Author |
Topic: cos(1 deg) = irrational (Read 28972 times) |
|
ultrafilter
Newbie
Gender: 
Posts: 3
|
 |
Re: cos(1 deg) = irrational
« Reply #25 on: Sep 1st, 2003, 2:33pm » |
 Quote  Modify
|
Actually, I'd say that my proof is pretty good if you already have the results it depends on. If not, then it is pretty bad.
That cos(no) can be expressed as a rational polynomial of cos(1o) is a fairly standard result, isn't it? If not, it shouldn't be too hard to do by induction, using the formula from mathworld.
(btw, is the discussion in this forum limited to actual Putnam problems, or is any theoretical math ok?)
|
|
 IP Logged |
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #26 on: Sep 1st, 2003, 3:15pm » |
Quote Modify
|
I agree, your proof is original, clever, and subtle, in that it is true to say that cos(1o) being rational implies that a cosine, known to be irrational, is rational; which is a contradiction, hence cos(1o) must be irrational. However, the opposite: the assumption that cos(1o) is irrational, tells us nothing about other angles.
Excuse me digressing, but the previous objections I made were rooted in an issue that I have with mathematics and the way that we teach it. After my students present proof, I sometimes complain that they cannot assume a particular aspect to be true, as it is not a trivial result. They normally ask the question, "Well, what can I assume is true?" Good question!
As I have worked with intelligent school students for many years now, you begin to conclude that part of the problem is not with their solution, but the question. The proofs we learn, as mathematicians, and the proofs that I, in turn, expect my students to reproduce are often nothing more than jumping through hoops like performing animals.
For example, one of my objections to your proof, that cos(20o) is irrational is by no means trivial, is reasonable. But why can't you assume it? As I mentioned in a previous post, is it any more reasonable for me to use Euler's formula, e[smiley=i.gif]x=cos(x)+[smiley=i.gif]sin(x), without proof?
Equally, why couldn't I have proved the original problem by saying, "cos(xo) is rational iff x=90k, 90k[pm]30, where k is integer. Therefore, it follows that cos(1o) is irrational, as 1 is not a member of x." This is a well known fact, and is taught to most students as they are first introduced to the cosine graph; not in such explicit terms though. Of course, the formal proof of this 'fact' is definitely not trivial!
Anyway, back to my new challenge...
Prove that cos(20o) is irrational.
(In reference to your question, I assume by the fact that this section of the forum has recently had "pure math" appended to the title implies that general mathematical problems (of a pure nature) are equally encouraged now – thereby widening the scope of posted challenges – but I am no authority. Perhaps one of the moderators could clarify?)
|
| « Last Edit: Sep 1st, 2003, 4:32pm by Sir Col » |
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: cos(1 deg) = irrational
« Reply #27 on: Sep 1st, 2003, 6:49pm » |
Quote Modify
|
It seems to be the standard here that any problem of ~ Putnam level would be acceptable. Simpler math problems would be better suited to the main forums (a large number of them, as well as several that are harder than the Putnam level are to be found there).
on Sep 1st, 2003, 3:15pm, Sir Col wrote:| Equally, why couldn't I have proved the original problem by saying, "cos(xo) is rational iff x=90k, 90k[pm]30, where k is integer. Therefore, it follows that cos(1o) is irrational, as 1 is not a member of x." |
|
Say WHAT?? The reason you can't prove the original problem in that way is because your premise is false! Cosine takes on rational values for infinitely many angles not included in this set. It's possible that these are the only INTEGER-degree angles for which it is rational. But I would hardly call that "well-known". I doubt many people are even aware that these are the only multiples of 15o for which the cosine is rational, though it is easy to deduce from the angle difference formulas and the standard values.
Quote:| This is a well known fact, and is taught to most students as they are first introduced to the cosine graph; not in such explicit terms though. |
|
 why would students be taught this when introduced to the cosine graph? I taught trig for many years and never saw any reason to bring in rational vs irrational values. For graphing you need to plot enough values to give them a feel for how the curve is shaped. (We had to do it all by hand, since we had no computers in the classrooms, and graphing calculators were still hideously expensive.) And of course you teach them the standard values (0, 30, 45, 60, 90...), but why bring up esoteric information about rational versus irrational values?
