FWIW, catching up on old threads.

Penny, if Einstein was a mathematician so am I. I meant the ability to research math including theorems. Presumably this was his reasons to work with Grossman. For the rest, see my previous comment.

Einstein, as a teen, also was wondering what the world might look like if he could ride a light bean–a thought experiment,
NOT an experiment.

The most brilliant minds are conceptual–they have deep
IDEAS. Often, long before any experiment shows the necessity
of those ideas.

If you read his paper on SR ” On the electrodynamics of moving bodies”, you will see that his math proofs are very elegant–and similarly in his work on Brownian motion. He claimed he had NEVER heard of Brownian motion until he had submitted the paper. Similarly, for the Michaelson-Morley experiment.

His main ideas “SR and GR” were based on group theory–
the Lorentz group, and the diffeomorphism group–with a lot
of differential geometry in GR.

The motivation for SR originally (aka before the paper, according to Einstein) was that Maxwell’s equations were not
invariant under Galilean transformation–if one looked at
a moving conductor in a magnetic field in two reference frames the MATH predicted incompatible answers.
NO EXPERIMENT NEEDED.

His principle in GR was that fundamental physics was basically….
Geometry.

And, indeed, Hilbert showed that if one extremized the simplest possible geometric energy invariant –the integral of the scalar curvature–one gets the Vacuum Einstein Equations.
Thus, this would have happened anyway as mathematicians were exploring such things. In fact, there is an eight day difference ( and a priority conflict–which I hold in favor of Einstein) in publication.

Adding the stress energy tensor to get a divergence free
right hand side if you are not in a vacuum was a standard thing–even then–already familar from Maxwell, and
in solid and fluid dynamics.

When asked if he would wait up for the results of the Eddington eclipse experiment he replied, ” If they UNDERSTOOD my theory, they wouldn’t need to do an
experiment”.

The concept of Space-time ( sometimes called Lorentz geometry) was actually due to Einstein’s math prof
Herman Mink. No experiment was needed to motivate this.

The theory of causality in Lorentz geometry–largely due to Penrose ( who is a phd in math, a student of the same thesis advisor–Hodge–as Michael Atiyah) is another triumph of the math approach to physics.

Similarly for Penrose’s theory of Twistors ( Which predated him in pure algebraic geometry–a Robinson congruence is simple the Hopf Fibration), was not motivated by experiments.

And, thus we see again that :

Some physicists have done as you say, but some have not.
When, Feynman was working on a version of the four vertex
theory, it didn’t match the experimental data—like Newton he
said: ” The experimental data must be wrong, because my theory is too mathematically elegant to be wrong.” Feynman was incidentally a math Putnam scholarship winner!

Consider the Penrose singularity theorem, the Hawking Penrose singularity theorem, the work on Black hole area
thermodynamics–no experiments motivated this work.

The string theorists were motivated purely by math–in the spirit of Einstein–and it all started with an addition formula
for Beta functions.

We didn’t get Black holes, neutron stars, Einstein rings,
from experiments. The MATH came first–the experiments came later–often decades later.

In the same way, Maxwell’s equations ( yes, Faraday’s experiments had input–but the main idea was Maxwell’s pure thought idea of the displacement current–missed by Gauss etc.) PREDICTED Radio waves decades before they were observed. Maxwell, the ….mathematician.

In Quantum theory consider for example ” The Casimir effect”, which was predicted mathematically by the physicist
( who did enough rigorous pure group theory, like Eugene Wigner, to have an independent stellar career in math–geniuses!) Casimir decades before any experimental
confirmation.

One might say the same for the currently popular theory of
“wormholes”.

In the end, if several experiments disagree with predictions of math, the theory is discarded–as well it should be.

But, many of the absolutely top level theorists, are far more motivated by mathematical ideas than our education process
in physics makes clear. In fact, history is rewritten to make it look like the opposite is true in textbooks.

