Bill Hobba

Nice response. BTW he did get it wrong as pointed out by his mentor Bohr. Its reason is the commutation relations as advanced books on QM explain (eg page 225 of Ballentine) BTW my account of QM, still leaves some things open e.g. the implications of Bell's theorem, but I think pursuing that needs a new thread. Thanks Bill

It's just so well-verified that it looks self-evident. Of course, more detail than that is required to explain why it is true. The only thing I take as self-evident is mankind is rational by its nature; doubting it would lead to prehuman barbarism without any of the great benefits of modern society, such as the internet we are using to discuss this. It is exactly why I think postmodern philosophy, which does doubt it, is going down the path of irrational nonsense. The practical consequences of which you can read about in many sources these days. My favourite is a book by Sokal: https://www.amazon.com/Beyond-Hoax-Science-Philosophy-Culture-ebook/dp/B006TC2EIO Also, Feynman had some wise words. When talking about flying saucers, he said - 'Listen, I mean that from my knowledge of the world that I see around me, I think that it is much more likely that the reports of flying saucers are the results of the known irrational characteristics of terrestrial intelligence than of the unknown rational efforts of extra-terrestrial intelligence.' The reason I believe it is my own - I don't think it is a common explanation. I don't know the reason objectivists give. There is a very powerful theorem called Noethers theorem - explained by Brian Greene in the attached video. It says that when we have symmetries, things are conserved. For example, suppose we do something at a certain place and do the same thing at another place; we expect the same thing to happen. If not, something must be different about that other place affecting the result. But suppose that the results are always the same. Then Noethers Theorem says momentum is conserved. This means if a particle accelerates and gains momentum, some other particle must have lost momentum - in other words, whatever lost momentum caused it. So the situation is this. To make sense of the world, if something, otherwise the same, is different in different places (times, directions etc.), being rational means we do not accept when something is different; it is just the whim of the gods etc. There is a reason - and here, the only thing different is place it so must be the cause - it can't be anything else. Reason forces us to ask why - the only answer being the difference in place (time, direction etc.). If it is the same, then that implies something is conserved. If that changes, something else must have changed to cause it since it is conserved. Some may bring up Quantum Mechanics (QM). I have started a separate thread about what QM is that the reader may wish to peruse. You get different results for quantum objects simply because we can only know anything about them by interaction. We cannot know if we are dealing with the same situation, so cause and effect cannot be checked. We do have a powerful theorem called Gleason's theorem that allows the prediction of probabilities. Still, we cannot tell if things are the same because we do not know the details of subatomic objects until we interact with them. The law of large numbers applies to everyday systems composed of many quantum objects, and there is no issue. It does not say they are not real, created by human consciousness and other rubbish that some have written about QM. They are perfectly real - we can't know their properties until we interact with them.

1. They are conserved - relativity demands it. In fact, the world lines of each particle must be straight. For charged particles that is not the case - hence the difficulty. For momentum to be conserved the field is needed and the field has momentum. The particles and field have momentum conserved. 2. It is conserved as per point 1. Thanks Bill

I read a lot when young, but my interests turned more to math/physics as I got older. Enrolling in the Masters naturally has made me much more interested. Thanks Bill

Yes, but at this stage of my philosophical journey, just some simple philosophical considerations, eg reality exists and is not observer dependent as in some interpretations of QM. Even just analysing QM, so it is in that form, has taken me decades - what I wrote in the QM post is the result of many years of struggle. I find the going tough when confronted with more complex philosophical issues. Due to my background being in math, I will often resort to mathematical arguments. They are logically sound. But, great physicists can see the 'substance' behind the equations. That represents a deep philosophical insight. They do it intuitively, These are people like Feynman, who, on the surface, was very anti-philosophy but was deeply philosophical. It's just that this substance behind the equations came so naturally to him he saw no reason to take it any further formally. This can lead to confusion. For example, in arguments that QM is whacko, Feynman's path integral approach is often bought up and, correctly attacked, as irrational - how can particles take all these different paths simultaneously? They forget that it is just a heuristic suggested by writing the equation in a certain way. Particles do not take all paths at the same time. Such people have not progressed to even looking at the superficial substance behind the equations. Thanks Bill