Quote:| Of course, the formal proof of this 'fact' is definitely not trivial! |
|
Nor do I see that it is reasonable to use it (assuming we are refering to the restriction to integer x) in proving a statement that is a key part in proving it. This smacks of circular reasoning.
But Ultrafilter's proof does not require circular reasoning, in that proving the non-constructability of cos 20o does not involve using the irrationality of cos 20o or of other cosines (it does involve some other things from which the irrationality of cos 20o is a trivial consequence, such as that it is algebraic with characteristic polynomial of degree > 1. But this is shown without reference to irrationality.
As to what can or cannot be used in proving a theorem. I would say it depends on 3 factors:
(1) Is the result you are trying to prove used in proving the theorem you want to use?
Clearly, if so - you can't use it in the proof!
(2) Do you have some understanding of the theorem you want to use? Have you seen it's proof? Do you have any idea of the logic involved in proving it?
If you answer no to any of these, you should at least be very hesistant to make use of it.
(3) Is your audience aware of the theorem you want to use?
Again, if the answer is no, you should either look for other ways, or aquaint your audience with the result as much as is reasonable before making use of it.
Quote:Anyway, back to my new challenge...
Prove that cos(20o) is irrational. |
|
Why stop there: For which integers [smiley=n.gif] is cos [smiley=n.gif]o rational?
Is it just [smiley=n.gif] [equiv] 0, [pm] 30 mod 90 ?
More generally, for which rationals [smiley=r.gif] is cos [smiley=r.gif][pi] rational?
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
Barukh
Guest
|
|
Re: cos(1 deg) = irrational
« Reply #28 on: Sep 2nd, 2003, 7:02am » |
Quote Modify
 Remove
|
on Sep 1st, 2003, 6:49pm, Icarus wrote:
Why stop there: For which integers [smiley=n.gif] is cos [smiley=n.gif]o rational?
Is it just [smiley=n.gif] [equiv] 0, [pm] 30 mod 90 ?
More generally, for which rationals [smiley=r.gif] is cos [smiley=r.gif][pi] rational? |
|
Icarus, your guess is right. There exists a theorem stating that if [smiley=r.gif] is rational, then only rational values of cos([smiley=r.gif][pi]) are 0, [pm]1/2, [pm]1. If you (or somebody else) are interested, I can try to find or reproduce it - the one I saw looked quite elementary.
|
|
IP Logged |
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #29 on: Sep 2nd, 2003, 8:47am » |
Quote Modify
|
I'd definitely be interested, Barukh. Although I made a mistake, it's actually, n[equiv]0 mod 90 or n[equiv]90[pm]30 mod 180.
on Sep 1st, 2003, 6:49pm, Icarus wrote:| why would students be taught this when introduced to the cosine graph? ... And of course you teach them the standard values (0, 30, 45, 60, 90...), but why bring up esoteric information about rational versus irrational values? |
|
The way that I teach trigonometry, and I am sure many other high school teachers nowadays, is to begin with construction; for example, a 30o right angle triangle with hypotenuse length, say, 10 cm, and get them to measure the opposite side. Either by further construction or by talking about the enlargement/scaling principle we would discuss, and work towards, the idea that an n degree right angle triangle with hypotenuse length 1 would be useful in determining the length of the opposite side in a triangle with any length hypotenuse. Hence we talk about the definition of sine, with reference to sine tables (which I used in school) or the button on scientific calculators, and define it as: the length of the opposite side in a unit right angle triangle. It is a small step then to introduce cosine, as the adjacent side, and finally to consider the unit circle and, of course, the sine/cosine graph. Natural questions relate to the 'nice' answer we got for the opposite side in a 30o triangle (and the adjacent side in a 60o triangle). The teacher would then usually reinforce the idea that other answers (generally not multiples of 30) are going to be approximations, because... the cosine/sine of those other angles will be irrational. Don't misunderstand me, I do not teach this explicitly, nor do I labour this point, but I would like to believe that my students take with them a vague impression of this 'fact'.
If I asked one of my newly initiated trigonometres if cos(1o) is irrational, I would be delighted if their answer was, "Yes, becasue it's only multiples of 30 that give nice answers and not all of them work anyway; even cos(30o) is irrational, but 0, 60, 90 are nice (rational)."
Is that a bad answer? I don't think so, as it conveys a much more general truth than a specific reference to one known irrational result. However, mathematicians become more sophisticated and recongnise the importance of rigour. This means building on smaller truths. But my question still stands, how far back do we need to go before it becomes a 'complete' proof?
