Energy and mass are. not equal. The equation says mass is a form of energy. The question is what is energy. For that see Noether's theorem. Bit busy right now, will expand on it with links a bit later.

Good point. Me, despite years of study of physics I still have no idea. I think like Wienberg we are closing in, but as of now beats me

Thanks
Bill

I like the answer that says its just an attempt to clarify what is. 30 years of reading books on physics, especially QM, but GR etc as well, and philosophical writings on it by people like Wittgenstein (conventionalism), Poincare (conventionalism as well), Turing (applications were paramount. But had a magnificent debate with Wittgenstein about one of the most fundamental of things - math - https://www.britishwittgensteinsociety.org/wp-content/uploads/documents/lectures/Turing-and-Wittgenstein-on-Logic-and-Mathematics.pdf . Its ironic that before being a philosopher Wittgenstein used applied math all the time as an aeronautical researcher), Weinberg (realist - science is progressing towards something), Kuhn - well I am sure you get my drift - I still have no firm idea. I think its one of those things you need to read and form your own view - if you can - like I said I can't. I recently found a little known discussion between Dirac and Heisenberg that helped me quite a lot:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.485.9188&rep=rep1&type=pdf

I side with Dirac in that what is, is never really known, we just continually advance theories about it. Weinberg thinks we are advancing towards something, I think he is right, but bowed if I can justify it.

To me it's maddening we cant even pin it down well.

Thanks
Bill

What I was pointing out is virtually anyone can learn QM so there is nothing conceptually hard about it, although misconceptions abound. Now, as you correctly point out when you ask questions like is QM casual you immediately run into issues, even issues as to what casual means - I suppose its a quibble on my part about conceptually understand and able to discuss using normal philosophical discourse.

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Bill

In Quantum Mechanics physicists generally think so, as do I, but when looked at carefully it has issues eg decoherence is likely what gives quantum things its properties independent of 'observation', which of course means you do not have an object by itself. I would say, more carefully, further research is required.

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Bill

Actually there are no conceptual difficulties in QM - really its just what in math is called a generalized probability model:
https://www.scottaaronson.com/democritus/lec9.html

Now what it means - that is another matter.

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Bill

He certainly did have a heart and that impresses me as well. Maybe that's it.

I thought Hugo was a Republican. I am centre maybe a bit centre right politically. Did Hugo go through phase's. Many certainty start left and become more to the centre later. I was strange, I started on the right and then became of the centre. I read Hugo when I had changed to the centre and certainly had no issues with the political views in his books.

There was one author that I did like not of the science fiction etc variety. I do not remember who but someone who knew me, was an avid reader of novels etc, suggested I read Victor Hugo. I gave it a try (The Man Who Laughs) and have since read other novels by him. I know he is a famous novelist, one English LIt graduate said to me one Victor Hugo is worth 10 F Scott Fitzgerald. I have read the Great Gatsby and thought - bla - but not Victor Hugo - he grappled with what I thought were genuine issues. I simply know they are both famous novelists. Is there something about Victor Hugo that sets him apart?

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Bill

I think the style of teaching literacy etc is the important thing.

I read a huge amount but mostly technical stuff in math/physics and some politics and philosophy. I wasnt always like that. At 12 years of age I read hardly at all, but was very interested in electronics so read a bit on that. I do not know why but I picked up my math book called Algebra and Geometry about then. I did a couple of problems, got more confidence, did more and so on. I finished it in about a week and was hooked. I went town to the local library and got more advanced books, teaching myself calculus at 13-14. I then applied it to electronics and a shocked how it made things I couldn't understand a snap. I basically went my own path forgetting what I as taught a school. My English suffered badly because it simply did not interest me, but did start reading a bit of science fiction. My English was so bad they had to call my parents in and had to do a IQ test. It was 151 so IQ was not an issue (its really about 130 - the test was biased heavily towards logic etc which I am good at - not the 'soft' areas like comprehension etc which was not my thing - I think it was to see if I had anything actually wrong rather than to get a real IQ). Anyway this is a lead up to what I think is the real problem in teaching literacy. My English teacher started asking my opinion on texts like Animal Farm in class. I said what I thought but was dismissed as if I as some kind of moron. My math and science teachers even had to speak to the English teacher to give me a fair go and explain why I was wrong rather than being dismissed. That sealed it - I switched off English/Humanities entirely and failed HS (except in math/science where I did well). I got into university though based on my good math/science marks and overall all was fine.

