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Icarus
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Re: cos(1 deg) = irrational
« Reply #50 on: Oct 2nd, 2003, 7:50pm » |
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on Oct 2nd, 2003, 10:24am, Sir Col wrote:| 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. |
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I do not claim that I independently rediscover anything here. My abilities of reasoning and discernment were indeed trained into me. But on the other hand, I kept at that training because I could follow the sense of it myself. I never accepted "because that's the way it's done!" as the answer to the question "why do it that way?"
I have great respect for those who first uncovered these results I find so interesting, who first searched laboriously through unmapped country to find the things of beauty that I am privileged to come to by well-built roads. But I trace the path of the roads myself to see if they are leading in the direction I was told.
Turning aside from towr's well-stated comments on the philosophy of science back to the foundations of mathematics, maybe tonight I can finally figure out how to say what I want before it gets too late.
What was learned from the discovery of hyperbolic geometry, and was driven home by the development of formalism was this: any mathematical theory is dependent upon its axioms and primatives. And these are NOT statements that are "self-evidently true", but simply the assumptions - the definitions upon which the theory is based. By choosing any set of primatives and axioms, you can develop a mathematical theory from them, and the only measure of the validity - the "truthfulness" of the resulting theory is whether or not it is contradictory. All non-contradictory theories are equally valid.
Is the sum of the angles of a triangle equal to 180o? In Euclidean geometry, yes. In Lobatchevskian geometry, no. Without reference to the geometry in use, the question is meaningless. Usually, that reference is implicit, but it still must be there for the question to even make sense.
If I say "the sum of the angles of a triangle is 180o", it should now be evident that what I am really saying is "if the axioms of Euclidean geometry are used, then the sum of the angles of a triangle is 180o". Which is true.
If I say "the sum of the angles of a triangle is less than 180o", what I am really saying is "if the axioms of Hyperbolic geometry are used, then the sum of the angles of a triangle is less than 180o". Which is also true.
Since, when we first teach students (or whatever is was that towr calls them) about geometry, we do not want to muddy the waters with needless complications, we teach them Euclidean geometry. Everything we say is within the context of Euclidean geometry. So when we say that the sum of the angles of a triangle is 180o, what we have said is wholly true.
And when we say that V+F = E+2, what we say is also true, provided we didn't pretend it holds for more than can be shown.
It follows that there is no such thing as an "unprovable axiom". Axioms are where proofs start. Within the theory all axioms are true by definition. (If the theory is contradictory, they are also all false. But such theories are worthless.)
Your charge of circular logic is one that needs to be backed up. Any proof involving circular logic is invalid, and mathematicians go to great lengths to avoid this. Search carefully, and if you can find an example, bring it to light. But if you do, I suspect further investigation will show that the result can and has been proven without the use of results dependent on it.
I am surprised any one would think the continuum hypothesis "self-evident". It is rather surprising that it apparently independent of the standard axioms of set theory, but that is because it involves the real numbers, a set with very strong structure built into it. The idea that even with all that structure, one can either assume to be true or to be false that there exists sets strictly larger than the natural numbers, but strictly smaller than the full reals, is hard to believe.
However, there is nothing "self-evident" about the hypothesis itself. Without knowing its history, I would assume when first confronted with it that one could either prove it to be true, or to be false, but there is nothing to suggest to me which would be the case.
(The axiom of choice is another matter: when stated properly, it seems so obvious that it must be provable.)
If you would like to come to a better understanding of the foundations of mathematics, and the basis of proof, I would recommend that you study formalism. I personally am very fond of Bourbaki's Elements of Set Theory, though some find it very dry, and others pan his approach as cumbersome. But any well written formalism will do.
Of course you have already stated Lakatos has raised objections to formalism, and others have been critical of the formalist movement as well. I do not disagree (I can't entirely agree either because I only have a limited knowledge of their objections). That is beside the point. Formalism has limitations, but it is a wonderful taskmaster for defining the basis of mathematical theory and of proof.
