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Alvin Chan
Hong Kong
407 Posts |
 Posted - Mar 11 2006 : 02:23:32 AM
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Hi Frank and Ether,
Ether, the "self-evident" part of geometry are what called "axioms" in mathematics. They were considered as self-evident by the Greeks. But as you (very cleverly) noticed, the notion of being "self-evident" is not convincing/explanatory. It simply avoid the problem of looking for "absolute truth" altogether by stopping somewhere.
That's why mathematicians nowadays have abandoned such thing. They consider "axioms" as having no truth value in their own right. The axioms (say, of geometry) are not "true" nor "false" statement about ANYTHING. They are some conditions, some membership criteria. (technically, it's the basics of naive set theory. The term "naive" to be explained, if you would like to hear. It's somehow due to Russell) So the axioms of Euclidean geometry are not telling you anything. And they're not part of the statements of mathematics. Mathematics deal with the deductive part, which see the consequences of such axioms. To make it simplier, they would like to see what properties the members of this membership must have. And that part is absolute. Well, almost. It based on logic and set theory. And former is really absolute, but the second one----people come up with "axiomatic set theory" which constitutes the "foundation" of modern mathematics. Is this absolute? So far people don't find any trouble with it. (as oppose to the "naive set theory" which has proved to be problematic after Russell's famous paradox) And it's likely that it has no problem. But people aren't sure what "absolute" means, and so the birth of a branch of philosophy called "philosophy of mathematics".
If you want a very simple example of the thought behind "axiomatic mathematics", here is one:
Alvin is a boy.
In the standard of mathematics, this is not absolute, even if I've taken 10 medical test to confirm it!!
If X has 3 hands, then X is called a "3-handed stuff".
Theorem: A 3-handed stuff has 3 hands.
I know this is really stupid, but the latter example is how axiomatic mathematics (which is the whole of mathematics)works. It's really absolutely true because it tells you nothing. And its truth has nothing to do with whether there are 3-handed stuff or not.
Oh, I'm going too far, although the story shouldn't have finished without mentioning David Hilbert's scheme, and Goedel who killed this scheme. They're really beautiful things.
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Frank, I agree. Math is most self-evident, second only to logic. It's more natural than "natural science". In fact, it's like the living ground of science. The term "natural science" is just a bad convention.
But on the other hand, Math (Math alone) tells you the FEWEST thing about our real world. (second only to logic, which tells you NOTHING at all). Math is too self-evident that it tells you almost nothing.
Note that by Mathematics, I am not refering to its application, which certainly have great contributions. The tools of General Relativity are all mathematical in nature. But what really matters are the interpretations inherented in the theory of relativity.
In short, Mathematics alone tells you really nothing. But it REVEALS many things which you should know but just couldn't see clearly. So if you can give a mathematical theory an interpretation, suddenly it opens things up for you. Note that the interpretation part is, for most people, even easier than the "proper" mathematical part.
It is weird that math symbols can handle so many patterns of our world. But when you realize that such and such mathematical theorems are considering patterns in our world, you are already giving it an interpretation----a nice thing, and the development of math does need such motivation. But there are also good reasons to separate interpretations from the formal, logical part of math.
Alvin |
Edited by - Alvin Chan on Mar 11 2006 10:07:28 AM |
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Etherfish
USA
3615 Posts |
Posted - Mar 11 2006 : 09:14:35 AM
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Thanks Alvin, I love this stuff!
Do you know any easy to understand websites about
"David Hilbert's scheme, and Goedel who killed this scheme."
or "naive set theory and the story of Russell?
If not I'll do a search sometime.
I know we've stepped over the topic line here. I can delete this message later, but Alvin's post seems relevent to what we were discussing. |
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Alvin Chan
Hong Kong
407 Posts |
Posted - Mar 11 2006 : 10:01:16 AM
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naive set theory is basically (just basically, I've to simplify things) the theory (or better put, the use of language) which classify things into sets (of objects) by some charaterising properties. For example, the set {x|x is a male on the earth} is something like a membership which all male creatures on earth are in. No more and no less.
One more example:
{x|x is a human and x is not a human}
This set contains nothing, called an empty set. There are many operations which you can do with sets, which I'll not go into. Anyway, people believe that any well-defined charaterising property (or properties) would determine a set definitely. May be an empty set, like the above. But still you get a set, which know who is a member of the set and who is not, with no problem, right? This has been the basics of mathematics for a while.
But surprisingly, Russell discovered some hidden problems with this naive set theory, and this lead to the 3rd crisis of mathematics. (finally settled down, temporarily at least, by axiomatic set theory) His "set" is as follows:
Y= {x|x is a set and x doesn't contain itself as a member}
Now Y is a set. And you and me certainly are not member of this set, since we are not set! We are human. And we don't contain ourselves as a member. But is Y a member of Y. A moment of thought will tell you that "Y is a member of itself and Y is not a member of itself" which is a contradiction!
If you can't follow, which is natural if you first read such things as set, here is a everyday language version:
I say: "I will cut the hair of those who don't cut their own hair".