Now why did this happen - teachers should never dismiss comments - they might seem wrong or silly, but it should be explained why. These days I am a believer in teaching by a Harkness table where everyone contributes and all ideas are subject to critical analysis. I think that is the key. You must learn to think for yourself with the teacher as a facilitator.

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Bill

There is quite a bit wrong with your post eg acceleration does depend on time as does velocity by their very definition. But let's start with the principle of least action, it comes from QM namely Feynman's path integral approach. That's why its more fundamental than Newtons laws.

'So for instance, Newton's laws assume that mass doesn't depend on velocity, that's a weakness.'

Newtons laws have many weaknesses I will not go into, but they can be rectified and is now the modern theory of Classical Mechanics as you will find in a proper treatment like Landau - Mechanics. The laws are replaced by a critical assumption - The Principle Of Least Action. What is the reason for that assumption? - I will let people think about that - there is a reason - but its not what you would think. Mass in that treatment is shown to be a constant independent of velocity:

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Bill

I am a mentor on Physics Forums and the reason philosophy is off topic there is it once was its own sub-forum. It was moderated by someone who was basically qualified in both physics and philosophy and kept it under control. They left and it got out of control - people could not tell the difference between speculations (and mostly gibberish at that) and philosophy. The physicists over there kept the physics in line - by forum rules you basically cant discuss 'junk' only actual physics as in papers, textbooks etc. As a mentor part of my job it to help adjudicate on stuff that's 'borderline' science. A philosophy mentor would need the same and we lost ours, so it was ditched.

Regarding the relationship of physics to philosophy it took a big hit with the story of Kant and Gauss regarding non-euclidean geometry - but that is a thread in itself. The view of today's physicists is probably summed up by Feynman 'Philosophers say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong.'. An example would be what is energy. Philosophers probably argued that one for yonks really getting nowhere. But along came Noether and all was clear, but in a way no philosopher would ever have thought of - its merely a consequence of symmetry in time. As a matter of fact we now know much of physics is about symmetry - but most philosophers don't know it, and of those that do they probably argue about it like what was done about energy. Scientists accept we do not know, and another Noether (who was a mathematician that moonlighted in physics) is needed to resolve it. Its humbling admitting that's the way it is. Heisenberg and Dirac had an interesting discussion about a Kuhn like view of science and its just steady progress:
http://philsci-archive.pitt.edu/1614/

Myself and I would say most physicists side with Dirac eg Weinberg:
https://www.physics.utah.edu/~detar/phys4910/readings/fundamentals/weinberg.html

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Bill

Maybe its because they are one of the majors least likely to earn the big bucks. Nobody particularly likes constant self sacrifice, but as a philosophy major you likely will not earn much that will be taken from you and given to others, nor are your wants particularly great. One of the purest forms of socialism is the Kibbutz's in Israel. Evidently they produced good citizens but most left when old enough - a life of constant self sacrifice is hard to sustain.

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Bill

Read Feynman's the Character of Physical Law. There are videos on it as well I will post later along with a few other observations

There are a lot of misconceptions about QM and virtual particles is one of the more insidious
https://arxiv.org/pdf/quant-ph/0609163.pdf

Basically its just a name for terms in a Dyson series when represented pictorially in a Feynman Diagram. They are no more a particle in the classical or quantum sense than squiggles on some paper are.

