Why is Pi, the number that it is?
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6/15/2013 10:22:36 AM Posted: 6 years ago Why, when you draw a straight line, rotate it on a 2D plane around one of its ends and continue to do this until it reaches it's original position, does the length of the line created if you join up each point that the other end passed through, does that new line (the circumference) equal a length 2 * Pi longer than the original line? Why is the area in side this new line equal to Pi * (the length of the original line)^2?
Why, if you create a straight line, and rotate it through every position possible in 3D space around one of its ends, join all the points the other end passed through to create a surface, why is the internal volume inside this surface equal to 4/3 * Pi * (the length of the original line)^3 ? Why is the surface area 4 * Pi * (the length of the original line)^2? On a maybe unrelated note, why do there seem to be three spatial dimensions? Anyone have any hypotheses? bladerunner060 | bsh1 , 2014! Presidency campaign! http://www.debate.org... http://www.debate.org... - Running for president. http://www.debate.org... - Running as his vice president. May the best man win! |
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6/15/2013 3:18:05 PM Posted: 6 years ago Your first set of queries are mathematical. They may be variously answered with mathematical demonstration (i.e. "showing") or proof. Interestingly, there are people who advocate another "circle constant" called "tau", which is just the proportion of the circle's circumference to its radius. So: tau=pi/2. They argue that tau should supplant pi in its current role.
So none of this stuff is mystically ordained; it all pertains to convenience and convention. That the circle does have certain constants which characterize its proportions completely is due to various unique traits of the circle: it is the only planar curve of constant curvature; despite being embedded in 2-space, it's actually a 1-dimensional manifold, etc. The sphere is fully analogous to the circle in all of these respects. The last question (which you rightly suppose is less related) is of a physical nature, and of a much deeper quality. I don't know "why" space seems to have the number of macroscopic dimensions that it does. Some working physical theories posit a greater number of dimensions than the three. But the same questions may be asked of those numbers. Why 11? Why 23? Etc. Have I hit the right pressure points of your inquiry? "The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker |
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6/16/2013 12:06:07 PM Posted: 6 years ago I understand that is has constants, I was asking if anyone had a reason for why they were the specific ones that they are. Not how we've worked them out, or that Pi is the answer, but why it is Pi.
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6/16/2013 3:31:55 PM Posted: 6 years ago I have been told that if you divide the circumference of a circle by it's diameter no matter how big or how small the circle is, you will get an answer of approximately 3.14 or there about.
Beware of the people who are in your circle but are not in your corner. And with the stroke of a pen people 18 to 21 who own a gun became criminals and public enemy #1 having committed no crime and having said nothing. Just like the Jews in Germany during WW2. Must be a weird feeling. When I hear people crying and whining about their first world problems I think about the universe with everything in it and people in wheelchairs and all of their problems go away. |
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6/16/2013 5:10:11 PM Posted: 6 years ago At 6/16/2013 12:06:07 PM, AlbinoBunny wrote: But the method for working out Pi is the definition of Pi. To ask why it has the value it does is to ask why the method doesn't come to a different answer or why we didn't choose a different definition. The methods of working it out all rely on intuitive mechanisms for constructing a circle. Thus we are confident they give us the right answer because we can see how a circle arises from the construction of Pi. What more do you want? |
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6/16/2013 5:12:25 PM Posted: 6 years ago I'd also add that we could easily have used a different number to do the same job as Pi (like tau as mentioned above) it would just require all our formulas to be different. So the value of Pi is definitional only. In fact I could pick any number I wanted, but all the equations would be translated.
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6/17/2013 8:18:26 AM Posted: 6 years ago At 6/16/2013 5:10:11 PM, the_croftmeister wrote:At 6/16/2013 12:06:07 PM, AlbinoBunny wrote: Yes, why doesn't it come to a different answer? or why we didn't choose a different definition. The methods of working it out all rely on intuitive mechanisms for constructing a circle. Thus we are confident they give us the right answer because we can see how a circle arises from the construction of Pi. What more do you want? Thought on the question above. bladerunner060 | bsh1 , 2014! Presidency campaign! http://www.debate.org... http://www.debate.org... - Running for president. http://www.debate.org... - Running as his vice president. May the best man win! |
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6/17/2013 8:26:50 AM Posted: 6 years ago I'm no mathematician, but I'd say that it's the same reason as any other shapes have the properties they do; without using those values, you do not end up with those shapes. It is then a matter of definition that certain shapes require certain constants, because those shapes are in some sense an expression of those constants. They aren't really distinct from them, so it's a question of identity; a circle is a circle because if it wasn't, it wouldn't be. Geometric anthropic principle, in a way.
