Lost in Math is a recently published book by Sabine Hossenfelder that argues that elegance, beauty and naturalness are vague gauges of the plausibility of theories in physics even though many physicists, including prominent ones, to varying degrees subscribe to them or at least consider them useful guides. Her critique is bracing and thoughtful. On the other hand, if the universe is constructed out of simple principles that combine in a large variety of interesting ways, shouldn’t a reductionist approach reveal that simplicity in ways that physicists characterize as elegant, beautiful or natural? Your views?
Most beauttiful seems to be the best generalisation of known phsical laws.
Consider two statements:
1. Universe is 4% matter, 20% dark matter and the rest dark energy.
2. Universe is 100% suffering.
The answer to your question is available in
L. Smolin, “The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next”, Houghton, Boston, 2006.
Dear Robert
Your question is right. The best theory is always the simplest one, which of cause must be verified by experimental results.
I'm sure that today's theories are all superficial, not reaching the underlaying action of nature. Just look at the mathematical complexity of those theories. I'm sure that the reality is much simpler. Maybe the problem comes from the use of analytical mathematic to obtain useful formulas. Nature is note necessarily working with nice formulas.
Stellan
As you see the beautiful and elegant proposition "Universe is 100% suffering" is useless.
"Do elegance, beauty and naturalness have a role to play in theories in physics? "
No, not in the real sense! It is in fact the exact opposite - the lack of symmetry, the negation of beauty is the important case for the theories of physics in the REAL world! Math (or could it be Meth!) intoxicated theoretical physicists should worship anti-symmetry (the dialectical negation of symmetry), because this is what enabled them to practice their powerful profession in the first place!
According to their own account; it is the “spontaneous symmetry breaking” at various successive steps, starting from the so-called Big Bang creation of the universe; which made any EXISTENCE (at all) possible, in the first place! In an universe with “naturalness” and “perfect symmetry” of matter and antimatter; no physicist could be there to worship “beauty” etc.! Hegel would call this phenomenon the “portentous power of the negative!”
To the researcher there is emotional support of long hours and hard work when the result is something of elegance, beauty and appreciation of nature. It can be misleading to choose the pretty one and reject the ugly science. Elegance and beauty are not the tools for selecting answers in science. Science is looking for the truth. Elegance and beauty are appreciated, but not always achieved.
It is better to have recognition from peers, but more difficult to obtain than the artistic view of a new idea that is becoming science. When reaching beyond the peer group sometimes researchers must be content with elegance and beauty.
Symmetry has a role in that if the problem has an inherent symmetry, then so must the equation that describes it, however f the problems does not, then neither must the equation. In my view, Sabine is correct. Forcing your work to accommodate a symmetry, or basing your work on the assumption there is a symmetry, is just plain wrong. Anyone that considers the Lagrangian of the standard model to be beautiful has a really weird idea of beauty. Elegance tends to be merely a tool by which the person forming the theory shows off.
The one defining aspect of theory is whether when you make predictions nature agrees, or if you correlate sets of data, the set is big.
Most beauttiful seems to be the best generalisation of known phsical laws.
It is very difficult to deal with theory, which has no elegance, beauty and naturalness.
A good example is St.Hawkings theorie of causality in general Space Time manifolds, which can be made more beautiful by topogical treatments.
I think we would all agree that “elegance, beauty and naturalness” certainly have a role to play, but that the math can become somewhat inelegant, depending on the subject matter. Rizzi et al.’s “Synchronization Gauges and the Special Theory of Relativity” (2008) is a case in point. Shifting from Einstein’s “special case” isotropic/constant light-speed to more fundamental anisotropic light-speed is definitely a step away from mathematical beauty—as a particular, new/additional terms in the Maxwell equations emerge to retain predictive/explanatory capability. This development is absent laboratory measurement or (immediate) predictive capability, and accordingly outside of the bounds of science.
However, should (now) decades long work by numerous researchers (besides Rizzi, Ruggiero, and Serafini we can note Franco Selleri, Mordehai Milgrom, Stacy McGaugh, Lee Smolin, …) yield deeper space-time theory that explains too-fast galaxy dynamics and cosmic acceleration, we may find that less elegant mathematical physics moves us forward.
I do not think "naturalness" is at all useful. Thus nature, by definition, has to be natural, so ANY procedure that correctly describes nature must also, by definition be natural. For those who demand elegance and beauty, write down a mathematical description for a strong snooker break. Anyone who thinks the resultant maths are elegant or beautiful, well I can't agree. They are horribly complicated, but the physics are clear. Those looking for elegance and beauty, in my opinion, are mainly seeking very simplified situations to describe.
If the solution to the problem can be presented in a graphic form, then from two possible solutions, the one whose chart is more beautiful will be the one to rule. So says my experience solving geometric problems.
According to Einstein's viewpoint you can formulate a set of axioms of your own, considering subjective notions such as elegance, beauty, naturalness etc. etc. But then starting from this very set of axioms you have to make physical predictions. Physical measurements will ultimately establish or refute these axioms.
`Do elegance, beauty and naturalness have a role to play in theories in physics?'
Do any of these assist in evaluating the plausibility of the many worlds theory of Hugh Everett? Are they even applicable? The many worlds approach avoids problems raised by the role of the observer in the collapsed wave function (arguably elegant and clever — beautiful), but at the cost of having an unbounded number of universes (not very efficient and in that sense inelegant). The solution of having each possibility is symmetric in the sense of all being equally probable, but asymmetric in that each world differs. These observations suggest the subjectivity of elegance, beauty and naturalness.