In QM particles are emitted and absorbed without cause all the time:
http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf

Of course science is only provisional - a cause may be found one day - but in principle it certainly is possible and in the current theory it is like that. Note according to the QM you likely learned at school electrons in a hydrogen atom is in a stationary state. But because electrons are charged they are coupled to the quantum EM field that modern physics thinks permeates everywhere. That means the electron is not quite stationary and will, according to the theory, change state and emit a photon unpredictably according to the math of the Fermi Golden Rule (the coupling to the EM Field is viewed as a perturbation):
http://staff.ustc.edu.cn/~yuanzs/teaching/Fermi-Golden-Rule-No-II.pdf

So yes in principle you can have effects without cause (ie when a photon will be emitted) but we know the cause of it being emitted in the first place. So its a bit nuanced - I will let those better than me at philosophy discuss that one.

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Bill

Whats a greedy capitalist? - someone who by legal/moral means maximizes profit? Or is it someone that does it by any means possible? If a God like the Christian God exists I think his/her reaction will differ depending on which group you belong to.

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Bill

He has really goofed it up - look at equation 2.26 instead. If it was to the power 0 then you would have 1+1+1+1..... which in fact is -1/2 and his answer is wrong. I would need to do the math correctly from 2.26 to see what he is getting at. The point is 2.26 contains the Hurwitz Zeta Function with s = -1:
http://mathworld.wolfram.com/HurwitzZetaFunction.html

As you can see from the above the equation he wrote later connecting it to the zeta function is correct and that is how the Zeta Function at -1 or 1+2+3+4..... enters into it. I just hate it when a supposedly peer reviewed paper has errors like that - but it happens a lot more frequently than people think.

My view is as I said. When writing something like 1+2+3+4...... because you are using real numbers you think you are confined to the real number system. But there is nothing stopping them being from the complex number system in which case you have analytic continuation that allows the summation to be defined. A similar trick is used in a number of summation methods in order to interpret whats going on (the following also contains a correct derivation of the Casmir force using the Zeta Function - I checked it this time):
https://arxiv.org/pdf/1703.05164.pdf

Take Borel Summation as an example. Deriving it is simple as well as seeing its a natural extension of normal summation. Its always true if you can reverse the sum and integral. For normally convergent sums there is a point where for all practical purposes the rest of the terms are zero so you can to any accuracy replace the sum by a finite one hence the order can be reversed. If not normally summable you have a method for summing series you normally can't sum. Whats really going on? Although it only uses real numbers by going to the complex plane you can see its a hidden method of analytic continuation which, as explained in the link is mainly what divergent series summation is.

Math is funny - looking at exactly the same problem in a different context, often one that is more general, allows a deeper understanding.

BTW the above paper is by the same lecturer that did a series of lectures containing a lot of the above plus all sorts of puzzling mathematical physics stuff worth philosophical consideration. I will admit I am not enough of a philosopher to nut it out.:
https://www.youtube.com/watch?v=LYNOGk3ZjFM&list=PLOFVFbzrQ49TNlDOxxCAjC7kbnorAR1MU

Thanks
Bill

Ok - first an example of its use in calculating the Casimir Force:
https://pdfs.semanticscholar.org/90a0/5e9d920f271f20704406828d94d0890c6608.pdf

See equation 2.31.

The above uses a naive summation, a deeper justification based on renormalization can be found here:
https://www.iopb.res.in/~sjp/sjp_past_issues/sjp_aj17/new/Torode_Casimir.pdf

The idea of re-normalization is this. You do calculations, but end up with infinity. To get around this one writes it in a form with a new parameter called a cutoff that gives a finite answer - a wrong one of course because you have a cutoff in there - but an actual answer depending on the cutoff. Give this cutoff dependent answer the name re-normalized value - a fancy name for a simple concept. Rewrite the equation in a different form by some algebra so it contains the re-normalised value but no explicit cutoff (the cutoff is now contained in the re-normalized value). That way you can do calculations that give finite answers. Now take the limit as the cutoff goes to the correct value. But the cutoff is now contained in the re-normalized value and we know what that is in the limit because we can measure it. Tricky - here is a paper on it:
https://arxiv.org/abs/hep-th/0212049

Now in the Casimir force you do a slightly different variation of the trick. You write the infinite sum in terms of the Zeta Function with s as the parameter in the Zeta Function. Its only defined for s > 1 but we want it for s = -1. Its the same trick as above but here its much easier. You simply let s approach -1 which can be done by a powerful technique called analytic continuation that can be done with complex numbers - but not real numbers unless considered as part of the complex plane.

