Tom Mattson

Good, glad to hear it. Don't mean to be cheeky, but I see you poking around the General Math Forum at PF. My thread is in Tensor Analysis and Differential Geometry, in case that's what you are looking for.

Yes' date=' it should work that way. In this case you will have [imath']k^2=\frac{2mE}{\hbar^2}[/imath]. You mean X(x)T(t), right?

I think I know the part of the book to which you are referring. It's where he describes the graph of the 1-form, and he says that it lives in TpR2XR. Note that the dimension of the space is not (2-1)*1. Rather, the dimension is 3. From your remarks above, I think that you may be holding 2 misconceptions. 1. That the dimension of TpRn is n-1. As you can see from the discussion on pp 48-49, this is not the case. On p. 48 he explicitly writes down the basis of the tangent space to the plane P, and there are 2 basis vectors. And on p 49 he draws a picture representing TpR2, and as you can see there are 2 coordinate axes. 2. That the dimension of a space RnXRm is n*m. That is not the dimension; rather it is n+m. By the way, I've just revived my own thread on this book at PF, which has been dormant for about 1.5 months. When I first started it I got Bachman himself to come and answer some of our questions. I'll drop him an email and see if he'll come back.

I can see that I was unclear last night. What I mean by this the predicted experimental outcome of a theory need not be "physically possible" for the theory to be considered scientific. Insisting that it does need to be possible is tantamount to demanding that a theory be true to be considered scientific. But I would agree that it must be possible to test the theory, at least in principle. I would not agree that the theory must be testable with current technology, as I explained in my last post.

I don't see the sense in classifying a theory as "scientific" or "unscientific" according to the state of the art of engineering. If we do that, then what is "unscientific" today will become "scientific" tomorrow, and this change takes place without any change in what string theorists actually do. Perhaps you could explain more? Well I'm sorry you don't like it, but your personal opinion is in fact mistaken. There are a number of predictions of string theory that are testable in principle. Besides, I'm sure all the hard working scientists who do string theory wouldn't care too much for you contradicting their personal opinion that their life's work is scientific. Thanks, but no thanks. If you are going to use proprietary definitions of words that have well-known standard meanings, in an academic forum no less, then you will have to live with being perpetually misunderstood. It's up to you to use the terminology properly, rather than me figuring out what you mean by it.

Ben Son, I would also like to remark on your comments quoted in the opening post. You are of course free to make up your own definition of "scientific", but there is good reason to reject the definition you imply here. As I said in my first post, a theory is considered "scientific" if it meets the condition of falsifiability. That means that it does not matter if the technology is currently available to test it, and it does not matter if the theory is actually true. All that matters is that the theory makes claims that subject it to being proven wrong. Since string theory does that, I do not see how it can be classified as unscientific. And as for bracketing string theory with philosophy and religion, that is not right. Philosophy for the most part is conducted a priori, as opposed to science which is conudted a posteriori. And religion is unfalsifiable, even in principle. In order for this to make any sense at all, I suppose you would have to define what you mean by "philosophy" and '"theory". There are philosophical theories and there are scientific theories. The expression "for a philosophy to become theory" has no more meaning that the expression "for physics to become a theory".

No' date=' you aren't getting it. In order for a theory to be considered scientific, it is [b']not[/b] necessary that the predictions be "physically possible". Indeed, how do you even know what is "physically possible" apart from experimentation? No, what is necessary is that the claims made by the theory be such that if the experiment goes against the prediction, then the theory is proven wrong. More succinctly: In order for a theory to be considered scientific, it must make risky claims by which it can be proven wrong, if the theory is in fact wrong. Proving all of the claims wrong is not necessary. It is only necessary to prove one of the claims wrong. And as for "what experiment?", I don't know. It is not technologically feasible yet. But that has not one whit to do with whether the theory is scientific or not. A theory need not be true to be scientific. It only needs to make predictions that are experimentally contingent. Sorry, but I cannot make sense of this.

This really has nothing to do with quantum mechanics per se. It is really an issue in the philosophy of science. Near the turn of the last century the main question in that subject was, "How do we distinguish metaphysics from physics?" Karl Popper's brilliant answer to that question was to abandon the philosophy of the day: confirmationism. Confirmationism says that any experimental evidence that agrees with a theory should be taken as evidence that the theory is scientific. The trouble is that this school of thought is riddled with logical problems, and not without good reason (I'll refer you now to Hempel's Paradox of the Ravens; Google it and I'll fill you in on more details later). Popper turned all that on its head by reasoning that what makes a theory "scientific" is not that it can be confirmed, but rather that it can be falsified. That is, a theory is "scientific" if it makes claims that, if they don't go "the right way", will prove the theory false. This is quite independent of whether or not it is technologically possible for the theory to be tested. It has to do with the logical structure of the theory itself. Since string theory makes claims that are contingent on experimentation (whenever that experimental technology may become available), and by obtaining falsifying results the theory may in principle be proven wrong, I say that according to our best understanding of "what science is", yes, string theory is scientific. And that's my $0.02.

Don't forget this:

I'm not a math expert, but I'll tell you what little I do know. Knowledge of calculus will be about as helpful to you as knowledge of French art during the Rennaisance. What you will want to be grounded in is set theory. Also, geometry and topology are not the same thing. Both subjects can deal with the same objects (lines, circles, etc..) but they ask and answer different questions. Loosely speaking geometry deals with shape and topology deals with structure. As for a book, the Schaum's outline called General Topology is pretty friendly. You can also dig around the following Link Directory, which I maintain. I know there are some free lecture notes there.