One of the reasons, in my view, that many logical and intelligent people reject mathematics is because they find the subjective choice of assumptions that can be made, and those which can't be made, illogical. I have learned the 'rules', and enjoy solving mathematical problems, but it doesn't necessarily make it right.
|
| « Last Edit: Sep 2nd, 2003, 8:53am by Sir Col » |
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #30 on: Sep 2nd, 2003, 2:28pm » |
Quote Modify
|
I am really struggling with this latest problem... can it be done without an exhaustive approach?
We have already demonstrated that there exists a polynomial for cos(x) in terms of cox(x/n).
If cos(x/n) were rational, so too would be cos(x). If it is known that cos(x) is irrational, it follows that cos(x/n) is also irrational. In other words, the cosine of all the factors of x will also be irrational.
As cos(45), cos(30), and cos(20), can be shown to be irrational, it follows that cos(x) is irrational for x=1,2,3,4,5,6,10,15,20,30,45.
That leaves an awful lot of gaps.
Playing with addition formula seems to be fruitless. Consider cos(59+1)=cos(60)=cos(59)cos(1)–sin(59)sin(1); it is hardly intuitive that the RHS is rational.
It would seem that the irrationality of cos(x), will have to investigated for quite a few special cases. 
|
| « Last Edit: Sep 2nd, 2003, 2:30pm by Sir Col » |
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: cos(1 deg) = irrational
« Reply #31 on: Sep 2nd, 2003, 3:58pm » |
Quote Modify
|
My method of teaching was similar, except I never saw any need to bring in questions of rationality. The natural occurence of 30o, 45o, and 60o in applications were sufficient to show why they need to learn these values. They are sufficient to get a fair idea of how to draw the graph of an affine function of cosine or sine. I know you are blessed with teaching a "higher grade" of student than I was (I was teaching mostly people who put off taking high school level mathematics until they were in college - which should give you a hint about their abilities), but if I brought up rational versus irrational values to my students I would of just confused them even more.
Barukh - I vaguely recall having seen that result somewhere, but if I ever investigated the proof, I have completely forgotten it.
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #32 on: Sep 2nd, 2003, 4:41pm » |
Quote Modify
|
on Sep 2nd, 2003, 3:58pm, Icarus wrote:| if I brought up rational versus irrational values to my students I would of just confused them even more. |
|
Who says I don't confuse them!? After all, it's part of my job description as a teacher of mathematics.  Imagine being at a party and the conversation going like this. Guest: "So what do you do?" Me: "Oh, I teach mathematics." Guest: "Excellent, I really enjoyed maths at school – it made a lot of sense to me." It just wouldn't be natural, would it? 
By the way, I think that Barukh's result is the only (or best) way to solve this problem.
|
| « Last Edit: Sep 2nd, 2003, 4:50pm by Sir Col » |
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: cos(1 deg) = irrational
« Reply #33 on: Sep 2nd, 2003, 5:04pm » |
Quote Modify
|
Well - Don't show it yet! Let us have a chance to work it out ourselves!
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
SWF
Uberpuzzler
Posts: 879
|
|
Re: cos(1 deg) = irrational
« Reply #34 on: Sep 2nd, 2003, 7:49pm » |
Quote Modify
|
Elementary proof that cos(20o) might be irrational :
cos(60o)=1/2=cos(3*20o). Use identity for cos(3x) (Sir Col gave it earlier) to get:
8x3-6x-1=0, when x=cos(20o).
Try to find relatively prime integers p and q, such that x=p/q:
8p3-6pq2-q3=0
The first two terms are even integers, so q must be even, and p must be odd, since it is relatively prime to q. Substitute q by 2*u where u is some integer, and divide by 8:
p3-u3=3*p*u2
Remember that p is odd. If u is odd, left side of equation is even, and right side is odd which is not possible. If u is even, left side of equation is odd, and right side is even which also is not possible. Therefore, x cannot equal p/q.
|
| « Last Edit: Sep 2nd, 2003, 7:52pm by SWF » |
IP Logged |
|
|
|
Barukh
Guest
|
|
Re: cos(1 deg) = irrational
« Reply #35 on: Sep 3rd, 2003, 1:03am » |
Quote Modify
Remove
|
on Sep 2nd, 2003, 5:04pm, Icarus wrote:| Well - Don't show it yet! Let us have a chance to work it out ourselves! |
|
Icarus, as I didn't quite get whether your exclamation was for the last problem, or for the proof of the general theorem, I will present here the latter in the hidden form.
