I Know the Math Says so, but Is It Really True?

[PeterDonis]

the people who actually fund and hire scientists cannot evaluate the fruits of their investment?

In many cases, no, they can't. Many people who fund and hire scientists have no idea whether or not the science those scientists are doing makes accurate predictions. They are not funding the scientists to get accurate predictions from them; they are funding the scientists for other reasons, such as prestige or politics.

Good science journalists must have PhDs in physics to knowledgably write on the topic?

No. I have nowhere claimed that having a PhD is either necessary or sufficient for having an informed opinion about a scientific theory, nor has anyone else in this thread. You are attacking a straw man.

That said, I think it is true that most science journalists do not have informed opinions on the topics they write about; they are just accepting the word of their favorite scientists as authoritative.

 

[MikeeMiracle]

Going back to the original post I have a slightly different viewpoint. As someone who doesn't understand the maths, I am perfectly happy accepting that I cannot have a "full picture" understanding of the events at hand without a detailed maths knowledge. What I try and do is compile information from various sources, learning concepts from pop-sci videos (PBS Spacetime / Science Asylum type level)and then reading about them on PF to fill out the picture as much as possible which as worked very well for me. In my opinion I feel I have a "better than most" people's understand of concepts based on the "plain english" explanations.

What frustrates me are the posts where we have people asking questions on concepts. They appear to struggle to understand the concepts so they receive replies stating that "if they knew the maths it would make sense" only for them to reply that "they have studied and understand the maths, can perform the required calculations"...but struggle with the concepts still.

I would love to get a grip on the maths behind the concepts and feel some envy that they understand the maths where I do not yet they don't seem to understand the concepts.

I just don't get it!

 

[bhobba]

I just don't get it!

Susskind is doing his best with the Theoretical Minimum series of books. He has done 3 already - but none for a while. I do hope he finds the time to finish them. Here the general public will find a correct account, but they must be willing to put in the effort to understand a bit of calculus. At the end though they will understand things far better than the usual pop-sci accounts. As to what the concepts mean, when using the language of math, you will find most of the time that is all that is needed. Going beyond that can, and often is, hard.

Thanks
Bill

 

[BWV]

How many hours of study does it take to become proficient in, say, QFT at a professional level? Thinking one can casually learn these topics is like believing that tinkering on the piano a few hours a day can make you a concert pianist. I watched the Susskind Stanford videos years back and they were great, but after that, I did not 'know' these topics, but I understood better what I did not know

 

[gmax137]

Thinking one can casually learn these topics is like believing that tinkering on the piano a few hours a day can make you a concert pianist.

Well that's an interesting observation. I have enjoyed music my whole life without ever playing piano, guitar, or anything. I believe musicians appreciate some of the details more than I do. But, nobody is promoting the idea that concert-going is a waste of time for non-musicians.

 

[BWV]

Well that's an interesting observation. I have enjoyed music my whole life without ever playing piano, guitar, or anything. I believe musicians appreciate some of the details more than I do. But, nobody is promoting the idea that concert-going is a waste of time for non-musicians.

Indeed, the analogy is the more you learn about music as a listener or amateur musician, the more appreciation you have for people who do it at a very high level

 

[bhobba]

How many hours of study does it take to become proficient in, say, QFT at a professional level?

You mean after you have done something like Susskind? Well he claims to take you to a level where you at least understand the jokes. To take you to more advanced QM that would be about 1 university course which is about 150 hours. Then 2 subjects on QFT or a further 300 hours. But it depends on how hard you find Susskind. If it was about average difficulty for you then I would say that 450 hours is reasonable. Let's be generous about this and you study about an hours per day with weekends free - then I would say about 2 years.

But if you are interested in professional level understanding you will be better going to university for a graduate degree like the following:
https://my.uq.edu.au/programs-courses/plan.html?acad_plan=PHYSCX2321

That will also take about 2 years part time - but with about 2-3 hours study each night.

