Thursday, April 30, 2020

Book Review: “The Dream Universe” by David Lindley

The Dream Universe: How Fundamental Physics Lost Its Way
By David Lindley
Doubleday (March 17, 2020)

Let me be honest: I expected to dislike this book. For one because it looked like a remake of Lindley’s 1993 book The End of Physics which I already disliked. Also, physics didn’t end. Worse still, if you read the description of his new book, you can easily mistake it for a description of my book Lost in Math. On the website of Lindley’s publisher you find, for example, that The Dream Universe is about “how theoretical physics is returning to its unscientific roots” and that physicists have come to believe
“As we investigate realms further and further from what we can see and what we can test, we must look to elegant, aesthetically pleasing equations to develop our conception of what reality is. As a result, much of theoretical physics today is something more akin to the philosophy of Plato than the science to which the physicists are heirs.”
However, after reading Lindley’s book, I changed my mind. It is a good book and while I think that Lindley in the end draws the wrong conclusions, it is well worth the read. Let me explain.

First of all, The Dream Universe is dramatically better than The End of Physics. The latter struck me as a superficial and, ultimately, pointless attack on some trends in contemporary physics just because the author had other ideas for what physicists should do. There really wasn’t much to learn from the book. The Dream Universe is instead a historical analysis of the changing role of mathematics in the foundations of physics and the growing divide between theory and experiment in the field. In his new book, Lindley makes a well-reasoned case that something is going badly wrong.

Lindley’s book of course has some overlap with mine. Both discuss the problem that arguments from mathematical beauty have become widely accepted among physicists even though they are unscientific. But while I wrote a book about current events with only a short dip into history, and told this story as someone who works in the field, Lindley provides the perspective of an outsider, albeit one who is knowledgeable both about physics and the history of science.

As Lindley tells the reader in the preface, he started a research career in physics, but then left to become a science writer. The End of Physics was his first book after this career change. He then became interested in the history of science and wrote several historical books. Now he has taken on the foundations of physics again with a somewhat more detached view.

The Dream Universe begins with some rather general chapters about the scientific method and about how scientists use mathematics. You find there the story of Galileo, Copernicus, and the epicycles, as well reflections on the conflict-loaded relation between science and the church. Lindley then moves on to the invention of calculus, the development of electrodynamics, and the increasing abstraction of physics, all the way up to string theory and the idea that the universe is a quantum computer. He lists some successes of this abstraction – notably Dirac’s prediction of anti-matter – before showing where this trend has led us: To superstrings, multiverses, lots of empty blather, and a complete lack of progress in the field.

Lindley is a skilled writer and the book is a pleasure to read. He explains even the most esoteric physics concepts eloquently and without wasting the reader’s time. Overall, he maintains a good balance between science, history, and the lessons of both. Lindley also doesn’t leave you guessing about his own opinion. In several places he says very clearly what he thinks about other historians’, scientists’, or philosophers’ arguments which I find so much more valuable than pages of polite tip-toeing that you have to dissect with an electron microscope to figure out what’s really being said.

The reader also learns that Lindley’s personal mode of understanding is visualization rather than abstraction. Lindley, for example, expresses at some point his frustration with a professor who explained (entirely correctly, if you ask me) that “a tensor is an object that transforms as a tensor” with a transformation law that the professor presumably previously defined. Lindley reacts: “Here is how I would explain a tensor. Think of a cube of jellylike material.” It follows two paragraphs about jelly that I personally find entirely unenlightening. Goes to show, I guess, that different people prefer different modes of explanation.

In the end, Lindley puts the blame for the lack of progress in the foundations of physics on mathematical abstraction, a problem he considers insurmountable. “The unanswerable difficulty, as I hope has become clear by now, is that researchers in fundamental physics are exploring a world, or worlds, hopelessly removed from our experience… What defines those unknowable worlds is perfect order, mathematical rigor, even aesthetic elegance.”

He then classifies “fundamental physics today as a kind of philosophy” and explains it is now “less about a strictly rational understanding of the universe and more about finding a scenario that we deem intellectually respectable.” He sees no way out of this situation because “Observation, experiment, and fact-finding are no longer able to guide [researchers in fundamental physics], so they must set their path by other means, and they have decided that pure rationality and mathematical reasoning, along with a refined aesthetic sense, will do the job.”

I am sympathetic to Lindley’s take on the current status of research in the foundations of physics, but I think the conclusion that there is no way forward is not supported by his argument. The problem in modern physics is not the abundance of mathematical abstraction per se, but that physicists have forgotten mathematical abstraction is a means to an end, not an end unto itself. They may have lost sight of the goal, alright, but that doesn’t mean the goal has ceased to exist.

It is also simply wrong that there are no experiments that could guide physicists in the foundations of physics, and I say this as someone who has spent the past 20 years thinking about this very problem. It’s just that physicists are wasting time publishing papers about beautiful theories that have no relevance for nature instead of analyzing what is going wrong in their discipline and how to make progress.

In summary, Lindley’s book is not so much a competition to Lost in Math as a complement. If you want to understand what is going wrong in the foundations of physics, The Dream Universe is an excellent and timely introduction.