On the other hand, elegance, beauty and naturalness, if nature proceeds by looking for conceptual geodesics (ideas that find the most energy efficient path) must be present in some sense. The tough part is finding out which of the available instances of elegance, beauty and naturalness works best.
This relates to parsimony: what choice does Ockham’s razor lead us to?
This conundrum arises in connection with the 4/3 law. The 4/3 law appears to account for a wide variety of phenomena, and may be a universal, scale-invariant law. On the other hand, it in effect requires two universes contemporaneously existing occupying a similar or the same space, one with three dimensions one with four. Is that elegant or inelegant?
Thank you to all so far for your answers, all interesting.
Bob Shour
Beauty and elegance go hand in hand with a theory widely accepted and appreciated. Also, it depends on the perspective of the person reading a theory. A theory is called beautiful and elegant when it is able to explain the intricacies of a complex system in to something meaningful and understandable and most importantly worth being appreciated.
Naturalness on the other hand depends a lot also on how intuitive one is about the subject they are dealing with. This intuition develops in time and the ability to look at a theory with an eye for naturalness comes for every researcher at different stages.
So beauty and elegance go hand in hand. Because it is possible to find a theory in physics beautiful and elegant but may be the naturalness aspect comes with time.
No wonders Hawking always wanted one beautiful elegant equation to describe all the physical aspects of universe.
Theories are nothing but stories to "explain" "observations". Theories are accepted if they work otherwise they are set aside for a while but not rejected. As theories are nothing but constructions they depend on the cultural beliefs of the theorist. Beauty and elegance are also relative to the culture.
Evolution of physical theories have demonstrated that a complete physical theory is invariably elegant (i.e. consisting of a small number of simple assumptions or postulates, involving very few free parameters) and beautiful (enjoys a great deal of symmetry), predicting several astounding effects. There are many examples to support this. For instance, before Maxwell formulated his four equations of electromagnetic theory, electricity and magnetism were explained in terms of seemingly unconnected rules like Coulomb's law, Faraday's law, etc. Maxwell's theory not only unified electricity and magnetism, but also led to local conservation of electric charge and prediction of electromagnetic waves. It reflected great deal of symmetry like Lorentz covariance as well as duality symmetry (for the vacuum case, and generally true if magnetic monopoles are discovered).
Similarly, special relativity with just two postulates led one to Lorentz transformations, time dilation, length contraction, unifying energy and mass via E=m c^2, twin paradox, brought space and time together, etc. and finally leading to the most beautiful theory - general relativity (GR), a relativistic theory of gravity. GR is a tensorial generalization of Maxwell's electromagnetism that unified geometry with gravitation. In short, I agree with Robert Shour.
Yes and no! Feyman said:
"You can recognize truth by its beauty and simplicity."
but he also said
"It does not make any difference how beautiful your guess is ... if it disagrees with experiment it is wrong."
In other words, the correct answer is usually beautiful, elegant, simple and natural; but wrong answers can be too.
If physical theories don't deliver the right results... no elegant beauty can help... it's a nonsensical demand... beautiful is the theory that gives all the right answers... no matter how ugly deemed by a delusional mathematician, or physicist
Our scientific history, until recently, has shown us that nature is a simple, elegant, beautiful, and natural combination of principles and actions. We have missed something and "gone astray" in the last century or so. We have established theories and fervent beliefs which are clouding our vision and not allowing us to decipher the simple elegance of nature. The results are a strange combination of the unexplained we call the standard model.
There is a view which can simply and naturally describe particles, space, and time, but we have not found it in the standard model.
`Do elegance, beauty and naturalness have a role to play in theories in physics?'
Maybe elegance, beauty and naturalness should be not treated as absolute standards but as comparative standards: which is MORE elegant, beautiful, natural?
Ptolemaic epicycles in the Almagest might be considered elegant, since only circles are used, and beautiful and natural, because of the perfect symmetry of circular paths. But Newton’s theory of gravity might be considered more elegant because a single equation not only gives good approximations to all planetary orbits with no need for prescribing specific epicycles, it also accounts for the revolution of the Moon around the Earth, and projectile paths from the Earth’s surface, that is, because of its universality. The general applicability of Newton’s gravity model is elegant because one equation applies to a large variety of different phenomena, and so compresses the amount of mathematical modeling required into that one equation. One equation that accomplishes the same as many for the same phenomena is arguably more elegant.
Ptolemy’s modeling is elegant and beautiful based on its geometric appearance. Newton’s modeling is elegant based on the concision and generality of his formula for gravity. The criteria applied to Ptolemy’s model is aesthetic visual representation. The criteria applied to Newton’s model is the aesthetics of a mathematical equation of universal applicability. From this one can infer that the playing field affects evaluation of elegance, beauty and naturalness. Is it mathematical, visual, or energy efficiency used as criteria for elegance, beauty, naturalness?
If to model a given set of phenomena n - 1 equations suffice without any loss of accuracy compared to n equations, we could say the n-1 equations are more elegant than the n equations. Does that make the idea of elegance more precise?
Bob Shour
These are metaphysical notions and depend on the person-so it doesn't make sense discussing them in any other capacity.
The standards of ``beauty'' and ``elegance'' aren't fixed notions, either.
One shouldn't confuse personal opinions-that refer to such notions-to impersonal issues, like whether a calculation or experiment is correct or not.