Check these out: Indeterminate Undefined That would be quite a neat trick, considering that division has nothing to do with taking a limit. The former is an issue of algebra, the latter an issue of analysis. I think that the function is "undefined", not "indeterminate" at z=1, but I would defer to a mathematician. The reason I say this is that in my (limited) experience with pure mathematics, I have not seen the term "indeterminate forms" used except to refer to the result of a limit. But otherwise, yes. The limit is 2 as z approaches 1, and the function is not defined at z=1.

The definitive classic is The Theory of Spinors by Cartan. http://www.amazon.com/exec/obidos/tg/detail/-/0486640701/qid=1118280520/sr=1-1/ref=sr_1_1/102-4889131-5900916?v=glance&s=books

Actually, the real question is: What difference does it make? As I said, we aren't talking about limits when we address the question, "Is f(1)=2?". We would be talking about them if we were addressing the question, "Does the limit of f(z) as z approaches 1 equal 2?" Can you bring the theorems about limits to bear when you take the derivative of sin(x)? Of course, because the derivative is a limit. Can you bring the theorems about limits to bear when you want to know if a certain point is in the domain of that function? No, because the definitions of "domain" and "function" make no reference whatsoever to limits.

Which process are you referring to? There is not just one.

Ah' date=' of course. Yes, you can do simplification operations on functions when you take limits. But that's only because a theorem was proved that says that, if f(z) and g(z) agree at all but one point, and that point is z=c, then: [math] \lim_{z\rightarrow c}f(z)=\lim_{z\rightarrow c}g(z) [/math] But no one asked about the limit of f(z) as z approaches 1. The question "Does the limit of f(z) as z approaches 1 exist?" is completely different from "Does the function f(z) exist at z=1?", and the theorem I just mentioned for limits cannot be brought to bear.

How did "z-1" turn into "1"?

The operations you refer to are defined on the reals. You have no justification for using those operations on objects that are not real numbers, such as 0-1.

I say just what I've been saying all along: There's no problem with making that identification as long as B-1 exists.

Well if you mean the field axioms then you should not be using division at all. You should be using multiplicative inverses. Then we would have the following: f(z) = XY-1 X= AB f(z) = ABY-1 Y=B f(z) =ABB-1 But if B=(z-1) then B-1 is not defined for z=1.

LOL' date=' no need for experiments then. It must be nice to know the answers ahead of time! Just a note: The speeds need not be anywhere close to light speed. This effect has been tested on an airplane. Unfortunately for you, the universe is under no duty to conform to your a priori arguments. These questions are decided by experiments, not by what sounds ludicrous or credible to you. And what do you mean by "return to the same time that the observer is in" anyway? Once again, for emphasis: There's no reason to consider "at" or "near" light speed. This effect occurs at any speed. On to your question: It's possible that there are forces at work. It's also possible that there are little gnomes who set the clocks back, or that the clocks conspire to play a trick on us as I joked earlier. But I think the reason you are fixated on this is that you do not understand what a big deal it is for clocks that operate on completely different principles to respond in exactly the same way in time dilation experiments. On the contrary, we know a whole lot about the dynamics of atomic and subatomic systems. But it is not necessary to bring that to bear here. I've already made my points without going into those theories. It has already been demonstrated that it does happen. You have reverted to talking nonsense about time reversal again. towjyt, with each post you make I become more convinced that you aren't interested in learning nearly as much as you are interested in being a pain in the rear end. I have asked you some pointed questions that you have completely dodged. Here they are again: 1.) Why don't you try to explicitly show your reasoning regarding the apple? Then it will be clear why you think that SR predicts this "apple paradox"? 2.) What exactly do you think physicists are doing that is wrong, regarding the interpretation of their experiments? And why do you think they are wrong? 3.) What can you point to, apart from the "effects of time", that is this thing called "time"?

I determine what the domain of f(z) is by looking at where it is defined. That is' date=' for which z does f(z) return a real number? I'm assuming of course that z is a real variable and that f is a real valued function. I already answered that: No. The two functions agree at all but a single point.

1 is not in the domain of the original function because the original function does not map 1 onto a real number.

No, I mean what exactly do you think they are doing that is wrong. Simple incredulity does not suffice. Because all clocks react the same way in twin paradox experiments. You are trying to argue by analogy, and I am showing you where the cases differ in a crucial detail, and hence are not analogous. It would have to be done with actual human beings before we could say for sure. What we do know is that atomic and subatomic clocks that operate according to the electromagnetic, weak, and strong interactions. If we had as precise a clock that were based on the gravitational interaction and tested it, then we would have covered all the bases. So the answer to your question above is: There is presently no reason to think that the effects would be different. When you define what a "time frame" is, then we can talk meaningfully about it. Still waitin' for this. I asked it once before, now I'm asking again: What's the difference? What can you point to, apart from the "effects of time", that is this thing called "time"? Nothing, that's what. So if all experiments designed to test the so-called "effects of time" conform to the predictions of SR, then there simply is nothing you can point to and say, "Ah, but this hasn't been affected!" So, what exactly grounds your faith?