Whose hair will be cut by me? You can determine that. Except that when it comes to me. Should I cut my own hair? Either way I can't do what I promise, even in principle.
This is called Russell's paradox, which give set theory a new name: NAIVE set theory. And it initiates many development of philosophy of mathematics, all hoping to find an "absolute" foundation of mathematics.
In theory, mathematicians should not use naive set theory because it contains constradiction which is not acceptable. They should use axiomatic set theory instead, which tells them what they can do with sets, and what they can't. But in reality most mathematicians don't care. Naive set theory is so much nicer and easier to use. And usually (>99.999%) there would not be any problem-- the two theories of set actually coincide in practice.
So much for "set theory" and "Russell's paradox". To know more, just search with the underlined words. But you will get too much results! SO better search with both terms together.
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Try using "Hilbert's scheme" . But Hilbert is a great guy who has too many contributions, and this term may mean something else. Also, this is a stuff more difficult to explain in detail.
Very roughly, Hilbert, like others, wanted to make mathematics "absolute". His way (just in theory, never come up in practice) is to extract all meaning from mathematics, and do "mathematics" by setting something like axioms and rules of inference. But this time not with logic in the usual sense. Logic in usual sense involve understanding of meanings. Not in Hilbert's scheme! He wanted to make mathematics like a game, do-able even by computers! You don't need any understanding about the symbols, you just have to know what you can do with them!! "Theorems" in this case are those which you can get within a finitely many steps by those rules of games (again meaning is not involved in the process).
This is just a scheme which he wanted to achieve, and never really come true. And now we know it won't come true, because of....
Goedel!! (a German name, "oe" means "o" omlaut. May be you can't search him out...) A genius in mathematical logic! He proved that such scheme (when useful enough. Precisely, when it is consistent and includes natural number manipulation) will contain a statement which is true (and provable in a tranditional way) but is not verfiable within the stupid, meaningless Hilbert's scheme.
Goedel is a much misused and misunderstood guy. Many people (even some books) think that he show that "there are always in mathematics something which is true but is not (even in principle) provable". Well, this is wrong. It's just not provable when you deal with a system (Hilbert's scheme) which ignore meaningful. We human have the insights which works, and which computers doesn't have.
An insightful reading is "the emperor's new mind" by Penrose. It talks about physics and math (including Goedel) at a philosophical level.
Alvin |
Edited by - Alvin Chan on Mar 11 2006 10:11:58 AM |
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Alvin Chan
Hong Kong
407 Posts |
Posted - Mar 11 2006 : 10:16:17 AM
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They are not really relevant to yoga. Not directly. The philosophy of mathematics has stimulated various thoughts on analytic philosophy. The other inspiration of this is the philosophy of science, which deal largely with the question of "what is scientific". So it's slightly related....
Hibert and his method is an inspiration for the theory of computation. Without him we may not have modern computer. Goedel told us that our computer, as long as it still use the basic principles we're using now, will never be able to "see" something which we human find obvious. (even when it get a lot faster!!) |
Edited by - Alvin Chan on Mar 11 2006 10:43:12 AM |
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Frank-in-SanDiego
USA
363 Posts |
Posted - Mar 11 2006 : 8:32:30 PM
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Hari Om Tat Sat
~~~~~~~~~~~~~~
quote:
Originally posted by Alvin Chan
Math is most self-evident, second only to logic.
Hello Alvin and thx for the time you have spent positioning the whole conversation. You have made good points. This is not a rebuttal to any, just some observations.
Even if we say |Alvin is a boy| , using the mathematical symbols of | | to define Absolute, will not be sufficient? << >> just some math humor.
I see math as a tool... if the axiom fits, and gives use, then its worth the use. Whether its pure to the core, well...its a tool
and I see your point and would not 'waste the breath' to argue a different point.
Yet, as I see it, Science and purity of reflecting reality evolves. Just as Newtonian physics was superseded by Einsteinian physics, and the perhaps may be superseded by string, super-string theory. We a humans are so new at being knowledgable we need some growth time.
Going back thousands of years to thouse that started math, applied it, etc of history is a millisecond in time. We need time to develop and get it right...not a few decades, but some serious time. That is why I am a big fan of Ritam - that level of consciousness that only knows the truth. One becomes the uptimate scientist - his lab is his ability to discrimiate and see the world as it is ( e.g. Sat-Chit-Ananda Truth-consciousness-bliss).
agnir satyam rtam brhat
Frank in San-Diego
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Etherfish
USA
3615 Posts |
Posted - Mar 11 2006 : 11:36:05 PM
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Yes that is why I got into the "proof" thing in a yoga forum. because there is only one path to finding the truth. One direction, rather.
But I love science and the ways of man, as a diversion. . .entertainment. |
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david_obsidian
USA
2602 Posts |
Posted - Mar 12 2006 : 5:51:54 PM
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Alvin said:
Concerning natural science, the "disprovability" (in another name) is actually one of the most frequently used criteria for what is "science". Again there's an elaborate philosophy behind.