See: https://arxiv.org/pdf/1601.03616 Thanks Bill

I really should have started with the no interaction theorem. This states particles are unable have any interaction if the principle of relativity is satisfied. Take two charged particles - we know they do interact in violation of the theorem. To circumvent this something else is added than just two particles. This something is a field and when one writes the Lagrangian of the system it contains not only the particles, but the field as well. You then derive the equations of motion and get Maxwells equations and the Lorentz Force Law. The reason the no interaction theorem is not violated is because you also have more than two particles - you have a field as well. In fact from Noether this field has energy and momentum. Because of that, while I suppose you could mount an argument, they are just mathematical abstractions; if they have energy, due to E=MC^2 the most reasonable assumption is the field are real. To me this is a very profound result. Interestingly using Coulomb's Law and relativity one can derive Maxwells Equations: http://richardhaskell.com/files/Special%20Relativity%20and%20Maxwells%20Equations.pdf Hence you get the standard Lagrangian and that the field introduced in the above as a mathematical abstraction must have energy and momentum. The argument works just as well for other fields such as gravity. Here is a conundrum to think about. Exactly the same argument in the link above can be used to derive similar equations for gravity. They are wrong - but why? Have a think and we can discuss it (hint - gravity gravitates and 'gravitational charge' is not a scalar but the stress energy tensor). Thanks Bill

Mathematics is grounded in reality simply because history shows that no area of math does not eventually find application. One of the purest of pure mathematicians, Hardy, who defended mathematics purely on the grounds of its beauty, once claimed 'nothing he had ever done had any commercial or military usefulness'. How wrong he was. Just take one area he was interested in, the divergent series. I have a book sitting in front of me called Advanced Engineering Mathematics that has a whole chapter on it because it is so useful in solving differential equations that occur in engineering. Even in physics, calculating the Casimir Force (by one method, anyway) requires summating a divergent series. Hardy's claim is the excrement of the male bovine - with all due respect to Hardy, whose mentoring of Ramanujan was a great service to mathematics. Others have remarked on this - even what looks like the most useless branch of math always seems to find application. The reverse sometimes occurs as well - sloppy math, sometimes used in engineering or physics, can inspire other mathematicians to sort out what is happening. An example is the Dirac Delta function, the solution to what it means, requiring a whole new branch of math called Distribution Theory. This has led to all sorts of views, e.g. those of Penrose, which I once agreed with, but now don't. The most straightforward answer is - mathematics is about the real world. I will give an example. Imaginary numbers were introduced to solve any quadratic equation. But we now know that looking at numbers as part of reality made their existence obvious. Draw a line and mark it with the distance from its start. These are the positive real numbers. Now let's continue the line in the other direction. We can think of -1 as an operator that rotates a point on the line by 180% so that -1 = -1*1 is one rotated 180%. We can extend this further to i being an operator that rotates whatever is after it by 90%. If nothing is after it, take it as 1 follows it. So i rotates the number 1 by 90% anticlockwise (by convention). Hence i^2 = -1. We can think of the imaginary number line as the real line rotated by 90%, (i.e. with i applied to all points on the real line), and you can specify a point in the plane by a number on the real line (a say) and a number on the complex line (b say). This is written as a + i*b or a with b rotated by 90% added to it. In other words, all complex numbers are, is a way to describe points on a plane. You can do it in other ways of course (eg vectors) - but all that shows is there are several ways to skin a cat - each with its own advantages and disadvantages. To see its power, consider the operator f(x) that rotates whatever is after it by the angle x. Well f(x) = f(n*x/n) = f(x/n)^n. But if n is large, the rotation by angle x/n = 1+(i*x/n) to a good approximation, getting better as n becomes larger. So to good approximation f(x) = f(x/n)^n = (1+ i*x/n)^n, and we expect this to be exact as n goes to infinity. But from calculus, we know the e^x = (1+x/n)^n as n goes to infinity. Substituting i*x for x, we have e^ix = (1+ ix/n)^n as n goes to infinity. So you get e^ix as an operator that rotates by an angle x. Take the derivative of e^i*x, and you get i*e^i*x - try it - it's easy. This makes proving the trig identities, derivatives etc., a snap compared to what is usually done. For example, a line of unit length rotated by angle x from the real line is cos(x) + i*sine(x). So e^i*x = cos(x) + i*sine(x). Hence taking the derivative we have cos(x)’ + i*sine(x)’ = i*cos(x) - sine(x) or cos(x)’ = -sine(x) and sine(x)’ = cos(x). Compare that to other proofs; you will see it is much easier. Now try it on e^i*(x+y), and you get the standard trig identities without further ado. IMHO, before doing more advanced trig, calculus etc., you should study complex numbers - things are more manageable. Math is about reality - but different mathematics can describe the same reality, each with its advantages and disadvantages. Thanks Bill 20 views

