets where 6+4 = 10 under either operation or something about a more general case.

Sets and operators and how they work are of particular interest to me since I'm interested in Computer Science and many sets (or types to be precise) have operators that appear to work like in your example, but in reality work more like my example. In this case intuition leads to bugs, which leads to problems (often security problems which is where I started from). hey, if I can reason about these in a formal way and forget about intuition perhaps I can specify this thing with better safety checks, or I can create an engine that can catch the bugs!

(on that note, 1+1=0, haven't you heard of ecksor? )

I would appreciate if you would not try to break down the points of what I have said now to refute and negate in some kind of mathematical procedural way. That will just show you have completely missed or misunderstood the fundamental message of this argument. If people cannot still come to terms with what I'm saying, then I have failed, I give up.

I suppose I wasn't supposed to do that. Anyway, I can't come to terms with what you're saying, but I hope that I've explained why.

Sev

I did not say that maths was subjective.

Originally posted by ecksor
I suppose I sound condescending again, but bear in mind that you sound equally condescending when you tell us what to think about things you don't appear to understand.

Now that.. that is subjective.

Giblet

Infinity cannot be used as a value in a formula such as
pi.r^2
r= d/2 (= infinity)
Which means the radius equals the diameter, which cannot happen, so the whole thing is invalid.

Sev

Originally posted by ecksor

But, again, I come back to my usual questions "what set is this" and "how do you define +". Supposing that the set is all the integers from 0 to 7 inclusive and that + is the operation of modulo arithmetic, then 10 doesn't even exist in that set, and most certainly isn't the correct answer. You might feel that this is irrelevant, but perhaps the triviality that such a proof is investigating can illuminate something about the nature of sets where 6+4 = 10 under either operation or something about a more general case.

(on that note, 1+1=0, haven't you heard of ecksor? )

Regardless, you just proved (unintentionally it seems) that maths can be subjective.

(I hope you considered that the word can have a number of definitions and interpretations)

ecksor

No. There is a convention in certain texts to assume that one is talking about the set of real numbers under the normal arithmetic operators if those details are omitted, and 6+4=10 makes perfect sense within those parameters. I believe we were on the same page, discussing the same idea, so there was no subjectivity unless you implicitly assumed a different set. Even then, a misunderstanding is unlikely to affect the argument.

I explicity offered a different set to try to make a point. I wasn't attempting to suggest that what you said didn't make sense, I was merely trying to give a rationale for providing a proof of such a result, by saying that a proof might lead to something that gives a more general result about the nature of sets where 6+4=10 or where 6+4 != 10. Perhaps someone else can offer a better reason.

Sev

I hope you realise, that by 'subjective', in the quote which you have paraphrased, I was referring to a specific use of mathematical "language", language which in general can have different interpretations, uses and meanings depending on the conditions in which it is used and the goal of its application. You appear to clearly agree, as you have shown in your previous post, which would mean that you are either now contradicting yourself, or could not understand me the first time.

Obviously I know you know exactly what you're talking about, there was never any doubt, and this argument has been reduced to a game of semantics (was it ever not?). I was just turning your post around to show that your original point, that you feel I don't know what I was talking about, was entirely uncalled for, and if you'll allow me to use that ridiculously vague and equivocal word, which you seem to center the core of that argument around... subjective.

ecksor

Originally posted by Sev
Regardless, you just proved (unintentionally it seems) that maths can be subjective.

Hm, a few minutes ago this read something like "Didn't you just prove", and then you deleted it so you could assert "you just proved" when I did no such thing. Unintentional was it? How clever of you.

(I hope you considered that the word can have a number of definitions and interpretations)

Good grief, now I'm starting to think you're just trolling again. Which word are you talking about? 'subjective' ? I suppose it can have different meanings, fine, and you can leave it open to interpretation if you wish, and perhaps even claim this as a basis for your claim that mathematics is subjective. Oh, sorry, you said 'can be', not 'is', not that you'd illuminate your point, only point out the difference in wording.

Here are the possible motives that I think you could have for not trying to agree on a definition.