Disclaimer: Free review copy.


  1. I'll have to check it out!

    By the way—"an end to a means" should be "a means to an end"

    1. Thanks for spotting this, I have fixed this now.

  2. "physicists have forgotten mathematical abstraction is an end to a means.."
    I must suppose that was deliberate phrasing, but still don't understand..

    1. That was a typo, sorry. I have fixed this now.

    2. I disagree. Turning the phrase on its head mimics the mistake of writing down a beautiful equation and then trying to force nature to conform.

  3. Possible typo: "but that physicists have forgotten mathematical abstraction is an end to a means, not an end unto itself"

    Did you mean to write, "a means to an end?"

  4. I loved physics in high school. But the last teacher referred my questions of understanding to ... philosophy. To my question "what is an electric field" he answered "it is a vector field" (probably as opposed to a scalar field). Instead of explaining the concept invented by Faraday and implemented by Maxwell. Perhaps what David Lindley criticizes is also this way formal and void of understanding, where mathematical structures replace an picture of reality to which they are applied.
    Speculative physics of our time has not only privileged formal beauty over logical and conceptual consistency. By devaluing understanding it devalued the explanation of nature, the first goal of classical science. For example, it is well known that Newton was the first to criticize the lack of physical significance of his law of instantaneous attraction at a distance. Who now thinks that good physics should be able to explain the laws?

    1. Let me cite Heisenberg (from Physics and Beyond):
      ,,‘Understanding’ probably means nothing more than having whatever ideas and concepts are needed to recognize that a great many different phenomena are part of coherent whole. Our mind becomes less puzzled
      once we have recognized that a special, apparently confused situation is merely a special case of something wider, that as a result it can be formulated much more simply. The reduction of a colorful variety of phenomena to a general and simple principle, or, as the Greeks would have put it, the reduction of the many to the one, is precisely what we mean by ‘understanding’. The ability to predict is often the consequence of understanding, of having the right concepts, but is not identical with ‘understanding’.”

      An electric field is just a concept, a part of greater structure that is called Classical Electrodynamics.
      Are we forced to think using this concept? No, but we used to and not without a reason.

      ,,privileged formal beauty over logical and conceptual consistency''
      You are on dangerous ground, logical consistency of mathematics itself is something we can postulate and belive in, but not something we can prove (to be more specific: consistency of ZFC ca not be demonstrated in ZFC, at least if ZFC is consistent).



    2. If mathematical aesthetics are a poor replacement for measurement, how much worse are narrative aesthetics!

    3. Jean-Paul,

      Your teacher did not serve you well.

      Here is Dick Feynman's famous comment (section 20-3) on how to think about electric and magnetic fields:

      >"I have no picture of this electromagnetic field that is in any sense accurate. I have known about the electromagnetic field a long time—I was in the same position 25 years ago that you are now, and I have had 25 years more of experience thinking about these wiggling waves. When I start describing the magnetic field moving through space, I speak of the E- and B-fields and wave my arms and you may imagine that I can see them. I’ll tell you what I see. I see some kind of vague shadowy, wiggling lines—here and there is an E and B written on them somehow, and perhaps some of the lines have arrows on them—an arrow here or there which disappears when I look too closely at it. When I talk about the fields swishing through space, I have a terrible confusion between the symbols I use to describe the objects and the objects themselves. I cannot really make a picture that is even nearly like the true waves. So if you have some difficulty in making such a picture, you should not be worried that your difficulty is unusual."

      In some purely logical sense, it is just a fact that every electrically charged object, if placed at a certain position in space, will feel a force proportional to its charge and in the same direction as any other electrically charged object were it to be placed at that point. We take the direction of the force, and assign it a magnitude equal to F/q, and we call it the "electric field."

      Pure logic.

      But no one really thinks that way.

      Humans are visual thinkers, and we like to anthropomorphize things (notice that a bit earlier I said the charge "feels" a force!).

      So we come up with mental visualizations to help guide us. Every good physicist I know does this. I don't think it is possible to grasp physics without doing it. (I get the feeling that some mathematicians do not do this, though I am sure that Michael Atiyah, for example, did.)

      So we "see" atomic orbitals, we use spacetime diagrams in relativity, etc.

      The problem is how to train and control these visualizations while still making effective use of them.

      So, in the end, the electric field just is. But as humans, we have to do the sort of thing Feynman describes or we will never see it at all.