Thanks
Bill

 

[BWV]

[

You mean after you have done something like Susskind? Well he claims to take you to a level where you at least understand the jokes. To take you to more advanced QM that would be about 1 university course which is about 150 hours. Then 2 subjects on QFT or a further 300 hours. But it depends on how hard you find Susskind. If it was about average difficulty for you then I would say that 450 hours is reasonable. Let's be generous about this and you study about an hours per day with weekends free - then I would say about 2 years.

But if you are interested in professional level understanding you will be better going to university for a graduate degree like the following:
https://my.uq.edu.au/programs-courses/plan.html?acad_plan=PHYSCX2321

That will also take about 2 years part time - but with about 2-3 hours study each night.

Thanks
Bill

but that would assume a good foundation in basic physics and math, if your only introduction to classical mechanics was Susskind, that would likely not be a good enough foundation. Also there is a large gap between understanding the math conceptually, which is really what Susskind requires, vs being proficient at using the math to solve problems. Then once you get there, its ‘use it or lose it‘ otherwise you will just forget it all

 

[bhobba]

but that would assume a good foundation in basic physics and math, if your only introduction to classical mechanics was Susskind, that would likely not be a good enough foundation. Also there is a large gap between understanding the math conceptually, which is really what Susskind requires, vs being proficient at using the math to solve problems. Then once you get there, its ‘use it or lose it‘ otherwise you will just forget it all

Susskind covers a lot more than just classical mechanics. Of course it depends on your preparation and what you recall. If you just read Susskings 3 books then yes I think one hour each night for 2 years is enough. If you did a degree with not much math, or do not remember much, then a prior diploma in math would be of value:
https://my.uq.edu.au/programs-courses/plan.html?acad_plan=MATHEX2321

That would take it to 3-4 years. But you are asking for professional level knowledge. Really is that want you want - or just to get the jokes as Susskind says. That's all the general citizen really needs.

Thanks
Bill

 

[swampwiz]

I had thought that someone falling into black hole would have time being so dilated from the mass that he would observe the universe evolving very rapidly, even to the point of seeing stars form, go into giant phase, supernova, etc. very quickly; OTOH, anyone observing him would observe his watch as almost not ticking at all.

 

[PeterDonis]

I had thought that someone falling into black hole would have time being so dilated from the mass that he would observe the universe evolving very rapidly, even to the point of seeing stars form, go into giant phase, supernova, etc. very quickly

No, this is not correct. An observer hovering at a constant altitude above the hole's horizon, not falling in, will see what you describe. But an observer free-falling into the hole will actually see the outside universe redshifted, not blueshifted, with the redshift increasing as he falls.

OTOH, anyone observing him would observe his watch as almost not ticking at all.

This is correct (at least as long as the observer is outside the horizon, so light signals from him can get back out to distant observers). But that is because, for an observer free-falling into the hole, the time dilation between him and an observer who is far away is symmetric, like time dilation in SR--each observer sees the other's clock running slow. This is a key difference between this case and the case of a hovering observer.

 

[LieToMe]

Why are people making this weird assumption that "math" says anything? People are the ones making arguments. No matter what you write it will only ever be an approximation of relative observations. The only reason we take an interest in it is for practical applications in being able to predict the evolution of systems to some arbitrary degree of certainty in some arbitrary consideration of variables.

People happen to choose some level of tolerance in the statistical correlation of their measurements and there happens to be a relativistic space-time model with the highest correlation to patterns we observe in astronomical objects. According to our best understanding of the standard model, it is astronomically unlikely that the pattern of observations can be explained by a capacity to measure "infinite" time, so the point was moot to begin with, just like asking for the frame of reference of a photon.

 

[romsofia]

Why are people making this weird assumption that "math" says anything?

Why do people make this weird assumption that "English" says anything? People are the ones who give words meaning. No matter what you write it will only ever be an approximation of how society interprets it. The only reason we take an interest in it is so for practical applications in being able to communicate with others to some arbitrary degree of certainty in some arbitrary consideration of intent.