For example, Bourbaki defines a proof as being a list of statements satisfying the condition that every statement in the list is either an instance of an axiom, or is a statement "B" which is preceded somewhere in the list by statements of the form "A" and "A [implies] B" (syllogism). A theorem is any statement that appears in a proof. As you can see, this clearly prevents the possibility of circular proof.
Of course, such a "proof" would be prohibitively long for real mathematicians. So he then proceeds to develop metamathematical methods of proving the existance of full proofs from less strenuous requirements. For instance, if you have a list wherein each statement is either a theorem from some pre-existing proof or a syllogism, then there exists a full proof containing all the statements in this list. This is fairly easy to justify: just replace each instance of a theorem in your list with its proof.
Bourbaki continues in this vein to validate all the "standard" practices in proofs - from proof by contradiction (he follows standard logic, so this is possible) to trans-finite induction.
By the formalist ideal, if two mathematicians are in dispute over whether or not a particular "proof" is valid, all they need do is convert it into a full proof. If this is possible, the "proof" was valid. If not, it wasn't. (In practice, even the formalists admitted that it isn't so easy.)
Kronecker & co. may have felt the irrationals to be an abomination, but while they could rightfully turn their backs on analysis and apply their energies to the parts of mathematics they liked, I heartily disagree that they could intelligently reject analysis and the infinite as false or unfit for mathematics. To do so, they would have to show that these ideas lead to actual contradictions. This they and their successors have entirely failed to do.
Quote:| but... what are we building our mathematics on? Can we really be sure that the foundations are firm? |
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The foundations of a mathematical theory come in three parts: the logic on which it is based, the basis of proof, and the axioms.
While other logics are possible, the standard symbolic logic of Aristotle is both simple and powerful. Other logics are studied using it, so I have no fear of it being baseless.
The concept of proof as given by Bourbaki (but which in essence was first established by Thales of Miletus) is easily justified as logically sound, being only an expression of Aristotlean logic itself. The more sophisticated concepts of proof actually used have been carefully tied back to this basic concept and have thus been justified as well. A proffered proof needs to be examined carefully to make sure that it is correctly constructed, but if so, there is no need to fear that the method itself is invalid.
This leaves the axioms. But there is no need for axioms to meet some nebulous requirement to be "true". Within their theory, all axioms are by definition "true". The complete truth stated by a theorem is not contained in the theorem statement itself, but rather in stating that the theorem follows from the axioms.
The only condition to be placed on axioms is that the theory they produce not be contradictory. That is, that is impossible in the theory to both prove a statement "A" and its negation "[lnot]A". This is the place where modern mathematics is most vulnerable to attack. Just because we know of no contradictions now does not mean there isn't one out there, waiting to pounce. The best approach we have for insuring that a theory is not contradictory is to build a model of it within another theory. But this only works if the new theory isn't contradictory either.
However, with that said, I (like most other mathematicians) am not worried. The lion has already pounced once, and it was just a matter of redefining to avoid his claws. Should he pounce again, I do not doubt that another work-around will be possible. I look forward to what we will learn from the experience. Last time was a real eye-opener in several areas, even if the cure has left us unable to do some of the things we would like to do.
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Sir Col
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Re: cos(1 deg) = irrational
« Reply #51 on: Oct 3rd, 2003, 1:52pm » |
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I have re-read your post three times now, Icarus, to give it the respect that it deserves.
I quite agree with most of what you say. I do, however, still have some reservations.
Quote:| My abilities of reasoning and discernment were indeed trained into me. But on the other hand, I kept at that training because I could follow the sense of it myself. |
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One of my concerns is that our ability to judge the methods of reasoning we are taught is already too entrenched in earlier learned methods. In other words, when we reach the stage where we are able to critically consider the methods we use to critically consider ideas, we can only use those taught methods. An eye can only, at best, see a mere reflection of itself.
Let me explain... I accept that the system of logic and mathematics is consistent; what we can ever know within that system is another question. Our axioms are simple, self-evident truths, created by earlier intelligent generations. A young child, because of their lack of mathematical experience, is not able to make a well informed judgement about whether or not we have truly selected the optimum axioms, and ones which will lead us to the best generalisations. By the time we reach a point where we can make a truly intelligent judgement about their optimality, our world has been shaped by them; it is all we know.