And therein lies what is going on. If I write 1+2+3+4......... I make an assumption not implied by the the equation - that the numbers are real. However they can equally be taken as complex numbers. In the complex plane the powerful method of analytic continuation is available to use and it can be summed. Math is not just about logical constructs - its more than that - its about concepts. In order to do math we must choose the right concepts and interpret your equations using those concepts.

It took me a while in my mathematical studies to nut that one out. My professors used to get annoyed at me because I asked questions about logical issues. But gradually I realised while math requires rigorous logical proof it is not done in a vacuum - but has a context and that context has a big impact.

Thanks
Bill

Some interesting points here. What I would like to do is have a look at a concrete example. Consider 1+2+3+4......... It's obvious to most its infinity. But consider the so called Zeta function:
https://simple.wikipedia.org/wiki/Riemann_zeta_function

Let s = -1 and you get 1+2+3+4......... . and it is known that at -1 the Zeta function is -1/12 or 1+2+3+4......... = -1/12. Even stranger is calculations such as the Casmir Force uses 1+2+3+4......... = -1/12, and it is in accord with experiment. A very strange phenomena - or is it? I have my view but would be interested in what others think.

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Bill

Mathematics is the natural language physics is written in. The validity of a physical theory is an experimental matter - nothing to do with the math.

Here is an interesting example:
https://arxiv.org/abs/1507.06393

It's mostly just math - but there is that word - almost - in there. That almost is the existence of magnetic mono-poles which is an experimental matter - and far from easy to experimentally look for. None have yet been found - but may in the future in which case Maxwell's equations are not quite correct - however its easily accommodated in the math of the linked paper.

Another example is Lovelock's Theorem that shows in dimension 4, the Einstein equations are the unique second-order field equations generated by an action. i.e., you can derive Einstein's equations by requiring that they are generated by an action and that they are of second order in the derivatives of the metric. Normally actions contain only first order derivatives but its impossible to construct a first order one in GR, so you go to second order. The interesting thing is it turns out when you calculate the GR equations the second order derivatives in the action do not matter, and you get normal field equations that second order actions would not usually give. Now the question is why do we have actions (I will not go into why you would usually only want first order derivatives in the action)? That requires QM to explain (it follows from Feynman's path integral approach). Without the supporting experiments who would come up with QM?

The relation of math and physics is in fact quite subtle.

I do however find the ending of the link in the first post amazing - its true - but virtually never pointed out except by those like me that have read it:
'A teacher of mathematics who has not got to grips with at least some of the volumes of the course by Landau and Lifshitz will then become a relic like the person nowadays who does not know the difference between an open and a closed set.'

The first book in the series, mechanics, is simply beauty beyond compare as reviews on Amazon attest to:
https://www.amazon.com/Mechanics-Course-Theoretical-Physics-Landau/dp/0750628960

BTW it explains, amongst other things why actions only (usually) contain first derivatives.

Could this be taught at HS? Well its deep and requires multi-variable calculus but IMHO it can - generally where I am in Australia calculus is taught later than it should (for good students its taught by some schools in grade 10 - so it is possible for them - normally one waits until grade 11 and 12). Then books that authors think well prepared HS students can handle like Morin's is possible:
https://www.amazon.com/Introduction-Classical-Mechanics-Problems-Solutions/dp/0521876222

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Bill

I'll shrug my shoulders and move on after I've given my feedback and made my case


Don't worry If I do not agree my 'feedback' is given in spades. The point is beyond that you are better off just moving on. And it is a hard often thankless job. I cant recall the number of private emails I have had about shut down threads. I may or may not agree with it, but explaining a group decision to a sometimes angry 'member' tests my virtually non existent diplomacy skills to its limit. Just another highlight of that 'wonderful' job of Mentor. Seriously like all tasks it has good and bad aspects - obviously for me the good outweighs the bad.

