QuoteLike I said, but you deleted, this was a simplistic summary of important features, sufficient for the issue of whether mathematics involves beliefs -- and that was all. Now if our resident Stalinist-apparatchik-wannabe-oligarch can tell us what sort of axioms don't depend on definitions and how they become beliefs, I'm sure we'll all be impressed.I thought i was classified as laissez faire capitalist and your enemy in class struggle, now i am your friendly Stalinist apparatchik? Well, i guess it's all the same for a guy who doesn't see a difference between an axiom and a definition. You are such a thoroughly and well-roundly confused individual i am surprised you learned not to walk around with your underwear on your head. I suppose my point was that one needs to resolve some trivial issues before you get to "mathematics and beliefs". Definition is what an angle is, axiom is parallel postulate in Euclid elements. In natural sciences definition is what a system of references is and an axiom is that systems of references moving with constant velocities relative to each other have same laws of physics. Definition is a name, a language. Axiom contains non trivial information formed about the constructs of the language. I know, it's all the same to you, comrade. Keep plugging in some keystrokes. Workers of the world, unite!

QuoteOriginally posted by: Traden4Alpha1. Scientists believe math can describe nature (i.e., that certain mathematical axioms are true in the real world)No. They hypothesize ways in which math axioms can be applied and test them.Quote2. Scientists believe their application of logic and use of mathematics is logically correct (e.g., long chains of logic can be flawed)No. They hypothesize this and good scientists correct errors when they find them.Quote3. Scientists believe that their experiments are proper tests of their working hypotheses (e.g., that their control group is valid)No. They hypothesize this -- it becomes part of the experiment. Sometimes they make mistakes and good scientists correct them.Quote 4. Scientists believe that the data from their experiments (i.e, that no mischievous deity has intentionally skewed the outcomes)No. They hypothesize this. If an experiment provides a questionable result, they may review either the theory or the data and a good scientist will construct other experiments and/or theoretical extensions to seek to resolve the issue.QuoteThe first and last beliefs are probably the most interesting. First, in the real world, the circumference of a circle is not 2*pi*r in the strictest mathematical sense. A host of quantum, physical, and gravitational distortions disrupt the exactness of the radial and circumferential distances. Pi may be a damn good approximation (to the first dozen or so digits), but that's all.Already addressed. QuoteThe last belief could be called the "fossils are real" belief. If one or more deities synthesized the fossil record (and other cosmological traces of the Big Bang), then many scientific theories concerning the past dissolve into falsehood. Now some might counter-argue that hypotheses of mischievous deities are not falsifiable hypotheses and thus not scientific. And yet it doesn't change the fact that scientists must have faith in their data, especially any data coming from uncontrolled, natural experiments. At a deeper level, no hypothesis about the past is falsifiable using a properly controlled experiments because we cannot truly rewind the clock (at least not yet).Theories that relate to history are just as falsifiable as theories of the present. Hypotheses about the significance of data are just another part of the theory and involve no faith. This may make both theories and tests more complex, but if the results don't agree with the theory then at least some part of it is falsified. Science is always hard.

QuoteOriginally posted by: zerdnaQuoteLike I said, but you deleted, this was a simplistic summary of important features, sufficient for the issue of whether mathematics involves beliefs -- and that was all. Now if our resident Stalinist-apparatchik-wannabe-oligarch can tell us what sort of axioms don't depend on definitions and how they become beliefs, I'm sure we'll all be impressed.I thought i was classified as laissez faire capitalist and your enemy in class struggle, now i am your friendly Stalinist apparatchik?Moving from Stalinist apparatchik to capitalist oligarch is trivially easy -- as so many of your compatriots have demonstrated.QuoteWell, i guess it's all the same for a guy who doesn't see a difference between an axiom and a definition. You are such a thoroughly and well-roundly confused individual i am surprised you learned not to walk around with your underwear on your head. I guess you were talking to a mirror.Quote I suppose my point was that one needs to resolve some trivial issues before you get to "mathematics and beliefs". Definition is what an angle is, axiom is parallel postulate in Euclid elements. In natural sciences definition is what a system of references is and an axiom is that systems of references moving with constant velocities relative to each other have same laws of physics. Definition is a name, a language. Axiom contains non trivial information formed about the constructs of the language.Axioms depend on definitions. Axioms are not beliefs but are postulated. Further distinction (like you) is just as irrelevant now as it was in my original comment.

