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    1. #26
      Consciousness in the Void Universal Mind's Avatar
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      Quote Originally Posted by Xei View Post
      I'm sorry UM but you are not following the conversation; I can only recommend that you reread. I already responded to your supposed contradiction (which I had also posted beforehand), that's what this whole bit of the conversation was about in the first place. To recapitulate: there is no contradiction because sqrt(a)sqrt(b) is not sqrt(ab) in general, only when a and b are non-negative reals. There is no issue of 'making rules up', the proof works for non-negative reals and it doesn't work in general; these things come from the axioms.
      I'm sorry, Xei, but you are not following the conversation. I said I thought the contradiction in my proof is related to what you just addressed, but in my proof, i^2 is -1, not 1. As for what you discussed concerning sqrt(ab), a new rule did have to be made up because using imaginary numbers did not fit into the sytem of rules as it had been known. That's what creationists do.

      Quote Originally Posted by Xei View Post
      I don't know how there's any ambiguity in which proof I'm referring to. We are talking about sqrt(a)sqrt(b) = sqrt(ab) for a, b non-negative. The proof of this fact is therefore the one I am talking about; I recommend you try to prove it and examine every step very carefully to see how general it is. The failure of this identity for numbers other than a, b non-negative does not somehow show that complex numbers are inconsistent any more than the failure of the identity for matrices shows the inconsistency of matrices, or the failure of 1/[email protected] = [email protected]/[email protected] for @ = 0 shows the inconsistency of 0.
      Ha ha, I know you are talking about such a proof. I just assumed you had more to post because that equation plus your following commentary alone shows nothing more than the fact that a new rule had to be made up when imaginary unicorn numbers got introduced into the system. Also, my proof I asked you about (I should have been clearer.) was the long ass one I posted first. In that one i^2 = -1, not 1. I said I think the proof's flaw may be related to the issue of products of square roots being square roots of radicand products. That was an abstract statement I can't make concrete at this point.

      I know that exceptions exist in math. I am talking about an exception that had to be introduced when something imaginary was introduced.

      Quote Originally Posted by Xei View Post
      These terms are all undefined. Please define exactly what you mean by 'it's made up'. In particular you need to explain how this differs from the premise that 2 has a square root is made up. Refer very carefully to my latest post which is about this exact issue.
      Do you know how the Flying Spaghetti Monster was made up? Imaginary numbers and their square root product exception were made up in a similar fashion.

      Quote Originally Posted by Xei View Post
      Yes, this is all covered in that post. There can't 'be i of anything', but by your own admission there can't 'be -3 of anything' either. Different numbers have different contexts; a context for negative numbers is displacements, like you said, rather than quantification of numbers of objects. A context for pi or sqrt2 are lengths. Put a mug on the floor in front of you. Now turn it around by 90 degrees anticlockwise. You have just multiplied that mug by i. Turn it by 180. You've just multiplied it by i^2. That's a good context for imaginary numbers. Rotating a mug certainly seems to me to be just as 'quantifiable as it relates to the tangible' as the circumference of the mug or the number of mugs.
      There can be -3 of something, such as dollars and feet. There can't be i of anything whatsoever, so it seems, but you are getting me interested. What do you mean you have multiplied the mug by i when you turn it 90 degrees counterclockwise? How do you multiply a mug by something? Perhaps you are onto a good point, but I don't know what you mean. You have turned the mug 90 degrees and therefore pi/2 radians, and what you have multiplied its position on the coordinate plane or unit circle by depends, but I don't (at this point) see how what it is multiplied by would be something imaginary.

      Quote Originally Posted by Xei View Post
      It's a $1 bill. It is not a 1.
      I agree. However, it illustrates the reality of 1. I never said 1 is an actual object. It is a principle involved in the existence of an actual object.

      Quote Originally Posted by Wayfaerer View Post
      A collection of atoms that I've made up to be 1? Atoms being a collection of particles that I've made up to be 1? Particles being temporary phases of a process that I've made up to be 1? Though I'll have to do more research about how likely could elementary particles be eternally absolute. In that case, and in that case only, could 1 actually exist in nature.

      If you can say the square root of a unit of measurement has a tangible use in reality and can be considered "real", what's the difference in reality of subtracting the square root of the unit?
      The dollar bill is not the number 1. It is 1 bill worth 1 dollar.

      I am not sure what you are asking in the second paragraph. Subtracting square units is fine. It is getting the square root of a negative number I am taking issue with.

      Quote Originally Posted by PhilosopherStoned View Post
      Frankly, these are the only people worth talking to about math and I've been considering the same thing. Trying to teach people that don't actually care would cause me to go insane.
      If I am insane now, it is from doing that shit.
      Quote Originally Posted by really View Post
      God cannot destroy himself because He is Omnipotent.


    2. #27
      LD's this year: ~7 tommo's Avatar
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      You should both do it.

    3. #28
      Rational Spiritualist DrunkenArse's Avatar
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      Quote Originally Posted by Universal Mind View Post
      That's what creationists do.
      That's what you're doing. The rules in math say that you have to prove things. If it can only be proved that taking square roots is multiplicative over non-negative reals then taking square roots is only multiplicative over non-negative reals.

      Hence the square root function is only multiplicative over non-negative reals.

      Hence if you want to claim that the square root function is or should be multiplicative over all numbers, then you're the one making shit up.

      Did you say that you're a math teacher?

      If I am insane now, it is from doing that shit.
      I don't know if you're insane or not but you certainly don't know anything about how math operates.
      Last edited by PhilosopherStoned; 03-25-2012 at 02:53 AM.
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      Quote Originally Posted by tommo View Post
      You should both do it.
      Do it yourself.

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    6. #31
      Xei
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      Quote Originally Posted by Universal Mind View Post
      I'm sorry, Xei, but you are not following the conversation. I said I thought the contradiction in my proof is related to what you just addressed, but in my proof, i^2 is -1, not 1. As for what you discussed concerning sqrt(ab), a new rule did have to be made up because using imaginary numbers did not fit into the sytem of rules as it had been known. That's what creationists do.

      Ha ha, I know you are talking about such a proof. I just assumed you had more to post because that equation plus your following commentary alone shows nothing more than the fact that a new rule had to be made up when imaginary unicorn numbers got introduced into the system. Also, my proof I asked you about (I should have been clearer.) was the long ass one I posted first. In that one i^2 = -1, not 1. I said I think the proof's flaw may be related to the issue of products of square roots being square roots of radicand products. That was an abstract statement I can't make concrete at this point.

      I know that exceptions exist in math. I am talking about an exception that had to be introduced when something imaginary was introduced.
      Things are getting messy so I am going to recapitulate again.

      We are talking about the identity sqrt(a)sqrt(b) = sqrt(ab) for a, b non-negative.

      This statement is not an axiom of arithmetic; it is proven from the axioms of arithmetic, and the proof only works for when a and b are non-negative reals. When a and b are not non-negative reals, the proof does not work and the identity is in fact incorrect.

      Therefore, I reiterate my reiteration: there was no creation of a special case. The truth for a and b non-negative and the falsity otherwise are both deduced from the axioms of arithmetic.