Let [alpha] = [smiley=r.gif][pi]. Because [smiley=r.gif] is rational, the sequence of angles [alpha], 2[alpha], ..., 2n[alpha],... has repeated terms, and so does the sequence
cos([alpha]), cos(2[alpha]), ..., cos(2n[alpha]), ... (*)
Let cos([alpha]) = p/q (the fraction is reduced). Then cos(2[alpha]) = 2cos([alpha])2 - 1 = (2p2 - q2)/q2. Because gcd(p,q) = 1, we have gcd(2p2 - q2, q2) <= 2. Thus, if q > 2, the sequence (*) will consist of (reduced) fractions with strictly increasing denominators, and therefore won't have repeated values. This contradiction shows that q must be equal to 1 or 2, and completes the proof.
A really simple and brilliant argument, I must admit (I don't know who's the author).
|
|
IP Logged |
|
|
|
Barukh
Guest
|
|
Re: cos(1 deg) = irrational
« Reply #36 on: Sep 3rd, 2003, 1:31am » |
Quote Modify
Remove
|
on Sep 2nd, 2003, 7:49pm, SWF wrote:| Elementary proof that cos(20o) might be irrational |
|
SWF, your proof is very good. I just wanted to note that in the last part of your proof the following criterion could be used: The rational roots of the equation with integral coefficients are also integral; and they divide the free term.
This immediately shows that your derived cubic equation doesn't have rational roots (although it makes the proof less elementary... )
|
|
IP Logged |
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #37 on: Sep 3rd, 2003, 10:06am » |
Quote Modify
|
Great proof, SWF! I love the reasoning on the last part; my version was so much more tedious. Although worked slightly differently, I wrote the equivalent of: p3=u(u2+3pu)=ku, and argued that because HFC(p,u)=1, all the factors present in p must be in k, therefore, u=[pm]1. Similarly, from u=p(p2-3u2), we deduce that p=[pm]1. As this does not satisfy the equation, there can be no rational solution, p/q.
Barukh, I'm obviosuly being really stupid, but could you, or someone, please explain the central part of the proof?
::
Because gcd(p,q) = 1, we have gcd(2p2 - q2, q2) <= 2 (why?).
Thus, if q > 2 (why?),
the sequence (*) will consist of (reduced) fractions with strictly increasing denominators (why?)
::
|
|
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Barukh
Guest
|
|
Re: cos(1 deg) = irrational
« Reply #38 on: Sep 3rd, 2003, 11:20am » |
Quote Modify
Remove
|
on Sep 3rd, 2003, 10:06am, Sir Col wrote:
Barukh, I'm obviosuly being really stupid, but could you, or someone, please explain the central part of the proof? |
|
Let me try:
1) gcd(2p2 - q2, q2) = gcd(2p2, q2) (the fundamental property of gcd, the basis of Euclid's algorithm for finding gcd).
gcd(p,q) = 1 implies gcd(2p2, q2) = gcd(2, q2) <= 2.
2) We assume that q is greater than 2, and show that this leads to a contradiction.
3) Because of 1), the fraction (2p2 - q2)/q2 cannot be reduced by a factor greater than 2. Therefore, if q > 2, the reduced fraction will have the denominator at least q2/2 > q. Obviously, this relation will hold for every term in the sequence.
|
|
IP Logged |
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #39 on: Sep 3rd, 2003, 3:28pm » |
Quote Modify
|
Feels even more stupid now you've explained it so well and made it sound so straight forward. 
Many thanks, Barukh – you're right, it is a really simple and brilliant argument and adds some closure (formal proof) to one of the results I've had for many years. So, one down, only a few hundred more to go on my list now...
|
|
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: cos(1 deg) = irrational
« Reply #40 on: Sep 4th, 2003, 5:24pm » |
Quote Modify
|
I've resolutely avoided Barukh's posts, because I want to figure this out myself. So far, this is what I've come up with.
