John 165 Posted April 23, 2015 (edited) I think it's more accurate to say that certain physical systems can be modeled using the mathematics of complex numbers. Whether complex numbers (or indeed, real numbers) exist in their own right is a matter of philosophy. Edit: One representative discussion can be found here: http://philosophy.stackexchange.com/questions/451/do-numbers-exist-independently-from-observers. Edited April 23, 2015 by John 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 I think it's more accurate to say that certain physical systems can be modeled using the mathematics of complex numbers. Whether complex numbers (or indeed, real numbers) exist in their own right is a matter of philosophy. Edit: One representative discussion can be found here: http://philosophy.stackexchange.com/questions/451/do-numbers-exist-independently-from-observers. It is not a matter if philosophy if you consider our label of numbers to be different than the inherent values themselves. Obviously, there are finite amounts of real objects. 0 Share this post Link to post Share on other sites

Strange 3814 Posted April 23, 2015 It is not a matter if philosophy if you consider our label of numbers to be different than the inherent values themselves. That sounds like a philosophical argument. (Actually, I don't understand what you are trying to say, there.) Obviously, there are finite amounts of real objects. That is not obvious at all. The universe may be infinite, in which case it is filled with an infinite number of atoms. 0 Share this post Link to post Share on other sites

StringJunky 2098 Posted April 23, 2015 ... Obviously, there are finite amounts of real objects. What is 'real' and what is an 'object'? To the second question: the boundary of what constitutes an object, and how many there are, is arbitrary. Is a wall a plural of objects (bricks) or a single object? 0 Share this post Link to post Share on other sites

studiot 1712 Posted April 23, 2015 (edited) Buych778 Do numbers actually exist? According to the site information the OP has not been since since posting this. Upon rereading my post#3 I find I owe him, or her an apology for taking this thread down imaginary avenue towards complex street, fascinating though the scenery en route has been. Post#1 clearly did not envision complex numbers, indeed it is not totally clear if even the full real number system was meant. I am backtracking because in other mathematics forums there are currently debates going on as to whether numbers actually exist or can be realised in the real world. That seems to be the thrust of post#1 and the thread title. My answer is yes, any real number can, in principle, be given a place in reality. Edited April 23, 2015 by studiot 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 (edited) That sounds like a philosophical argument. (Actually, I don't understand what you are trying to say, there.) Assuming that you do not regard all of humanity's knowledge as an illusion, it in fact is not, because we can measure amounts from empirical observation. A physical amount is different than a concept of a symbol. That is not obvious at all. The universe may be infinite, in which case it is filled with an infinite number of atoms. Well, if you don't have eyes then I suppose it is more difficult to see. Regardless of whatever you may think, we empirically measure amounts of objects. To say otherwise would be to discard all knowledge that exists in humanity. What is 'real' and what is an 'object'? To the second question: the boundary of what constitutes an object, and how many there are, is arbitrary. Is a wall a plural of objects (bricks) or a single object? Objects that are real occupy nonzero quantities of dimensional space. If an object does not exist, it will not have any dimensional coordinates or capacity for empirical measurement. Take this apple I'm holding. It has 0 length, 0 width and 0 height, and no one can observe it, not even me. As a scientist, would you tell me that apple exists? Edited April 23, 2015 by MWresearch 0 Share this post Link to post Share on other sites

Strange 3814 Posted April 23, 2015 Well, if you don't have eyes then I suppose it is more difficult to see. Regardless of whatever you may think, we empirically measure amounts of objects. To say otherwise would be to discard all knowledge that exists in humanity. You said "there are finite amounts of real objects" not "there are finite amounts of real objects that [have] measured". If the universe is infinite, there are an infinite number of real objects. Whether we can measure them or not. 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 (edited) You said "there are finite amounts of real objects" not "there are finite amounts of real objects that [have] measured". If the universe is infinite, there are an infinite number of real objects. Whether we can measure them or not. Regardless of semantics, it is a fact that people have measured finite amounts of objects. Whether or not you want the parameters of measurement to encompass the whole universe is arbitrary. If you want you can say all measurements are an illusion, but then you would have no basis to say all measurements are an illusion. Edited April 23, 2015 by MWresearch 0 Share this post Link to post Share on other sites

MonDie 141 Posted April 23, 2015 I give up. I've never evaluated inequalities between vectors. We can also induce a partial order on the complex numbers using the concentric circles of magnitude mentioned earlier, counting clockwise around each circle before moving to the "next." It's probably easier to think about this in terms of polar coordinates, in which we can say that [math](|z_1|, \theta_1) < (|z_2|, \theta_2) \iff \left(|z_1| < |z_2|\right) \textnormal{ or } \left(|z_1| = |z_2| \textnormal{ and } \theta_1 < \theta_2\right)[/math]. However, this is a bit unsatisfying since it means, among other things, that 1 < -1. 0 Share this post Link to post Share on other sites

studiot 1712 Posted April 23, 2015 (edited) mondie I give up. I've never evaluated inequalities between vectors. Complex numbers and geometric vectors are not the same. Edited April 23, 2015 by studiot 0 Share this post Link to post Share on other sites