I think that William Wordsworth summed it up very well.
Quote:| Mathematics is an imaginary world created out of pure intelligence. |
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As I said, I take great pleasure in 'doing' mathematics. Using the definitions, rules, and theorems, to manipulate ideas and solve problems. However, I would still contest that we are doing nothing more than building towers with the toy bricks we have been presented with. Hence my rhetorical question, "What are we actually proving?"
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Icarus
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Re: cos(1 deg) = irrational
« Reply #52 on: Oct 3rd, 2003, 5:46pm » |
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on Oct 3rd, 2003, 1:52pm, Sir Col wrote:| Our axioms are simple, self-evident truths, created by earlier intelligent generations. |
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Apparently, my wordiness has utterly obscured my point, if you still think this after three readings.
Our axioms are not, nor need to be, "simple self-evident truths". Nor do they have to be those handed down from earlier generations. In the last ~125 years mathematics has moved away from the concept that its theories are expressions of "truth" -at least directly.
The truths of mathematics are no longer such things as "vertical angles are equal", but rather "Under the rules of syllogism, the axioms of Euclidean geometry imply that vertical angles are equal." (Yes, the vertical angle theorem holds in other geometries, but that is another truth.)
Any statement may be taken as the axiom of a mathematical theory. In that theory it is true. In other theories, it is false. In some it is both - but these theories are worthless and discarded.
So while mathematics still is engaged in a hunt for truth, it is not the theorems that are the real truth we find, but rather their dependence on the axioms.
This is not "handed down". We can can consider: what if it took 3 points to determine any line? Suppose our geometry possesses an asymmetry, so that vertical angles in one direction are larger than their opposite? Such ideas would have been scoffed at as preposterous not long ago. Now the only reason not to study them is simply that the results may not be worth the while.
Quote:| However, I would still contest that we are doing nothing more than building towers with the toy bricks we have been presented with. Hence my rhetorical question, "What are we actually proving?" |
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I suggest you look more carefully at these "toys". They appear to be stronger than titanium with perfectly tight non-slip interlocking edges. You may call them toys, but I would trust any building built with them by a contractor who uses them with care.
And I believe your rhetorical question has been non-rhetorically answered quite well.
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Sir Col
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Re: cos(1 deg) = irrational
« Reply #53 on: Oct 4th, 2003, 9:04am » |
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on Oct 3rd, 2003, 5:46pm, Icarus wrote:| Apparently, my wordiness has utterly obscured my point, if you still think this after three readings. |
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Icarus, I deeply respect your expertise. In fact, and I hope you don't find this embarrassing, you are one of the most experienced mathematicians I have ever met – albeit via the internet. Apart from the internet, and books and journals I read, I have no face-to-face contact with passionate mathematicians. However, much of what we have been discussing lies outside of the objective security of mathematics. We have been philosophising concepts of truth, both axiomatic and intuititive. Consequently, it is likely that we will differ on some counts.
For example, two fundamental questions that challenge those of us serious about what we study are, (i) do numbers exist, or are they creation of man? (ii) does the continuum exist?
I'm sure that both of us, like most mathematicians, agree that numbers exist independently of man. Question (ii), is where many mathematicians begin to disagree. No one can prove the existence of the continuum, or the infinite. You can define it, but that is not the same. Physicists still hypothesise the divisibility of space ad infinitum; confer tiny strings and Planck unit.
The work of Gödel and Cohen, especially, has demonstrated that the starting point in mathematics (axiomatic truth), will never be sufficient to prove everything.
There is little doubt that the infinite, in analysis, produces results that are too accurate to be ignored, but no one will ever prove the existence of the infinite.
Kroncker et alii, for all their disgraceful treatment of Cantor, should be respected as mathematicians for starting with the axiom: "God made the integers, all the rest is the work of man."
The question that motivated Gödel throughout his life was, "What is proof?" He recognised the importance of this question, he showed the world that the answer will stupify.