People happen to choose some level of tolerance in the statistical correlation of their definitions, and there happens to be a book with the highest amount of definition to words we observe in society. According to our best understanding on English, one cannot define "the point of LieToMe's post" so the point was moot to begin with, just like asking for the definition of the word "obagooba".

 

[LieToMe]

Why do people make this weird assumption that "English" says anything? People are the ones who give words meaning. No matter what you write it will only ever be an approximation of how society interprets it. The only reason we take an interest in it is so for practical applications in being able to communicate with others to some arbitrary degree of certainty in some arbitrary consideration of intent.

That's something I've actually found to be true. Even now, all we have are statistical measurements, so to a certain degree, you can't be entirely certain about anything. That inability to be entirely certain is how scientific models appear to perpetually evolve.

You can make an argument that logic is independent of observable results, but then such an assumption would only be relative to an individual's construct of reality as a reiteration of past interactions.

 

[Imager]

As one of the people who haven't taken the time to understand the math, I think Peter's points are correct. When I first came to this site, one of my big problem was not getting simple answers like "Youtube" videos.

My other big problem was terminology. For example, using the term universe when they mean observable universe. It took me a while to figure out that those where different things. It's hard for us to ask the right questions, when we don't even understand the differences in terminology.

What makes this forum Great is that people like @PeterDonis, @Dale and many others showing massive amounts patience to walk people like me through the basic concepts!

 

[joeh3rd]

The mathematical models you refer to in other disciplines are still subject to the same test as mathematical models in physics: either they make predictions that match the data, or they don't. Models that don't make predictions at all aren't the kind of "math" I am talking about in the article.

Also, your post implies that mathematical models in physics don't have the characteristics you describe--simplifying complex processes, modeling domains where inputs are not fully knowable. That is quite wrong. There are plenty of domains in physics where the same issues arise. In fact, it's hard to find a domain even in physics where those issues don't arise.

Perhaps there is an example of predictive models containing non-existent or at least non-intuitive features. In electrical engineering Fourier transforms and convolution are used extensively in radio transmitter and receiver design. The heterodyning used for frequency shifting of modulated signals up for transmission and down for audio routinely use the negative frequency component of frequency shifting. I don’t believe that there have been tests for the existence of such however in application they are definitely present as symmetrical side bands of upshifted modulation.
Here we have a mathematical construct that emerges in transformation from the time domain to the frequency domain and back again that works mathematically but has a seemingly non-detectable real world component. Are we sometimes taking a mathematical model that when used may have non-existent internal characteristics yet when used in a series of operations yields measurable results and mistakenly assuming it is what occurs in reality?
Perhaps “virtual particles” are the same just as for modeling purposes we assume the atomic model and things called particles which may be side-effects of our means of measurement and perception — no tiny balls that behave similar to things we perceive by immediate sight and other senses on a much larger scale?

The positive frequency side of a modulated signal. More properly shown, it is symmetric when along the zero frequency axis.

The result when frequency shifted by multiplying with a sinusoid centered at 30kHz. Note the formerly negative frequency lower side and.

 

[PeterDonis]

I don’t believe that there have been tests for the existence of such however in application they are definitely present as symmetrical side bands of upshifted modulation.

You're contradicting yourself. First you say there are no tests for the existence of such, then you say they are definitely present. They can only be definitely present if we have a way of testing for their presence, and the test says they are. If the test is "symmetrical side bands of upshifted modulation", then if those things are present, "such" is present.

Are we sometimes taking a mathematical model that when used may have non-existent internal characteristics yet when used in a series of operations yields measurable results and mistakenly assuming it is what occurs in reality?

In order to even ask this question, you must assume that you can somehow distinguish "non-existent internal characteristics" from internal characteristics that aren't "non-existent". But that would require some kind of experimental test for such characteristics, and if such a test exists, the characteristics aren't "internal" to begin with.

 

[Thelamon]

Continue reading...