Going back to the polynomial expressions for [smiley=c.gif][smiley=o.gif][smiley=s.gif] [smiley=n.gif][smiley=x.gif], we see that they are all integer polynomials in [smiley=c.gif][smiley=o.gif][smiley=s.gif] [smiley=x.gif] with leading coefficient 2[supn][supminus][sup1] and constant term 0 or [pm]1.
let [smiley=x.gif] = ([smiley=r.gif]/[smiley=s.gif])[pi] with [smiley=r.gif] and [smiley=s.gif] relatively prime integers. Because they are relatively prime, [exists] [smiley=n.gif] such that [smiley=n.gif][smiley=r.gif] [equiv] 1 mod [smiley=s.gif]. Applying the polynomial formula tells me that [smiley=c.gif][smiley=o.gif][smiley=s.gif] [smiley=r.gif][pi]/[smiley=s.gif] is rational iff [smiley=c.gif][smiley=o.gif][smiley=s.gif] [pi]/[smiley=s.gif] is.
Applying the polynomial formula to -1 = [smiley=c.gif][smiley=o.gif][smiley=s.gif] [smiley=n.gif][pi]/[smiley=n.gif] gives me a polynomial equation in [smiley=c.gif][smiley=o.gif][smiley=s.gif] [pi]/[smiley=n.gif] with leading coefficient 2[supn][supminus][sup1] and constant term 0, 1, or 2. The rational root theorem says that if [smiley=c.gif][smiley=o.gif][smiley=s.gif] [pi]/[smiley=n.gif] is rational, it must be of the form [pm]2[supminus][supi] for some [smiley=i.gif] with 0 [le] [smiley=i.gif] [le] [smiley=n.gif] - 1.
If [smiley=n.gif] is even, then [pi]/2 - [pi]/[smiley=n.gif] is a multiple of [pi]/[smiley=n.gif], and so its cosine must be rational as well. But
[smiley=c.gif][smiley=o.gif][smiley=s.gif]([pi]/2 - [pi]/[smiley=n.gif]) = [smiley=s.gif][smiley=i.gif][smiley=n.gif] [pi]/[smiley=n.gif] = [sqrt]( 1 - 4[supminus][supi] ) = [sqrt](4[supi] - 1) / 2[supi],
which is clearly irrational if [smiley=i.gif] > 0.
Hence, for the cosine of a rational multiple of [pi] to be rational, it must be 0 or [pm]2[supminus][supi] for some [smiley=i.gif] [ge] 0, and the denominator of the argument must be 2 or odd.
This is where I'm at so far. I just have to figure away to eliminate odd denominators > 3.
|
| « Last Edit: Sep 4th, 2003, 7:56pm by Icarus » |
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: cos(1 deg) = irrational
« Reply #41 on: Sep 29th, 2003, 7:11pm » |
Quote Modify
|
I finally gave up hope that I would have time to return to this one and figure the rest of it out myself. So I have read Barukh's post. This is certainly a lot easier than the method I was following!
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
BNC
Uberpuzzler
Gender:
Posts: 1732
|
|
Re: cos(1 deg) = irrational
« Reply #42 on: Sep 29th, 2003, 10:52pm » |
Quote Modify
|
on Sep 2nd, 2003, 8:47am, Sir Col wrote:
However, mathematicians become more sophisticated and recongnise the importance of rigour. This means building on smaller truths. But my question still stands, how far back do we need to go before it becomes a 'complete' proof?
|
|
On a side note, you may want to check the origins of proof.
|
|
IP Logged |
How about supercalifragilisticexpialidociouspuzzler [Towr, 2007]
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #43 on: Sep 30th, 2003, 9:50am » |
Quote Modify
|
Thanks for the link, BNC; it was an interesting article.
I must say that I still remain unconvinced and skeptical about what we are actually proving mathematically. I don't know if anyone has read the excellent book, Proof and Refutations, by Imre Lakatos? He uses the example of Descartes-Euler polyhedral formula: V+F=E+2, and he employs a Socratic method in which a teacher becomes involved in a discussion with some of his students about the validity of this result. After the students attempt to 'prove' the result, their ideas are continually refuted until most of them reluctantly come to the conclusion, along with the rest of us when we first discover, that the relation does not hold for all polyhedra.
It is a result we teach, but it is not true.
We teach that the angles in a triangle are equal to two right angles, but that isn't true.
We teach that 0.999...=1, but that... only joking! 