QuoteFermion:Axioms depend on definitions. Axioms are not beliefs but are postulated. Right, axioms are nothing more than the antecedent "p" in a conditional "p ----> q", where "p" and "q" themselves may be compound statements (or propositions). And we may think of the consequent "q" as theorems (or the results). There is no need for believing the axioms as true (or false). In fact, one of the easiest and most utterly useless of all deductive systems is one in which the axioms are patently contradictory, within which any result can be derived. A mathematician (anyone in fact) can do both Euclidean and Non-Euclidean geometry without ever worrying as to which of the two has the believable (or true) set of axioms. This is why mathematics is independent of reality of the world, existence! But, axioms do not at all depend on definitions. Actually one can do math without any definition. The device of definition in math serves the same purpose as it does in ordinary speech; namely, to make communication more manageable. In an axiomatic-deductive system, there are always certain elementary or primitive terms that will forever remain not defined. For example, in euclidean geometry we take the 'notions' of "set", "point", "line" as undefined. Using these notions, we formulate the axioms as stating a relationship between these primitive terms. For example, we have the following axiom: Given any two distinct points, there exists a unique line containing these two points. So far we have not defined anything. Now, let's say we were interested in a certain subset of a line. The subset consisting of all points of the line that are 'between' two given points of the line. Now, in all our future consideration of such a subset, either we can restate our idea expressed in those many words, or we can just agree to give a special symbol or name to the idea expressed by all those words. We commonly use the name "line segment" to define (or symbolize) the idea of a subset consisting of all the points (of the line) between the two given points. So, by defining we never create any new knowledge. In fact some of the most amazing definitions apply to certain things that don't even exist! But since the idea of a (hitherto unrealized) impossible thing does exist, one can always define the idea by the choice of an appropriate word (or symbol).

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QuoteTraden4Alpha:First, in the real world, the circumference of a circle is not 2*pi*r in the strictest mathematical sense. A host of quantum, physical, and gravitational distortions disrupt the exactness of the radial and circumferential distances. Pi may be a damn good approximation (to the first dozen or so digits), but that's all. In the real world there is no such thing as a circle. Circle is an idea, a mathematical construct. The real world consists of things, stuff. Things in the real world may be circle-like, but they are never circles. Therefore, the issue is NOT whether or not pi is a good approximation for anything, including its use in the circumference formula. It is after the assumption that an object under consideration is close enough to a circle that we then use the circumference formula. Mathematics is an idealization in relation to the real world. Nonetheless, mathematics -- being neutral to the true nature of reality (the existing world) or any part of it -- helps (us) explain the real world better than any other tool does. The circumference formula cannot be 'chipped' yet, or rather never. And pi is perfect, and perfectly fits in the circumference formula.

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QuoteOriginally posted by: dunrewppQuoteTraden4Alpha:First, in the real world, the circumference of a circle is not 2*pi*r in the strictest mathematical sense. A host of quantum, physical, and gravitational distortions disrupt the exactness of the radial and circumferential distances. Pi may be a damn good approximation (to the first dozen or so digits), but that's all. In the real world there is no such thing as a circle. Circle is an idea, a mathematical construct. The real world consists of things, stuff. Things in the real world may be circle-like, but they are never circles. Therefore, the issue is NOT whether or not pi is a good approximation for anything, including its use in the circumference formula. It is after the assumption that an object under consideration is close enough to a circle that we then use the circumference formula. Mathematics is an idealization in relation to the real world. Nonetheless, mathematics -- being neutral to the true nature of reality (the existing world) or any part of it -- helps (us) explain the real world better than any other tool does. The circumference formula cannot be 'chipped' yet, or rather never. And pi is perfect, and perfectly fits in the circumference formula.That's an excellent description of the relation of math to reality. It's also a perfect description of why math is not a belief system -- because there is nothing to believe!