      There is nothing wrong with this and it is not unusual.

      To help show this, here is another identity:

      a^2 = a*|a| for a non-negative.

      |a| means the absolute value of a, which means its size irrespective of whether it is positive or negative. You can write it as sqrt(a^2) if you want.

      This is exactly analogous. The identity is not an axiom of arithmetic: it is proven from the axioms of arithmetic, and the proof only works when a is non-negative. If a is negative, the proof fails, and the correct identity is in fact

      a^2 = -a*|a|

      What your argument says is that we have invented a special case because the identity doesn't work for negative numbers; and therefore, negative numbers are somehow fake. As you don't believe that negative numbers have the same problem as complex numbers, hopefully you now see why your argument doesn't make sense.

      There can be -3 of something, such as dollars and feet. There can't be i of anything whatsoever, so it seems, but you are getting me interested. What do you mean you have multiplied the mug by i when you turn it 90 degrees counterclockwise? How do you multiply a mug by something? Perhaps you are onto a good point, but I don't know what you mean. You have turned the mug 90 degrees and therefore pi/2 radians, and what you have multiplied its position on the coordinate plane or unit circle by depends, but I don't (at this point) see how what it is multiplied by would be something imaginary.
      'How do you multiply a mug by something?'; it depends on which abstraction you are using. Again this goes back to my exposition of the second pitfall.

      One abstraction you could be using is abstracting from the number of mugs. This abstraction is encapsulated by the natural numbers. You can multiply the mug by 10; the result is 10 mugs.

      A different abstraction could be the orientation of the mug. This abstraction is encapsulated by the complex numbers. The centre of the mug is at 0 and the handle is pointing in the direction of 1. If you multiply the mug by i, the handle now points towards i, which is 90 degrees counter clockwise. If you multiply by it by i again, the handle points towards -1, which I'm sure you can see represents a 180 degree turn. Thus we have an entity which squares to give -1.

      That's just one example. There are plenty of other examples of complex numbers being in one-to-one correspondence with some kind of intuitive physical basis. If you know basic stuff about matrices, you will know that 2*2 matrices represent linear transformations of the plane (rotations, reflections, enlargements, skews), with addition representing the addition of the transformations and multiplication representing one transformation followed by the other. Surely you grant that these are fine. Well, complex numbers are just a special type of linear transformation.

      There is also the issue of formal construction which you didn't respond to. Here is the crux of it again: if you grant that real numbers, along with addition, multiplication, and ordered pairs all 'exist', and that anything made out of them 'exists', then you are in fact compelled to concede that complex numbers 'exist', because you can construct the complex numbers using only these objects.

      Quote Originally Posted by Wayfaerer View Post
      Didn't you get my pm a while back? I'm basically trying to find out how true new empiricism/psychologism is as the foundation of mathematics. Philosophy of mathematics - Wikipedia, the free encyclopedia
      Oh man, I think I remember putting a response on my to-do list and forgetting. Sorry. Well, it's my holiday now. Maybe you could make a thread about this, and/or post your essay when it's done? I think I have a fairly solidly formed opinion about this, I just need people to chuck what other philosophers have said at me for me to decide if it's right or wrong.

      If both are consistent with all observed phenomena then I guess you couldn't really say until new observations or a better model shows up.
      I was really talking about the case where the two models are empirically indistinguishable; it's a question about the ontological status of the underlying workings, related to 'hidden variables' I suppose. When I thought about it, I essentially decided it was literally meaningless, according to my radical empiricist / analytic way of doing things. I emphasize that this is very different from the problem being undecidable.

      Quantum mechanics is such a conceptual mess right now that I don't think anything could be rightly called it's true "language" yet. We can do with what we have and I suppose something we made before was bound to be the best one to describe such unfamiliar things. Why they work could hide something profound, we just won't know until we make more sense of everything.
      Being a mathematician though, I have to say that the complex field isn't really just some random object we found and put on a pile with a thousand other mathematical objects, and we only found that it fit by virtue of our having so much random stuff that something had to: it is a very fundamental thing and a natural language for vast swathes of math, which made everything much more beautiful and simple.

      Particle physics doesn't have this mystery: indeed, it is a great example of the opposite. We really did have a big pile of mathematical objects (groups) and one of them happened to fit.
      Last edited by Xei; 03-25-2012 at 03:49 AM.
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    7. #32
      LD's this year: ~7 tommo's Avatar
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      Quote Originally Posted by Wayfaerer View Post
      Do it yourself.
      I'm not good at maths.

    8. #33
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      Quote Originally Posted by tommo View Post
      I'm not good at maths.
      Oh, that's ok I doubt that disqualifies you from being a teacher. lol

      Quote Originally Posted by Xei View Post
      Particle physics doesn't have this mystery: indeed, it is a great example of the opposite. We really did have a big pile of mathematical objects (groups) and one of them happened to fit.
      I'd be glad to share my essay when I'm done. I still have a lot of paper finding to do, it's kind of funny how difficult supposedly highly influential papers can be to find.

      What aspect of particle physics are you referring to? Something in the standard model?

    9. #34
      Rational Spiritualist DrunkenArse's Avatar
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      Xei is referring to the use of gauge groups in quantum field theories, so yes, this includes but is not limited to the standard model. See here. Essentially, bosons arise as the generators of certain groups. So the dimension of a group describing a force will be the amount of gauge bosons that the theory predicts.

      The gauge group of electro-magnetism is U(1) which has one generator so only one boson, the photon, needed to describe the force.

      The strong force has SU(3) symmetry. SU(3) has eight generators and so has eight bosons. These are the gluons.

      I'm kinda over my head here with quantum chromo-dynamics so I may not be able to be too helpful.
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    10. #35
      Consciousness in the Void Universal Mind's Avatar
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      Quote Originally Posted by PhilosopherStoned View Post
      That's what you're doing. The rules in math say that you have to prove things. If it can only be proved that taking square roots is multiplicative over non-negative reals then taking square roots is only multiplicative over non-negative reals.

      Hence the square root function is only multiplicative over non-negative reals.

      Hence if you want to claim that the square root function is or should be multiplicative over all numbers, then you're the one making shit up.

      Did you say that you're a math teacher?

      I don't know if you're insane or not but you certainly don't know anything about how math operates.
      Uh, kid... Are you okay? You suddenly, out of the clear blue, seem to have turned into an absolute shit-head. Bad day? Do you need a hug? Did I step on your tail somewhere along the way? I'll mail you a tissue if you'd like.

      Since you are being so rude, I will be completely blunt with you. I think this conversation might be over your head. You are accepting claims as gospel and challenging me to prove a negative. Just like with the existence of God, the burden of proof is not on the disbeliever. It is on the claimants. The claimants have not done a satisfactory job. You see, in the beginning and for a very long time, mathematicians did not talk about imaginary numbers. It was accepted that the square (and any other even power) of a positive and the square (and any other even power) of a negative are both positives. Then, mathematical liberals started talking about hypotheticals called "imaginary numbers," which are even roots of negative numbers. Later, mathematicians started claiming that they are actual realities, yet kept the name "imaginary numbers." THEY claimed that imaginary numbers exist. I am challenging their claim. I don't think they have proven the reality of imaginary numbers. I have argued to the bone why I don't think imaginary numbers are true numbers, and your rude ass has yet to counter my points. Do you think you can? Then give it to me, son. Let's see what you have to say. To be honest, I am not expecting much.