Physics and math are impersonal, not personal-their content can be checked and understood by anyone.
The reductionist approach (which, also, doesn't mean anything by itself) doesn't have anything to do with any notion of elegance, beauty or simplicity-that are words without any impersonal meaning, by definition. It's just a term that's used in a social context and, therefore, also, can't have anything to do with the question of whether a calculation is correct or not and whether an experiment is correct or not.
The only things that matter are consistent mathematics, that define the framework of any description of natural phenomena and consistent experiments. The consistency and the relevance of the calculations and of the experimental setup can be checked by anyone, and don't require adhering to any standards of what's elegant, beautiful or simple. The explicit realization of the experiment is, of course, a different issue, altogether. However the consequences of either a calculation or an experiment can be checked as to their conclusions without having to enter into the details. There are many ways of obtaining the same result and it's the result only that matters.
One shouldn't confuse the social behavior and/or social status of people that work in physics and mathematics with the content of these subjects. More explicitly, what people write as technical articles and what they write as opinion pieces. The former only have anything to do with physics and mathematics-the latter don't. And it is unfortunate that the latter are given more attention than the former.
That's why it doesn't make sense arguing over what any person considers simple/elegant/beautiful (or not).
Sociological critiques are pointless because they can't decide whether a mathematical description is consistent or not. Its content matters, not the way it's presented or perceived.
Hi Stam Nicolis
One problem we face in physics is the fact that technical papers are often not just impersonal and factual. They reflect the opinion and point of view of the writer.
Whether a single calculation is correct or not is not the larger concern here. We are seeking an entire theory with all requisite calculations, which contains no mutually exclusive conditions, and is comprehensive and coherent. We are far from such a theory, and our theories are becoming more complex as we try to "fix" their shortcomings.
It is reminiscent of the complexities we have historically introduced to try to explain our beliefs. For example the geocentric beliefs and the epicycles of long past. Once the subject was understood, the complexities diminished, and we saw the simplicity of the answer. The motions of the planets was not so complex as previously thought. we saw the causes and effects more clearly. While we have no metric for this simplicity and elegance, it is still a real phenomenon which we have seen and understood.
This reminds me of a poignant few words from the past...
… “It is my opinion that everything must be based on a simple idea. And it is my opinion that this idea, once we have finally discovered it, will be so compelling, so beautiful, that we will say to one another, yes, how could it have been any different.”—John Archibald Wheeler.
It's their technical content that matters. The opinions and views they reflect aren't relevant. So it's pointless paying any attention to the latter and not focusing on the former, when discussing physics. It doesn't matter, what the opinions and views of the author of a technical paper are-about anything.
Hi Stam Nicolis
The "technical content" is dictated by the beliefs of the writer, there is therefore no separation of opinion from content.
While the "technical content" of the epicycles of history was mathematically correct, it did not actually describe the cause for the motions of the planets or stars as we observe them. This universe has proven itself to be one of cause and effect. Until we can actually understand the causes in a coherent manner, we may not have all the pieces of the puzzle that we think we have.
Again, the larger problem is a much more fundamental issue than just taking the existing math and discarding that which we individually feel is the author's opinion.
In my opinion we need to revisit the foundations of our beliefs in order to find the correct path to this deeper understanding of what is actually going on.
Of course the technical content doesn't have anything to do with the beliefs of the writer-and the example offered shows this, since no writer is mentioned on epicycles at all!
The personal opinion of any one person doesn't affect whether a mathematical statement is correct or not-and that's the only thing that matters when describing any effect in nature-the way a planet moves is completely oblivious to what the people that monitor its motion believe in-and it moved the same way when Galileo was observing it as it does now.
Absent any impersonal definition of what ``actually is going on'' means, no useful conclusion can be reached, beyond personal satisfaction or frustration. But neither can affect the description itself. The equations don't describe these effects.
Hi Stam
You are quite correct in stating that " The personal opinion of any one person doesn't affect whether a mathematical statement is correct or not..."
But the important issue is the the physics, the cause and effect are the things which matter. The math is then just a convenient and disciplined way of presenting the actual physics, once we discover what the actual physics is.
Of course a mathematical formal can be correct mathematically, and still not accurately describe the actual physics. The epicyclists derived the math empirically, based on their foundational beliefs, by watching the motions observed and creating math which mimicked those motions.
Prior to Galileo, many authors wrote quite complex math to try to describe the motions of planets and stars observed. But they did not take the bold move and rethink their foundational beliefs like Galileo. So their quite correctly formatted and worked math, was actually useless and misleading.
Conditions indicate that this is where we are, to some extent, with the Standard Model. We have derived the math empirically, based on our foundational beliefs, which mimics the things we observe in Quantum physics. But we find ourselves with more unanswered questions, dichotomies, conflicting mutually exclusive conditions, math that does not work and has to be fudged, and strangeness which defies any cause and effect explanation.
Note: Some personality types seem to thrive on this strangeness, but I feel that true scientists would rather have the real answers, and not be asked to accept the magical.
The indicators are that we have missed something along the way, and been diverted from the real answers by the things which we have come to accept and believe.
Just as the writers on epicycles, we may well be off track due to our preconceptions.
There aren't any ``real'' answers, because physics provides, always, an approximate description of natural phenomena. What does occur is that questions that were once thought to make sense, turn out not to; and questions that couldn't be framed can now be asked and their answers lead to further questions. Incidentally, the same holds true in mathematics.
