DrRocket

ahem (polishes nails on sweater)

This is rather too vague. Belief, as with religion or mysticism, can arise in the absense of empirical data or logic. Immortal has been deprived of neither in forming what you propose to be his conclusion. That conclusion would therefore be more accurately described as a deduction rather merely a belief.

If everyone had to rely on their elementary and high school education in mathematics for either interest or content prior to entering a university, there would be very few professional mathematicians.

The order is alright. However, you can put probability and statistics anywhere, and it might help to understand a bit of linear algebra in conjunction with a study of ordinary differential equations (say as is done in Braun's book). I have no idea how they differentiate pre-calculus from algebra and trigonometry or why anyone would want to, but I would guess that it is just more algebra and trig and you do need to be proficient in those subjects to be able to learn calculus efficiently. I suggest moving "Vi Hart" to the trash can. What I saw thee is cute but would tend to just confuse someone trying to learn the subject.

Reply. We won't do your homework for you. You must show a reasonable attempt on your own, and then we will provide comments, suggestions and leading questions. You learn mathematics by doing mathematics. Learning is not a passive exercise.

This ignores the empirical evidence that the moon is made of green cheese. "It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong." -- Richard P. Feynman

Maybe physicians ought to stick to medicine. The situation is this: If one assumes a minimal amount of matter in the universe, consistent with observation, and applies the general theory of relativity, then Hawking and Penrose have shown that the existing universe was once in a very dense state and that there is an initial spacetime singularity. Now, singularities in general relativity are very subtle things. Spaceetime itself cannot, essentially by definition contain any singular points. So the meaning of the initial singularity is this -- it is not possible to extend timelike geodesic curves indefinitely in the past direction. That in common parlance means that there is no such thing as "before the big bang". Some future theory that goes beyond general relativity might have something to say about this situation. But as of this moment science is not able to do that. There is no such thing as Planck's Wall, in any currently viable theory. Contrary to the pap one sees in the popular literature, neither string theory nor loop quantum gravity has as yet even been clearly defined, let alone made any sensible statement regarding the big bang. Neither are really theories in the sense that the word is used in physics. They are avenues of research, speculative ideas with some promise, but not theories. Maybe someday one or the other of these attempts at a theory wil be able to attain the status of a real theory and allow someone to make such a statement. But that time is not now and I suggest that you not hold your breath in anticipation. No one has a clue, and this is not a question for science anyway. Science addresses the question of HOW nature behaves. It is rather good at that. The question as to WHY nature behaves as it does is a questoin for philosophers and theologians. They are notorious for never reaching a conclusion.

It is not a matter of rigor. It is simply a matter of notation. 1. The function [math]x\mapsto y=\sqrt{x}[/math] for [math]x \in \mathbb R^+[/math] is well-defined only by the convention that [math]\sqrt x[/math] selects the non-negative square root. 2. The requirement that for every x there is one y simply makes the function a function . To be a bijective function one also requires that no two values of x map to the same y-value. This is true of the square root function only because of the convention above and the fact that the domain of the function and range are taken to be the non-negative real numbers. 3. In the more general case of negative real numbers and complex numbers the situation requirs a bit more subtle touch. The square root function and the use of exponents in general is dependent on use of the logarithm. In complex analysis, the logarithm is not a simple well-defined function, but requires that one "choose a branch of the logarithm", reflecting the fact that the complex argument is defined only modulo [math] 2 \pi [/math]. Different chooices of that branch result in different values for the square root function. In older times the logarithm was viewed as a "multi-valued function", which of course violates the modern definition of the term "function" itself. So one must deal with a branch of a function that is actually defined on an appropriate Riemann surface rather than on the complex numbers itself. Yes and that expression manages to accomodate all of the ambiguity that is incurred in the expression [math]\sqrt{-\Delta}[/math] since both the negative and positive square roots are included in the complete expression.

Frequency is a rather loose term when applied to functions given as a fuction of time. Even a "pure sinusoid" when presented as a physical signal, because it started at a definite time in the past and has run only as far as the present time, is not a pure sine wave. ALL physical signals have a frequency spectrum that is much broader than a single point. To make these notions precise one simply must resort to Fourier transforms and formulate questions and answers in the "frequency domain". Radio receivers do not simply "tune" to a single frequency. Rather they accept a frequency band, centered on the frequency to which we say that they are tuned. The modulated signal is limited to that band and decoded in the receiver to produce the signal that drives a speaker which is then heard as sound. In fact, since electronics does not create perfectly accept/reject bands the situation is slightly more complex and some distortion and noise enters the picture, but the general idea is adequately explained by the simplified picture. If you wish to understand this technology in detail there is simply no substitute for learning the relevant mathematics, which will allow you to read textbooks on the subject such as those previously noted. A slightly more ambitious alternative is to do what I did when, as a high school student I also wanted to understand how a radio works. That alternative is to go get a university degree in electrical engineering. There is no royal road to understanding. You will have to put in some serious effort in order to gain serious understanding.