Philosophically, so much of our reasoning is circular and is rooted in unprovable axioms – self evident, really? It was thought that Cantor, or someone after him, would prove the 'self-evident' continuum hypothesis; but it is now known to be unknowable in our system of logic and understanding.
Kronecker was no dummy, and he denounced the irrationals as an abomination in the face of the beauty of mathematics. He, and his peers, could intelligently justify a rejection of the infinite and methods of analysis.
Don't misunderstand me, I love mathematics. I enjoy proving results and solving problems, but... what are we building our mathematics on? Can we really be sure that the foundations are firm?
|
|
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: cos(1 deg) = irrational
« Reply #44 on: Sep 30th, 2003, 8:49pm » |
Quote Modify
|
hoo-boy! What a can of worms to open, and it's late...
First of all, how much of this was intentional and how much was the result of limited understanding on the part of the author, I don't know, but the article takes a very simplistic and ill-defined approach to a number of things. (I expect that most if not all of it was an intentional simplification in order to make her particular point more readily available to her audience. But if so, I think she did them a disservice by stating the final paragraph as she has. And Sir Col's remarks are a clear indication of why it is a disservice. Perhaps she means to clarify the matter in the promised later articles.)
I'll start off with a point that I hope you are familiar with, but which she totally ignores: Euclid's first few definitions are nothing of the kind. There is indication in the Elements that Euclid himself knew this, and was only offering them as "heuristic descriptions" rather than true definitions. The problem is: how do you define "point"? Euclid defines them as "that which has no part". But this only begs the question of what does "having no part" mean? In order to define anything, you have to use terminology which is already defined. But if you are to avoid circularity, at some point you end up with some terms for which no predefined terms exist. These are called "primatives". Everything else is defined in terms of them, but for them, there is no definition. In Geometry, "point", "line", "plane", "between", "closer together than" are usually taken as primatives (though only "point" and "closer together than" need be - the rest can be defined from these two and the "common notions" of set theory).
The same thing hold true for theorems. The starting points this time are the axioms: A collection of relationships between the primatives. A more sophisticated view of primatives and axioms is that they are, taken together, a definition of themselves: that is, the axioms provide meaning to the primatives by describing how they relate to each other.
The article, and Sir Col, raise the question: are these axioms really true? While I suspect the author knows the answer, she leaves the waters muddied. This question has occupied the attention on many great mathematicians. When Gauss first realized the existence of hyperbolic geometry, he surrepticiously surveyed the angles formed by three well-separated peaks, to see if he could detect any difference between the sum of the angles and 180o. He was trying to find out which was the "true" geometry. Hyperbolic (or Lobatchevskian) geometry was slow in gaining acceptance because mathematicians doubted it's "truthfulness".
Then Weierstrauss discovered the pseudosphere: a shape in Euclidean geometry which shows the same behavior as a strip cut from the Hyperbolic plane. He was followed by analytic descriptions from Poincare and Dedekind which exactly model hyperbolic geometry in its entirety. A simpler model for Euclidean geometry had already been found within hyperbolic geometry. This makes the answer to the question "which is the true geometry" clear: They are either both true, or both false. The truth of one implies the truth of the other!
But the two are identical in their axioms except for axiom 5. One demands it to be true, the other demands it to be false. How then can the truth of one imply the truth of the other? Isn't this contradictory?. The problem is not in either Euclidean or Hyperbolic geometry. It is not in the concept of proof. Where the problem lies is in the concept of "true".
The usage of "true" in both the article and in Sir Col's post is ill-defined. I have intentionally kept with this ill-defined usage in the statements above.
Obviously, I have more to say (when have I ever been short-winded)? But it's late and I have to work tomorrow, so I'm leaving it for now with this question: what does it mean - mathematically - for a statement to be "true"?
[ Sir Col - One last parting shot: I have never read the book by Lakatos, but quite frankly from your description I am not impressed. He pretends that a famous formula is applied more broadly than the theorem actually states, and then uses this artificially expanded scope to "disprove" it? This is chicanery, not mathematics! If his purpose is to show that you should be careful about exactly what it is you think you know, then I applaud. Descartes and Euler never claimed that particular formula applied to all polyhedra. Only to polyhedra topologically equivalent to a sphere (they stated it differently of course, but this is how I know it). But your description seems to indicate that he is claiming it shows a weakness in the idea of proof, and that it most certainly does not! ]
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #45 on: Oct 1st, 2003, 9:20am » |
Quote Modify
|
I apologise, as I have clearly done a major disservice to Lakatos's work. You can read a little about him at St. Andrew's History of Mathematics.