Yeah, there is nothing more powerful than half knowledge, i think Nietzsche commented about it. It always helps to keep you convinced in your superiority. Ever heard of something like this Godel? "Gödel's theorem shows that, in theories that include a small portion of number theory, a complete and consistent finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program. Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent."

Your mention of Godel is very apt, and one that we should be all lucky to never forget! However, my comments (assuming your post was in response to) do not contradict Godel's incompleteness/inconsistency theorems. If you think my post says something in contradiction to Godel's, please let me know.Quotezerdna:Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom.It would be interesting if a physics (or objective measurable) experiment can be devised that would give a result that, because of Godel's theorems, cannot be predicted through the use of any existing mathematical theorem. Such an experiment can help develop a new mathematical theory that would include an axiom that can help predict the said result. Also, Another theory can be built that would predict the 'negation' of the result, in some sense. Then the question that could come to mind is whether there is another universe in which the latter mathematical theory can help explain, including the prediction of the negation of the result of the experiment. Just a thought ... QuoteOriginally posted by: zerdnaYeah, there is nothing more powerful than half knowledge, i think Nietzsche commented about it. It always helps to keep you convinced in your superiority. Ever heard of something like this Godel? "Gödel's theorem shows that, in theories that include a small portion of number theory, a complete and consistent finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program. Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent."

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dun, i think your comments are totally reasonable, i was responding to Fermi-bable .

QuoteOriginally posted by: zerdnadun, i think your comments are totally reasonable, i was responding to Fermi-bable .Which comments of Fermion do you disagree with?In his latest post he agrees with what I said. Just curious as to what you find incorrect in his latest statement(s). As I have said elsewhere previously, I do not agree with everything or most things Fermion says. In fact there are probably a few (an understatement) foundational areas in which I might disagree with him. But that's for another day. What I want to say now is that this guy - Fermion - disliked as he is by a few others on these fora - has a lot to offer to all those who are willing to listen even if they disagree viscerally with him. I think Fermion and his detractors/contenders should reach a workable truce within which they can be more productive ... and polite.

QuoteOriginally posted by: zerdnaYeah, there is nothing more powerful than half knowledge, i think Nietzsche commented about it. It always helps to keep you convinced in your superiority. Ever heard of something like this Godel? "Gödel's theorem shows that, in theories that include a small portion of number theory, a complete and consistent finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program. Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent."None of which contradicts (or has anything to do with) anything I wrote. The existence of true statements that cannot be proved in no way mean that any statement has to be believed. I guess you like the ways your words appear on the page.

QuoteAnd that is a really stupid question that only someone like you could pose.One of the private pleasures in discussing things with you is the Ciceronian wit that we've all come to know and love.I do apologize though for being away from the OT Forum for so long, as I've been tied up on a work project that willprobably keep me busy until midweek. A quick scan, though, shows that others have been keeping you busy so I don'tfeel too bad.TTFN - I shall return to more of your rhetorical warmth and affection midweek.

QuoteOriginally posted by: HamiltonQuoteAnd that is a really stupid question that only someone like you could pose.One of the private pleasures in discussing things with you is the Ciceronian wit that we've all come to know and love.I do apologize though for being away from the OT Forum for so long, as I've been tied up on a work project that willprobably keep me busy until midweek. A quick scan, though, shows that others have been keeping you busy so I don'tfeel too bad.TTFN - I shall return to more of your rhetorical warmth and affection midweek.I'm glad you like my "warmth". I've always found it preferable to pomposity.

Underwear Bomber Abdulmutallab: Proud to Kill in the Name of Allahlife in prison

Feds arrest man allegedly heading to U.S. Capitol for suicide mission after sting investigationQuoteThe man, in his 30s and of Moroccan descent

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