      As for your EXTREMELY absurd claim that I "CERTAINLY don't know ANYTHING about how math operates," let's discuss it. That means that all of the claims I have made in this thread are bullshit. That means pi is not the ratio of the circumference of a circle to its diameter. It means that the square of a positive is not a positive. It means that the square of a negative is not a positive. It means that not a single line I accept in the proof I typed up is correct. It means that my definition of "imaginary numbers" is false. It also means I don't know anything at all about mathematical principles I have not discussed in this thread. I don't even know that 2 + 2 = 4? Do you really want to stand by ALL of that? If so, you are an imbecile. Also, yes, I said I was a math teacher. You could go back and re-read the post that gave you the idea instead of asking me if I said what you think I might have said. Now either counter my points or go sit in the corner and don't say anything.

      Xei, you are on a much higher intellectual level than that *#@%, and your post has a lot of actual specific counterarguments in it. Plus, I am drunk. I will respond in detail to your post tomorrow. Peace.
      Last edited by Universal Mind; 03-25-2012 at 08:28 AM.
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    11. #36
      Member Olysseus's Avatar
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      Don't know if this will help or not...but hope so.

      Would it be correct to say that another way to see why the "paradox" -1 = i*i = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1 is that each number has two square roots?

      Thus the step that is wrong is the step that says sqrt (1) = 1, because it really needs to be written as +/-(1). Its really as simple as saying that if x^2 = (16), for example, it is not correct to say the answer is 4, rather 4 is an answer, while the answer is +/- 4.

      This way you can resolve the problem without appearing to negate an established rule just to make things fit. (And please note I am using the word "appearing" intentionally here, I am not saying there is anything wrong with other's explanations.)


      Another thing that has helped me explain the "reality" of imaginary numbers to students is the following:
      -Imagine a number line, what happens if you multiply each number on the line by two? by three?
      A: The number line will contract.(What was once 1 will now be 2, what was once 2 will be 4 and so on if you multiply each point by two.)
      -What happens if you multiply each point on the number line by 1/2?
      A: The number line will dilate (what was once 2 will now be 1, what was once 4 will now be 2.)
      -What happens when you multiply each pint on the number line by (-1)?
      A: The number line will rotate 180 degrees! (What was once 1 will now be -1)
      -If multiplying by a negative implies rotating 180 degrees around the number line, what happens if you multiply by the square root of (-1)?
      A: You will rotate by 90 degrees.

      I know this is not a formal proof of anything. But it gives me an intuitive idea that if we think of numbers as transformations, i is as "real" as any other number. The concept of i simply requires us to consider numbers in a new way. Rather than being thought of as static objects, numbers have to be thought of as something else. I would readily concede that i does not correspond to anything "out there" in the world of objects. But it does correspond to a "real" transformative action on the number plane.

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      Last edited by Olysseus; 03-25-2012 at 09:57 PM.
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    12. #37
      khh
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      Quote Originally Posted by Universal Mind View Post
      You see, in the beginning and for a very long time, mathematicians did not talk about imaginary numbers. It was accepted that the square (and any other even power) of a positive and the square (and any other even power) of a negative are both positives. Then, mathematical liberals started talking about hypotheticals called "imaginary numbers," which are even roots of negative numbers. Later, mathematicians started claiming that they are actual realities, yet kept the name "imaginary numbers." THEY claimed that imaginary numbers exist. I am challenging their claim. I don't think they have proven the reality of imaginary numbers. I have argued to the bone why I don't think imaginary numbers are true numbers, and your rude ass has yet to counter my points. Do you think you can? Then give it to me, son. Let's see what you have to say. To be honest, I am not expecting much.
      I think the reason for this argument is that you have a fundamentally different view on what exactly math is. Seem UM is arguing from the standpoint that math is a natural science, describing the real world, while Xei is arguing from the standpoint that math is a tool that can be used by science. So when Xei says something is "real", he means that it can be used to correctly model occurring phenomenon (which it certainly can: There's a reason all engineers have to learn how to work with complex numbers).
      So perhaps which is the correct interpretation of math is what should be argued about? Either that, or I've completely misunderstood what you're trying to achieve.
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    13. #38
      Xei
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      I'm not really sure what the difference is between those two. The fields of the natural numbers, integers, real numbers, and complex numbers, with their natural operations, are all in one-to-one correspondence with various physical situations. I'm not sure what else could be meant by saying a number is 'real'. Obviously numbers aren't physical things... there are no number 3s floating around. They are abstractions of various phenomena with something in common. What else could be meant by the reality of such an abstraction other than its consistency?

      Quote Originally Posted by Olysseus View Post
      I readily concede that i does not correspond to anything "out there" in the world of objects. But it does correspond to a "real" transformative action on the number plane.
      Hey, welcome to DV!

      I'd ask the question above of this, as well; does 3 correspond to anything "out there" in any sense that i does not?
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    14. #39
      khh
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      Quote Originally Posted by Xei View Post
      I'm not really sure what the difference is between those two. The fields of the natural numbers, integers, real numbers, and complex numbers, with their natural operations, are all in one-to-one correspondence with various physical situations. I'm not sure what else could be meant by saying a number is 'real'. Obviously numbers aren't physical things... there are no number 3s floating around. They are abstractions of various phenomena with something in common. What else could be meant by the reality of such an abstraction other than its consistency?
      That's exactly my point. I myself and apparently you find it almost nonsensical to talk about what numbers are real, because they're obviously abstractions and concepts rather than actual entities. It seems to me that UM have a different opinion, though I may be wrong.
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    15. #40
      Consciousness in the Void Universal Mind's Avatar
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      Quote Originally Posted by Xei View Post
      Things are getting messy so I am going to recapitulate again.

      We are talking about the identity sqrt(a)sqrt(b) = sqrt(ab) for a, b non-negative.

      This statement is not an axiom of arithmetic; it is proven from the axioms of arithmetic, and the proof only works for when a and b are non-negative reals. When a and b are not non-negative reals, the proof does not work and the identity is in fact incorrect.

      Therefore, I reiterate my reiteration: there was no creation of a special case. The truth for a and b non-negative and the falsity otherwise are both deduced from the axioms of arithmetic.

      There is nothing wrong with this and it is not unusual.

      To help show this, here is another identity:

      a^2 = a*|a| for a non-negative.

      |a| means the absolute value of a, which means its size irrespective of whether it is positive or negative. You can write it as sqrt(a^2) if you want.