But this description doesn't have anything to do with the personalities of the people involved.
Incidentally-the statement that ``math doesn't work'' regarding quantum physics in general or the Standard Model, in particular, is, simply, wrong.
The statements about ``strangeness'' and ``magical'' are, just, expressions of personal taste, with no effect on anything.
There's nothing to ``accept and believe'' about physics, in general, precisely, because it doesn't depend on anything personal. Its sociology, of course, does-and many people do tend to identify the sociology with the technical content. But there's no reason for this to occur.
The statements about elegance, beauty, naturalness, simplicity, have to do with how questions and problems in physics and mathematics are addressed. What matters, however, are the answers, themselves, to such questions-and they are, essentially, insensitive to how questions are addressed-indeed there are many ways of addressing questions in physics and mathematics, that lead to equivalent answers. It's the appreciation of the answers that's more relevant, since learning how to frame questions is part of understanding any subject. And these words have technical meaning. So, as long as the answers are the same, it's a question of personal taste, how they're obtained. Cf. for instance, https://www.youtube.com/watch?v=VW6LYuli7VU&t=15s
Until we come to understand, accept, and believe the underlying mechanisms of a system, we cannot frame the pertinent mathematics. To understand, to know, is also to believe. We cannot separate our beliefs, no matter how we wish to, from the process. The universe, the "physics" does not care whether we have the right approaches or not. But we should care, and double check, so that we can obtain more valuable progress.
You may believe that the mathematics, with renormalization, is a reasonable solution, but to others it may be seen as an indication that we simply do not yet understand what is actually going on. You may be OK with the incompatibilities of SR and Quantum mechanics, but to others it may indicate that we do not yet understand what is really going on. And of course there are literally dozens of similar examples which can indicate that we are not yet on the right track. You might think that the Lagrangian for the standard model is the simplest way to express the reality of nature. But the evidence says to me that this is simply not the case.
In this exchange you have written of your beliefs.
We are human, and in a quest for knowledge. We cannot separate our belief from that process. But we can recognize that we are driven by our quest for knowledge and our belief system to ask specific questions. However if we use that as a tool it can help us to reexamine the landscape and look for questions which might lie outside our current belief system. If we are to find the correct answers we must do this. This is what Galileo did. This is the path to real discovery which has proven itself many time in our history.
So, while you discount the importance of the notions of simplicity, beauty and elegance, we see examples of these all around us and in many of our past discoveries.
You are correct in that the universe does not care what we believe, and that is specifically why we have to question our belief, our theoretical basis, so that we might stay on the path to discovery.
I don't ``believe'' that renormalization is ``reasonable''-I don't understand what the statement even means. I can prove how it works, in a way that doesn't depend either on my mood, or the mood of anyone else: it provides the construction of all terms of the series that describes the perturbation about free fields of any field theory. Any other method better include this, else it's useless. So far for the mathematical part; and this, in turn, seems sufficient to describe experimental data, in all known situations. The mathematical issues involved in going beyond the individual terms of the expansion don't invalidate the procedure-they complete it. And it is known how to describe renormalization operationally, in a way that doesn't rely on perturbation theory at all, in fact. The mathematical techniques involved in controlling finite size and finite volume effects, in this formulation, similarly, don't invalidate the approximations-they enhance their scope.
There aren't any ``incompatibilities" between special relativity and quantum mechanics, that aren't resolved within relativistic quantum field theory, of which the Standard Model is one example.
Instead of making vague statements, it would be more useful to produce concrete examples of any ``incompatibilities"". That's the difference between history of physics and physics. Once upon a time such incompatibilities did exist-now they don't, because the correct formulation, that does have the correct non-relativistic limit, is understood.
It doesn't matter whether one uses a Lagrangian description or any other, when performing calculations in physics-the only quantities that matter when describing any physical system, are those that are invariant under the transformations, that define the symmetries of the system. How these quantities are calculated doesn't matter. Indeed the strong interaction backgrounds that must be computed for discovering the Higgs are not computed by summing the individual Feynman diagrams, but using a totally different approach. It doesn't matter, though-that they could be computed quickly enough and correctly, does.
That's why it doesn't make sense going on and on about how to compute something-but to actually compute it. (Or prove it, if it's a mathematical statement-the methods used don't matter; that the proof is correct, does.) The necessary techniques and understanding are developed by solving exercises of increasing complexity, as is the case for any other activity-nothing new here.
Yes, the disciplines of physics and mathematics are largely free from subjective perceptions. But notions of elegance and beauty in these subjects are not subjective either. Most practitioner of theoretical physics would be unequivocal in stating that Maxwell's formulation of electromagnetism (EM) in terms of four differential equations is more elegant and beautiful than the combination of disjointed rules like Ampere's law, Coulomb's law, Faraday's law, etc. In fact, the differential geometric formulation of EM (F = d A and d*F= 4 pi J/c) is even more elegant and beautiful than the Maxwell's version. There is a deep reason for it. Mathematicians in their pursuit to prove nontrivial theorems are driven by elegance and beauty. And since physics is based on logical analysis of outcomes of measurements of various observables (that yield real numbers) and their mutual correlations, it is not surprising that mathematics is the preferred language of physics. Therefore, fundamental laws of Nature tend to be beautiful and elegant. I had contemplated on these points some time back in my article: https://arxiv.org/pdf/1508.06988.pdf
Statements about ``elegance'', ``beauty'' and ``depth'' or ``non-triviality'' are subjective, precisely because they focus on the sociology and not on the technical aspects.