For example–Faraday is portrayed as a mathematical incompetent who didn’t know calculus and was primarly a tinkerer–meanwhile, Faraday invented a theory of flux tubes that contained many of the ideas of topology and led him to
conjecture the radio wave ( proved more standardly by Maxwell’s partial differential equations method). The radio wave was math before it was experiment.

p.s. Einstein himself always was falsely modest–saying he wasn’t much of a mathematician. His letter from his
Realgymnasium teacher said he was a “mathematical
genius”, his work in the statistical mechanics of Brownian
motion etc., say he was a very talented mathematician–real proofs.

He was also a large believer in the use of ” thought experiments”–another name for ” elegant mathematical
proofs”. In this, he was inspired by Galileo–who faked his
experimental data–but, as a mathematican–trusted his proofs.

In the “false” history of textbooks, Galileo dropped a heavy and a light object from the leaning tower to show that they fall at the same rate in a vacuum. In fact, Galileo gave a mathematical proof–a thought experiment–involving an imaginary line dividing an object into heavy and light objects.
He never dropped things from the leaning tower.

The impossibility of a pepetual motion engine was proved by
Simon Stevin–a medieval mathematician–by a geometric proof involving a weighted chain and an inclined plane.
Stevin was also led by this to the concept of momentum.

I have given plenty of examples to support my point. You have simply pontificated and reiterated yours.

Enough. It’s been fun.

This was however a minor nitpick, which won’t lead to an agreement in any case. But I accept your (or mine argument against string theory.

As I said: I gave examples: The work of Newton, Einstein, Dirac, Hawking,
etc.

I agree, this has been fun, but I have said enough.

Penny

String theory is not the only natural road to supersymmetry. Again, if string theory does work–aka, some prediction will made and tested–that it
would be YET another example of the mathematical approach to physics–for it has sustained its creation, and thirty to forty years of work,
by thousands of “Physicists” who have been willing to work on mathematical theories without having ANY experimental justification to do so. They follow Einstein’s philosophical lead in this.

And, since I have some small piece in it. It would please me, if string theory were correct.

As I said, my argument against it is basically your “physicist” argument–that it makes no solid experimental predictions.

Supersymmetry is independent of string theory–though picked up later by string theorists.

Yes, but AFAIU it gets a more thorough and natural physical motivation there.

The black hole entropy is a retrodiction

My point was that it will be a prediction if confirmed by observation. I see that is your criteria as well.

The method of physics lets physicists cut corners when they can do experiments–but, they can’t always–consider string theory.

There will be no difference there, when they get observations.

Lots of mathematicians have played around in labs.

Yet they want to use math instead of physics methods in physics. (Gauss made it work, which is why I give him kudos.) Which remains my point.

I’m sorry, but I don’t see that we can come to an agreement here.

It is obvious to me that one should use physics methods in physics, since in science math is a tool but not the essence of the observational method or of all of theory. Others may not agree, which is fine with me but when they need to demonstrate that alternate methods works. That is all I have to say.

Einstein:
The theory of Brownian Motion by kinetic molecular methods
The theory of the photoelectric effect
Special Rel
General Rel
Bose-Einstein Stats
A correct unification of Electric fields and Gravity ( different from the one
done by O.Klein) using torsion connections and teleparallel geometry
The first paper on “amplification by stimulated emission of radiation”

GOOD GOD!!!!

Thinking of Fermi–I am also very impressed. By Einstein just blows my mind.

The math in string theory, as used by mathematicians, is pretty fancy—for example my coauthored paper uses geometric measure theory, Global Calculus of Variations, Differential Geometry, The topology of fiber bundles,
Sobolev Spaces, Harmonic Analysis. And, this is just for the classical version of the non–supersymmetric theory.

People like Yau ( who is a bonefide mathematical Genius and Fields laureate) do even fancier math related to String Theory.

My argument with String Theory isn’t a math one. It is that no
decent predictions have been made—or verified in this theory yet.

So, there I am agreeing with your criterion.

Parsimony is one of the ingredients of mathematical elegance.