Again, I've also failed to express myself clearly. One of my main areas of interest is the history of mathematical thought and philosophy. I am no expert and I do not claim to understand the all of the mathematics – in that respect, I am an enthusiastic novice – but with regard the life and works of the mathematicians I am well versed. I selected the Descartes-Euler relation as one example of many: interestingly, Euler abandonded the formula when he was unable to deal with the counter-examples.
My point, which I made badly, was that if there is one thing we can learn from history, other than we never learn from history, is that something remains true until it is no longer true.
That sounds like I'm stating the obvious, but it is in fact quite subtle and has everything to do with perception. I believe ideas to be true, because I respect the minds behind the concept, and I can (eventually) be persuaded of their truth. I depend on the insights and brilliance of the giants of intellect that appear so rarely: it is estimated that there are around 100 revolutionary great thinkers in each century. I do not decide truth, because I am an intellectual pygmy on the grand scale of things.
Once in a while someone, who gains respect from his/her peers arrives on the scene, and they challenge our thinking. Among many of the great thinkers I have had the privilege to meet, through their written works, Lakatos is one who has obtained special respect.
What Lakatos does, through his book, is challenge the methodology of mathematics. He presents a case for a system of observation, proof, and refutation; constantly refining and improving ideas through creativity and criticism. He challenges the ideas of formalism and the dogmas of logical positivism, and questions the notion that mathematics is merely the accumulation of established truths; ultimately, he calls for a more organic approach to the way in which we learn and discover mathematics.
|
|
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: cos(1 deg) = irrational
« Reply #46 on: Oct 1st, 2003, 8:48pm » |
Quote Modify
|
on Oct 1st, 2003, 9:20am, Sir Col wrote:
Without reading his work myself, I cannot say one way or another.
Quote:| Again, I've also failed to express myself clearly. One of my main areas of interest is the history of mathematical thought and philosophy. I am no expert and I do not claim to understand the all of the mathematics – in that respect, I am an enthusiastic novice – but with regard the life and works of the mathematicians I am well versed. I selected the Descartes-Euler relation as one example of many: interestingly, Euler abandonded the formula when he was unable to deal with the counter-examples. |
|
I am not intimately familiar with Euler's life, but I do not believe that this is correct. Euler did not "abandon the formula". He had no need of doing such a thing. It was proved quite well for the polyhedra for which it was stated. I find it far far far more likely that what Euler did was either give up trying to extend the class of polyhedra on which it was known to hold to the fullest extent, or he gave up trying to generalize the formula to apply to all polyhedra.
"Abandoning the formula" because there are polyhedra for which it does not hold would be like abandoning the formula 0x = 0 because you have the case 00 = 1. You don't simply abandon the whole thing because of exceptions. Instead you state explicitly where it is known to hold: [forall] x [in] [bbr] with x [ne] 0, 0x = 0.
Quote:My point, which I made badly, was that if there is one thing we can learn from history, other than we never learn from history, is that something remains true until it is no longer true.
That sounds like I'm stating the obvious, but it is in fact quite subtle and has everything to do with perception. I believe ideas to be true, because I respect the minds behind the concept, and I can (eventually) be persuaded of their truth. I depend on the insights and brilliance of the giants of intellect that appear so rarely: it is estimated that there are around 100 revolutionary great thinkers in each century. I do not decide truth, because I am an intellectual pygmy on the grand scale of things. |
|
I do not believe ideas to be true just because someone who is considered an "intellectual giant" says they are. I accept things as true only on the basis of evidence. (This does not mean that I reject what people say until I have understood it myself. I have neither the time nor the capacity to follow every line of reasoning. But I hold all results I have not understood at "2nd level". I accept them, but not without reservation.)
I am not asking anyone to decide what is true or what is not. I am asking for what it means to be true. Why is 1 = 1 true and 1 = 2 not true?
You say that the Descartes-Euler formula is false. I say it matters first of all exactly what you are stating as the Descartes-Euler formula. If you say "For all polyhedra, V+F = E+2", then yes, that is false. It is also not the Descartes-Euler formula, as you have applied it more broadly than they ever did. This falsehood is the result of bad teaching, not the result of broken mathematics. If you say "For all polyhedra homeomorphic to a sphere (or a different, still workable restiction), V+F = E+2", then you are wrong in saying it is not true.
