What caused the shift of interest in quantum cosmology?

[Sardano]

The physical reason is that Bojowaldian LQC is the first place anywhere in LQG-related stuff that something like the Einstein equations have appeared in a principled way - I can make a theory with one fairy so it's simple, and have it predict a bounce so it's testable, but it's unprincipled so no one will be interested in it.

Does anyone find it ironic that background independence was discarded for a hint of the Einstein equations to appear?

Great comments. I had never though about it, but now that you mention it, yes it is ironic. It evenis more ironic if you recall that "background independence" is one of the criticism to String Theory that the LQG people uses. It is useless to explain them that String Theory is background independient, but we just know how to "do String Theory" in some particular consistent backgrounds. Its like saying that General Relativity is not background independient because I am working with a concrete solution and trying to obtain physical information from it.

 

[Sardano]

Anyway, the tendency in both fields is the same: try to work in something feasible with as direct as possible physical implications. As a result the LQG people goes to LQC and the String Theorist tend to work in things like ADS/CFT and related.

 

[marcus]

...
Because if there is really a growth in the number of LQC papers is because the people that used to do LQG is now doing LQC, since LQG is obviously a dead end. You should compare your numbers with the number of LQG papers among the most cited ones. You will see that if LQC is going up, LQG is going down in more or less the same amount. I haven't checked, its a prediction ;) .

I think you must be joking, because this is so obviously wrong. Both sets of numbers are going up: Loop cosmology and also the other Loop gravity research (that is not about cosmology).

If you seriously meant that as a prediction, you can falsify it easily with a Spires search.

To get you started here is a search for Loop overall (including cosmo). Type or paste this into the search box at Spires:

FIND DK SPIN,FOAM OR QUANTUM GRAVITY, LOOP SPACE OR QUANTUM COSMOLOGY, LOOP SPACE AND DATE = 2005

and then repeat, each time increasing the date. You will get

Loop gravity research overall
2005    41
2006    81
2007   121
2008   142
2009   141
2010   154

You draw a false analogy between trends in the two communities. People have not "discarded" background independence and moved OUT of developing the main LQG theory in favor of working in the cosmo application. Both lines of research (main and applied) are growing and attracting new people.

I would say that Atyy is wrong if he means that the main evidence that the theory recovers Gen Rel is in "Bojowaldian" LQC. The bulk of the evidence concerns the main (background independent) theory, not the application.

No one in the Loop community, that I've seen, talks about "Bojowaldian" LQC and I'm not sure what Atyy means. That word is his alone. Bojowald's original formulation of LQC was
flawed and was replaced by other people's in 2006. If he really means Bojowaldian then most of what he says seems to be wrong or misleading because it is not the current (post 2006) mainstream.

In any case several recent Atyy comments don't make very much sense. And yours, Sardano, seem based on a fairly general deficiency of information.

I think we are, however, getting closer to understanding why there has been a growth of interest in Loop cosmology. There are physical reasons for this surge of growth.*

There also seems to have been a decline of interest (activity and citations) in the String program proper---actual work with string and brane models---but these two things may very well be unrelated. They may have occurred for a separate set of physics reasons.

*simple, concrete, testable, based on spinfoam QG (a background independent theory.)

No time to edit, I'll get back to this later today.

 

[Sardano]

I think you must be joking, because this is so obviously wrong. Both sets of numbers are going up: Loop cosmology and also the other Loop gravity research (that is not about cosmology).

If you seriously meant that as a prediction, you can falsify it easily with a Spires search.

To get you started here is a search for Loop overall (including cosmo). Type or paste this into the search box at Spires:

FIND DK SPIN,FOAM OR QUANTUM GRAVITY, LOOP SPACE OR QUANTUM COSMOLOGY, LOOP SPACE AND DATE = 2005

and then repeat, each time increasing the date. You will get

Loop gravity research overall
2005    41
2006    81
2007   121
2008   142
2009   141
2010   154

Marcus, you do not have to choose the words in the search in order to obtain the values at your will: you only will obtain fake results. It is very simple. If you are looking for LQG related papers, you write like this (in Inspire, of course):

LOOP QUANTUM GRAVITY AND DATE=YEAR

If you want to look for LQC papers, you have to write like this:

LOOP QUANTUM COSMOLOGY AND DATE=YEAR


Results:

LQG

2006 104

2007 106

2008 134

2009 128

2010 132

As you can see, in the last years the number of papers has no significatively changed.