Gauss was a poor kid, one of the first to get a scholarship to support his
education. But, that ran out when he finished his Phd, and the way he chose to get some kind of job was to calculate the orbit of Ceres using
celestial mechanics. Along the way, he had some jobs doing surveying.
They, he got a job as director of the Gottingen Observatory— a minor, poorly paid job that allowed him to sleep in the unheated observatory.
As soon as proved some more major theorems, he got a decent job as a professor of pure math.

His major interest was number theory–which he called the “Queen of the Sciences”.

Lots of mathematicians have played around in labs. As a young teen, I built a betatron, and a Gas laser, and an interferometer etc.–and many similar toys, mostly from the Scientific American
Amateur Scientist column.

When I was eight or nine my birthday present was bottles of Meth. Blue,
and Wright’s stain.

I built reflecting telescopes, and ground mirrors.

Then we discover our math ability, and we change directions. It’s a bit like being possessed. And that is ok.

Some physicists do both theory and experiment too–few nowadays. But
Fermi is a good example. Fermi–interestingly enough–started out
wanting to be a mathematician–and his first publication was on differential geometry–Fermi Walker coordinates.

The black hole entropy is a retrodiction–the formula was already known from GR work ( Beckenstein, Hawking, Yau).

When Einstein did GR, he predicted important effects BEFORE they were observed. That is the difference between science and curve fitting. GR predicted gravitational time dilation, Greater light deflection by gravity,
Gravitational lensing, Einstein Rings, and Black holes all BEFORE they were observed.

The method of physics lets physicists cut corners when they can do experiments–but, they can’t always–consider string theory.

Gauss would have been insulted to be called a physicist.

That doesn’t matter. What matters is that he, among other mathematicians of course, have actually made physical observations (regarding Earth size IIRC).

Not bad extracurricular activity for such a giant among mathematicians!

the defining difference is that mathematicians prove rigorous theorems.

I don’t agree. The defining difference is that scientist use the scientific method of observation and theory.

There is a flaky calculation to get the black hole entropy out of string theory–but…

With respect; if they predict the entropy, with whatever technical procedure that works with their theory, and it is tested as correct it is valid science, if not valid math. [Actually, I would argue that if it is algorithmically sound it is math of a kind, albeit possibly not yet axiomatic math as AFAIU second quantization is not. But I'm no mathematician.] That is all we need.

It can’t be all flaky, as AFAIU they get the same entropy as semiclassical derivations. Now remains the actual test against observation…

It is many of today’s physicists who cut lots of corners.

Sure, and as the method lets them, what is the harm? Physics doesn’t equal math, and there is no reason to believe everything can be formalized. No such expectations exist in biology, yet everyone agrees that it is testable science.

It would nice to do an experiment at the next few decimal places to verify that the Cartan-Einstein torsion connection theory is what replaces GR at that level of accuracy.

Sure, GR is an effective theory. But surely string theory the strongest contender? It predicts a lot, inclusively that pesky black hole entropy, supersymmetry, and what not. And I hear the math is hard as well.

If experimental accuracy were the main measure, it would have been enough. In the same way, do virtual particles really exist?

I suspect you know very well that theories are constrained by other conditions put on them, until the best contender can be picked among those who passes the most critical set of tests.

Parsimony is often chosen for reasons of likelier to be correct, less number of reversals if wrong, as a part of “beauty” et cetera. And here it explains why epicycles went out of fashion, I believe.

Btw, isn’t this an argument against math as the only measure of science? Epicycles or ellipsoids, equally valid math objects. But I see that you use “mathematical elegance” yourself.

Interestingly, his book on Natural Philosophy “Principia Mathematica N” NEVER uses any calculus–except the implicit use of limits.

Thanks for the math history, it is interesting. (And needed, obviously.)

It neglects to mention that Newton also invented the Newton Polygon method for summing fractional series, and the calculus of variations.