It doesn't matter what anyone feels about Maxwell's equations-what matters is setting them up and solving them. There are many ways of doing this and, as long as they all lead to the same result, which way one chooses doesn't matter; any more than what coordinate system one chooses. What does matter is focusing on the quantities that are Lorentz invariant and gauge invariant. The other quantities only are useful as building blocks of the invariant quantities.
Mathematics isn't the ``preferred'' language of physics-it's the only language. Of course not any mathematical structure is appropriate for describing any natural phenomenon-the corresponding invariants are.
Hi Stam
I must differ with you on the notion that mathematics is the only language of physics. Math is a tool to help us identify and specify action and relationships. But the understanding of how and why the math takes the specific form it does is based on an understanding which transcends the math.
We can create a set of complex mathematics which predicts the path of an object, but that does not mean our math has anything to do with the actual forces acting on the object. When we do finally get the math correct (so that it represents the underlying physics) that math is generally far simpler than any other mathematical solution we might find.
Of course an example of this is the epicycle mathematics we discussed earlier compared to the math of Galileo and Newton.
So just because we have created a set of complex math which predicts something, does not mean we have discovered the real physics behind the action. This applies especially in circumstances where the math we have created does to relate to or answer a broader range of related questions. If we have not tied all the pieces together then the theory is still incomplete, or worse yet, inaccurate and misleading. Which is the contrary to our goal as physicists.
`Do elegance, beauty and naturalness have a role to play in theories in physics?'
Simplicity may sometimes induce ascribing beauty to a perception or concept. Simplicity in turn may arise from energy being utilized in the most efficient of available ways. Efficient deployment of energy can result in natural phenomena such as cosmic microwave background radiation as well as to artifacts devised by humans; it can apply to the evolution of the seeing eye and to development of the ideas and technology leading to cameras. The analogy between the outcome of nature’s principles applied in nature and the artifacts of human creation is described by William Paley in his Natural Theology disregarding the theological point of view.
Perhaps given our experience and observation of efficient energy distribution in nature we are predisposed to consider the efficient deployment of energy in modeling nature as elegant or beautiful. Simplicity may be a feature of nature that we by habituation to it in our living environment ascribe elegance or beauty to a scientific theory; it is perhaps a learned aesthetic sense.
Sabine Hossenfelder in her book Lost in Math appears to argue that sometimes the aesthetic sense gets ahead of the evidence.
Bob Shour
`Do elegance, beauty and naturalness have a role to play in theories in physics?
The word wit usually is thought of as describing a use of words that is clever and humorous or entertaining. Scientists who perceive elegance or beauty in a mathematical model of natural phenomena seem to me to be appreciating Nature’s wit.
Many dictionary definitions seem to define wit as a form of word play. But wit can by analogy apply to anything that can be characterized as clever. This appears in one of the definitions of witty in The American Heritage® Dictionary of the English Language, Fifth Edition: Entertainingly and strikingly clever or original in concept, design, or performance: a witty sculpture; witty choreography.
Similarly, one might say that Nature is witty in its exhibits of natural laws. That induces a similar appreciation for the wit of a mathematical model of such a law.
One might then say that Newton’s universal law of gravitation presents Nature’s wit; so do Maxwell’s equations for electromagnetism.
Along these lines, a scientist’s sense that beauty or elegance applies to a theory in physics shows the scientist having developed an appreciation of Nature’s wit.
I have the experience of encountering a problem that seemed initially to present a significant (sometimes I suspected an impossible) hurdle to mathematically model. If as a result of persistence and luck I happen to light upon a mathematical model that seems to work my initial sensation is: how extremely clever of Nature to proceed that way, I would never have thought of doing things that way. That seems to be Nature showing how much wittier it is than its human interlocutors. I suspect other people have had similar experiences.
Bob Shour
Once more: There's no such notion as ``complex math'' (beyond what's called ``complex analysis''). When discussing how mathematics describes physical phenomena, the only mathematical concepts that are relevant are the invariants of the transformations that are symmetries of the physical system: that map solutions of the equations of motion to other solutions of the same equations and have corresponding effects on the space of states, more generally.
The symmetries and their invariants are all there is and the only thing that matters is calculating them. Within any particular framework what these invariants are can be determined. Much of the discussion that is a waste of time and effort revolves around quantities that aren't invariant and/or on the different ways of how to do the calculation-which doesn't affect the result, however. Any way that computes the invariant quantities is as good as any other, so which one to use is a question of personal taste.
So, once more, it would be much more useful to provide concrete examples of ``incompleteness'', rather than making vague claims, that lead nowhere.
Any mathematical description of natural phenomena, inevitably, provides an approximate description-but, within its framework, it is consistent and complete, otherwise it doesn't make sense.
While people may talk colloquially about ``Nature's wit'' and so on, such terms only make sense in defining a community. People within that community know what the terms mean-people outside that community don't. That's all. The community, however, doesn't confer magical powers in solving the technical problems-anyone can do that on their own. The social recognition of the solution, conferred by belonging to a community, can be independent of its technical content; something many people tend to forget. That's why, when discussing physics, or mathematics, it doesn't make sense bringing the social issues in the discussion. They do have their interest-but they don't have anything to with whether any calculation is correct or not. They only measure its ``usefulness'' for the particular community.