LQC

2006 33

2007 58

2008 62

2009 72

2010 76

In the last years the number of LQC related papers has growth in a few number of papers. Not significant. ;). Speak about "more general interest" in LQC is therefore false, since if that were true, I should expect an increase in the number of papers of a few hundred. In fact, the number of LQC papers is ridiculous.

STRING THEORY

2006 1513

2007 1506

2008 1545

2009 1412

2010 1453

No significant changes.

 

[marcus]

I tend to suspect you don't have valid DESY keyword searches there, Sardano.
Let's see the links you are using for, say, 2007.

Oh! I see you are using Inspire. You may have forgotten to set it to do a keyword search, so it is just looking for occurrences of words in various fields, like the title, abstract etc.

I've had a lot of experience watching the Loop literature over the years and find that at least after 2006 or so the DESY librarians' classification is pretty reliable. I trust the librarians cataloging rather than a computer just picking out occurrences of words.

You may just be clowning. Your numbers for String Theory are way way off (much too low).
I don't know if I should take you seriously. The others numbers you get I would imagine are meaningless. (I expect I could immediately spot false positives, and if I wished to take more time could discover omissions as well, I imagine).

 

[Sardano]

Oh! I see you are using Inspire. You may have forgotten to set it to do a keyword search, so it is just looking for occurrences of words in various fields, like the title, abstract etc.

No, I havent. And the numbers are the real results. You can check yourself.

 

[marcus]

No, I havent. And the numbers are the real results. You can check yourself.

Then please provide the links that get your searches for, say year 2008.

I may have figured out which you used, I get 134, the same number you listed for 2008, with this search:
http://inspirebeta.net/search?ln=en...Search&sf=&so=d&rm=citation&rg=100&sc=0&of=hb

It gets this string theory paper:
http://inspirebeta.net/record/796131
and this philosophy paper:
http://arXiv.org/abs/arXiv:0804.3742
Maybe that's all right, not too bad. My Spires searches may pick up a few false positives too.

Now I do exactly the same search (using the beta version of Inspire, as you did) but for 2005:
http://inspirebeta.net/search?ln=en...Search&sf=&so=d&rm=citation&rg=100&sc=0&of=hb
I get 79.

Again, same search but for 2006, I get 104. Same number you got. Very likely these are the settings you used, then.
I noticed this false positive:
http://arXiv.org/pdf/hep-th/0602256
but again it doesn't seem way off.

 

[atyy]

See post #114 for what I mean by "Bojowaldian".

 

[marcus]

@Sardano,
I see what your trouble was! You only put in "loop quantum gravity" and you left out the keyword "spin foam"! After 2008 or so much LQG is formulated with spin foam dynamics (instead of the old canonical/hamiltonian formulation)

I still don't fully trust the beta version of Inspire. It was unstable and buggy when I tried it a month or so ago. But I see what you were doing and I can correct the numbers!

Put in "Loop quantum gravity AND spin foam AND year = xxxx"
and now let's see what numbers we get starting 2005

and then repeat, each time increasing the date. You will get

Loop gravity research (using Inspire beta)
2005    96
2006   118
2007   125
2008   159
2009   164
2010   173

Here for example is the link for the Inspire 2009 search in case anyone wants to see the settings or run down the list.

 

[marcus]

@Sardano,
I was able to reproduce your LQG numbers just by leaving out the "spin foam" key word. So I think I understand how you are doing Inspire searches.

I will now do the same thing for LQC---just put in "Loop quantum cosmology AND year = xxxx"
and now let's see what numbers we get starting 2006 when the Bojowald version was discarded and replaced by new dynamics.

Loop cosmology research (using Inspire beta)
2006    33
2007    58
2008    62
2009    72
2010    76

Your "prediction" was that LQG would decline to balance increase in LQC---that is no net increase in total activity. But actually both LQC and total LQG were increasing. However it does seem that the non-cosmo level stayed roughly flat! If we go by what Inspire says, then the growth was mostly in the cosmology part.

Loop gravity research (using Inspire beta) minus the LQC contribution
2006   118-33    85
2007   125-58    67
2008   159-62    97
2009   164-72    92
2010   173-76    97

That is not exactly what you predicted, but it kind of parallels it and so I have to give credit for a good hunch.