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Icarus
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Re: cos(1 deg) = irrational
« Reply #54 on: Oct 4th, 2003, 7:06pm » |
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Here is definitely where we part ways: What exactly do you mean by "exist"? Do you think that these intellectual concepts have some sort of nebulous existance beyond the conception of those who think of them? (I include not only man, but animals and any other thinking beings who may exist.) If so, please point out to me where "one" may be found.
Do numbers exist? I say yes, but not apart from the conception of those who think of them. Does the continuum exist? Of course it does - in the exact same way! There is only one test for existance for any constructed mathematical object: The theory in which it appears must not be contradictory. If this holds, then it is foolishness to deny its existance! Within that theory, it exists. And as long as that theory is non-contradictory, the theory has as much truth and "vitality" as any other.
Kronecker et al. had every right to avoid the infinite in their mathematics. But they were fools in demanding others should avoid it as well. "Infinity" (your pick as to which) has every bit as much as existance as "one".
I did not need Gödel to tell me mathematics cannot prove everything. It is obvious that mathematics can only prove things about mathematics. Thus any great and eternal truths of the Universe must come from another source. Alas that none exists with absolute certainty (for those of a religious bent (everyone is, but some deny that their religion is a religion), this uncertainty is rooted in the human condition).
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Barukh
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Re: cos(1 deg) = irrational
« Reply #55 on: Oct 5th, 2003, 8:09am » |
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on Oct 4th, 2003, 7:06pm, Icarus wrote:| Kronecker et al. had every right to avoid the infinite in their mathematics. But they were fools in demanding others should avoid it as well... |
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Wow, it's too much, I think. Although I generally agree with Icarus (the content, not the way), I wouldn't call Kronecker et al fools in any respect. Maybe, they were not too far-sighted and did not anticipated that mathematical foundations would not be univeral anymore (how many axiomatic set theories there exist today?).
Let me also remind that Hilbert - who was the founder of the Formalists school - for the proof of the consistency of arithmetics (the central part of his famous programme) suggested to use the "finitary" methods that were just the methods of Intuitionsits, whose forerunner was Kronecker.
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Sir Col
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Re: cos(1 deg) = irrational
« Reply #56 on: Oct 5th, 2003, 8:35am » |
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It's funny that you should mention Hilbert, as I was thinking about metamathematics and his system of absolute proof of consistency (through finitistic methods), when I mentioned Kronecker. However, like so many things in mathematics, I don't know enough about it to speak with any authority.
I hear, and have certainly been moved on in my thinking by the persuasive arguments, but I still remain quite befuddled with all of this. I think my fundamental objection is the demand for consistency even at the cost of detachment from meaningful concepts.
As my level of mathematics is unlikely to ever go beyond recreational/elementary/applied fields, I am probably never going to fully understand abstract higher mathematics.
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Icarus
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Re: cos(1 deg) = irrational
« Reply #57 on: Oct 5th, 2003, 1:53pm » |
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I make no aspersions on their intelligence. And I am not sure to what extent they were actually guilty of attempting to prevent other mathematicians from working in areas they disparaged. But to whatever extent that was, I would call them foolish. Wisdom and Intelligence are not the same thing.
I do not know if Kronecker and his disciples ever engaged in this sort of a practice. I do know that this foolishness has reared its ugly head before and since. Nicholas Bourbaki was not an actual mathematician, but rather the pen name used by a group of young french mathematicians to publish mathematical results that would have resulted in repression had they been published in their own names, because the mathematical "elite" in France at the time did not feel these were fit areas for research.
The appellation of "foolish" is not an indictment against their mathematics, but rather against their personal behavior.
Consistency is not an external demand in mathematics. It is a requirement fundamental to all we do. To be inconsistent is to be completely meaningless for the mathematician, because anything can be proved from a contradiction.
On the other hand, just because we are able to create theories on any consistent foundation, even when the foundation of one theory may be directly in opposition to the foundation of another, does not mean that these concepts we introduce are meaningless. Rather, these theories allow us to explore new worlds of possible meanings.