Makes me think of what Feynman sais here:



(Of course there are theoretical physicists and mathematical physicists though. A lot of physics can be explained by using just words, but none to apply it.)

 

[Thelamon]

Makes me think of what Feynman sais here:



(Of course there are theoretical physicists and mathematical physicists though. A lot of physics can be explained by using just words, but none to apply it.)

I meant this as a reaction to the article, I didn't knew there was a whole conversation about it where this comes from out of nowhere I image. (Thought I might had to clarify that just in case.)

 

[zdcyclops]

Continue reading...

Interesting.
Einstein is reported as having once said " If you can't explain it to a six year old, you don't understand it." Mathematics is no more special than any other creation of the human imagination.
The universe does not calculate, does not use mathematics, yet everything seems to work quite well without it.

 

[Dale]

Einstein is reported as having once said " If you can't explain it to a six year old, you don't understand it."

It is probably misattributed: http://en.wikiquote.org/wiki/Albert_Einstein#Misattributed

But even if it were correctly attributed it is wrong. Teaching a concept clearly and understanding it are two different skills

 

[strangerep]

The universe does not calculate, does not use mathematics, yet everything seems to work quite well without it.

Heh, the universe is one big analog computer.

 

[Klystron]

Mathematics is no more special than any other creation of the human imagination.

Arguably incorrect.

Tinker Bell is a special creation of the human imagination. As much as I like Tink and clapped my hands hard so she would get well, mathematics remains far more useful and special.

The universe does not calculate, does not use mathematics, yet everything seems to work quite well without it.

Of several logical fallacies inherent in this statement, an existential 'least' refutation may be simplest.

1. Human beings exist in (are part of) the universe.
2. Humans count, calculate and use mathematics.
3. Therefore, (part of) the universe calculates and uses mathematics.

 

[HAYAO]

Excellent article.

Yeah, I was on the both ends of this. When I was young and had no good mathematical intuition of a physical phenomenon, I was really frustrated. Now, as an assistant professor, I get frustrated when people don't understand the mathematical representation of physical phenomena.

I still do think that almost all, if not all, physical theories are based on certain postulates. These postulates are based merely on physical intuition, although a good one. For example, one of the postulates of special relativity is that speed of light is constant in vacuum. No one can really "prove" this, but it is based on a good reasoning and observation if anything else. And since the theory of special relativity reflects the real world and it works, we "assume" that the postulates are correct. Similar thing for wave representation of particles. We assume that the particle behaves like a probability (amplitude) wave function, but the theory based on this actually works and reflect what we'd expect to observe. The more we do experiments and the more we have stronger mathematics to dig deeper into this, the more it reinforces the theory or provide a more generalized theory that encompasses all the other theories (e.g. QM and QFT).

In many cases, mathematical representation of physics is a well-thought-out reasonable modeling/interpretation of the observed phenomena. If any student is doubting the math, it's not necessarily because they don't agree with math, but because they don't agree with the modeling/interpretation, due to lack of intuition (which can be trained to a certain extent; it's why people learn to accept it).

 

[Hornbein]

The math for general relativity was already known. The difficulty was discovering and showing that it related to the real world.

 

[bhobba]

The math for general relativity was already known. The difficulty was discovering and showing that it related to the real world.

As Hilbert, who discovered the GR equations slightly before Einstein (by five days), said: "Every boy in the streets of Gottingen understands more about four-dimensional geometry than Einstein. Yet, despite that, Einstein did the work, not the mathematicians."

Thanks
Bill

 

[weirdoguy]

If you can't explain it to a six year old, you don't understand it.

As a teacher, I think that is one of the stupidest things I have ever heard in my whole life.

 

[Hornbein]

As Hilbert, who discovered the GR equations slightly before Einstein (by five days), said: "Every boy in the streets of Gottingen understands more about four-dimensional geometry than Einstein. Yet, despite that, Einstein did the work, not the mathematicians."

