1.  Originally Posted by Universal Mind I reiterated something very relevant because it also applies to my bigger point that you just quoted. A complex plane in the physical realm is nonsensical, period. That's an assertion, not an argument. The real line in the physical realm is nonsensical. Period. (Not true, of course). It is two dimensional in concept. It is a plane in concept. It does not work out to anything sensical when you try to physically create one (not just a representation on paper or a computer, etc.) in the actual universe. You just have a representation that does not fit reality. If you walk the distance from (0, 0) to (0, i), you have not really traveled i units. As you said, that cannot happen because imaginary numbers cannot denote distance. And yet again this is exactly the same as the real plane. Like you say, if you walk the distance from 0 to i, you have not travelled i units. You have travelled the norm of the difference, which is 1. If you walk the distance from (0, 0) to (0, 1), you have not travelled (0, 1) units. You have travelled the norm of the difference, which is 1. These two are literally in exact analogy. In fact they are exactly the same statement. The existence of complex numbers does not somehow entail the existence of complex distances any more than the existence of ordered pairs entails the existence of ordered pair distances. Your belief that they did was a mistaken one. That has been thoroughly explained now and there is no argument left here. Complex numbers have real norms, just like ordered pairs. (3, 1) is not a number. It is a point. 3 is a number, and 1 is a number. (3, 1) is an ordered pair of numbers. You can't type your full address and call it a distance either. It is not a number, though it has numbers in it. 'Number' is a new and undefined term. Up to now we have been talking about what makes a mathematical object 'real'. You defined this to be conditioned on the object corresponding to something physical. (3, 1) and 3 + i are 'real' by this definition. Now you're talking about numbers, which is something different. You need to give a precise definition of what you mean by 'number', and then we can easily determine what kind of thing is one. But non-numberhood will not imply non-reality. Rotations are actual, but if they happen on the complex plane, they have mathematical fiction labels. You could just as well rotate a mug on a Star Wars blanket and call the new position R2D2. They don't 'happen on the complex plane', they happen on the real plane. You define a rotation operation on the real plane and that makes it the complex plane; the complex plane is a special type of real plane, 'complex' referring to the way in which the objects in it act, not what they are, which is just ordered pairs. I think you're only rejecting this because of some vague, mistaken stereotype that 'complex' means something weird and magical. If we called it the 'real plane with rotations' instead I don't think you'd actually have a problem with it.

3.  Originally Posted by Universal Mind I thought it might be something like what you said at the end. If it is the case, my view is that what would make the imaginary figures less real is that I cannot conceptualize sqrt(-1) as anything logical beyond its value as a hypothetical. Does the existence of -1 (represented by the squaring of i) as 180 degrees have anything to do with the fact that the x-coordinate of -1 on the unit circle is at 180 degrees? Yes. But I guess it's easier to say that it follows directly from Eulers formula eiφ = cos(φ) + i∙sin(φ).

6.  Originally Posted by Universal Mind By "logical" I mean it follows the laws of logic/reality. -1 has no true square root. It's just a concept with a symbol. It is based on absurd premise. That is why there is no such amount. Why are you assuming that a real number coordinate system isn't just a conceptual diagram?

7.  My main point is really that with both mathematical and philosophical clarity, your issues with complex numbers (which refers to any number a + bi rather than imaginary which only refers to bi, by the way) should be solved. Philosophically, different number fields are abstracted from different things, and having a single physical basis in mind can lead you astray, as it led the Pythagoreans astray. And mathematically, the complex plane is really a very benign formal construction, a fact which is somewhat obfuscated by the language and the notation. Something I must remark on looking at your final response is that in the careful modern approach, we do not simply add a new entity 'i'; rather, we literally use i as a shorthand notation for points on the plane. Namely, we write (a, b) (where a and b are both real) as a + bi; they are literally exactly the same, it just makes shunting the symbols a little simpler and clearer. Don't think of i as a special entity, just think of it as a marker. You could just as well write a' + b". That is my summary and I hope it turns out to be edifying. Beyond this discussion, I really recommend you look into the maths of complex numbers even if just on a provisional basis, because being familiar with what something is and what it does is usually important for the 'meta' considerations, and because a lot of the most beautiful mathematics is found here. In fact it contains what is generally considered to be the most beautiful result in all mathematics, e^i*pi + 1 = 0; that justifies enquiry for aesthetic considerations alone. I also really really recommend you look at the theory of abstract algebra. It will be totally different from anything you've seen before; it's really awesome and I think you'll like it. It will also provide far greater insight into what maths actually is, especially through ideas like actions and isomorphisms. Groups and rings is the place to start; I have notes for the former on my laptop which I can email you if you'd like.

8.  Originally Posted by Wayfaerer Why are you assuming that a real number coordinate system isn't just a conceptual diagram? See post #55. Originally Posted by Xei Beyond this discussion, I really recommend you look into the maths of complex numbers even if just on a provisional basis, because being familiar with what something is and what it does is usually important for the 'meta' considerations, and because a lot of the most beautiful mathematics is found here. In fact it contains what is generally considered to be the most beautiful result in all mathematics, e^i*pi + 1 = 0; that justifies enquiry for aesthetic considerations alone. I also really really recommend you look at the theory of abstract algebra. It will be totally different from anything you've seen before; it's really awesome and I think you'll like it. It will also provide far greater insight into what maths actually is, especially through ideas like actions and isomorphisms. Groups and rings is the place to start; I have notes for the former on my laptop which I can email you if you'd like. I know a good bit about complex numbers and have taught them on a high school level (algebra II and pre-calculus), but there is always more to learn. Abstract algebra sounds like fascinating stuff, and I would love to look into it. I might request your personal notes after my initial research. Thanks.

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