Euclidean geometry was the valley we grew up in. We wandered it and became intimately familiar with all of its glades and dells. But we kept exploring its boundaries until one day we came to the top of a rise. Lo & behold! Beyond the rise lay the new valley of hyperbolic geometry, very similar to the one we knew, but with distinct differences as well. And so we proceeded to explore it.
But its existance and our new knowledge of it does not effect in any way the original valley. Euclidean geometry is still there. Still the same strong useful theory it has ever been. The only change is now we know that it isn't everything there is.
And by continuing to explore this metaphorical world, we discover that there are other areas of it that look nothing like our comfortable little valley: barren deserts, frozen wastelands, majestic mountains, wide oceans. Because these are like nothing we have seen before does not mean that they should be rejected, or that they are worthless. To abandon my burgeoning metaphor for a simple simile, I live in a region which 150 years ago was called "the great American desert". As explorers of European descent first came to the region, they noticed that it got significantly less rainfall than did lands in Europe or the eastern parts of North America. The flora mostly consisted of tough drought-resistant grasses. It was quickly written off as unfit for habitation. Today, of course, it is known as the most productive farmland anywhere in the world.
The point is, just because we invent theories that have no grounding in our current experience does not mean that these theories are meaningless. It merely means that we must explore them before we can comprehend their meaning.
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Sir Col
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Re: cos(1 deg) = irrational
« Reply #58 on: Oct 5th, 2003, 2:52pm » |
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on Oct 5th, 2003, 1:53pm, Icarus wrote:| I make no aspersions on their intelligence. And I am not sure to what extent they were actually guilty of attempting to prevent other mathematicians from working in areas they disparaged. But to whatever extent that was, I would call them foolish. |
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I would add the word, evil, to their description. The extent, rather, the depth to which Leopold Kronecker, and his colleagues, sunk to stop free thinkers is abominable. After attempting to destroy Karl Weierstrass's career, and failing due to his strength of character, Kronecker turned all his destructive energies on the life of the more vulnerable Georg Cantor. He ensured that Cantor would never achieve his lifelong dream of joining the teaching staff at the University of Berlin, he rubbished Cantor's reputation and thwarted every opportunity to publish in respected journals, and continued to undermine the brave mathematician who was seeking to help mathematicians to move into new exciting territories.
I could strongly recommend a book, called The Mystery of Aleph, by Amir D. Aczel. It is partly a biographical account of the life of Cantor and his works, but equally an historical tour of the concepts of actual infinity (as opposed to potential) leading to the continuum problem and beyond.
Quote:| The point is, just because we invent theories that have no grounding in our current experience does not mean that these theories are meaningless. It merely means that we must explore them before we can comprehend their meaning. |
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I know what you're saying. Before non-Euclidean geometries became established, no one really suspected, or could be sure, that they related to real space. It was a purely abstract adventure, and it taught us a valuable lesson.
I'm almost there, Icarus. I am beginning to see/appreciate the benefits of abstract exploration. Actually I'm reading a super book at the moment, called Gödel's Proof, by James R. Newman and Ernest Negel (with a new forward by Douglas Hofstadter). Coincindentally, it has taken me amazingly close to everything that has been discussed here.
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Icarus
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Re: cos(1 deg) = irrational
« Reply #59 on: Oct 5th, 2003, 8:26pm » |
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on Oct 5th, 2003, 2:52pm, Sir Col wrote:
I would add the word, evil, to their description... |
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Yes, I agree. Both foolhardy and evil.
Quote:| I'm almost there, Icarus. I am beginning to see/appreciate the benefits of abstract exploration. |
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The important thing is to understand the position. Then even if you still disagree, you will be disagreeing on the basis of what you know, not on the basis of not following the argument.
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david martinez
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Re: cos(1 deg) = irrational
« Reply #60 on: Dec 15th, 2004, 6:42pm » |
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this result is a consequence of the irreducibility of the n-th cyclotomic polynomial
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Icarus
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Re: cos(1 deg) = irrational
« Reply #61 on: Dec 16th, 2004, 6:02pm » |
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True, but when you replace the jargon with the actual proof, this is a rather difficult way to go about it. The proofs offered by Sir Col and I in the first & second replies are far more elementary.
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