So all the discussion about what ought to be understood as ``elegant'', ``beautiful'', ``deep'', ``non-trivial'' (or not) is just a way of showing that someone belongs-or doesn't belong-to the particular community. It doesn't have anything to do about the impersonal issues, that only matter when deciding whether a calculation is true or false and whether an experiment has measured what has been claimed to have been measured-or not, by itself. Unfortunately too many people tend to identify social recognition with technical content. They're distinct attributes.
Back to the question...Do elegance, beauty and naturalness have a role to play in theories in physics?
Our history of scientific discovery has indicated that nature... the universe, works on quite simple principles. Anything we view as complex can be broken down into a set of very simple relationships. That is a reflection of the efficient simplicity of nature.
So my answer to the question is a resounding yes!!!
Math serves as an efficient tool to communicate those simple actions of nature, but math does not, by itself, disclose the basis, meaning the underlying physics and mechanisms. It takes a comprehension of the cause, the reason the math takes that specific form, in order to understand the physics.
Nature does not care whether we believe in her robust simplicity. Thankfully, nature, the universe, does not change to suit our misconceptions.
But what we believe, our theoretical basis, guides our research, whether we recognize that fact or not. The questions we ask, our interpretation of experimental data, our hypotheses, are all influenced by this set of beliefs we have each built. We do of course think that everything we believe to be true has been proven. However we have been guided by specific interpretations of experimental data, theories.
One mistake we have made historically, and probably continue to make, is the adoption of theory as if it were fact, or a law of nature. This has likely deterred us, and slowed our progress.
When we finally do have that theory which is comprehensive and explains all of what we observe, it likely will not closely resemble our existing theory. And nature has shown us that it will be a simple and robust set of principles which underlies the causes for what we observe.
So I concur with John Archibald Wheeler...
“It is my opinion that everything must be based on a simple idea. And it is my opinion that this idea, once we have finally discovered it, will be so compelling, so beautiful, that we will say to one another, yes, how could it have been any different.”—John Archibald Wheeler.
All physical theories express facts of nature-by construction. And there's no such notion as a ``final'' theory, whatever may have been written, in non-technical form, on the subject. While Maxwell's equations do provide a complete description of classical electromagnetism, the constitutive relations that describe the different ways known matter interacts with electromagnetic fields is a very active area of research.
General relativity is the complete description of all gravitational phenomena of classical matter with classical spacetime-but it's an active area of research, nonetheless.
Classical, Newtonian mechanics, indeed, also, is an active area of research in the domain of chaotic dynamics.
Quantum mechanics was formulated in a complete way in the 1920s-it is a very active area of research in the present time as technology made thought experiments possible.
Physics won't stop when it's understood how the strong interactions are unified with the electroweak interactions or when a theory of quantum gravity is formulated-the questions only will change, as previous questions will be understood to be approximations or meaningless and new questions will emerge.
That's why the notion of a final theory doesn't make any sense.
Simplicity, elegance and beauty don't determine that Maxwell's equations are invariant under Lorentz and gauge transformations, not Galilean transformations; and neither is ``simpler'', ``more graceful', or ``more beautiful'' than the other. Nor that general relativity is defined by general coordinate invariance makes it more or less ``elegant''. That, once technical issues have been understood, this understanding can lead to feelings that can be expressed as a sense of beauty is good for individual appreciation and a sense of community, nothing more. Science isn't a spectator sport, however. And artistic appreciation is no substitute for learning.
What a ``simple idea'' is, means something for some person at some moment in time-and nothing to some other person. The mathematical description, however, means the same thing to everyone at any time. Some people may, of course, seek out a sense of exclusiveness-they won't find it in the technical issues, however.
That's why it doesn't make sense, when discussing technical issues, to focus on the non-technical writings of scientists-but on their technical work. Science isn't scripture, either; that's why the words used in conveying an idea are less important than the idea itself. Appealing to authority might make sense in understanding social structures; not in understanding technical content.
There's no compelling reason, based on aesthetic criteria, why photons, whose superpositions are classical electromagnetic waves, are quantum objects and not classical objects; which is the property that distinguishes electromagnetic waves from all other known waves; while the microscopic degrees of freedom, whose collective behavior is described by classical thermodynamics, do have classical properties and can be identified within bulk matter as atoms, that in that context behave as classical objects.
It's possible to provide consistent mathematical descriptions of all cases that are equally valid and can be understood as elegant according to some criterion; and it's the experiments, based on the mathematics and the technology, that lead to the distinctions.
Esthetic criteria come way after the fact. They express the tautology that understanding a concept makes it beautiful and/or simple to those who think they have understood it and ugly or indifferent to those who haven't.
The claim, however, that because someone feels that a formulation is ``beautiful'' implies it is correct-or because that person feels it is ``ugly'' implies the formulation is incorrect, is, simply, meaningless. Unfortunately, non-technical publications seize on such, personal, statements and confuse them with technical statements. While the experts do know the distinction, the non-experts don't.
As mentioned, it's the other way around!
The beauty of the equations or of the physical models mentioned by Dirac or Salam, among other physicists, was that of 'symmetries', those associated with well-defined transformation operations.
If the problem has a symmetry, then so must any equation that will represent it. That does not mean that any equation with a symmetry will represent nature. In my opinion supersymmetry is starting to look like a mathematical theory based more on "beauty" than on fact.
perfect symmetry does not exist; it is an artificial wangle. natural things seem to aim perfect symmetry
"Naturalness", "beauty" and "elegance" in any case are by no means scientific terms. What is perceived as natural, elegant and beautiful, in most cases, lies in the eyes of the beholder.