1. You're either being deliberately vague in an effort to make it look like you can't be pinned down as you twist and feint with your logic.
2. You are trying to claim that it all depends on interpretation. You may have read something by a mathematical philosopher on this subject that appears to lend credence to the idea.
3. Somehow the subjectiveness of the definition of the word subjective (which is only subjective because you deliberately refuse to provide an objective definition) lends weight to the idea that mathematics is (can be) subjective. Circular logic.

Sev

Originally posted by ecksor
Hm, a few minutes ago this read something like "Didn't you just prove", and then you deleted it so you could assert "you just proved" when I did no such thing. Unintentional was it? How clever of you.

I have an obsessive compulsive disorder with posting and in general too. I tend to delete and repost about 3 times before I'm totally happy with what I have said. Either way both get the message across. The latter seemed more definite.

You might want to check up the meaning of the word "rhetorical question" when you look up "subjective".

My previous post, that I take it you havn't read yet, should clear up your other queries.

bonkey

Originally posted by Sev
The fact that there would need to be a proof for why 1+1=2, simply annoys me. The concept is so unbelieveably strikingly obvious, and the proof is nothing but an attempt to try to put into mathematical logic what any normal brain would automatically assess without second thought.

This ties back to something I made a passing reference to earlier. The mathematics that most people have been taught is a simplification of "real mathematics" which is a formal system.

The proof of 1+1 = 2 (in all bases above base 2) falls into the category of "formal" mathematics, where it is not sufficient to state that something is "intuitive" or "obvious", but rather has to be shown that it is always true. For everyday mathematics, it is sufficient to accept it.

However, when you start discussing infinities, then you go outside the generalities covered by everyday maths, and instead head towards the land of formal mathematics again. Here, the "intuitiveness" of normal mathematics leads to apparent contradictions (infintiy + 1 = infinity for example). This is why proof is required, and at that point, the "intuitive" answer is often the wrong one (e.g. subtracting infinity-1 from each side would appear to give us 0 = 1, but in reality we should actually end up discovering that infinity - infinity = infinity - again an unintuitive result which is actually correct if Im remembering my formal maths correctly).

A simple example of the distinction between "real world" and "formal" is that most people who know a bit about maths accept that the radius of a circle is pi times the radius or that the sum of the angles in a triangle will only equal 180 degrees.
However, if you could measure a circle or trianglea angles accurately enough in the real world, you will discover that this is not actually true!

Now, without resorting to formal mathematics, the best someone can typically tell me is that this is either incorrect (and I assure you it isnt), or counter-rintuitive (I assure you it isnt when you know why its happening), or simply proof that mathematics is flawed (once more - I assure you that this is not the reason).

Once you explain why these things happen, the general response is "but thats stupid.....thats something comletely different". Of course its something "completely different", but the point is that intuition only works when dealing with an incredibly simplified subset of mathematics....were we accept that certain truths will hold for no specific reason. This subset, in terms relative to the current argument, should not include infinity, as it is not a number in the sense that we understand numbers intuitively, nor is it subject to the rules that we intuitively accept as being "true" (infinity - infinity = infinity). Thus, the moment we include such concepts (as happened in the first post), it should become intuitively evident that intuition is no longer enough.

Again, take 1+1 =2. We know this is intuitively true? Why? well because its obvious that it will always happen and so on. We've never seen 1+1 not equal 2 in base 10. Then again, by the same logic of observation, I can conclude "intuitively" that only other people die. It doesnt make me right, and thats where formal proof comes in.

jc

ecksor

Originally posted by Sev
I hope you realise, that by 'subjective', in the quote which you have paraphrased, I was referring to a specific use of mathematical "language", language which in general can have different interpretations, uses and meanings depending on the conditions in which it is used and the goal of its application.

The point of Mathematical language is to remove ambiguity.

I think the point of putting double quotes around the word "language" is to introduce ambiguity.

You favour the ambiguity. Am I correct? If so, how can you complain that you are misunderstood?