The case for 180o in a triangle is similar, but it is again late, so I will expound on it another time.
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #47 on: Oct 2nd, 2003, 10:24am » |
Quote Modify
|
on Oct 1st, 2003, 8:48pm, Icarus wrote:| I do not believe ideas to be true just because someone who is considered an "intellectual giant" says they are. I accept things as true only on the basis of evidence. |
|
Where did you acquire the art of reasoning and discernment? I believe it is a cultural indoctrination. We think in words, pictures, and numbers, based on our experiences, and this methodology is disseminated to future generations. To think outside out of how we think, is almost a contradiction. Our standards, absolutes, and truths, are determined by the conditioning of our mental faculties.
Although I am passionate about mathematics, I share an equal love for language and etymology. Sad as it may sound, I get quite excited when I discover the origin, or root, of a particular word. I'm generally satisfied to know that a word like, geometry, comes from the Greek, geo=earth, and metron=measure; that is, earth measure, which helps us understand the original practical aspect of the science. However, how far back do we go? Where did those Greek words come from? Am I interested? If not, why not? This is analogous with my concern with what we do in mathematics. As I keep saying, I enjoy what I do, but I recognise that I am deluding myself with a form of semi-truth. I accept on the one hand that I am not really getting to the heart of the matter, but on the other hand, I don't care. If I asked someone to explain where the word, geometry, came from, and they answered, "From the dictionary," should I be concerned? To reject that type of response, and accept the, geo-metron, response is still falling short of what is actually going on.
I am not seeking approval, but I am trying to explain where my apparent anti-establishment attitude is coming from. It is not to get a reaction, neither is it a primitive naivity, it is by considering mathematics and learning on a philosophical level. Unfortunately for me, it is one of those revelations that fascinates on the one level, yet abhors on another.
|
|
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: cos(1 deg) = irrational
« Reply #48 on: Oct 2nd, 2003, 1:53pm » |
Quote Modify
|
I don't really see what the problem with proof would be in mathematics. You have primitives and axiom, and everything what is proven in mathematics is true given those primitives and axioms.
The problems with logical positivism arises when you deal with physical reality, and observation, and you suddenly find yourself lacking both the necessary primitives and axioms to proof things. You can only guess at things to model reality with as best you can.
As soon as you abstract from the reality that problem disappears, you can just examine where your (assumed) primitives and axioms lead you. Which is quite valid, and can help you discover more about similar systems, which can in turn give new insights, which you may, if you so wish, apply to new models of reality.
One of the problems with positivism in science is that you can't proof general statements, since you would need to exhaustively search your domain to check your model against it. The only meaningfull thing you can do is try to disprove general claims, by finding counter examples. But then you still won't end up with models which are true, just models which haven't been proven false.
So the problem you have is, how much credence should you give such models? None of them can be proven, but some are instinctively 'better' than others. What gives a good measure that drives you closer to the truth?
One of the popular ways to prove a model/theory is better than another these days is to find the most unlikely event implied by the theory, and then try to disprove it (which should be easy if it's an event that seems quite unlikely to be true).
One of the things Lakatos (and Kuhn) are well known for in this context is that they state that there are research programs, with a hard core that doesn't change much for long times, and looser hypothesis that get adapted over time. And every once in a while there's a scientific revolution that sparks rivaling research programs which may drive the other to extinction.
(To be honest, "philosophy of science" by Bechtel explains it much better, and more thoroughly. And it's thin enough for even me to read )
|
| « Last Edit: Oct 2nd, 2003, 2:17pm by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
Sir Col
Uberpuzzler
impudens simia et macrologus profundus fabulae
Gender:
Posts: 1825
|
|
Re: cos(1 deg) = irrational
« Reply #49 on: Oct 2nd, 2003, 3:27pm » |
Quote Modify
|
Thanks for your thoughts and book recommendation, towr. After a tricky search on the internet I've located copy of Bechtel, and one should be with me within a couple of days.
I know what you mean about radical thinking, or being the better orator, driving one methodolgy to extinction; as you cited, Kuhn vs Popper is a classic example. The right person doesn't always win, or sometimes the right person wins, but for the wrong reasons.
|
|
IP Logged |
mathschallenge.net / projecteuler.net
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print