Also, even terms like "simplicity" are relative only to the way things are worded and formulated. Take Einstein's equations, for example. The more abstract you write them down, the simpler they look. Are the resulting diff-eq's elegant? Are they simple? I wouldn't say so.
And apart from the fact that say supersymmetry or string theory are completely unfounded by experimental data or by measures of predictability: are they elegant theories? I don't think so...
What about the standard model of elementary particle physics as of today? Is it a "beautiful" theory in the sense that it is even easy to write down the Lagrangian with all the field components? I would say no.
What about "naturalness"? Is 1/137, the approximate value of the fine structure constant, a "natural" number? Schwinger might have thought so, but isn't somehow 1/12 even more natural (albeit wrong)? Is it not just a completely arbitrary concept that implicitly assumes that certain quantities are related, so that easy-to-memorize numbers arise? But then again, is "pi" a "natural" value?
Like the "anthropic principle", I have the feeling that all those vague mental concepts are just due to the lack of a deeper understanding, aka theoretical ignorance.
dear Oliver, you are right, but the eye doesn`t see ALL. (not nearly). i.e.: the shortness of visible light spectrum
Absolutely! The problem is that all other senses really don't help either. Theories don't smell (but they might stink), equations have no taste, and no haptics either...so it is even more difficult outside the visible spectrum to speak of beauty or elegance in physics...
The concepts of Beauty, Elegance, Naturalness, Symmetry and so forth are our mental creations which we impose on Nature/Cosmos to help us have an understanding of it. They are like the grid of latitudes and longitudes we "cover" our globe to have some understanding of it's surface which we then can use for our goals. This "Law of our Mind" is the result of our evolution and has served us very well in our survival. We find symmetry to be beautiful and elegant as it helps us to predict what the other half looks like when can see only one half. Our bodies are outwardly symmetrical (or almost) but internally they are not. Even our brains which is probably the most symmetrical organ is not when we take a closer look. a simple example is an area called the Planum Temporale. Then we have most of the right side of our body being controlled by the left side of our brain and vice versa! Let us not forget that our vision uses a specific band of the EM spectrum. Imagine if our vision used ultra-violet or the infra-red parts of the EM spectrum? We have found Symbolic Logic to be most helpful and also the decimal system. Just look at our "beautiful and elegant" physics formulae using the Binary system. Hence, we must not forget the hidden Anthropomorphism when we consider our views, theories and so on about ourselves and the cosmos we happen to live in. thanks.
I think a theory has to be understandable by humans to be a theory. A perfect predicting machine, which could be built by us, is still not a theory if we don't grasp what it's doing. And I think everything we can make sense of can also be considered beautiful or natural.
My intuitive answer, I'll refrain from thinking it over again :)
This question was posted on July 6, 2018. The question mentions the book Lost in Math by Sabine Hossenfelder. In the September 2018 issue of Physics Today, at page 57, is a book review of Hossenfelder's book Lost in Math, by Frank Wilczek of MIT. (The Nobel Prize in Physics 2004 was awarded jointly to David J. Gross, H. David Politzer and Frank Wilczek "for the discovery of asymptotic freedom in the theory of the strong interaction" : https://www.nobelprize.org/prizes/physics/2004/summary/). Anyone following this question might be interested in Wilczek's comments in his book review.
The science journalist John Horgan observed in a Scientific American blog on October 23, 2018 that `Beauty Does Not Equal Truth, in Physics or Elsewhere’. He was prompted to consider the connection between truth and beauty after hearing Sabine Hossenfelder speak about her book Lost in Math which is mentioned in the comments above following the question Do elegance, beauty and naturalness have a role to play in theories in physics? Horgan’s online piece links to an article by Gregory J. Morgan (2013), The Value of Beauty in Theory Pursuit: Kuhn, Duhem, and Decision Theory. Morgan argues that `beauty can play a significant role in the logic of pursuit’: the beauty of a theory might make it worth investigating. One might also argue that the apparent aesthetic beauty of a hypothesis (even if it is subjective) might sometimes at least in turn be based on simple fundamental physical principles, not arising simply or only as a subjective aesthetic opinion.
When studying a physical problem (theoretically or experimentally), you look for the most "natural" variables and assumptions. That is an obvious way in which naturalness appears in our work. This should be reflected, also, in the final product.
What does "natural" mean? For example, suppose I were to develop a theory in which electromagnetism played a role, it would be "natural" for me to include permittivity and permeability terms. However, that is merely modest competence, and nothing remarkable. S what part of "natural" should guide a theory? Similarly, with "beauty". Take the Lagrangian of the standard model. Anyone who thinks that is beautiful has to have an aesthetics upgrade.
theoreticians take for the beginning simple conditions (such as homogenity, periodicity, symmetry,...) and than they abort them one by one; theories become more complex, but they are still simplified (compared to what really happens)
What if Georges Lemaître was an expert in Jain, Hindu or Buddhist philosophy? Would the current main stream Big Bang theory model be different?
I think it would.
Unlike Sir Fred Hoyle, I do not dismiss a good theory, just because it came from a Non-scientist. But in the case of the Big Bang, it looks suspiciously like the "Only Once" creation of the Universe, found in Abrahamic Religions.
It is possible that Lemaître was influenced by his philosophy (not simply being a Christian, but a Jesuit Monk), and found it thus Aesthetically appealing.