As to “G-invariance”–the language wasn’t there, but if you read the Principia–he does discuss the issue, and he even discusses the issue of universal reference frames–choosing reluctantly to use the fixed stars as
a stand-in for a fixed reference frame.

Interestingly, his book on Natural Philosophy “Principia Mathematica N”
NEVER uses any calculus–except the implicit use of limits. All of the book is written in the Language of synthetic Euclidean Geometry.

Newton published Calculus later, after he was concerned about priority conflicts.

You might also be interested to know that Newton’s prof–Issac Barrow
already knew the concept of the differential triangle, and that calculus
was being done –before Newton–in the Indian State of Kerela–which was a major spice trader with Europe. This can be found in the Cambridge University Press book ” The Crest of the Peacock”.

Indeed, this used to be done. Does that mean that it is good physics? Do angels pull the planets? Does the high accuracy prove that Angels exist?
If experimental accuracy were the main measure, it would have been enough. In the same way, do virtual particles really exist?

What it really is–in the epicycle case– is the fitting of data to a Fourier series.

Newton replaced angels by the mysterious inverse square law–a law imposed–he thought–by God. What makes Newton’s method work is
differential equations. That is why it is better. One differential equation
–One constant to measure ( G), and that’s it.

His experimental motivation was that Kepler had deduced–based on experimental observations of Tycho Brahe that the planets move in ellipses—Kepler was a mathematician, by the way.

Newton solved the math problem of finding the right central force.

It’s not about curve fitting and accuracy–it is about good math. Good math should NOT require that every prediction be confirmed by an experiment. Newton only need one match–elliptical orbits.

Before the Bohr atom, there was a theory of Nicholson–that was far superior–it predicted spectral lines accurately for many complex atoms.
It even included Bohr’s predictions for the two simple atoms for which he could get results.

It is on the dust heap–because Bohr’s elegant mathematical approach
( which was already implicit in Nicholson ( cited by Bohr)) was better
math.

Indeed, Nicholson’s work was called ” fitting epicycles” by his detractors.

I just disagree with you–in the case of very expensive experiments in a
field ( GR) that has never had a single experiment that disproved its predictions. If we had gobs of money–of course, we should do the experiment. But, we don’t.

Newton, Maxwell, Einstein etc, may well have been wrong about their approach–but I think their successful results in physics give some support for their point of view.

p.s. It would nice to do an experiment at the next few decimal places to verify that the Cartan-Einstein torsion connection theory is what replaces
GR at that level of accuracy. It’s my math guess ( and Cartan’s–who was the greatest geometer of his age) that it does. If I had to do a billion dollar GR experiment–that would be the one.

When he was young, he was employed as an astronomer–which lasted
until he proved a few more theorems. His Phd degree was in mathematics.

If proving theorems about physics makes you a physicist–then,
all the mathematicians that I mentioned would also be physicists–heavens, even I would be a physicist.

But, the defining difference is that mathematicians prove rigorous
theorems.

Oddly, that would –for me–qualify Einstein as a mathematician, because in special relativity he proved rigorous theorems. He didn’t think of himself as a mathematician-but worked with Grossman and many other mathematicians during his career. Einstein’s degree was in physics.

Quantization by the mathematical method of geometric quantization works just fine. It is quantum field theory that is a problem–that is because QED-as accurate as it is, is mathematically inconsistent–the only QFT satisfying the Whitman axioms in four dimensions is the free scalar field.

Math needs to done there—or better axioms found.

Oh, and it turns out that Brian Greene has a joint appointment in math and physics.

LQG, I agree. What has string theory actually predicted? There is a flaky
calculation to get the black hole entropy out of string theory–but…

If you read my comments carefully, you will see that I said that the
mathematicians that I named –who contributed to physics–didn’t cut corners because they were doing their work with a mathematicians perspective–aka Newton’s “natural philosophy”. It is many of today’s physicists
who cut lots of corners.

( Of course, rigor in math has become greater in the last two centuries, then it was in the days of Gauss. Still, Gauss was VERY interested in rigorous proof, and was excellent at that by the highest standards of his day.)