You appear to clearly agree, as you have shown in your previous post, which would mean that you are either now contradicting yourself, or could not understand me the first time.

I don't think I've contradicted myself anywhere. The fact that I couldn't understand you the first time has been the recurring theme here. The heart of the debate isn't so much the word 'subjective', but my requirement for rigour and your distaste of it. I've tried to give examples that show why it is necessary.

this argument has been reduced to a game of semantics (was it ever not?).

I won't dispute that.

I have quite consistently tried to be precise and figure out precisely what you are talking about. You have not cooperated in helping to extract the meaning of what you are trying to say in an understandable manner. The semantic game was entirely of your creation.
You may claim that I made it necessary, but I have already explained why I feel that rigour is necessary to discuss a mathematical idea (such as infinity) and also seeked clarification from you in the event that you were not discussing a mathematical idea.

that you feel I don't know what I was talking about, was entirely uncalled for, and if you'll allow me to use that ridiculously vague and equivocal word, which you seem to center the core of that argument around... subjective.

I don't argue that that is subjective, my opinions are all subjective.

I don't think the argument is centred around the word 'subjective', I think it is centred around necessity of rigour. The definition or lack of of that word is just a particular instance of the argument.

For the sake of clarity, I will state exactly my standpoint:
You give the impression that you think that people demanding rigour in mathematics are dimwitted narrow minded pedants with no imagination who are constrained by unnecessary rules and conventions. (correct me if I'm wrong). Annoyingly you seem to be trying to 'educate' us of the folly of such ways.

I think that mathematics requires rigour to be useful and it gives confidence of correctness, that proofs of seeming trivialities reduce the need for unnecessary axioms and therefore help us to generalise, that stating your assumptions are essential since a proof in one system may be meaningless in another.

Sev

Originally posted by ecksor
The point of Mathematical language is to remove ambiguity.

I don't agree. But I don't have to.

Originally posted by ecksor
I think the point of putting double quotes around the word "language" is to introduce ambiguity. You favour the ambiguity. Am I correct? If so, how can you complain that you are misunderstood?

Not a particularly ambigous word that one. Although you have seem to hit on a core point I made earlier. You are right, I do tend to delve into ambiguity for the purpose of explaining something that I might find difficult otherwise. But I tend to expect people to understand regardless, a mistake I have come to dread. Sorry. Some read sentences, some read into sentences. It ironically reflects the subject matter of my debate. Something I have been trying to hint at.

I don't think I've contradicted myself anywhere. The fact that I couldn't understand you the first time has been the recurring theme here

I gave reasons why that could be the case.

The heart of the debate isn't so much the word 'subjective', but my requirement for rigour and your distaste of it.

I really only ever used the word "subjective" in very subjective circumstances. I wasn't referring to the debate in general. But yes you are correct, the debate boils down to what you have said. I think bonkey summed things up quite nicely. I guess I have a severe alergy to formal proof.

I have quite consistently tried to be precise and figure out precisely what you are talking about. You have not cooperated in helping to extract the meaning of what you are trying to say in an understandable manner.

Now I would have to totally disagree with that. I feel, although I may be wrong (either way you wouldn't admit it), that you seem to take advantage of this suggestion as to dismiss my arguments.

I don't argue that that is subjective, my opinions are all subjective.

As are mine.

You give the impression that you think that people demanding rigour in mathematics are dimwitted narrow minded pedants with no imagination who are constrained by unnecessary rules and conventions. (correct me if I'm wrong). Annoyingly you seem to be trying to 'educate' us of the folly of such ways.

I did at one stage make a suggestion to that effect, it was supposed to have been done in a more playful, humorous manner. I never tried to educate. I initially gave an opinion in gest, in a sense trivialising the application of very specific mathematics. Something I realise you see as inappropriate. Everything I have said as follows, has only been an attempt to defend my position from ridicule. I hope you realise that there are many who would agree with me, they just dont seem to make their voices heard on this forum.

I think that mathematics requires rigour to be useful and it gives confidence of correctness, that proofs of seeming trivialities reduce the need for unnecessary axioms and therefore help us to generalise, that stating your assumptions are essential since a proof in one system may be meaningless in another.