      This is exactly analogous. The identity is not an axiom of arithmetic: it is proven from the axioms of arithmetic, and the proof only works when a is non-negative. If a is negative, the proof fails, and the correct identity is in fact

      a^2 = -a*|a|

      What your argument says is that we have invented a special case because the identity doesn't work for negative numbers; and therefore, negative numbers are somehow fake. As you don't believe that negative numbers have the same problem as complex numbers, hopefully you now see why your argument doesn't make sense.
      I am not disagreeing with your stance that axioms existed and conclusions followed from them. What happened is that axioms existed, and then conclusions based on them were added to by the introduction of imaginary numbers. The conclusion that there is a negative number exception to the radicand product rule is based on the original axioms PLUS the assumption that imaginary numbers exist. I don't think imaginary numbers exist, so I don't agree with the exception rule. However, if I did believe in imaginary numbers, I would agree that the exception is legitimate. There would have to be that exception if imaginary numbers were real.

      Quote Originally Posted by Xei View Post
      'How do you multiply a mug by something?'; it depends on which abstraction you are using. Again this goes back to my exposition of the second pitfall.

      One abstraction you could be using is abstracting from the number of mugs. This abstraction is encapsulated by the natural numbers. You can multiply the mug by 10; the result is 10 mugs.

      A different abstraction could be the orientation of the mug. This abstraction is encapsulated by the complex numbers. The centre of the mug is at 0 and the handle is pointing in the direction of 1. If you multiply the mug by i, the handle now points towards i, which is 90 degrees counter clockwise. If you multiply by it by i again, the handle points towards -1, which I'm sure you can see represents a 180 degree turn. Thus we have an entity which squares to give -1.

      That's just one example. There are plenty of other examples of complex numbers being in one-to-one correspondence with some kind of intuitive physical basis. If you know basic stuff about matrices, you will know that 2*2 matrices represent linear transformations of the plane (rotations, reflections, enlargements, skews), with addition representing the addition of the transformations and multiplication representing one transformation followed by the other. Surely you grant that these are fine. Well, complex numbers are just a special type of linear transformation.

      There is also the issue of formal construction which you didn't respond to. Here is the crux of it again: if you grant that real numbers, along with addition, multiplication, and ordered pairs all 'exist', and that anything made out of them 'exists', then you are in fact compelled to concede that complex numbers 'exist', because you can construct the complex numbers using only these objects.
      Okay, now I see what you are saying. I think you are saying something along the lines of what Olysseus said. I boldfaced the three comments that highlight where I disagree with you. I agree that you can multiply by imaginary and complex numbers and get imaginary and complex figures. You can multiply coordinates and matrix figures by i and numbers that involve i, and from there you can get other numbers involving i. However, I see that as using fiction to get more fiction. I can multiply 3 by unicorn and get 3 unicorns, but that doesn't mean the product represents something actual. If you throw in a fictitious factor, you can get a fictitious product. You cannot prove the existence of something fictitious when the premise involves something similarly fictitious.

      The solutions to equations can be imaginary or complex, but that is only because the numbers are used as hypotheticals. 3i and -3i might be hypothetical x values that would make an equation work, but that does not mean they are actualities. They are fictitious principles that, if actual, would be actual values of x. I don't think x really can be 3i or -3i. There is only the hypothetical scenario that x would be 3i or -3i if there were such thing as i. I can see where you would say that such a situation proves that 3i and -3i are actual in the sense that they work as solutions (which they can) and it proves that they are actual in that way. What I am saying is that (to use an example and illustrate my overall point) such numbers can be used only in a hypothetical sense. The premise that such numbers are actual is what I think is false.

      This is a complicated and sticky issue, but I will try to clear up my position with an analogy. Suppose a detective is given information about a crime. He can come to the conclusion that there are three characters who fit as suspects. Now suppose that a hypothetical person who does not exist is one of them. That fictitious person would be a viable suspect if he were real. In that way, he is a "solution" to the crime scenario but not an actual suspect. If the detective assumes the existence of a fictitious character named Orzog, and the defining characteristics of Orzog make him a person who fits the scenario, the detective could say, "There are three suspects-- Bob Smith, Al Johnson, and Orzog. Orzog fits the scenario just like Bob and Fred, but that does not make him an actuality. He is just a hypothetical that fits the situation. Equations can work the same way. 3i and -3i may be solutions to an equation, but that does not make them actual. They exist only as fictitious concepts that fit scenarios as hypotheticals.

      Quote Originally Posted by Olysseus View Post
      Don't know if this will help or not...but hope so.

      Would it be correct to say that another way to see why the "paradox" -1 = i*i = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1 is that each number has two real square roots?

      Thus the step that is wrong is the step that says sqrt (1) = 1, because it really needs to be written as +/-(1). Its really as simple as saying that if x^2 = (16), for example, it is not correct to say the answer is 4, rather 4 is an answer, while the answer is +/- 4.

      This way you can resolve the problem without appearing to negate an established rule just to make things fit. (And please note I am using the word "appearing" intentionally here, I am not saying there is anything wrong with other's explanations.)
      That was my initial hunch when I came across the general issue, but I don't think that's the resolution. 1 does have two square roots, but it does not mean that if you have 1 = 1 you can get the sqare root of each side and put +/- a the beginning of one side. It only makes sense to do that when a variable is involved. If x^2 = 25, x can be 5 or -5. However, if you have the equation 25 = 25, you cannot get the square root of each side and come to the conclusion that 5 possibly equals -5. You can only come to the conclusions that 5 = 5 and =5 = -5. +/- only makes sense when an unknown is involved.

      Quote Originally Posted by Olysseus View Post
      Another thing that has helped me explain the "reality" of imaginary numbers to students is the following:
      -Imagine a number line, what happens if you multiply each number on the line by two? by three?
      A: The number line will contract.(What was once 1 will now be 2, what was once 2 will be 4 and so on if you multiply each point by two.)
      -What happens if you multiply each point on the number line by 1/2?
      A: The number line will dilate (what was once 2 will now be 1, what was once 4 will now be 2.)
      -What happens when you multiply each pint on the number line by (-1)?
      A: The number line will rotate 180 degrees! (What was once 1 will now be -1)
      -If multiplying by a negative implies rotating 180 degrees around the number line, what happens if you multiply by the square root of (-1)?
      A: You will rotate by 90 degrees.

      I know this is not a formal proof of anything. But it gives me an intuitive idea that if we think of numbers as transformations, i is as "real" as any other number. The concept of i simply requires us to consider numbers in a new way. Rather than being thought of as static objects, numbers have to be thought of as something else. I would readily concede that i does not correspond to anything "out there" in the world of objects. But it does correspond to a "real" transformative action on the number plane.
      I mentioned you where I responded to Xei. What I said there is my response to your above point. You can mulitiply anything by i and potentially get a product with i, but that does not mean i is real in the first place. The symbol is real, and the concept obviously exists because we are talking about it, but it is a fictitious concept.