The book Lost in Math was referred to when this question was posed. There is a new book review of Lost in Math in American Journal of Physics 87, 158 (2019); doi: 10.1119/1.5086393 by physicist Moira Gresham.
I think we researchers sometimes needlessly ignore outliers in x-y charts. Is it because outliers are Not elegant or beautiful? Maybe this practice should be rethought.
I haven't fully finished Sabine Hossenfelder's book, but I follow her blog and I know what she is saying about her book herself. One of her main points is the same points that -- nearly 15 years ago -- people like Lee Smolin or Peter Woit already complained about: the detachment of a certain strain of "modern" physicists from traditional principles that go back to centuries ago, namely: the notion that the job of natural science like physics is to describe and explain nature by means of models, the language being spoken mathematics. These models need to be constantly put under scrutiny by comparing theoretical predictions and implication with experimental results.
This standard principle unfortunately somehow got lost when string theory gained strength back many decades ago.
What Sabine and also above-mentioned physicists complain about is the complete lack of experimentally verifiable prediction and the detachment from the above-mentioned principle, the bottom line even being that string theory does not even have any predictiveness at all. They complain about the detachment from the above principle of dialectic synergy between theory and experiment, and the replacement of it by a new principle which puts "naturalness" or "beauty" on the throne, and "mathematical elegance" supposedly being one of the guding principles now.
In general, I fully agree with this viewpoint. One must bear in mind however that there seems to be some difference in the pursuit of string theory in the physics community: string theory and supersymmetric models seem to be focussed on primarily in the U.S., whereas my perception is that in Europe, string theory as a field of research does not really play a big role (any more).
With regards to terms like "beauty", "elegance" etc: We must also reflect that physical laws can be formulated in several ways. Depending on the level of abstraction, they can be made to look either "elegant", which means you can't calculate anything at all, but you might be able to see a bigger picture. Or, alternatively, they can be made to look hellishly complicated, which usually is the case when you want to start calculating and finding out results.
I also do not see however that new physics has always been triggered by mathematical elegance in the past. In most cases, mathematics needed to be adapted or even newly created to match the theoretical requirements. Hilbert space theory, distributions, even the whole field of calculus itself has actually been invented by a symbiosis of theoretical physics with mathematics. But as I see it, beauty and elegance were not guiding principles then.
They reasonably play a role. And in a situation like modern physics, where we have no reason to expect that experiments and observations will give much hints for the developments of physics, it is important to discuss what can be reached by fundamental physics at all without its most important guide - the experiment.
The begin of this discussion is, certainly, that the simple, naive, intuitive methods used up to now, where wrong conclusions have been corrected by experiments, have to be criticized. A methodology of natural philosophy - which would be the more appropriate name for physics in a region where experiments are no longer accessible - is essentially something which has to be developed from the start.
I think there is a lot to develop and to argue about. We have, first of all, Popper's criterion of empirical content. It can be applied even where the experiments which would distinguish different theories cannot be done in reality. We have simplicity (Occam's razor). Some aspects are easy, some not. Is it the Lorentz ether (which needs only a 3D reality changing in time) or the Minkowski spacetime (with its 4D block universe) the interpretation which multiplies entities without necessity? There is compatibility with other principles of physics. Like realism, causality in favor of realist interpretations of quantum theory. Or local energy conservation in conflict with GR, or preference for theories with Lagrange/Hamilton formulations. Which of the two is preferable? What about compatibility with common sense?
So, Hossenfelder's book may become a good start for discussing all this.
I think Hossenfelder is wrong about naturalness. It is not really a useful general principle. But we have a quite special situation, where we have QFT which works as an effective field theory, but with the critical distance being unknown, and GR, where the critical distance is known, Planck length. So, we have a plausible candidate theory which works down to Planck length, but can access only much much larger distances. Thus, we have a class of candidate theories at Planck length, and can test only some large distance approximation. In this situation, it makes sense to talk about naturalness.
Absolutely. Nothing is better than the elegance and the beauty in physics. Symmetries are the best thing physics have given.
Symmetries are Good as an Art form, but are they the TRUTH? The REALITY?
Maybe All of Science and Mathematics is a Sort of Art Form, Conveniently Disguised as "SCIENCE".
Maybe it is Time to Discard that Disguise, and Recognize Physics and Science as an Art-form (the Best there is).
With Regard to Symmetries, we have the "Positive" Side, so Where is the "Negative" of that?
Where is NEGATIVE Kelvin Temperature ? (Colder than Absolute Zero)?
Where is Negative Mass and Negative Matter ?
Where is Negative Length ?
How do you Demonstrate to the Five Senses (without any Mathematics) that those Negative quantities REALLY Exist ?
perfect symmetries do not exist in nature- even when they talk about broken symmetries it is flase.
One has to note that physics has - almost unregarded - two major parts: The world of rules and the world of "noise" (hazard): Our real world is dominated by "noise": the shape of leaves, of clouds, of waves on the sea, of stars in a galaxy.
(I learned this by a tete-a-tete with Benoit Mandelbrot). The simple, clear conditions are the exception, even in astronomy. But of course it is the playground where physics came up. The variety of the shape of leaves (of one tree) or of a stony desert may rise beauty feelings and naturalness, but it is not elegant. The laws of deep physics may have some beauty and elegance but they are probably not genuinely natural. I prefer the "oceanic feeling" - valid all over in nature.
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