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6/17/2013 12:07:11 PM Posted: 6 years ago At 6/17/2013 8:18:26 AM, AlbinoBunny wrote:Because a different answer would make the method false.At 6/16/2013 5:10:11 PM, the_croftmeister wrote:At 6/16/2013 12:06:07 PM, AlbinoBunny wrote: This is all akin to asking "Why is 1, the number that it is?" As opposed to what, the number that it's not? WOS : At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote: : Without nothing existing, you couldn't have something. |
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6/17/2013 1:37:54 PM Posted: 6 years ago At 6/17/2013 12:07:11 PM, tBoonePickens wrote:At 6/17/2013 8:18:26 AM, AlbinoBunny wrote:Because a different answer would make the method false.At 6/16/2013 5:10:11 PM, the_croftmeister wrote:At 6/16/2013 12:06:07 PM, AlbinoBunny wrote: No, it's asking why is the ratio of diameter to circumference 1:~3.14159. You're saying "it just is" is the answer which means we should never question it. I'm saying, that's not very philosophical of you. bladerunner060 | bsh1 , 2014! Presidency campaign! http://www.debate.org... http://www.debate.org... - Running for president. http://www.debate.org... - Running as his vice president. May the best man win! |
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6/17/2013 2:19:17 PM Posted: 6 years ago At 6/16/2013 12:06:07 PM, AlbinoBunny wrote: Concerning the number pi, as with all mathematical objects, its natural fact resides in its very symbol. Correspondingly, in the case of mathematical understanding, the reason for something being the case is not separate from any demonstration that it is; to see that something is so, is to see why it is so. Both of these states of comprehension are distinct from simply knowing about something being so, which is itself insufficient to cultivate any appreciate of that thing. These ideational mechanics may be the source of your psychological response to "pi" being the thing that it is. I'd encourage you to pore over some completely unsparing proofs and derivations of the surface area of sphere, etc. to see if you feel psychologically satisfied about this matter. "The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker |
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6/17/2013 2:31:20 PM Posted: 6 years ago At 6/17/2013 2:19:17 PM, Poetaster wrote:At 6/16/2013 12:06:07 PM, AlbinoBunny wrote: I just think it's interesting how a shape could have an irrational ratio, and also have a constant change in both the x and y axis. And that I'm sure there may be some mathematical explanations that are enlightening. I was just wondering if anyone had and insight. Just knowing that something is something, doesn't mean we know why it is that way. bladerunner060 | bsh1 , 2014! Presidency campaign! http://www.debate.org... http://www.debate.org... - Running for president. http://www.debate.org... - Running as his vice president. May the best man win! |
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6/17/2013 2:55:53 PM Posted: 6 years ago At 6/17/2013 2:31:20 PM, AlbinoBunny wrote: I think it's interesting too. And that I'm sure there may be some mathematical explanations that are enlightening. I was just wondering if anyone had and insight. Just knowing that something is something, doesn't mean we know why it is that way. In general this is true. But it is, I think, unique to this form of cognitive apperception (that is, mathematical understanding) that terminative reasons for a thing being so are to be found wholly embodied within the derivation of the fact that it is. Viewing a mathematical proof of a theorem generates an appreciation of why the theorem is true concomitantly to the realization that it is true. These sibling components of understanding are thus different passengers of the same vehicle. "The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker |
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6/17/2013 2:58:39 PM Posted: 6 years ago Well, if there is a God, I'm guessing it's because he wanted to screw with mathematicians with an obsessive compulsion to do long division.