I'm glad you mentioned that. 'Disprovability' is all about having a definite meaning, in a scientific sense. 'Disprovability' does not mean that it is possible in practice to prove a theory wrong, but possible in principle. Unless we can provide a scenario, or a finding, which would prove the theory false, the theory is not considered scientifically meaningful.
This is very nice, because it will generally be possible to produce findings which would prove the theory false, which are much easier to describe than the theory itself.
For example, one potential finding which would prove Newton's Law of Gravitiation false would be as simple as ' a body falling (under gravitational forces alone) at a constant speed'. That one in fact would also prove Einstein's gravitation theories false.
When Ajita says something like 'Electricity occurs when Shakti-energy prevails. ... Magnetism occurs when Shiva-energy prevails. ', if we want this to be a defensible and meaningful theory in Science, we are required to produce well-defined scenarios which would prove the theory false.
What are they? I don't believe such scenarios are forthcoming. Which is enough to say that it is not a 'Scientific' theory at all.
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Edited by - david_obsidian on Mar 12 2006 7:00:30 PM |
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Alvin Chan
Hong Kong
407 Posts |
Posted - Mar 13 2006 : 06:09:20 AM
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Though it's good to be careful to say "disprovability" in stead of "verfiability", (because, yes, yes, we can never "prove" a scientific theory; only disprove it), in practical situation (ie. when we are not arguing with philosophers) I think it's reasonable to say we "verify" a certain theory when its predictions agree with (a lot of) obervations. We didn't and we couldn't prove it for sure, but we convinced ourself that the theory is "true enough" when it passes our tests.
As an example, even though we say we didn't "prove" Newton's law (under "normal" situation, e.g. low velocity); everyone of us show our great respect to it; relying on it whenever we build tall buildings. But what about my theory on "the 3.56 ghosts living in our intestine"? We didn't prove it and we didn't disprove it, yet anyone will take this theory seriously? So although we don't admit that philosophically, we are accepting science as having a very good cognitive status. We "verified" it, at least to a certain extent.
quote:
'Electricity occurs when Shakti-energy prevails. ... Magnetism occurs when Shiva-energy prevails. '
May be very poetic, I don't know. Only good English speaker could comment on its aesthetic value.
But from a cognitive consideration, this is a typical piece of "statement" which you can either interpret it as trivially correct but useless (e.g. by "defining" Electricity TO BE Shakti-energy flowing, Magnetism TO BE Shiva-energy prevailing.), or interpret it in a more reasonable way (which would then make the statement incorrect or meaningless since eletricity is obviously different from what we consider as Shakti-energy, the later not even quite well-defined on its own)
From an educational consideration, you will either dismiss it as nonsense (if you know some science and are serious about it) or you will get nothing from the analogy since you may not even know what electricity and magnetism are, other than at the level of everyday language. |
Edited by - Alvin Chan on Mar 13 2006 06:21:03 AM |
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Jo-self
USA
225 Posts |
Posted - May 29 2011 : 12:25:59 AM
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Interesting insights about pseudo science in the guise of spirituality:
"You will never die".
Not just a critique, cause in some ways the "crazy" notions in Science do give ammunition for mystical concepts, but as the author points out, science itself if full of nonsense too.
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WoodDragon
USA
56 Posts |
Posted - Jun 01 2011 : 10:55:33 AM
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I have come late into this discussion, busy life.
I just want to say I love the sciences and mathematics, the works of Hilbert and Godel are among my favorites. Of course in the sciences the ideas of Heisenberg ("Atoms are not things!" so very true especially if you have followed any of the Bose-Einstein-Condensate, or BEC for short, materials) and Dirac.
There is just one thing I wanted to ask in terms of pseudo science followers and the feelings that have been expressed here concerning them: does it matter?
Who is right and who is wrong? What does it mean to be correct about the application of anything? If people want to apply dodgy metaphors based on science into any spiritual or mystical topic, why does it matter? Let people use what they may in their searching for Truth. Too many times we all get wrapped up in being correct about facts and opinions, theories and dogmas. None of these things encompass the Truth, none of them can and thus all can be seen as incorrect or limited in view. We all are. Use compassion and let it go, use stillness and focus on your path, never mind the path of others.
That being said, an interesting topic of discussion here! |
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HathaTeacher
Sweden
382 Posts |
Posted - Jun 01 2011 : 2:33:51 PM
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quote:
Originally posted by Alvin Chan And Russell didn't succeed to "completely" explain the nature of mathematics. (Nor did anyone else...)
That's correct. Interestingly, Gödel's theorem challenged "real" Big science, not pseudoscience; he pointed out a contradiction between provability/testability and the complexity in wide "system" initiatives by scientists (like Russell - Principia Matematica, I think ?). An important part of both science and yoga is their awareness of not having ready-wrapped off-the-shelf answers to everything ("this takes additional research" / "this takes a lot of practice") ... |
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the misuse of scientific concepts