 

[marcus]

Anyway I wish people would stop trying to talk away the growth of interest in Loop quantum cosmology and try to offer some physics reasons for it!

Sardano, I appreciate your showing how to put Inspire into use (I was still using Spires and still consider it more reliable.)

But in any case the growth in interest should be obvious and it is not so interesting just to tabulate the numbers. What do you think are the causes?

Do you really think that it is merely because it is easy, or as you say "feasible"?

Anyway, the tendency in both fields is the same: try to work in something feasible with as direct as possible physical implications. As a result the LQG people goes to LQC and the String Theorist tend to work in things like ADS/CFT and related.

Is drawing an analogy with the shift in String program to "things like AdS/CFT an adequate explanation?

Personally I suspect it has a lot to do with the prospect of testing, and the masses of early universe data which are coming in. If that is what you mean (I don't see the analogy with AdS/CFT but) you might be right, or we could be in agreement.

Also to be explained is the upsurge in citation numbers. I gave a table of those earlier, a few posts back. They multiply the effect, one might say, of the increasing numbers of Loop papers, and represent additional demand for explanation.

 

[atyy]

BTW, the physics reason I gave in post #120 is exactly what this advertisement says too:
http://www.aei.mpg.de/english/research/teams/quantumGravity/index.html

 

[marcus]

BTW, the physics reason I gave in post #120 is exactly what this advertisement says too:
http://www.aei.mpg.de/english/research/teams/quantumGravity/index.html

Then it should be interesting to quote the relevant part of that---it's the philosophy and mission statement of the Quantum Gravity and Unification division at Potsdam MPI Albert Einstein Institute, led by Hermann Nicolai. As you know, the division hosts LQG and Group Field Theory/Spin Foam as well as String research. It could help me understand your view of LQG better if we see what we can glean about Nicolai's view of it from this statement.
===quote===
Quantum Gravity & Unified Theories

Director: Prof. Dr. Hermann Nicolai
This division is concerned with the unification of general relativity and quantum mechanics into a theory of quantum gravity, which also provides a consistent framework for incorporating the other fundamental forces in nature.

Despite intense efforts over the last years it is far from clear at this time what a consistent theory of quantum gravity will look like and what its main features will be. In view of these uncertainties, the best strategy appears to be one which is both diversified and "interdisciplinary". For this reason, the division aims to represent the major current approaches to quantum gravity, in particular string theory and loop quantum gravity.

The canonical approaches to quantum gravity emphasize the geometrical aspects and appear well suited to deal with unsolved conceptual issues of quantum gravity, such as e.g. the "problem of time" or the interpretation of the "wave function of the universe". Important new insights have been gained over the past decade in the framework of loop quantum gravity, which is one of the division's main research directions. This approach, which complements and extends the old "geometrodynamics" approach, employs a non-perturbative and background independent framework allowing to describe the fluctuations of geometry itself, and leading to a discrete structure at the Planck scale. On this basis, it is now possible to study the full quantum dynamics of gravity. Most recently, these concepts have been successfully applied to the study of cosmological or black hole singularities, where classical general relativity breaks down, spawning an entirely new field of research, loop quantum cosmology.
===endquote===

BTW Nicolai's division of the AEI has a number of people working on Spin Foam QG although he does not stress that in the statement you linked us to.

I see a 2009 paper from them, a 2010 paper, and two spin foam/path integral papers already in 2011. The spin foam approach is just as "principled" as canonical and the two are closely linked. In the lexicon of Nicolai's outfit spin foam QG is one of the "covariant" as contrasted with "canonical". So you see these people listed at AEI:

Banisch, Ralf Canonical and Covariant Dynamics of Quantum Gravity

Benedetti, Dr. Dario Canonical and Covariant Dynamics of Quantum Gravity

Dittrich, Dr. Bianca Canonical and Covariant Dynamics of Quantum Gravity

Eckert, Frank Canonical and Covariant Dynamics of Quantum Gravity

He, Dr. Song Canonical and Covariant Dynamics of Quantum Gravity

Martin-Benito, Dr. Mercedes Canonical and Covariant Dynamics of Quantum Gravity

Ryan, Dr. James Canonical and Covariant Dynamics of Quantum Gravity

E.g. two 2011 papers on the "covariant" side, and another from 2010:
1. arXiv:1103.6264 [pdf, ps, other]
Spin foam models with finite groups
Benjamin Bahr, Bianca Dittrich, James P. Ryan
47 pages, 6 figures