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6/17/2013 3:27:32 PM Posted: 6 years ago At 6/17/2013 2:58:39 PM, Wnope wrote: Or show man that his cognitive abilities are limited. (Or he wanted to give man with what to keep busy.) I cannot write in English, because of the treacherous spelling. When I am reading, I only hear it and am unable to remember what the written word looks like." "Albert Einstein http://www.twainquotes.com... , http://thewritecorner.wordpress.com... , http://www.onlinecollegecourses.com... |
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6/17/2013 4:57:27 PM Posted: 6 years ago At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. You're saying "it just is" is the answer which means we should never question it. I'm saying, that's not very philosophical of you.Well, there are things one cannot call into question...regardless, it's axiomatic. WOS : At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote: : Without nothing existing, you couldn't have something. |
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6/17/2013 5:16:02 PM Posted: 6 years ago At 6/17/2013 4:57:27 PM, tBoonePickens wrote:At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. We can just leave it at that, and never question it or investigate further, I suppose. You're saying "it just is" is the answer which means we should never question it. I'm saying, that's not very philosophical of you.Well, there are things one cannot call into question...regardless, it's axiomatic. According to you, it's axiomatic. So you can not question, investigate, or ponder on axioms? bladerunner060 | bsh1 , 2014! Presidency campaign! http://www.debate.org... http://www.debate.org... - Running for president. http://www.debate.org... - Running as his vice president. May the best man win! |
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6/17/2013 5:19:07 PM Posted: 6 years ago At 6/17/2013 2:55:53 PM, Poetaster wrote:At 6/17/2013 2:31:20 PM, AlbinoBunny wrote: Poetaster, where did you learn to write like this? You express the same ideas I meant to but so much more elegantly and artistically. |
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6/17/2013 5:20:30 PM Posted: 6 years ago At 6/17/2013 4:57:27 PM, tBoonePickens wrote:At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. To be fair, axiomatic means we 'don't' call something into question, not that we can't. |
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6/17/2013 5:29:28 PM Posted: 6 years ago At 6/17/2013 5:16:02 PM, AlbinoBunny wrote:At 6/17/2013 4:57:27 PM, tBoonePickens wrote:At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. Pi having the value that it does is not axiomatic, it is derived from a much simpler set of axioms concerning the definitions of Pi, a circle, length and so forth. Of course you can question the axioms but all you can ever hope to do is to reduce it to a different set of axioms. In mathematics, questioning all your axioms will lead you to being incapable of answering anything since you remove any basis on which to prove things. The fact that pi is transcendental is quite interesting, though it is still computable. I think computability is by far the most interesting property of a number and would go far as to say that non-computable numbers aren't really numbers at all. Thoughts anyone? My chosen basis is computability due to the wide acceptance of the Church-Turing Thesis, it's unprovable but so much time has been spent on disproving it that I would rate it as one of the most objective non-trivial facts that we know. |
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6/17/2013 5:44:10 PM Posted: 6 years ago At 6/17/2013 5:29:28 PM, the_croftmeister wrote:At 6/17/2013 5:16:02 PM, AlbinoBunny wrote:At 6/17/2013 4:57:27 PM, tBoonePickens wrote:At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. And a non-computable number is? :P bladerunner060 | bsh1 , 2014! Presidency campaign! http://www.debate.org... http://www.debate.org... - Running for president. http://www.debate.org... - Running as his vice president. May the best man win! |
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6/17/2013 9:23:56 PM Posted: 6 years ago At 6/17/2013 5:44:10 PM, AlbinoBunny wrote:At 6/17/2013 5:29:28 PM, the_croftmeister wrote:At 6/17/2013 5:16:02 PM, AlbinoBunny wrote:At 6/17/2013 4:57:27 PM, tBoonePickens wrote:At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. The best examples arise from Specker Sequences. It's basically a number that we don't know all the digits to and never will. The problem is that while we can write a program that will eventually give the correct answer for each digit, we don't know when the correct digit will be chosen (and it is impossible to determine for all digits). It is important to distinguish between numbers for which there is no known algorithm and those for which such an algorithm is known to be impossible (non-computable). Computable numbers are those on which equality it decidable (there is an algorithm which can determine for any pair of such numbers if they are different or not). The non-computable ones do not have this propery (you may not be able to decide if two given numbers are equal or unequal). http://en.wikipedia.org... |
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6/18/2013 10:57:23 AM Posted: 6 years ago At 6/17/2013 5:16:02 PM, AlbinoBunny wrote:You can question it, but it's a fools errand as you'll be going around in circles, no pun intended. You can also choose not to accept the axiom and propose a different one, for example. There are some axioms that I do not accept.At 6/17/2013 4:57:27 PM, tBoonePickens wrote:We can just leave it at that, and never question it or investigate further, I suppose.At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. It's a waste of time, but you can do it. You could go about questioning A = A, but it would be foolish. You could reject it, but you wouldn't be able to go very far...at least not coherently in THIS case.According to you, it's axiomatic. So you can not question, investigate, or ponder on axioms?You're saying "it just is" is the answer which means we should never question it. I'm saying, that's not very philosophical of you.Well, there are things one cannot call into question...regardless, it's axiomatic. WOS : At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote: : Without nothing existing, you couldn't have something. |
