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Hose****.
You're kidding right? Well, first of all, you're mixing terms. "Hypothesis" is a term used in scientific method, to propose something that is observed, but isn't proven consistent. It doesn't exist in mathematics; proposals of mathematic properties are called "theorems". But I set that aside; this is casual conversation, after all. I did not mix terms - I used the term that someone else used and asked for elaboration. Not my confusion. Bear with me here, everyone. I'm going to make a pretty good point or two, in my opinion. Can't wait... Mathematical fundaments are composed of "Postulates", such as "A point is defined as a location in space", "A line is defined as the one-dimensional measure of distance between two points", and, "The shortest distance between two points is a line". Those are "postulates", specifically of Euclidean geometry. "Theorems" arise from logical conclusions of the interactions of the postulates. The ideas that triangles have certain properties, such as the sum of their angles equalling pi radians, are "theorems". Casually, these are sometimes called "laws", as in the "Law of Cosines". Non-Euclidean geometries, necessary for doing things like traversing the surface of a sphere (and none of us have *ever* done that, oh, no!), does *not* have, as a postulate, that the shortest distance between two points is a straight line; there are *no* straight lines in spherical geometries. Um, but the shortest distance between two points is STILL a stright line... Unfortunately you can't travel through the earth. For natural philosophers, people like physicists and mathemeticians, the discovery (or rediscovery) of alternate but valid geometric rulesets has resulted in several very useful discoveries, one of which being Einstein's body of thought on relativity, flawed as we now know it to be (but haven't come up with an all-encompassing replacement). One other result of the re-examination of Euclidean thinking has been the formulation of Theorems which deny the principal assumption of great works like the _Principia Mathematica_, Goedel's Theorem probably the most popular among them. The upshot of Goedel's Incompleteness Theorem is mathematical proof that "any self-consistent axiomatic system powerful enough to describe integer arithmetic will allow for propositions about integers that can neither be proven nor disproven from the axioms." [from the Wikipedia article on Goedel] Euclidean geometry is more powerful than integer arithmetic. That is, logical systems powerful enough to be useful will contain unprovable axioms. So the question, "Which [axiom or theorem] in mathematics can't be proven or shown false that is the basis for all other math?" is simply an utterly unanswerable question, given a powerful enough system. Goedel proved it years ago. What *can* be said is that "some axioms are unprovable, which doesn't mean they're false or true." I asked for which basic tenet was unprovable. My point was that the original poster of this math == religion thread was not making sense. There is nothing similar about them. Goedel (and Turing's equivalent with the halting problem) have nothing to do with this conversation. You still haven't answered the question - you have just tried to make the whole bit sound more complicated than it is. And I am sure we are all impressed with the disussion or Euclid, Theorems, incompleteness, etc. Mathematics itself is today in a state alongside physics and most natural science, of great uncertainty about the "Great Unknowables", therefore, while depending on mathematical fundamentals will be remarkably and consistently useful (can't compute a weight and balance and then observe performance, or watch your climb rate go down as altitude goes up, without noticing that), you just never know if your system will stand up to new stuff. Kind of like religion, that way, which works for most people. Until it doesn't. Except for mine, of course. :-) I still don't see how that is anything like religion. |
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On Sat, 22 Nov 2003 13:41:20 GMT, "Richard Hertz"
wrote: Hose****. Starting with fallacy. Not a good sign... I did not mix terms - I used the term that someone else used and asked for elaboration. Not my confusion. OK, Usenet attribution mea culpa. Um, but the shortest distance between two points is STILL a stright line... Unfortunately you can't travel through the earth. ....thus necessitating the use of non-Euclidean geometries. Don't forget that the point of philosophy really is to come up with useful stuff. I asked for which basic tenet was unprovable. My point was that the original poster of this math == religion thread was not making sense. There is nothing similar about them. [...] I still don't see how that is anything like religion. The single undeniable similarity between math and religion is that they are both philosophical systems, based on unproved (and maybe unprovable) axioms and definitions. Math: "A 'point' is defined as..." Math: "The set of 'Integers' is defined as..." (Aristotlean) Religion: "'God' is defined as..." (Aristotlean) Religion: "'Sin' is defined as..." You're right, of course, if you want to say that the similarity ends there. But IMO involving the Incompleteness Theorem when talking about complex axiomatic[1] systems is perfectly valid. The systems are axiomatic and complex, whether you use the language of religion or the language of mathematics to describe them. *Especially* orthodox Christianity, whose apologist Thomas Aquinas (I'm told), made enough of a significant case for basing scriptural understanding on Aristotlean philosophical underpinnings that the comparison is unavoidable. Mathematics is also based on Euclidean rules of reasoning, the same rules Aristotle used to build his thoughts. Therefore comparing the two is not invalid. you have just tried to make the whole bit sound more complicated than it is. So I have. It's because I believe that it is a far more complicated problem than a blanket dismissal of "religion" can solve. And I am sure we are all impressed with the disussion or Euclid, Theorems, incompleteness, etc. I hope so! It was more than a little bit of work. Rob [1] In theology I suppose they'd call it "dogmatic" -- [You] don't make your kids P.C.-proof by keeping them ignorant, you do it by helping them learn how to educate themselves. -- Orson Scott Card |
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"Richard Hertz" wrote in
straight lines in spherical geometries. Um, but the shortest distance between two points is STILL a stright line... At non-relativistic speeds, it's so nearly a straight line as to be inconsequential. But, at speeds approaching the speed of light, it isn't. Unless you don't believe that the speed of light is a constant in vacuo. le m |
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Happy,
Unless you don't believe that the speed of light is a constant in vacuo. What's there not to believe? Anyone using GPS cannot deny Einstein - it wouldn't work without relativity. -- Thomas Borchert (EDDH) |
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Thomas Borchert wrote:
Unless you don't believe that the speed of light is a constant in vacuo. What's there not to believe? Anyone using GPS cannot deny Einstein - it wouldn't work without relativity. Relativity has certainly been well-tested and (as you say) widely used. However, there is serious discussion taking place concerning the possibility of a variable speed of light as an alternative to inflation during the early phases of the big bang. See http://www.iop.org/EJ/abstract/1475-7516/2003/07/004 and http://www.arxiv.org/abs/gr-qc/0209014 |
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Peter,
during the early phases of the big bang. True. But that's been over for a while ;-) -- Thomas Borchert (EDDH) |
#7
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Peter wrote:
Thomas Borchert wrote: Unless you don't believe that the speed of light is a constant in vacuo. What's there not to believe? Anyone using GPS cannot deny Einstein - it wouldn't work without relativity. Relativity has certainly been well-tested and (as you say) widely used. However, there is serious discussion taking place concerning the possibility of a variable speed of light as an alternative to inflation during the early phases of the big bang. Which demonstrates the fundamental difference between science and religion - science is falsifiable, and it is correctable. Religion is not. Science can say "look here - here's something that can't be explained by relativity, maybe we need to modify our theories a little bit." When religion can propose a test that, if succesful, would disprove the existance of God, then will I be willing to grant it a status on par with science. Rich Lemert |
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