I totally agree, but as you can guess, there are just some things, that although I accept, understand or am willing to learn, simply push me over the edge and annoy me on a petty, insignificant level.

I hope I have layed this to rest now.

bonkey

Originally posted by Sev
I hope you realise that there are many who would agree with me, they just dont seem to make their voices heard on this forum.

So what?

If I said that the earth was flat, there would be many who would agree with me. It still doesnt make my assertion any more (or less) valid.

And another newsflash for you...it doesnt actually matter what you believe mathematical language was created for, no more than it matters if you believe that "1+1" must intuitively be 2. Belief and correctness are entirely seperate issues.


If you dont want to believe that mathematical language exists to remove ambiguity, then ask yourself how 1+1 can always equal 2, unless 1, 2, + and = are all unchanging and consistent in their meaning and behaviour...which means that they are entirely unsubjective, unchanging and unabmbiguous. The entire language behaves in the same manner, for ambiguity would destroy the entire foundation that mathematics is based on.

Whether or not you like this, believe this, or accept this doesnt change the facts. It may not even be intuitive to you....but mathematicians frankly wont care. They know and understand why it is not subjective, not ambiguous, and why it cannot be. Just because you desire to avoid admitting your wrong or genuinely believe otherwise still doesnt change anything.

You can argue your semantics all you like. You're still wrong.

jc

bonkey

Originally posted by Sev
I did at one stage make a suggestion to that effect, it was supposed to have been done in a more playful, humorous manner.

Interesting interpretation of playful and humorous :

he fact that there would need to be a proof for why 1+1=2, simply annoys me. The concept is so unbelieveably strikingly obvious, and the proof is nothing but an attempt to try to put into mathematical logic what any normal brain would automatically assess without second thought. Seriously, it makes me want to lash out and hurt people.

jc

Sev

Well bonkey, Its amazing how quick people jump to think that you are trying to refute an entirely unrelated argument with a simple, innocent point. I never disagreed with you, if anything I praised you. In fact, there is not much that anybody has said that I disagree with.

As for ambiguity, I would have thought mathematics was riddled with ambiguity, lets take a quick look at google.

Could you tell me, exactly, how many roots has any run of the mill 2nd order diophantine equation? You may want to do a little reading on Kurt Gödel's Incompleteness Theorem too.

In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules an axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

Apart from that, what exactly is it that im wrong about?

Victor

Originally posted by SOL
Okay a line is a circle of infinite diameter, So it follows that the line between two points is actually and arc not a line

So on which side of the (straight) line is the centre of this arc?

bonkey

Originally posted by Sev
As for ambiguity, I would have thought mathematics was riddled with ambiguity, lets take a quick look at google.

I was referring to the language of mathematics being unambiguous, rather than its ability to produce ambiguous statements. You will also find that this is the point Ecksor has been making from the start, which you have been disagreeing with.

You may want to do a little reading on Kurt Gödel's Incompleteness Theorem too.

Its ok thanks. I studied it fairly extensively about a decade ago.

I understand what the theory says, and what its implications are. I know why it isnt terribly relevant to the discussion at hand -for the reason stated above.

And I don't need to google for it either thanks. If I had to do that, I wouldnt know enough to be able to discss the point in the first place - just appear like I did until someone who actually knew what they were talking about decided to take me up on some point.

Apart from that, what exactly is it that im wrong about?

Go back and read my post. You'll find that I already specified what you're wrong about. I have no inclination of repeating myself.

jc

Sev

Originally posted by bonkey
I was referring to the language of mathematics being unambiguous, rather than its ability to produce ambiguous statements. You will also find that this is the point Ecksor has been making from the start, which you have been disagreeing with.

Originally posted by ecksor
The point of Mathematical language is to remove ambiguity.

I just realised, and I must admit I initially misread that line, and ommitted the word 'language' and understood just 'mathematics' when I first hastily inspected that sentence. Yes of course the point of mathematical language is to be unambiguous, I cant argue with that. That was most likely a source of confusion.