Your post is far too long and unintelligible to address line by line. You pretty much got everthing wrong, including the above, including the inane remark about Kip Thorne. You might want to take a closer look at Hawking's concession, which has some logical gaps, including the use of the Ads/CFT correspondence which itself is an unproved conjecture of Maldecena datting from about 1997. In short, you have no idea what in the hell you are talking about.

rubbish. There is no Incompleteness Theorem of Physics, except in your personal delusions, and those delusions have absolutely nothing to do with the Gödell Incompleteness Theorems. The only determinism in evidence is the apparent agreement between observation and the classical theories of general relativity and Newtonian mechanics in limiting cases at the macroscopic level. This is not fully understood, but it not particularly surprising in light of the theorem of probability known as the Law of Large Numbers. So far all experiments agree with quantum mechanics. The only need for repeated experiments is to show that the probability densities predicted by quantum mechanics are accurate -- again a matter of statistics and the Law of Large Numbers. But those results do not, in any meaningful sense, " converge towards those as predicted by quantum mechanics" -- each and every data point is consistent with quantum theory.

Per general relativity, time IS NOT exactly the same among all clocks.

Interesting, but also a bit shallow. He does not even touch upon the fact that there is quite a bit of pseudoscience that masquerades as real science and even supposedly "real scientists" are to be viewed skeptically, Michu Kaku leaps immediately to mind. One ought to demand not only "credibility" but in fact hard data and actual analysis from sources that one might ultimately accept. Moreover, one ought to be skeptical of classifications and labels and understand what they really mean. It is quite possible to use categorization of causes and effects to slant a conclusion towards a political bias without actually violating any basic scientific principles -- beware of the use of statistics and know what the statistics really means.

Yes indeed. That particular analogy has done at least as much damage as good.

Like many problems in mathematics I think that the importance of the problem lies at least as much with the anticipated new ideas that will be required to solve the Riemann Hypothesis as with the result in and of itself. This was true of Wiles proof of the Fermat theorem by way of proving the Shimura-Taniyama-Weil conjecture. It was also true Pereleman's proof of the Poincare Conjecture and the more general geometrization conjecture of Thurston using Hamilton's Ricci Flow. As you note the Riemann hypothesis is connected to the theory of L-funtions, and while the implications are primarily stated in terms of number theory, it is, of course, analytic number theory and I personally view it as an extremely difficult problem in complex analysis. So I anticipate that any proof will involve fundamentally new ideas and techniques in analysis. There is some thought, going back to Hilbert and Polya, that a successful proof might be found through the spectrum of some suitably constructed operator on a Hilbert space. No matter how one views the problem, I think it is safe to say that after so many years of determined yet unsuccessful to solve it, any successful attack on the Riemann Hypothesis will involve fundmentally new ideas and techniques and open up new avenues of inquiry into mathematics of which we cannot at this time even conceive. If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven? -- David Hilbert http://www.claymath.org/millennium/Riemann_Hypothesis/riemann.pdf http://www.claymath.org/millennium/Riemann_Hypothesis/Sarnak_RH.pdf

There is no such thing as a "correct heuristic proof". You either have a proof or you don't. If a "mathematician" told you differently then he is not much of a mathematician.

While the change in entropy does not depend on the path it most certainly does depend on the end points of the path (independence of path is a statement about paths in state space with the same end points). In an irreversible process energy is being lost somewhere or added to the system from an external source and the end points of the path will not both coincide with those of a reversible process.

What part of "no" don't you understand ? -- Lorrie Morgan

Are you referring to the widespread observation that some people seem to sit on their brains ?

Is this not an example where not even 1% is in productive use ? http://www.scienceforums.net/topic/63389-the-concept-of-truth/

The Pythagorean theorem is only periphally about area. What distinguishes Euclidean geometry from non-Euclidean (planar) geometries is the parallel postulate: Given a line and a point not on the line there is a unique line through that point parallel to the given line. It is not obvious, but the Pythagorean theorem is equivalent to the parallel postulate. It is quite proper to say that the Pythagorean theorem characterizes Eucllidean geometry in the plane. One can extend that to the fact that it is the Pythagorean theorem that gives us the usual notion of distance and it is that notion of distance that characterizes Euclidean geometries in higher dimensions -- and that is what distinguishes the Euclidean model that we learn in high school from the model that seems to actually describe the universe as one lerns in general relativity. So, no matter what specific proof and bag of tricks you use to prove the Pythagorean theorem, you can be sure that in any valid proof the parallel postulate will play a role somewhere, either directly or in the proof of a result that is in turn used to prove the theorem.

Good luck. That is likely to require a significant change in perspective on the part of both students and industry, and such a broad-based change can take a lot of time. A real problem is the lack of appreciation in industry for BS and MS degrees in the sciences, and the simple fact that academic departments in the sciences are geared to prepare students for the PhD -- programs are tailored for the hopelessly academic types. This ought not be surprising. While there are certainly good career paths for PhDs, it is not a degree that anyone in their right mind (this does exclude quite a few degree candidates anyway) pursues for economic reasons. It is simply not a good decision if money is your goal -- the only reason to pursue a PhD is an intrinsic interest in the subject (which is characteristic of "hopelessly academic types").

Any idea why ? Those cost figures are a bit breathtaking for a public school.

Not a problem. Most mathematicians can read French. There is quite a bit of literature that requires it.

gibberish