One could say the same of Maxwell.

@ penny:

Newton did realize that his equations were Galilean invariant–in fact that was part of his construction of the equations.

No, his theory (and so his equations) is galilei invariant – but he didn’t present his theory that way, as you yourself note.

He THEN, after he had invented his calculus, applied it to Natural Philosophy–what some may call physics.

Naturally, he needed it for the later. Ah, I see the history is that he invented the calculus early, published his physics later, and finished with publishing his calculus last. The last part sounds familiar, in fact.

However, Newton’s definition of Natural Philosophy was what we today call rigorous applied math–all about theorems.

Yes, but he was wrong. Science is empirical, and theories are tested. As for example when special relativity predicts what classical mechanics cannot.

Brian Greene is not a mathematician,

Which is why his opinion on physics science is informed. With respect, I find yours is not – for example when you complain that testable theories can “cut corners”. They are still right if they predict the observations. Formal theories aren’t everything, for example there is AFAIU no mathematically satisfying theory describing quantization, yet it works in physics theories.

Physics isn’t math. Get used to it.

PS. I find it amusing that you first object to physicists cutting corners, then claim that they are bona fide mathematicians. Gauss was both mathematician and hands on physicist. Another unique individual.

PPS. Let us forget LQG, as it can’t predict a harmonic oscillator or become lorentz invariant. But at least now I see where you come from.

I left out the OUT.

Sorry
Penny

It is a common misconception ( often taught in elementary physics classes by people with no historical background) that Newton invented
calculus to clarify his physics. In fact, he was a professor of Mathematics,
had done serious work on the classification of singularities of polynomial equations in several variables ( today that would be called Algebraic Geometry), learned about the differential triangle from his teacher–the pure mathematician and geometer Issac Barrow. He THEN, after he had invented
his calculus, applied it to Natural Philosophy–what some may call physics.

However, Newton’s definition of Natural Philosophy was what we today
call rigorous applied math–all about theorems.
The opening of his Principia ( translated from Latin),

” I, Issac Newton, Lucsian Professor of MATHEMATICS at Cambridge,
do here, with recourse to theology, ….. , and purely by the method
of GEOMETRICAL proof here set forth the system of the foundation of the world.”

( Newton did start his career with his experimental researches on the nature of color and prismed light, but his job was mathematician–just like Galileo–and he published quite a lot in pure mathematics). Look up Newton Polygon, for example.

Maxwell stated the invariance of the speed of light in his work,
and indeed derived a formula for that speed in terms of the electric and magnetic constants of the vacuum. He also was disturbed by the change of group invariance implied–but, it took a decade or two for others to
work hard on that problem.

Brian Greene is not a mathematician, and String Theory is dicey. That doesn’t mean that the math approach to physics is doomed or that “all the low hanging fruit are picked”. During the last century, Gauge
theory was formulated by theoreticians ( and mathematicians had already
looked at these equations in a classical setting in studying the theory of connections on fiber bundles) leading to electroweak unification, very serious advances in
General Relativity have been done by mathematicians, similarly in
optics, fluid mechanics etc, not to mention the theory of superconductors, and superfluids, and many other things.

Some mathematicians are also looking –in our slow plodding careful way–at
String theory.

A minor example: String Theory is the quantization of a classical theory
that uses the area of a low dimensional surface in a high dimensional space as an energy.

Already at the classical level there are serious issues about existence, and singularities. There is a math paper that studies that, and gets
careful results –in the general setting of fiber bundles—which, as in Gauge theory–is really the right setting.

I am a little happy about that paper: ” Regularity for Area Minimizing Rectifiable Sections
of Fiber Bundles”, as I am one of the two coauthors. There is a slight error in the arxiv version ( in the differential geometry section of arxiv) that is corrected in the published version.

It’s part of a short series of papers that took us maybe 17 years.

And, I didn’t even know it was related to String Theory until 16 years into the process!! That’s math for you.