      Quote Originally Posted by khh View Post
      I think the reason for this argument is that you have a fundamentally different view on what exactly math is. Seem UM is arguing from the standpoint that math is a natural science, describing the real world, while Xei is arguing from the standpoint that math is a tool that can be used by science. So when Xei says something is "real", he means that it can be used to correctly model occurring phenomenon (which it certainly can: There's a reason all engineers have to learn how to work with complex numbers).
      So perhaps which is the correct interpretation of math is what should be argued about? Either that, or I've completely misunderstood what you're trying to achieve.
      That is a good point. It goes along with what I said to Xei. I agree that imaginary and complex numbers can be used as good hypotheticals that can help people understand reality, but that is all they are. Hypotheticals can be used in law and other areas, but they are still just hypotheticals.
      Quote Originally Posted by really View Post
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      Rational Spiritualist DrunkenArse's Avatar
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      Quote Originally Posted by Universal Mind View Post
      Uh, kid... Are you okay? You suddenly, out of the clear blue, seem to have turned into an absolute shit-head. Bad day? Do you need a hug? Did I step on your tail somewhere along the way? I'll mail you a tissue if you'd like.
      You've been getting a kick out of me talking to creationists like that for some time now. Now you're being the reality denier. I would hope that you would do the same for me.

      Since you are being so rude, I will be completely blunt with you. I think this conversation might be over your head. You are accepting claims as gospel and challenging me to prove a negative. Just like with the existence of God, the burden of proof is not on the disbeliever. It is on the claimants.
      You're the one claiming that real numbers exist in some manner. That's a necessary presupposition to the claim that complex numbers are less real. Prove that first. You'll find that they're actually just defined. With that accepted, note that I'm not asking you to prove a negative as one can, once one accepts the definition (not the "reality") of the complex numbers, provide trivial counter examples to the multiplicativity of the root function on all of C.

      Hence that step fails. You are not operating with the complex numbers but with an inaccurate, inconsistent model of them. Hence of course your argument produces an inconsistent statement.

      As for your EXTREMELY absurd claim that I "CERTAINLY don't know ANYTHING about how math operates," let's discuss it. That means that all of the claims I have made in this thread are bullshit. That means pi is not the ratio of the circumference of a circle to its diameter. It means that the square of a positive is not a positive. It means that the square of a negative is not a positive. It means that not a single line I accept in the proof I typed up is correct. It means that my definition of "imaginary numbers" is false. It also means I don't know anything at all about mathematical principles I have not discussed in this thread. I don't even know that 2 + 2 = 4? Do you really want to stand by ALL of that? If so, you are an imbecile. Also, yes, I said I was a math teacher. You could go back and re-read the post that gave you the idea instead of asking me if I said what you think I might have said. Now either counter my points or go sit in the corner and don't say anything.
      You forgot to bold the part about "how math operates". You listed your ability to recite facts as a refutation of this claim. This is not how math operates. Thanks for reinforcing my point. Math operates by mathematical proof. You are claiming that the square root function is multiplicative on a set for which it can not be proven to be multiplicative. You are assuming that, because the proof doesn't go through, we should assume that the set "doesn't exist" rather than accepting the failure of the proof.


      As for Xei being on a higher intellectual level than me I'll just say that he's doing a commendable(?) job of tolerating your bullshit and is probably sharper at math than me right now. He probably will be from now on if he sticks with it. Or he may get rusty too. However any rust which I may have accumulated certainly hasn't been on display in this thread.
      Previously PhilosopherStoned

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      Quote Originally Posted by Xei View Post
      I'd ask the question above of this, as well; does 3 correspond to anything "out there" in any sense that i does not?
      Objectively no; I agree with you that 3 is an abstraction.

      Although conceptually 3 does have a correspondence to the concept of quantity that i does not have . I do agree with your mathematical reasoning that i has a reality in the broad sense of the term. But I would add that in order to understand the reality of i, we have to rethink what numbers correspond to. We even have to do this to understand the reality of negative numbers. For example, to get someone to understand the concept of negative 3, we can explain how numbers are not just quantities, but are relative positions. So in a sense I would say that there is a quantity that 3 corresponds to but i does not. But, none of this means, to me, that i is fictional.




      1 does have two square roots, but it does not mean that if you have 1 = 1 you can get the sqare root of each side and put +/- a the beginning of one side. It only makes sense to do that when a variable is involved. If x^2 = 25, x can be 5 or -5. However, if you have the equation 25 = 25, you cannot get the square root of each side and come to the conclusion that 5 possibly equals -5. You can only come to the conclusions that 5 = 5 and =5 = -5. +/- only makes sense when an unknown is involved.
      Yes, I thought someone might disagree with me for exactly this reason, so I perfectly understand what you are saying. But I'll explain why I think this is still the resolution. If we have 25=25 (as in your example) and we take the square root of both sides in the way suggested, getting 5 = +/- 5 , that does not suggest that 5 could possibly be either 5 or -5. Rather it means that either 5 is equal to 5 or 5 is equal to negative 5, only one of which can be true.

      You would agree that the following compound statement is true, yes?:
      Five is either equal to five or five is equal to three.

      So my claim is that -1 = i*i = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1
      really only proves that -1 is either equal to 1 or is equal to -1. It does not demonstrate that both identities can be true but that one of the two identities must be true.
      Feel free to let me know if you disagree.
      “Look at every path closely and deliberately, then ask ourselves this crucial question: Does this path have a heart? If it does, then the path is good. If it doesn't, it is of no use.” - Carlos Castaneda

    18. #43
      Consciousness in the Void Universal Mind's Avatar
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      Quote Originally Posted by PhilosopherStoned View Post
      You've been getting a kick out of me talking to creationists like that for some time now. Now you're being the reality denier. I would hope that you would do the same for me.
      No, I have not been getting a kick out of you talking to creationists like that. I make fun of creationists in general, but I never get personal with any of them unless they do something shitty first. I am generally polite in the way I talk to people. I have said this before... I don't like it when atheists initiate rudeness with individual creationists in the Religion forum. I have seen people get flat out bullied when they didn't deserve it, and I think it's sick. I say stuff in there that is highly offensive, but it is never personal, and it is always with the purpose of illustrating the truth of an argument.

      Quote Originally Posted by PhilosopherStoned View Post
      You're the one claiming that real numbers exist in some manner. That's a necessary presupposition to the claim that complex numbers are less real. Prove that first. You'll find that they're actually just defined. With that accepted, note that I'm not asking you to prove a negative as one can, once one accepts the definition (not the "reality") of the complex numbers, provide trivial counter examples to the multiplicativity of the root function on all of C.

      Hence that step fails. You are not operating with the complex numbers but with an inaccurate, inconsistent model of them. Hence of course your argument produces an inconsistent statement.
      I have discussed that a lot in this thread. I welcome your counterarguments. I have said more than once that the exception to the radicand product rule does consistently apply from the standpoint that imaginary numbers are actual. I even said it in my last post.

      Quote Originally Posted by PhilosopherStoned View Post
      You forgot to bold the part about "how math operates". You listed your ability to recite facts as a refutation of this claim. This is not how math operates. Thanks for reinforcing my point. Math operates by mathematical proof. You are claiming that the square root function is multiplicative on a set for which it can not be proven to be multiplicative. You are assuming that, because the proof doesn't go through, we should assume that the set "doesn't exist" rather than accepting the failure of the proof.
      The things I listed are all aspects of how math operates, and my mere mentioning of them shows that I know of them. Thus, I have shown that I know something about how math operates. I have also taught geometry and trigonometry, so I know about proofs. Algebra teachers can get away with never having to deal with proofs, but trig and geometry teachers cannot. This part of the discussion stems from a "proof" that 1 = -1. I just started another thread on that. So, you are incorrect.