It is said by Gleick that truly great physicists (eg Einstein, Landau, Feynman and likely Fermi), have a literally scary ability to see the substance behind the equations mere mortals can't. It's not like the great Von Neumann - a polymath (although as a physicist, he was in the top echelon) whose intellect was mind-boggling. But they somehow saw to the heart of things better - which is not to say Von Neumann was a slouch at it - he was right up there with the best. This was expressed well by Wigner: “I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Wemer Heisenberg. Paul Dirac was my brother-in-Iaw; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me. [...] But Einstein's understanding was deeper than even Jancsi von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of Jancsi's brilliance, he never produced anything so original.” Thanks Bill

That is a better way of expressing the situation than I did. Thanks Bill

Some recent work would suggest that while they may be successful initially, they will eventually run into problems: https://arxiv.org/abs/2101.10873 Thanks Bill

For classical mechanics, you are, of course, talking Noether and are correct. But when you go to relativity, things get a bit more complicated. Take two charged particles. If the momentum of one changes, i.e. goes from stationary to some velocity by momentum conservation instantaneously, the other particle should move. But it doesn't - from relativity, it happens a bit later. For a time, momentum is not conserved - in violation of Noether. To rectify this, Wigner said you need to introduce the concept of a field as a holder for the missing momentum. Wigner did interesting work in field theory. I tried to find more details from Physics Forums where I am a mentor. Here is the reply I got: https://www.physicsforums.com/threads/wigners-theorem-that-all-fields-must-be-tensors.984076/ BTW, rules are strictly enforced on that forum, and one rule, with a bit of leeway (as determined by mentors like me), is you can't discuss philosophy. I won't go into why other than to say it was once allowed but got out of hand. Added later: The result that the field must exist is called the no interaction theroem. Thanks Bill

I think you are correct. It is not a physical assumption - it is a consistency assumption. If we did not allow complex numbers then the poisson bracket would not be an observable. Thanks for that. Thanks Bill

It is true and the basis of a lot of 'rot' written about mathematics. Ethnomathematics seems to be the latest fad in this direction, as explained by James Lindsay https://newdiscourses.com/tftw-ethnomathematics/ He uses 2+2 =4, and its written truth depends on the base used. Those that argue this forget to emphasise in everyday discourse base ten is assumed. It is OK pointing this out - but the stuff like white supremacy etc., they read into it is utter rot of the first order. What surprises me is some believe it - the modern cultural disintegration seems to know no bounds. But what can you say about a view of the world that says doctors should be culturally representative of the populace they serve, e.g. if you have 10% people of Indian descent, then 10% of doctors should be Indian? Over 50% of doctors (and increasing) are female these days. So under their reasoning, we need more male doctors. What's the bet they won't touch that one because it brings their whole agenda down like the house of cards it is (it contradicts the feminist views they likely also hold)? Returning to the original comment, a better example, IMHO, is 1+2+3+4+5....... = -1/12. You should see what is written about that - even by some professional mathematicians (some do get it right). We naturally assume they are a subset of the real numbers; under that assumption, the answer is infinity. This is the context in everyday use. But they are also a subset of the complex numbers, in which case analytic continuation applies, and it indeed is -1/2. It's another example of context shifting - but more subtle than base 10. Also, interestingly it occurs in calculating the Casimir force. Sometimes, in our calculations, we must look carefully at our natural assumptions and change them if required - but only if required by reality - not some nonsensical WOKE idealism. Thanks Bill

Godel is not the profound thing it is made out to be. It is logically equivalent to a very practical problem in computer science called the halting problem. As a programmer, I would love a program that could accept my program as input and tell me if it will loop or not. A rather convenient thing to have. If such a program even exists is called the halting problem. The answer is actually no - you can't write such a program. Bummer. But it is nothing esoteric, weird or anything like that - simply a limitation on the tools computer science allows programmers to have. Also, interestingly the proof (at least the proofs I know) depends on the good old Cantor's diagonalisation argument. Here is a simple explanation: https://www.youtube.com/playlist?list=PLlwsleWT767dwRXyAyL0-63ON6cCOXY8E Another interesting fact, that makes Godel less of an issue than many think it is, is the real numbers are consistent and compete: https://math.stackexchange.com/questions/362837/are-real-numbers-axioms-a-consistent-or-complete-system In many scientific areas, real numbers are used - not integers. I will do a separate post on why I think mathematics is founded in reality.