( I discovered the relation–not mentioned in our paper–when I was asked to tutor someone in string theory–and went through a nice
set of notes from MIT.)

Not so flashy as splash covers in popular magazines about some
theory of everything by String Theorists.

In String theory, as done by physicists, most of the work is not about rigorous mathematical
theorems, and rigor corners are cut constantly. Thousands of papers were written at a breakneck pace. PERHAPS, the LESSON
here is that Newton had it right: We should be doing NATURAL PHILOSOPHY–with extreme care to prove rigorous theorems about nature. It’s slower, but it works.

Math research was never easy–even in Maxwell’s or Newton’s day.
That is why these people were geniuses.

Easy pickings? Gee, I have spent as much as 14-17 years working on a proof of a single theorem, and this is not uncommon in math. Newton spent 20+ years writing the Principia to try to get the math theorems correct, and he almost got it all correct. Maxwell had several Psychological exhaustion breakdowns–one after his work on the rings of Saturn. It was never easy. It fact, it is hard as hell. Even for those people: who are once in a century geniuses.

Penny

p.s. It is interesting to note how much of the great work in theoretical physics was done by people with math degrees and/or holding math professorships–who physics teachers often call physicists.

Some examples: Euler, Newton, Lagrange, Maxwell, Lord Kelvin,
Dirac ( who was also Lucsian Professor of maths with a maths Degree)… I include Max
Born, who was Hilbert’s postdoc assistant, Hilbert, Wiener, Pascal
Jordan, Hawking.

The only person to ever win two Nobel Prizes in physics was John Bardeen, who had a pure math Phd from Princeton.

Einstein wasn’t, but his first papers on his general theory of relativity–which had most of the ideas–were written with a co-author who was a mathematician–M. Grossman.

Pauli and Heisenberg were physics posdocs with Born, and not only were connected to Hilbert, but were disciples of the mathematician Sommerfield, who had great influence of the developement of QM.

p.s. Let us not forget that string theory has competitor theories such as loop quantum gravity.

Newton loved experiments–he invented the reflecting telescope–but he knew the power of math.

I’m not especially well versed in the history of science, but AFAIU Newton invented his infinitesimals to be able to describe his physics. Seems to me the above is putting the cart before the horse as earlier.

Did we really need a billion dollar experiment to check a simple prediction of his equations?-”frame dragging” aka non-diagonal metric terms.

That was debated AFAIU, but in truth every prediction of a theory is an opportunity for falsification and so further progress. As things stands now, there is a scarcity of empirical tests to further theoretical physics. Many theoreticians argue AFAIU that they need more data, not less.

The thing is, that many theoreticians (and others) have reasoned as penny does, after Einstein. But the absence of success with such an approach is evident. Even Einstein himself proved that, he didn’t believe in the stochastic nature of QM in spite of his early work on it.

Other theoreticians (IIRC; Brian Greene for example) have concluded for good reasons that this was a one off moment in history.

After all, Newton could have realized that his mechanics really implied galilean invariance. (Perhaps he would have if he hadn’t been so motivated to argue for a separation between objects and space that he had to make the latter “absolute” to get this point through.) Likewise, Maxwell could have realized that light speed should be a theoretical constant, implying lorentzian invariance.

These low hanging fruits remained; but today there are no easy pickings. To believe so is in all likelihood to chase a pipe dream.

The beginning of the movie is about the waste of war.
The middle is about the great engineering projects done with resources
freed from war.
The end is about the conflict between comfort and safety and the risks
of space exploration.

The book is better because it is clearer there that the technocrat leaders in the future are also repressive and authoritarian. Life is rarely one dimensional.

Got to go and do some math research.

This was all said very nicely by H.G. Wells, in his screenplay
“Things to Come”–based on his ” The Shape of Things to Come”
It is a great 1930′s movie–which is at the internet archive.

Get some rest CR, I very much enjoyed our conversation.

Same to everyone else, even Martin.