2. arXiv:1101.4775 [pdf, ps, other]
Perfect discretization of reparametrization invariant path integrals
Benjamin Bahr, Bianca Dittrich, Sebastian Steinhaus
22+8 pages

4. arXiv:1006.4295 [pdf, ps, other]
Simplicity in simplicial phase space
Bianca Dittrich, James P. Ryan
29 pages, 1 figure

Incidental info:
http://www.iem.csic.es/departamentos/qft/CV/CV_Martin-Benito.html

 

[marcus]

See post #114 for what I mean by "Bojowaldian".

Then I suggest you find a better word, like Hamiltonian LQC or canonical LQC.

Bojowald's formulation of LQC was seriously flawed. Bill Unruh among others pointed this out around 2004. Bojowald LQC was finally abandoned in 2006 when Ashtekar (with Corichi and Singh) came out with the new equations. The socalled LQC "new dynamics".

By now the Ashtekar et al version is just what people mean by canonical LQC.

You indicated you approved of that Hermann Nicolai statement. It uses the term "canonical", which says it clearly and is less likely to cause confusion.

 

[atyy]

Then I suggest you find a better word, like Hamiltonian LQC or canonical LQC.

Bojowald's formulation of LQC was seriously flawed. Bill Unruh among others pointed this out around 2004. Bojowald LQC was finally abandoned in 2006 when Ashtekar (with Corichi and Singh) came out with the new equations. The socalled LQC "new dynamics".

By now the Ashtekar et al version is just what people mean by canonical LQC.

You indicated you approved of that Hermann Nicolai statement. It uses the term "canonical", which says it clearly and is less likely to cause confusion.

I suppose you will not approve of the "Kadanoff-Wilson" renormalization group!

I used to say "Wilsonian", but every friend of mine who's in condensed matter (a small sample admittedly, perhaps those who hang around these fora can correct me) said well, Kadanoff had almost all of it.

Anyway, I bet it is agreed even outside of the LQG community that canonical LQC is neat and intriguing. It's a cute result, like CDT, that is looking for an explanation. I doubt there's similarly broad agreement on EPRL or FK.

If you recall, AS was thought to be the probable foundation for CDT some years ago. Now, other guesses have become prominent, although AS is still the leading guess. Similarly, I feel (yes, just a feeling) that EPRL and FK are not the way to go, and so they will not be the explanations for canonical LQC (I suspect some other spinfoams could do it).

 

[atyy]

BTW Nicolai's division of the AEI has a number of people working on Spin Foam QG although he does not stress that in the statement you linked us to.

Yes, my point is not that spin foams are uninteresting (they are very interesting). My point is that you asked why did interest in LQC increase. I think everyone except you has said that it's due to what's been achieved in canonical LQC itself. It has nothing to do with the non-existent decrease of interest in string, and little to do with spin foams. And what has been achieved? Well, canonical LQC is the first place, and only place so far in LQG that anything like Einstein's equations have popped out of a theory that is completely well-defined and thus deserves to be called QG. And this was achieved by discarding background independence.

So to use words from MPI's advertisement, I would say "LQC made it possible to study the full dynamics of quantum gravity, even near black hole and cosmological singularities".

 

[marcus]

Yes, my point is not that spin foams are uninteresting (they are very interesting). My point is that you asked why did interest in LQC increase. I think everyone except you has said that it's due to what's been achieved in canonical LQC itself. It has nothing to do with the non-existent decrease of interest in string, and little to do with spin foams. And what has been achieved? Well, canonical LQC is the first place, and only place so far in LQG that anything like Einstein's equations have popped out of a theory that is completely well-defined and thus deserves to be called QG. And this was achieved by discarding background independence.

That sounds like a fairly reasonable perspective on things, though I think you underemphasize (possibly ignore) some parts of the picture.

I don't recall saying that the decline of the String program was a cause of increased interest in LQC. I imagine that is something you thought I said. On the other hand I think the decline in citations in several areas is real---it is just an objective fact, not "non-existent", and helps to put things in perspective without being alleged as a cause.