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6/19/2013 5:36:07 AM Posted: 6 years ago At 6/18/2013 10:57:23 AM, tBoonePickens wrote:At 6/17/2013 5:16:02 PM, AlbinoBunny wrote:You can question it, but it's a fools errand as you'll be going around in circles, no pun intended. You can also choose not to accept the axiom and propose a different one, for example. There are some axioms that I do not accept.At 6/17/2013 4:57:27 PM, tBoonePickens wrote:We can just leave it at that, and never question it or investigate further, I suppose.At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. What makes you say questioning axioms is a waste of time? Questioning the law of identity is a bit hard, but I've seen it done. Non-deterministic valuation functions for instance. |
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6/19/2013 11:51:49 AM Posted: 6 years ago At 6/19/2013 5:36:07 AM, the_croftmeister wrote:Ah, I see your reading comprehension skills are selective.At 6/18/2013 10:57:23 AM, tBoonePickens wrote:At 6/17/2013 5:16:02 PM, AlbinoBunny wrote:You can question it, but it's a fools errand as you'll be going around in circles, no pun intended. You can also choose not to accept the axiom and propose a different one, for example. There are some axioms that I do not accept.At 6/17/2013 4:57:27 PM, tBoonePickens wrote:We can just leave it at that, and never question it or investigate further, I suppose.At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. Questioning the law of identity is a bit hard, but I've seen it done. Non-deterministic valuation functions for instance.Last I checked, functions REQUIRE the law of identity. WOS : At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote: : Without nothing existing, you couldn't have something. |
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6/19/2013 2:47:06 PM Posted: 6 years ago
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6/19/2013 3:38:33 PM Posted: 6 years ago At 6/17/2013 3:27:32 PM, 1Devilsadvocate wrote:At 6/17/2013 2:58:39 PM, Wnope wrote: I wonder if the above was a serious response. : At 5/13/2014 7:05:20 PM, Crescendo wrote: : The difference is that the gay movement is currently pushing their will on Churches, as shown in the link to gay marriage in Denmark. Meanwhile, the Inquisition ended several centuries ago. |
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6/19/2013 4:35:05 PM Posted: 6 years ago At 6/19/2013 11:51:49 AM, tBoonePickens wrote:Or perhaps your English expression skills are ambiguous.At 6/19/2013 5:36:07 AM, the_croftmeister wrote:Ah, I see your reading comprehension skills are selective.At 6/18/2013 10:57:23 AM, tBoonePickens wrote:At 6/17/2013 5:16:02 PM, AlbinoBunny wrote:You can question it, but it's a fools errand as you'll be going around in circles, no pun intended. You can also choose not to accept the axiom and propose a different one, for example. There are some axioms that I do not accept.At 6/17/2013 4:57:27 PM, tBoonePickens wrote:We can just leave it at that, and never question it or investigate further, I suppose.At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:I was addressing the quote "Why is Pi, the number that it is"; granted I did not take into account the rest of what you wrote in the OP. Regardless, I think this has been answered already: it so axiomatically. It is the relationship between a circle's diameter and its circumference.At 6/17/2013 12:07:11 PM, tBoonePickens wrote:No, it's asking why is the ratio of diameter to circumference 1:~3.14159. Questioning the law of identity is a bit hard, but I've seen it done. Non-deterministic valuation functions for instance.Last I checked, functions REQUIRE the law of identity. Check harder, non-deterministic functions don't. Also it can depend on whether you are using it for your meta or object language. |
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6/20/2013 12:34:53 PM Posted: 6 years ago At 6/19/2013 4:35:05 PM, the_croftmeister wrote:Only to the poorly educated.At 6/19/2013 11:51:49 AM, tBoonePickens wrote:Or perhaps your English expression skills are ambiguous. Check a little harder: ALL functions require it. If you feel you are correct then DEMONSTRATE it.Check harder, non-deterministic functions don't. Also it can depend on whether you are using it for your meta or object language.Questioning the law of identity is a bit hard, but I've seen it done. Non-deterministic valuation functions for instance.Last I checked, functions REQUIRE the law of identity. WOS : At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote: : Without nothing existing, you couldn't have something. |
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6/20/2013 2:21:35 PM Posted: 6 years ago At 6/17/2013 1:37:54 PM, AlbinoBunny wrote:At 6/17/2013 12:07:11 PM, tBoonePickens wrote:At 6/17/2013 8:18:26 AM, AlbinoBunny wrote:Because a different answer would make the method false.At 6/16/2013 5:10:11 PM, the_croftmeister wrote:At 6/16/2013 12:06:07 PM, AlbinoBunny wrote: Pi is Pi because the math says so. Philosophical arguments against it are pointless because any argument other than PI won't work mathematically. But you can pretend they do like they teach little Johnny in public school so his feelings won't get hurt. Beware of the people who are in your circle but are not in your corner. And with the stroke of a pen people 18 to 21 who own a gun became criminals and public enemy #1 having committed no crime and having said nothing. Just like the Jews in Germany during WW2. Must be a weird feeling. When I hear people crying and whining about their first world problems I think about the universe with everything in it and people in wheelchairs and all of their problems go away. |