      Also, I am not basing my stance of imaginary number nonexistence purely on the radicand product rule. My point on that is that the exception had to be made up when imaginary numbers were introduced. My stance that imaginary numbers do not exist is based on the great deal of stuff I have discussed in this thread.

      Quote Originally Posted by PhilosopherStoned View Post
      As for Xei being on a higher intellectual level than me I'll just say that he's doing a commendable(?) job of tolerating your bullshit and is probably sharper at math than me right now. He probably will be from now on if he sticks with it. Or he may get rusty too. However any rust which I may have accumulated certainly hasn't been on display in this thread.
      My bullshit? You just don't agree with me. That is fine. Perhaps you are really young and will grow out of the phase you are in, but you need to learn how to debate without making it personal. You, like a lot of college age people (I don't know if you are.), have the perception that a debate automatically involves a personality conflict. It doesn't have to. Try sticking to the debate topics and not making the discussion primarily about the person you are debating. I will do that in retaliation, but I tend not to initiate it. I said what I said about your intellect (mathematical intellect, to be specific) because you were acting like a douche instead of countering my very detailed points in response to a joke I made. I was expressing positivity toward you, and you came back at me with negativity. Xei has been debating me and doing a great job with it. Maybe you have the ability to do that too, but you have not shown it in the area of math. You have been really good at debating religion.
      Quote Originally Posted by Olysseus View Post
      Yes, I thought someone might disagree with me for exactly this reason, so I perfectly understand what you are saying. But I'll explain why I think this is still the resolution. If we have 25=25 (as in your example) and we take the square root of both sides in the way suggested, getting 5 = +/- 5 , that does not suggest that 5 could possibly be either 5 or -5. Rather it means that either 5 is equal to 5 or 5 is equal to negative 5, only one of which can be true.

      You would agree that the following compound statement is true, yes?:
      Five is either equal to five or five is equal to three.

      So my claim is that -1 = i*i = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1
      really only proves that -1 is either equal to 1 or is equal to -1. It does not demonstrate that both identities can be true but that one of the two identities must be true.
      Feel free to let me know if you disagree.
      Hmmmm, I agree that 5 is equal to either 5 or -5, and I agree that the +/- is an either/or. However, putting the +/- is not necessary, and only the + is correct. So, 1 is equal to either 1 or -1, but it is only equal to 1. Thus, sqrt (-1 * -1) cannot be both 1 and -1. Since it can only be one of them, the paradox is not resolved. There is still a conflict of rules, at least in the absence of the exception Xei and I have been discussing.

      The much longer proof I typed and just started a thread on is where your point is more relevant. There is the step of getting the square root of both side of the equation -1/1 = 1/-1. Putting +/- is not necessary on that step, but it would not be false. However, putting just a minus sign would be, and it seems to be what is what is necessary to resolve the paradox, aside from declaring the whole thing a reductio ad absurdum of imaginary numbers.
      Last edited by Universal Mind; 03-25-2012 at 11:18 PM.
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    19. #44
      Rational Spiritualist DrunkenArse's Avatar
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      Quote Originally Posted by Universal Mind View Post
      No, I have not been getting a kick out of you talking to creationists like that. I make fun of creationists in general, but I never get personal with any of them unless they do something shitty first. I am generally polite in the way I talk to people. I have said this before... I don't like it when atheists initiate rudeness with individual creationists in the Religion forum. I have seen people get flat out bullied when they didn't deserve it, and I think it's sick. I say stuff in there that is highly offensive, but it is never personal, and it is always with the purpose of illustrating the truth of an argument.
      How is saying something about some group X also not saying something about some element x of X? So why is saying something about X acceptable but not x? E.g., why is it acceptable to say that creationists are stupid, insane or ignorant but not to tell some specific creationist that he/she is stupid, insane or ignorant?



      The things I listed are all aspects of how math operates, and my mere mentioning of them shows that I know of them. Thus, I have shown that I know something about how math operates. I have also taught geometry and trigonometry, so I know about proofs. Algebra teachers can get away with never having to deal with proofs, but trig and geometry teachers cannot. This part of the discussion stems from a "proof" that 1 = -1. I just started another thread on that. So, you are incorrect.
      No. The things you listed are products of how math operates and not aspects of it. Also, the proof mechanisms used in high school geometry are irrelevant to the rest of mathematics and the proofs one finds in trig are essentially just computations, i.e. neither is impressive.

      Also, I am not basing my stance of imaginary number nonexistence purely on the radicand product rule. My point on that is that the exception had to be made up when imaginary numbers were introduced.
      There is no exception that was made up. If I say that P(x) holds for all x in X then saying that P fails for some y in Y when Y extends X is not making up an exception, it's just a failure of a rule to generalize to a larger set. This occurs all the time.

      My bullshit? You just don't agree with me. That is fine. Perhaps you are really young and will grow out of the phase you are in, but you need to learn how to debate without making it personal. You, like a lot of college age people (I don't know if you are.), have the perception that a debate automatically involves a personality conflict. It doesn't have to. Try sticking to the debate topics and not making the discussion primarily about the person you are debating. I will do that in retaliation, but I tend not to initiate it. I said what I said about your intellect (mathematical intellect, to be specific) because you were acting like a douche instead of countering my very detailed points in response to a joke I made.
      This is not a matter of disagreeing.This is a very basic point and you have made no points specific or otherwise. You are fumbling over essentially the same kind of points that creationists do. There is no debate and you are wrong. Period. For the record, I am thirty one.
      Last edited by PhilosopherStoned; 03-26-2012 at 12:32 AM.
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    20. #45
      Consciousness in the Void Universal Mind's Avatar
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      Quote Originally Posted by PhilosopherStoned View Post
      How is saying something about some group X also not saying something about some element x of X? So why is saying something about X acceptable but not x? E.g., why is it acceptable to say that creationists are stupid, insane or ignorant but not to tell some specific creationist that he/she is stupid, insane or ignorant?
      I don't tend to say those things. I say stuff like they use poor logic, and I use satire to illustrate it. My comments are generally about what they do, not what they are. Also, it is much more personal when you direct your comments at a specific individual, and that makes it ruder. I can say Democrats use self-righteousness too much in their arguments, I can say that Democrats are self-righteous assholes, and I can say that YOU PHILOSOPHERSTONED are a self-righteous asshole. Do you see the heirarchy of rudeness involved? Those are hypotheticals. I have never seen you act self-righteous.

      Quote Originally Posted by PhilosopherStoned View Post
      No. The things you listed are products of how math operates and not aspects of it. Also, the proof mechanisms used in high school geometry are irrelevant to the rest of mathematics and the proofs one finds in trig are essentially just computations, i.e. neither is impressive.
      They are principles involved in the operation, therefore knowing them is knowing something involved in the operation, which is therefore something about the operation. No matter what the relevance of high school geometry or the impressiveness of geometry and trig proofs are, they are separate issues. My point was that I have taught proofs and of course had to prove that I could do them to become a teacher in the first place. You said I know nothing about how math operates, and you were incorrect.