In my previous post, I was highlighting (as I have been trying to all along) the inherant ambiguity one meets when dealing with mathematics at the lowest axiomatic level.

Originally posted by bonkey
I know why it isnt terribly relevant to the discussion at hand -for the reason stated above

Clearly not. I'm not sure now what exactly you think my point is now, but Kurt Godel seems to have completely proven everything I have ascertained up to this point and that I instinctively know. As I said, I had an exception to the very low level proof of axiomatic statements. Those, I feel, are ludicrous and unecessary and that only reassert themselves in a circular way. The Incompleteness Theorem shows that my views are entirely justified. My line from the very beginning of this argument, was that some things must be taken on "faith", or assumed for the sake of mathematics, whether it was in gest or not.

If I had to do that, I wouldnt know enough to be able to discss the point in the first place - just appear like I did until someone who actually knew what they were talking about decided to take me up on some point.

Are you suggesting that people could not instinctively deduce such a fundamental notion before 1931? And this is not something that can be entirely self-evident to somebody, but that you would have to learn in university? And that if you had this debate with an esteemed mathematician from the thirties before the publication of said paper, then he would have no idea what hes talking about?

I believe how much you think you know about mathematics does not have any bearing on a subject of such fundamental truth. If you cannot accept this, then Im not going to bother arguing any longer.

bonkey

Originally posted by Sev
Ok, I was referring to both.

Well - that would explain the difference then....the language is clear and unamboguous. The statements it can produce can contain ambiguities. A simple example would be that we can formulate valid statements than can be neither true nor false...which means that they are ambiguous.

For anyone not well up on maths, one of the simplest analagies in the english language is : "This statement is false"

A slightly mathsier example is the idea of the set of all sets which do not contain themselves. If this set is a member of itself, it shouldnt be. If it is not a member of itself, it should be. Therefore, its membership is ambiguous.

As I said, I had an exception to the very low level proof of axiomatic statements. Those, I feel, are ludicrous and unecessary and that only reassert themselves in a circular way.

But they dont, really.

The Incompleteness Theorem shows that my views are entirely justified. My line from the very beginning of this argument, was that some things must be taken on "faith", or assumed for the sake of mathematics, whether it was in gest or not.

OK - you are correct in asserting that some things must be "taken on faith". More formally, some things are nothing more than the unproveable base definitions of our formal system.

Godel proved that any finite base of such definitions (or articles of faith) will lead to an incomplete or inconsistent system. (The obvious assertion which follows is that an infinite base may not lead to incompleteness or inconsistencies.)

However - it is important not to confuse this with the right to decide what articles must be taken on faith, and which must not.

so going back to the example we had in hand, you can "take it on faith" that 1+1=2, but the fact remains that this can be proven from the base axioms of our mathematics.

I agree that the proof is incredibly tedious, and more than irrelevant to the likes of you and me - or indeed almost everyone - but because it can be proven, it is not axiomatic.

Thus, it would be like saying "I take the truth of Fermats Last Theorem on faith, because its evidently true". Before it was proven, modern computers had searched to ridiculous lengths to verify that this was evidently true, but it still required the proof to be fully acceptable in mathematics. Any additional proof which used the results of Fermats' Last as part of its logic had to be qualified as being true "assuming Fermat's Last Theorem is true". Now, it was a reasonably safe assumption, but it still develops an element of doubt...especially when you start having theories which are true if and only if N other theories are proven to be true, where N is a large number.

So - the point I was making is that you, I, or anyone taking something on faith (or not) does not matter to the proveable correctness (or otherwise) of something.

The point you were making is that some things have to be taken on faith.

So, as the "ambiguity" being discussed by both sides was different (ambiguous even), it is fair to say that the point you were making is as valid from the context you had as mine is from the context I had. They are both, in effect, correct

jc

Sev

Uh oh, I edited my post in the time it took you to reply I hope I have still made peace.

However - it is important not to confuse this with the right to decide what articles must be taken on faith, and which must not.