The Einstein equations have not popped out of canonical LQC. The Friedmann equation is a radical simplification, so I look at that "popping" simply as one of several bits of evidence which encourage optimism that Loop is turning out to be right. I think you are mistaken to give heavy emphasis to the evidence from canonical LQC and ignore other types of evidence. It gives an impression of distorted vision. But that's how you see things, according to what you say. So be it.

I have been suggesting here that the increase of interest in canonical LQC can be attributed to intrinsic physics features (eg simplicity, concreteness, testability) which have been "achieved" in canonical LQC, to use your word.

So the main message in your post agrees with what I have been saying---and then says that I have not been saying it

Now the same features or "achievements" are beginning to be carried over to the full theory, but so far, as you know, there are only a few papers. These are the spin foam cosmology papers which show the possibility that the results from canonical LQC (radical simplification which boils the universe down to a couple of numbers) may carry over to a formalism which is NOT such a simplification and which is in fact background independent.

In that sense background independence was never discarded. No theory can properly be said to recover GR that doesn't explicitly have BI, so it was always on the agenda to work out a connection between LQC and the full theory. Now that is beginning to happen and I expect voices of belittlement and denial will simply call attention to the fact.

 

[atyy]

In that sense background independence was never discarded. No theory can properly be said to recover GR that doesn't explicitly have BI, so it was always on the agenda to work out a connection between LQC and the full theory. Now that is beginning to happen and I expect voices of belittlement and denial will simply call attention to the fact.

Very prescient. Let me be the first to fulfill your prediction! The full theory doesn't even exist. But if it does, it won't yield the Einstein equations.

 

[marcus]

The full theory doesn't even exist. But if it does, it won't yield the Einstein equations.

This question of the existence of theories is interesting. Partly it simply has to do with conventional expectations that one has in a mathematical-type (empirical) science. A theory is a human artifact that therefore does not exist until the equations are written down.
And you have to be able to derive predictions from the equations so the theory can be tested (or it has no content).

Also we've come to expect, at least in this area of physics, that a theory should rest on some basic principles, e.g. like background independence, general covariance, the finiteness of geometric information,...etc. Hard to be precise about that, but I think it's a case of "you know it when you see it": Open-minded people of good faith occasionally reach agreement on whether something is or is not based on principles, even though there might be no formulaic definition of what a principle is

There are some nebulous pie-in-sky things which don't yet exist as theories (and may never exist in the sense of some definite formulation.) Such a thing can even be named this or that "theory"---but that's just a hopeful misnomer.

 

[atyy]

Very prescient. Let me be the first to fulfill your prediction! The full theory doesn't even exist. But if it does, it won't yield the Einstein equations.

I see I forgot to add that by full theory I mean EPRL.

Actually, I was just looking at Vidotto and Rovelli's first spin foam cosmology paper. They seem to use KKL, so I guess they already agree with me;)

 

[marcus]

I see I forgot to add that by full theory I mean EPRL.

Actually, I was just looking at Vidotto and Rovelli's first spin foam cosmology paper. They seem to use KKL, so I guess they already agree with me;)

Aren't you just quibbling now?

It seems to me that you want at all costs to deny that Loop now has a definite formulation which (although not all the pheno is done) reasonable people can see is probably falsifiable.

What would it cost you to admit that? Would it be so terrible? After all, it could be falsified by observation in 5 or 10 years (depending on things like NASA/ESA budgets.)

 

[atyy]

Aren't you just quibbling now?

It seems to me that you want at all costs to deny that Loop now has a definite formulation which (although not all the pheno is done) reasonable people can see is probably falsifiable.

What would it cost you to admit that? Would it be so terrible? After all, it could be falsified by observation in 5 or 10 years (depending on things like NASA/ESA budgets.)

I don't agree. But ok, let's pick your preferred formulation of loops. It doesn't need any experimentalists to be falsified until it can produce the Einstein equations. Even CDT is closer to that. In fact CDT is probably the main reason to believe that something like loops could work.

 

[marcus]

It doesn't need any experimentalists to be falsified until it can produce the Einstein equations...

What do you mean by "need"? Do you mean it should not have? You are saying that people should be prevented or discouraged from testing UNTIL such time as your idea of a mathematical derivation criterion is met?

Don't understand. Some kind of moral injunction?

Just get out there and try to stop people from testing it .

