2.  You should both do it.

3.  Originally Posted by Universal Mind 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.

4.  Originally Posted by tommo You should both do it. Do it yourself.

5.  Or die.

7.  Originally Posted by Wayfaerer Do it yourself. I'm not good at maths.

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

11.  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. Thoughts? Flames? Disgust?

12.  Originally Posted by Universal Mind 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.

13.  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? Originally Posted by Olysseus 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?

14.  Originally Posted by Xei 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.

17.  Originally Posted by Xei 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.

23.  Originally Posted by Universal Mind 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.  Originally Posted by Xei 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. Originally Posted by Xei 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.

25.  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? Originally Posted by Universal Mind 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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