Thanks
Bill

Einstein and his collaborator (Rosen?) were stuck so he went to Gottingen for help. He was quite loathe to do this. He said Gottingen had a reputation for stealing the results of others. Nevertheless out of desperation he made the trip. He was informed that he was trying to do the impossible, that one of the conditions he thought essential actually wasn't, and he already had the answer. Hilbert published his solution, which greatly angered Albert. He got Hilbert to back down and recognize AE's priority. I imagine it helped that Albert had the backing of Berlin.

 

[bhobba]

He got Hilbert to back down and recognize AE's priority.

In public at least, and likely in private, Hilbert always gave Einstein credit for GR. Of greater interest was when Einstein, Hilbert and others discovered solutions that violated energy conservation. Both were stumped. But they knew one person they thought could tackle it - the great Emmy Noether. And her famous theorem was borne. It may even be a more important discovery than GR. IMHO, it was just a precursor to modern science, which is very collaborative. The idea of the lone genius that could revolutionise science was fast fading.

As Wigner said of Einstein:

'I have known a great many intelligent people in my life. I knew Planck, von Laue and Heisenberg. Paul Dirac was my brother-in-law; Leo Szilard and Edward Teller were among my closest friends; and Albert Einstein was a good friend. But none of them had a mind as quick and acute as 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 even than 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 extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity, and for all of von Neumann's brilliance, he never produced anything as original.'

Even the great Feynman, himself like Einstein beyond genius at the level of a magician, said knowing what Einstein did, he could not have invented Relativity. Ohanian has said such people are sleepwalkers. They did not know where they were going but were unerringly led there. Einstein was perhaps the greatest sleepwalker there ever was, except maybe for Newton.

Thanks
Bill

 

[Dale]

they knew one person they thought could tackle it - the great Emmy Noether. And her famous theorem was borne. It may even be a more important discovery than GR.

I agree 100%. IMO it is the single most important theorem in all of physics.

 

[PeterDonis]

Einstein and his collaborator (Rosen?)

Marcel Grossmann.

 

[fresh_42]

I agree 100%. IMO it is the single most important theorem in all of physics.

For anyone who is interested in the original paper (which can be translated if you use google's chrome, right-click + "Translate to English"):
https://de.wikisource.org/wiki/Invariante_Variationsprobleme

and here are the facsimile (pages 37 ff. and 235 ff.)
https://gdz.sub.uni-goettingen.de/id/PPN252457811_1918

 

[strangerep]

IMO [Noether's theorem] is the single most important theorem in all of physics.

Surely that can't be true.

(Try deriving Kepler's 3rd law using Noetherian symmetry/conservation techniques... )

 

[strangerep]

Marcel Grossmann.

Hmm. I thought it was just Einstein+Grossman who put together GR using Riemannian geometry (but I haven't read a full history). What exactly did they need help with from Gottingen?

 

[PeterDonis]

I thought it was just Einstein+Grossman who put together GR using Riemannian geometry

Einstein came up with the field equation (at least by one possible route--see below for another route that Hilbert took). Grossmann helped him to learn Riemannian geometry.

What exactly did they need help with from Gottingen?

Einstein was stuck regarding a particular aspect of the field equation. (It's been a while since I read up about this so I can't say off the top of my head exactly what aspect it was.) I don't know that his primary purpose in visiting Hilbert in Gottingen was to see if Hilbert could help him get unstuck, but it probably was at least in the back of his mind. I also don't know that the talks with Hilbert were the primary thing that got Einstein unstuck, although they might well have helped.

However, the greater impact of the Gottingen visit was not on Einstein but on Hilbert. After mulling over his talks with Einstein, Hilbert came up with a simple, quick route to the field equation for gravity using the principle of least action. He came up with what is now called the Einstein-Hilbert Lagrangian based on obvious and simple considerations, and then it was a simple matter to derive the Euler-Lagrange Equation for it, and boom! he had the field equation. As MTW say in their discussion of six different routes to the field equation, "no route to the field equation is quicker".