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Bill

'i disagree, at its base all math is, is a numerical model of reality.'


I think you should read the Turing-Wittgenstein debates on the issue. Turing had a view similar to yours, as do I, but Wittgenstein thought it was just a convention. Neither argument is easy to defeat - it comes down to personal preference.

My belief about Pi is its defined as the ratio of a circle's circumference to its diameter. As a logical system defined by Hilbert it has a precise meaning in that system. Are logical systems eternal or only exist once discovered? Well I will let posters here discuss the issue - on one side you have Penrose - on the other Poincare and Wittgenstein.

An illumining issue to discuss in this regard is the logical principle behind Cantors diagonal argument which can be used to prove some very counter intuitive things - even Godel's Theorem. That really brings the issues with infinity to the fore.

Regarding is the universe infinite and the infinite regress argument it uses the concept of before ie of time. The modern view is the big bang was the birth of space-time. Is the concept of before valid in an era when time itself did not exist?

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Bill

I don't know why that discussion was closed but I am a Mentor over on Physics Forums and can assure anyone deciding such things is both consensus based and exceedingly difficult. Much discussion with other mentors goes into it first. I do not agree with all closures, nor do I agree with some left open. Despite being a Mentor I have had discussions started by me shut down and at first its not nice. But after a while you realize - really is it the end of the world? Nowadays I personally just shrug my shoulders and say that's just the way it is. There is always plenty of other things to discuss.

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Bill

The same way I square it with tables, that consist of fundamental particles, not doing any thinking. Why anyone would think Spinoza's views imply fundamental particles think beats me. Now the beauty of the laws they obey suggest there may be a sort of Platonic realm that really determines their behavior - Roger Penrose believes mathematics is the only reality. I do not although I once did. Like an individual cell can not think, perhaps the universe as a whole can exhibit properties like thinking. Spinoza has a whole dialectic on this with subtle definitions of substance etc. I am not that formal - I just think there is some organizing principle that manifests in things like Noethers beautiful theorem.

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Bill

Photons are not conserved. So your proof is experimentally disproved.

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Bill

I like to think of myself as a scientist, I am into Quantum Mechanics and am a Mentor on Physics Forums. I believe in the God Of Spinoza and many scientists do. If this is the kind of God being talked about here, I cant quite tell. But a study of things like Noether's Theorem are just so amazing its hard to escape the idea something deep is going on.

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Bill

Please don't make me laugh. I haven't read it but can say a few things up front.

First when the IPCC released their report they said it plainly - some scientists say we have as little as 12 years. Well a little statistical knowledge of the Central Limit Theorem and The Normal Distribution shows this means the vast majority do not. But what did the climate change alarmists hear - we have only 12 years. It makes you wonder - it really does.

Secondly there is a tacit assumption - namely if catastrophe does occur we cant do anything about it. There are many engineers working on the problem that disagree - but that is generally not talked about.

Climate change is real, but a doomsday scenario it is not.

As a positive huge strides are being made in Fusion power and it is now thought it could be here about 2030. That will be a massive game changer.

Another thing to notice about the IPCC is the use of so called grey literature which is non peer reviewed literature. Anybody that knows anything about science knows that is a no no that cant be covered up by the public face of the IPCC that when pressed to justify what they say simply resort to - we are not scientists - we only report what they say. As far as I am concerned, while climate change is not a hoax, the way its reported to the public creates a lot of irrational alarmism.

So stop worrying, learn some basic statistical theory (you will be surprised at the misconceptions just doing that resolves) and spend a bit of time keeping up with the progress being made in fighting the global warming that is actually occurring.

Thanks
Bill