Strange 3814 Posted April 23, 2015 Regardless of semantics, it is a fact that people have measured finite amounts of objects. Whether or not you want the parameters of measurement to encompass the whole universe is arbitrary. I was simply pointing out that "obviously, there are finite amounts of real objects" is neither obvious nor necessarily true. If you want to move the goalposts, that's fine with me. If you want you can say all measurements are an illusion, but then you would have no basis to say all measurements are an illusion. I don't want to say that. But thanks for the offer. 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 I was simply pointing out that "obviously, there are finite amounts of real objects" is neither obvious nor necessarily true. If you want to move the goalposts, that's fine with me. Anyway, according to our models, there are a finite amount of objects in any finite space. Since we cannot measure outside of the hubble volume, our measurements of the universe are confined to finite space. I don't think we've ever encountered anything that we've "measured" as infinite. By it's very nature, it is impossible to count to infinity. 0 Share this post Link to post Share on other sites

studiot 1712 Posted April 23, 2015 I don't think we've ever encountered anything that we've "measured" as infinite. By it's very nature, it is impossible to count to infinity. But we can (and do) include infinity in our counting. There are as many real numbers between 0 and 1 as there are on the entire real number line. This is why ordering is an important concept. Because if we count reals in order from 0 to1 we have counted an infinity of numbers. 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 But we can (and do) include infinity in our counting. There are as many real numbers between 0 and 1 as there are on the entire real number line. This is why ordering is an important concept. Because if we count reals in order from 0 to1 we have counted an infinity of numbers. According to that logic if I add 0 and 1 = 1 then I add an infinite amount of numbers. But, the sum of all numbers between 0 and 1 is infinity, you are therefore saying infinity=1. There are infinite numbers between 0 and 1, but that doesn't mean we are distinguishing all of them in an infinite set just because we focus on the integers 0 and 1. The act of counting itself as I am sure you know is much more complicated, you are actually doing the opposite of what you describe, specifically seeking out certain numbers of a general infinite set, not necessarily creating another infinite set. 0 Share this post Link to post Share on other sites

John Cuthber 3642 Posted April 23, 2015 If numbers don't exist then how do these work? http://en.wikipedia.org/wiki/Periodical_cicadas And how did they choose primes? 0 Share this post Link to post Share on other sites

Strange 3814 Posted April 23, 2015 Anyway, according to our models, there are a finite amount of objects in any finite space. Since we cannot measure outside of the hubble volume, our measurements of the universe are confined to finite space. I don't think we've ever encountered anything that we've "measured" as infinite. By it's very nature, it is impossible to count to infinity. None of that is relevant to "obviously, there are finite amounts of real objects" being neither obvious nor necessarily true. That is all I was saying. If you want to modify that statement to "obviously, there are finite amounts of real objects in the observable universe" then I have no objection. 0 Share this post Link to post Share on other sites

John 165 Posted April 23, 2015 We can also induce a partial order on the complex numbers using the concentric circles of magnitude mentioned earlier, counting clockwise around each circle before moving to the "next." It's probably easier to think about this in terms of polar coordinates, in which we can say that [math](|z_1|, \theta_1) < (|z_2|, \theta_2) \iff \left(|z_1| < |z_2|\right) \textnormal{ or } \left(|z_1| = |z_2| \textnormal{ and } \theta_1 < \theta_2\right)[/math]. However, this is a bit unsatisfying since it means, among other things, that 1 < -1. Just noticed this should say "counterclockwise" rather than "clockwise." We could go clockwise too, I suppose, but then we'd need [math](|z_1|, \theta_1) < (|z_2|, \theta_2) \iff \left(|z_1| < |z_2|\right) \textnormal{ or } \left(|z_1| = |z_2| \textnormal{ and } \theta_1 > \theta_2\right)[/math], since angles increase in the counterclockwise direction. I give up. I've never evaluated inequalities between vectors. Well, in this case, the inequalities are just between magnitudes and angles, both of which are scalar values. 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 If numbers don't exist then how do these work? http://en.wikipedia.org/wiki/Periodical_cicadas And how did they choose primes? I'm arguing in favor of the existence of numbers, not against. Or at least, I did not make an effort to say they do not exist, I merely wanted to know where imaginary numbers fit into existence from a mathematician's point of view. 0 Share this post Link to post Share on other sites

Strange 3814 Posted April 23, 2015 Or at least, I did not make an effort to say they do not exist, I merely wanted to know where imaginary numbers fit into existence from a mathematician's point of view. At right angles to the reals. 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 At right angles to the reals. So does every single infinitesimal real value have a corresponding imaginary axis? How would that work in 3D space where you run out of ways to represent orthogonal dimensions? 0 Share this post Link to post Share on other sites

Strange 3814 Posted April 23, 2015 So does every single infinitesimal real value have a corresponding imaginary axis? How would that work in 3D space where you run out of ways to represent orthogonal dimensions? It was actually a joke. But complex numbers can be represented as points on a plane with the x axis as the reals and the y axis as the imaginary number line. http://mathworld.wolfram.com/ArgandDiagram.html 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 It was actually a joke. But complex numbers can be represented as points on a plane with the x axis as the reals and the y axis as the imaginary number line. http://mathworld.wolfram.com/ArgandDiagram.html And? 0 Share this post Link to post Share on other sites

Strange 3814 Posted April 23, 2015 And? And what? That was the answer to your question: where do imaginary numbers fit? (And also an explanation of my feeble "joke" of "at right angles"). 0 Share this post Link to post Share on other sites

MWresearch 40 Posted April 23, 2015 I was not asking anything akin to where imaginary numbers fit in mathematics, I was asking where they fit in physical reality. 0 Share this post Link to post Share on other sites

Strange 3814 Posted April 23, 2015 I was not asking anything akin to where imaginary numbers fit in mathematics, I was asking where they fit in physical reality. They can describe real things in the same way that real numbers can. For example, when describing signals the imaginary part describes the phase (actually, its a little more complex than that). 0 Share this post Link to post Share on other sites