Well, yes, I can understand that the proof of 1+1=2 is valid, despite how stupid it looks. I was just taking the example to the extreme to better express my sentiments.

ecksor

Originally posted by Sev
Are you suggesting that people could not instinctively deduce such a fundamental notion before 1931? And this is not something that can be entirely self-evident to somebody, but that you would have to learn in university? And that if you had this debate with an esteemed mathematician from the thirties before the publication of said paper, then he would have no idea what hes talking about?

Ah for crying out loud. Bonkey threw you a bone to try and save some credibility and you go and spout this.

If you feel that this was somehow 'instinctively' known by mathematicians and/or laymen up to then, then I invite you to review the work of Bertrand Russell, and in particular his Principia Mathematica

ecksor

When I say 'review', I mean review the rationale behind it, not go read it.

Sev

So are you agreeing or disagreeing?

ecksor

Originally posted by Sev
So are you agreeing or disagreeing?

I'm saying you're not credible.

Sev

Interesting way to dismiss an argument. Can you not answer? or are you afraid to damage your own credibility?

SOL

I am only ruining maths by an infinitly small ammount dont worry too much

ecksor

Which argument are you referring to? I'm not disputing Godel's proof or dismissing any arguments.

When you and bonkey finally found yourselves on the same page, you have to come along and imply that if you had been around in the late 20s you would have known instinctively what Godel proved.

Now, you might be some sort of super genius logician, but I really don't think so. The fact that Godel knocked the wind out of the sails of someone so eminent as Russell is what I'm using to back up my opinion (those subjective things).

Also, if this debate was being had with an esteemed mathematician from the 30s, I feel we wouldn't have been putting up with vague statements or claims of knowledge before proof.

Sev

Originally posted by ecksor
you have to come along and imply that if you had been around in the late 20s you would have known instinctively what Godel proved. Now, you might be some sort of super genius logician, but I really don't think so.

If you say so, I have no trouble coming to terms with such a notion (Incompleteness Theorem). I can only guess you're denying so in a bid to maintain the credibility of your argument.

Originally posted by ecksor
If you wish to try to explain why exactly mathematics is so unsatisfactory to you or whatever its shortcomings are in your eyes, then I think an interesting discussion could follow.

Are you backing out now? Have I pushed the debate beyond the grounds of conventional teaching that you are unsure you can rely on your rigid mathematical wisdom any longer? You might have to think for yourself.

But as I said...

I believe how much you think you know about mathematics does not have any bearing on a subject of such fundamental truth. If you cannot accept this, then Im not going to bother arguing any longer.

I give up.

SOL

I conclusion maths is as acurate as those who use it,
I win :cool:

ecksor

Originally posted by Sev
If you say so, I have no trouble coming to terms with such a notion. I can only guess you're denying so in a bid to maintain the credibility of your argument.

The notion that you're a super genius logician? If you are, then I envy you, and you're one of the smartest people I've ever encountered.

Which argument exactly are you talking about that I'm trying to maintain the credibility of?

Are you backing out now?

I'm not backing out. If you wish to discuss why mathematics is unsatisfactory for you, then please continue. In other words, since a mathematical system contains inconsistencies, discuss how that affects your usage or non-usage of it.

Have I pushed the debate beyond the grounds of conventional teaching that you are unsure you can rely on your rigid mathematical wisdom any longer? You might have to think for yourself.

I'm familiar with Godel's result. I don't think it can be regarded as unconventional teaching nowadays, I've seen it treated in several books of varying topics.

As to where it has pushed the debate (apart from where people have to think for themselves) you're saying that what you were trying to get across earlier was a less formal statement of what Godel proved. I'm not debating that result, so what part of the debate have I missed?

SOL

The point is that maths is only as accurate as you can be, like all the other sciences etc it is merely as good as the discoveries that make it and not the absoloute correctness of life
Anyways,
stop bitch fighting

DeVore

Oh dear God, dear Krishna, dear Buddha, dear Mohommad, dear L. Ron fnckin Hubbard....


MAKE THE BAD MEN STOP!!!!