      Quote Originally Posted by PhilosopherStoned View Post
      There is no exception that was made up. If I say that P(x) holds for all x in X then saying that P fails for some y in Y when Y extends X is not making up an exception, it's just a failure of a rule to generalize to a larger set. This occurs all the time.
      There is a difference. The rule regarding radicand products was established one way for a very long time, but it had to be given an addition when imaginary numbers were introduced to the system. The rule was generalized to the entire set for ages, though even roots of negative numbers were assumed not to exist. Then the new rule involved an exception.

      Quote Originally Posted by PhilosopherStoned View Post
      This is not a matter of disagreeing.This is a very basic point and you have made no points specific or otherwise. You are fumbling over essentially the same kind of points that creationists do. There is no debate and you are wrong. Period. For the record, I am thirty one.
      You are being flat out dishonest. Read the last post where I addressed Xei. Counter the specific points I made in it. Even if I were fumbling over the specific points creationists do, it would not warrant your immature rudeness, 31 year old.

      I challenge you to resolve the paradox involved in the "proof" I posted in the thread titled "1 = -1?" I think it is probably a reductio ad absurdum of imaginary numbers, but I welcome your resolution.
      Last edited by Universal Mind; 03-26-2012 at 12:57 AM.
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    21. #46
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      Quote Originally Posted by Universal Mind View Post
      I am not disagreeing with your stance that axioms existed and conclusions followed from them. What happened is that axioms existed, and then conclusions based on them were added to by the introduction of imaginary numbers. The conclusion that there is a negative number exception to the radicand product rule is based on the original axioms PLUS the assumption that imaginary numbers exist. I don't think imaginary numbers exist, so I don't agree with the exception rule. However, if I did believe in imaginary numbers, I would agree that the exception is legitimate. There would have to be that exception if imaginary numbers were real.
      As long as you are happy with why the failure of the identity for complex numbers doesn't affect their ontological status.

      Okay, now I see what you are saying. I think you are saying something along the lines of what Olysseus said. I boldfaced the three comments that highlight where I disagree with you. I agree that you can multiply by imaginary and complex numbers and get imaginary and complex figures. You can multiply coordinates and matrix figures by i and numbers that involve i, and from there you can get other numbers involving i. However, I see that as using fiction to get more fiction. I can multiply 3 by unicorn and get 3 unicorns, but that doesn't mean the product represents something actual. If you throw in a fictitious factor, you can get a fictitious product. You cannot prove the existence of something fictitious when the premise involves something similarly fictitious.

      The solutions to equations can be imaginary or complex, but that is only because the numbers are used as hypotheticals. 3i and -3i might be hypothetical x values that would make an equation work, but that does not mean they are actualities. They are fictitious principles that, if actual, would be actual values of x. I don't think x really can be 3i or -3i. There is only the hypothetical scenario that x would be 3i or -3i if there were such thing as i. I can see where you would say that such a situation proves that 3i and -3i are actual in the sense that they work as solutions (which they can) and it proves that they are actual in that way. What I am saying is that (to use an example and illustrate my overall point) such numbers can be used only in a hypothetical sense. The premise that such numbers are actual is what I think is false.

      This is a complicated and sticky issue, but I will try to clear up my position with an analogy. Suppose a detective is given information about a crime. He can come to the conclusion that there are three characters who fit as suspects. Now suppose that a hypothetical person who does not exist is one of them. That fictitious person would be a viable suspect if he were real. In that way, he is a "solution" to the crime scenario but not an actual suspect. If the detective assumes the existence of a fictitious character named Orzog, and the defining characteristics of Orzog make him a person who fits the scenario, the detective could say, "There are three suspects-- Bob Smith, Al Johnson, and Orzog. Orzog fits the scenario just like Bob and Fred, but that does not make him an actuality. He is just a hypothetical that fits the situation. Equations can work the same way. 3i and -3i may be solutions to an equation, but that does not make them actual. They exist only as fictitious concepts that fit scenarios as hypotheticals.
      As far as I can tell everything here is still an issue of ill-definition. You have never defined what you mean when you say a number is 'real' or 'not real', just lots of synonyms (actual, existent, fictitious, hypothetical...), and therefore it is impossible to discuss your position. I don't have any clear idea what you mean and we can't make any progress until you provide a clear exposition.

      What does 'x can't really be i' mean? An equivalent question is what does 'x really is a' mean, where a is a general number? Take x^4 + 1 = 0. 1 and -1 are both solutions for x. Are you saying that x 'really is' only one of these values? Which one?

      I don't actually have a clear understanding of your objection to the bolded bits. I don't see where you addressed them, unless they were supposed to be elucidated by your general meaning, but I've explained how that isn't so.

      If you multiply the mug by i:

      Again we have to be precise about definitions. Numbers and operations are abstractions from physical situations resp. physical actions upon them. In the first case the mug corresponds to 1 and 'lots of' corresponds to multiplication. In the second case the orientation of the handle corresponds to 1 and rotation corresponds to multiplication. This, and only this, is the meaning. Natural numbers are in exact correspondence with the behaviour of the mug in the first case and (unit) complex numbers are in exact correspondence with the behaviour of the mug in the second case. In fact any abstraction from the behaviour of the mug in the second case will be exactly the same thing as the field of complex numbers.

      As far as I can tell the only reason you are rejecting complex numbers is because you are conceptualising the incorrect physical bases. But you've already accommodated other number systems that had exactly this problem, as I already covered in detail. If you are conceptualising whole objects, you will reject any numbers that square to 2 as 'hypotheticals'. But with the correct conceptualisation, which is lengths, it is clear that there is no problem. If you are conceptualising lengths, you will reject any numbers that square to -1 as 'hypotheticals'. But with the correct conceptualisation, which is rotations, it is clear that there is no problem. If this is wrong please explain exactly where.

      Complex numbers are just a special type of linear transformation:

      This is simply a fact, there is a subfield of 2*2 matrices (and a corresponding subset of linear transformations) which exactly correspond to complex numbers.

      Anything made out of them 'exists':

      Why not? You don't have any problem with the ontological status of the abstractions of real numbers and their operations. So why not things constructed from them? For example, as far as I can tell you don't have any problem with matrices either, which are arrays of numbers with two operations called addition and multiplication defined on them in a certain way. Are 2*2 matrices 'actual'? What about 5*5 matrices?

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      Consciousness in the Void Universal Mind's Avatar
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      Quote Originally Posted by Xei View Post
      As far as I can tell everything here is still an issue of ill-definition. You have never defined what you mean when you say a number is 'real' or 'not real', just lots of synonyms (actual, existent, fictitious, hypothetical...), and therefore it is impossible to discuss your position. I don't have any clear idea what you mean and we can't make any progress until you provide a clear exposition.
      By "real," I mean "representative of possible physical realities; expressing an amount that can exist in the physical universe; an actuality and not a mere concept."