 

[atyy]

What do you mean by "need"? Do you mean it should not have? You are saying that people should be prevented or discouraged from testing UNTIL such time as your idea of a mathematical derivation criterion is met?

Don't understand. Some kind of moral injunction?

Just get out there and try to stop people from testing it .

My idea of a mathematical derivation? Have the Einstein equations been produced according to anyone's idea of a mathematical derivation?

 

[PAllen]

My idea of a mathematical derivation? Have the Einstein equations been produced according to anyone's idea of a mathematical derivation?

Has it been proven that they cannot be produced? If not, why such emphasis one criterion? So far as I understand, all QG approaches fail to meet goals it is hoped that they will someday meet (and, that if proved they cannot meet, will torpedo the approach). To the extent they can make predictions before having a satisfactory formulation, why not exploit this enthusiastically?

 

[marcus]

That's not the point, as I see it. There is a heap of evidence that the present formulation is a good theory of gravity. Regge. BF--TQFT. Recovering deSitter universe (like CDT does).
It's pedigree as deriving from the Ashtekar version of GR. Evidence from canonical LQC etc etc.

A reasonable unbiased judge can very well presume that Loop as currently formulated does OK or approximately so as a theory of gravity.

That's enough for a reasonable person to say "OK go out and test it!"

But it sounds like you want to say "Wait, don't test!" Is there some moral stricture?

 

[atyy]

Has it been proven that they cannot be produced? If not, why such emphasis one criterion? So far as I understand, all QG approaches fail to meet goals it is hoped that they will someday meet (and, that if proved they cannot meet, will torpedo the approach). To the extent they can make predictions before having a satisfactory formulation, why not exploit this enthusiastically?

My point of view is that LQC is a theory of QG (yields a symmetry reduced version of the Einstein equations, and is mathematically consistent), and makes predictions. The caveats to it are clear, so every thing is out on the table. LQC can be tested. But if LQC fails the test, that will kill LQC but not LQG.

I also believe LQG (in contrast to LQC) is, in its present state, possibly not even a consistent theory. Hence one cannot even talk about predictions. If you examine the "predictions", they come from taking only the first term of a possibly divergent series. Even Ashtekar has in questions following a presentation of ths current theory asked whether the theory exists. I do believe the theory is interesting enough to continue working on without it making predictions. I believe this for two reasons. First, the theory is linked to CDT, which computer simulations suggest has a ground state whose large scale structure is the de Sitter universe. Second, and in a different direction, spin foams are linked to lattice gauge theory, which by AdS/CFT probably contains gravity.

 

[marcus]

...
Even Ashtekar has in questions following a presentation of ths current theory asked whether the theory exists...

It would be interesting to hear that. Out of context paraphrase by you can make a difference. It sounds like you are referring to the 2009 discussion at ILQGS between Rovelli Ashtekar Freidel. Are those the remarks you mean?

 

[atyy]

It would be interesting to hear that. Out of context paraphrase by you can make a difference. It sounds like you are referring to the 2009 discussion at ILQGS between Rovelli Ashtekar Freidel. Are those the remarks you mean?

Yes.

 

[PAllen]

My point of view is that LQC is a theory of QG (yields a symmetry reduced version of the Einstein equations, and is mathematically consistent), and makes predictions. The caveats to it are clear, so every thing is out on the table. LQC can be tested. But if LQC fails the test, that will kill LQC but not LQG.

I also believe LQG (in contrast to LQC) is possibly not even a consistent theory. Hence one cannot even talk about predictions. If you examine the "predictions", they come from taking only the first term of a possibly divergent series. Even Ashtekar has in questions following a presentation of ths current theory asked whether the theory exists. I do believe the theory is interesting enough to continue working on without it making predictions. I believe this for two reasons. First, the theory is linked to CDT, which computer simulations suggest has a ground state whose large scale structure is the de Sitter universe. Second, and in a different direction, spin foams are linked to lattice gauge theory, which by AdS/CFT probably contains gravity.

I disagree that you can't talk about possible predictions from a not (yet - per its proponents) consistent theory. I see string theorists doing the equivalent all the time, and I applaud it. String theory is not yet a theory, but you say "if there is a consistent theory in here, we can argue it has certain properties, leading to the following predictions". Call them conditional predictions (based on some assumptions about how the theory will develop). I think it is highly advantageous to do such exercises while developing the theory, at all stages of its development.