1. No a line is not a curve. I can PROVE this. Ok, I will.

2. Mathematics is rigourous, its not perfect but it IS consistent within agreed parameters, much as physics is consistent within parameters.

In fact one of the FEW things maths has going for it is that it has well defined an agreed rules for what DOES make logical sense to do to equations, formula, etc and also whats ruled out (like division by zero etc).


Maths without adhering to that structure would leave you without a coordinate reference for logic and anyone could start to "feel" that any proof or equation was true.

But math isnt like that. Philosophy is. Go argue into the night with the philosophers if you want to argue that sort of stuff.

Next I'll feckin' *prove* that you're curve isnt a line.


DeV.

ecksor

Originally posted by DeVore
But math isnt like that. Philosophy is. Go argue into the night with the philosophers if you want to argue that sort of stuff.

Oh, that reminds me, Sev's posts remind me a bit of what I've read about Wittgenstein.

DeVore

At any point on this circle you can find the equation of the tangent by differentiating the equation of the circle. Accepted?

Move down the arc ( you can arbitrarily decide a direction which is "down") and redefine the equation of the tangent. On a normal circle the slope of the tangent (this requires you to differentiate a second time) will differ from the first slope because of the curvature of the arc.

The curvature of the circle (the very fact that its a second degree equation in the FIRST place) causes this difference in tangential slope.

Since the infinitely radial circle must STILL follow the equation of a circle (and since infinity IS just a constant and is therefore dropped during differentiation) the tangential slope at the two points MUST differ. Not by much you might say, but thats just a factor of HOW FAR THE DISTANCE BETWEEN THE TWO POINTS YOU TOOK TANGENTS AT ARE APART.

So, take the slope of tangents of two points infinitely apart and you will see a "BIG" difference in the slopes of the tangents.


On the other hand, take two points on a LINE infintely apart and take the slope of the line at that point AND IT WILL BE THE SAME BECAUSE STRAIGHT LINES ARE FIRST DEGREE EQUATIONS.


REAL mathematicians will note that I'm using a slight trick here but then all bets are off when you are dealing with infinity-arse.
Any self-respecting mathematician relegates such nonsense as either "stuff to waffle on about with your mates at 3am when you are stoned" or "stuff to distract the weak of mind".

DeV.

po0k

I haven't read through the thread, and my head is geared for maths today of all days (had two maths exams today, shotgunned).

I propose the following as one way of looking at a line:
x,y axes.
elipse centred at origin.
negligible length along the y-axis (lim x->0 of y component)
length across the x-axis = |legnth| of the "line".

The first post is just wrong in my opinion, for if the line is a circle of infinite diameter (as you look onto the edge (of negligible thickness I might add)), then the "line" the observer sees must also be of infinite length, from the observer's perspective.

That has more then likely been said already.

I saw devore mention tangents and I almost started hyposthesising whether or not my navel is best described as an infinite loop or square consisting of 720*.......

It's been a long day, I'm off to get wasted....

bonkey

Nice one dev,

Alternately, you could just easily prove that two circles of equal radius, where their centres are less than 1 diameter length apart, must intersect.

Now, for a line, the clear and obvious "equivalent" circle would be a parallell line. If we place two parallell lines, less than infinity apart, then either :

a) They never intersect, in which case a line is not an arc of a circle with infinite radius

b) They do intersect, in which case the lines were not parallell, unless we stated we were dealing with a curved space, in which case there is no question - lines are arcs.

we can prove a is true, and not b, which would then give us all the pieces we need to disprove the initial theory

I'm guessing that it would be more accurate to say that in a flat space the arc of a circle approaches (or is asymptotic) to a straight line as the radius of the circle approaches infinity.

jc

bonkey

BTW - theres an interesting read over at http://mtnmath.com/whatth/ which is at the very least tangentially related to the discussions here. The section on Formal Mathematics is a nice easy introduction to some of the stuff discussed here.

Whole thing is a pdf download too I think.

jc

SOL

My point is not actually any of the above, really I was just waffelling but also that maths is only as accurate as you are