      Quote Originally Posted by Xei View Post
      What does 'x can't really be i' mean? An equivalent question is what does 'x really is a' mean, where a is a general number? Take x^4 + 1 = 0. 1 and -1 are both solutions for x. Are you saying that x 'really is' only one of these values? Which one?
      Did you read my hypothetical concerning the detective?

      1 and -1 are both solutions to your equation, and both numbers fit into the definition of "real" that I just gave. If 3 + 7i and 3 - 7i were solutions, meaning that when put into the equation they make the equation true, they would have a metaphysical status that is different from 1 and -1. The complex solutions do not fit the definition of real.

      Quote Originally Posted by Xei View Post
      I don't actually have a clear understanding of your objection to the bolded bits. I don't see where you addressed them, unless they were supposed to be elucidated by your general meaning, but I've explained how that isn't so.

      If you multiply the mug by i:

      Again we have to be precise about definitions. Numbers and operations are abstractions from physical situations resp. physical actions upon them. In the first case the mug corresponds to 1 and 'lots of' corresponds to multiplication. In the second case the orientation of the handle corresponds to 1 and rotation corresponds to multiplication. This, and only this, is the meaning. Natural numbers are in exact correspondence with the behaviour of the mug in the first case and (unit) complex numbers are in exact correspondence with the behaviour of the mug in the second case. In fact any abstraction from the behaviour of the mug in the second case will be exactly the same thing as the field of complex numbers.
      Yes, I addressed the boldface comments with a broad explanation. With your mug rotation example, you are assuming the reality of the complex plane. It involves something that does not correspond to reality. Once you assume the legitimacy of complex numbers and start labeling things with them, everything may fit. I can call the position the center of my neighbor's porch "(5, -4 + 8i)," but that would just be a mere label, and one of its terms would be unreal. The complex plane works only in a fictitious (non-real) scenario. Like I said, you can multiply 3 by 1 unicorn and get 3 unicorns. That works only as a hypothetical involving something fictitious. Multiplying 3 by 1 mug results in 3 mugs. I can show you 3 mugs. If I asked you to show me 3 + 2i mugs, you would have nothing to show me, just like you wouldn't if I asked you to show me 3 unicorns. You could show me drawings of them, but you could not show me them. Complex plane coordinates work the same way.

      Quote Originally Posted by Xei View Post
      As far as I can tell the only reason you are rejecting complex numbers is because you are conceptualising the incorrect physical bases. But you've already accommodated other number systems that had exactly this problem, as I already covered in detail. If you are conceptualising whole objects, you will reject any numbers that square to 2 as 'hypotheticals'. But with the correct conceptualisation, which is lengths, it is clear that there is no problem. If you are conceptualising lengths, you will reject any numbers that square to -1 as 'hypotheticals'. But with the correct conceptualisation, which is rotations, it is clear that there is no problem. If this is wrong please explain exactly where.
      I explained the rotations example. As for the square root of 2, it is a real number; it exists on the real number line. It is an amount that corresponds to physical reality. It can be a distance, a weight, etc. If the two legs of a right triangle are 1 foot, the hypotenuse is sqrt 2 feet. That is a real distance. It is an actual amount and not a mere label. It corresponds to things in the physical universe without some of the principles involved not corresponding to the physical universe.

      Quote Originally Posted by Xei View Post
      Complex numbers are just a special type of linear transformation:

      This is simply a fact, there is a subfield of 2*2 matrices (and a corresponding subset of linear transformations) which exactly correspond to complex numbers.

      Anything made out of them 'exists':

      Why not? You don't have any problem with the ontological status of the abstractions of real numbers and their operations. So why not things constructed from them? For example, as far as I can tell you don't have any problem with matrices either, which are arrays of numbers with two operations called addition and multiplication defined on them in a certain way. Are 2*2 matrices 'actual'? What about 5*5 matrices?
      The problem is that the constructions from them involve fictitious principles. My unicorn example and neighbor's door step example illustrate my point on that.

      As for matrices, yes they are actual. I can make one out of wood. They are just constructions people use for representing mathematical principles. You might say that you can also make a complex plane out of wood, but that is different. A matrix is just a display of numbers, and the rules of addition and multiplication are involved in the rules for using it. A complex plane is more than just a drawing with rules involving numbers and operations. It is a principle that is represented by a drawing, and the principle involves numbers that are not real. A matrix can have numbers that are not real, but the matrix is still just the place where they are represented.
      Quote Originally Posted by really View Post
      God cannot destroy himself because He is Omnipotent.


    23. #48
      Xei
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      Quote Originally Posted by Universal Mind View Post
      By "real," I mean "representative of possible physical realities; expressing an amount that can exist in the physical universe; an actuality and not a mere concept."

      With your mug rotation example, you are assuming the reality of the complex plane.
      Cyclical fallacy. You say that something is real if it represents a physical situation. Then you disregard a physical situation represented by complex numbers on the basis that it is not real.

      An exactly analogous fallacy: irrational numbers are not real. You claim you can prove they are real because the diagonal of a unit square is an irrational number. But you are assuming the reality of irrational numbers to do this.

      As for matrices, yes they are actual. I can make one out of wood.
      I have no idea what being able to make a matrix out of wood refers to, nor do I see how it relates to the definition of real that you provided.

    24. #49
      Consciousness in the Void Universal Mind's Avatar
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      Quote Originally Posted by Xei View Post
      Cyclical fallacy. You say that something is real if it represents a physical situation. Then you disregard a physical situation represented by complex numbers on the basis that it is not real.

      An exactly analogous fallacy: irrational numbers are not real. You claim you can prove they are real because the diagonal of a unit square is an irrational number. But you are assuming the reality of irrational numbers to do this.
      No, the actual measure of something in the physical world is sqrt 2 units as proven by application of the Pythagorean Theorem. It is the logical conclusion derived from using something that has been established. The complex plane on the other hand involves assuming the existence of complex numbers from the get go. The triangle proves the reality of an irrational number. The complex plane is based on the existence of complex numbers.

      Quote Originally Posted by Xei View Post
      I have no idea what being able to make a matrix out of wood refers to, nor do I see how it relates to the definition of real that you provided.
      Then read my post again because I explained it there.
      Quote Originally Posted by really View Post
      God cannot destroy himself because He is Omnipotent.


    25. #50
      Xei
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      Read it again. Still no idea what it means. Mainly because you introduced new, undefined words like 'principle'. What does

      0 -1
      1 0

      look like when it's made out of wood?

      Quote Originally Posted by Universal Mind View Post
      The complex plane is based on the existence of complex numbers.
      Once again you only think this is an argument because you're assuming the consequent.

      The real number line is 'based on' the existence of real numbers. This isn't a problem, so long as we can prove that real numbers exist. The complex plane is based on the existence of complex numbers. This isn't a problem, so long as we can prove that complex numbers exist.

      We establish that real numbers exist because we can find a physical situation to which they exactly correspond (your definition of existence), for example, displacements along a continuous line in space.

      We establish that complex numbers exist because we can find a physical situation to which they exactly correspond, for example, rotations of objects.

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