    4. PhysicistDave,
      I know that since standard MQ, physicists consider the images they make of things as simple crutches convenient for intuition, and not real referents of concepts of theory. Considering all the conceptual problems of QM, this situation is probably not definitive.
      In RS, I would not say the same thing of the diagrams of ET which correspond to a precise concept: the relativity of the simultaneity. And in GR, the notion of space-time curvature is not a vague image, it is a concept supposed to refer to a reality: a space-time really curved by the masses. But of course, we can never explain "how physically" the masses curve space-time: it's just a geometric formulation of gravity. As said Einstein, GR is a theory in principle, not a constructive theory that is to say that mechanically explains nature. And Einstein said he preferred the seconds, just as Newton would have preferred to explain its gravity.
      For the notion of field, the fact that Feynman did not have a clear idea of ​​it does not imply that one must eternally stop there. In Lumière et Matière, Feynman brilliantly presents the rules of the EDQ by clarifying that he enjoys telling people that there is nothing other than these rules and that all this makes no sense. It doesn't make me happy.
      By contrast, Faraday, inventor of the concept of field, conceived it as a "modification of physical space" by the source of field. A change which in turn, far from the source charge, changed the behavior of the test charges. The modified space acted as a mediator of the interaction at a distance. As for Maxwell's electromagnetic waves, they were waves of modification of this medium. As the photons in FTQ have become excitations of the quantum vacuum.

  5. The bottom line is what the gate-keepers want - the journal editors, reviewers, university officials. I suspect that the interest in math is partly due to the amount of work involved. It's easier to sit at one's desk and do math than it is to go out and gather good data. The gate keepers benefit by maintaining the primacy of math because they need not work very hard to maintain their status.
    That seems to be happening in economics as well.
    A possible response to this problem is to require papers to detail the assumptions made when doing the math and to extensively analyze the likelihood that the assumptions are correct.
    In the end, though, a lot of math research has led to important empirical findings; the problem is that this is not happening as much as it used to.

  6. Sabine, the last sentence of your post, you give the title of the book as "The Dream University".

    1. hahaha, thanks for catching this, I have corrected this

  7. It's the wrong abstractions that are the problem, not abstractions per se. Science is fundamentally about abstraction of particulars. There's no need for physics to be aligned to naive realism or something like that. Our cognitive systems evolved to survive, not to understand fundamental physics.

    To me science is a profoundly "unnatural" activity. It requires a level of sustained surplus which wasn't around in our evolution. This means we have a natural tendency to fall back on more basic human propensities like being seduced as a means of seducing others with ideas. Staying on the science track requires discipline and that discipline is evidence.

  8. I very much liked Lindley’s The Dream Universe. The early chapters made me think of a lighter version of Boorstin's classic The Discoverers, still one of my favorite books. Lindley provided just enough human and historical anecdotes to keep the story interesting, without getting bogged down in minutia. At one point he just said "That's enough of that," and moved on to the next topic. I found his ability to avoid getting sidetracked admirable, and it kept the book moving forward at a nice clip.

    My enjoyment was enhanced by my agreement with many of the points Lindley made along the way. More than once I found myself making pointed mental exclamations of "Yes!" for his comments on weaknesses that too often are given a free pass by both the physics community and the scientific press.

    Given all that, I was beginning to think Sabine might have been exaggerating a bit about her disappointment with the final section of Lindley’s book. She was not. Here’s the last paragraph in the book:

    "As an intellectual exercise, fundamental physics retains a powerful fascination, at least for those few who are fully able to appreciate it. But it is not science. It’s not that I think such research should cease altogether. But I wish its practitioners would take the trouble to ponder where they are going, and to what end."

    Human minds are tribal minds, mortal minds, primate minds, often selfish minds. We live and perceive as parts of societies and subcultures that almost invisibly convey to us what is or is not allowable, plausible, acceptable, laughable, no matter what our mathematical and logical sides may say. The inhibiting impacts of such societies upon their even their greatest of minds are a powerful factor in why the shining islands of innovation in history are too often separated by multi-generation deserts of intellectual stagnation. But simply to give up, as Lindley seems to do in the above paragraph, is to choose to be the willing victim of the very trap that such stagnation creates.

    Far from being fatalistic, I am optimistic that after a half century of string-lash stagnation there are small signs we are about to land on one of those islands. If so, it won’t be new math that leads the way, but a renewed focus on the meaning and mysteries already contained in the huge set of very solid experimental data already available to us. We are fools indeed if we truly believe we have mined every bit of insight out of that massive set of data.

    We also have new tools. If nothing else, progress in artificial intelligence — in new machines that are slowing getting closer to human-level thinking — could provide new perspectives to help us break through. Radical insights are easier for AIs, since AIs do not worry about reputations, raised eyebrows, showing proper respect for leaders, funding, or even their own mortality. When properly programmed the one and only thing they do is look for the simplest, most compact programs capable of explaining validated experimental results. Even a few good AIs thus might help break fundamental physics out of its desert, and get it moving in new directions.

    I can even provide a specific example of this dissonance, of this odd mismatch between human and machine analytical approaches. If you feed the electric and color charge data for the fundamental fermions into a data-reduction AI, will you get SU(3) back out? You might expect so, but the answer is no. Driven solely by representational efficiency, an AI will instead combine unit color charges with and -⅓ electric charges to create three new, orthogonal axes of unified electric-color charge from which all known fermion charges can then be constructed, including that of electrons. Humans do not easily spot this unexpected simplification, since they know that the electric and strong are separate forces.

    But AI? AI don’t care.

    1. "Human minds are tribal minds, ..." etc.

      I have lived in Australia for some time, have been there also on various visits. I think "The Dreamtime" is a nice introduction to how it all began.

  9. I would love to have you read and review Tom Van Flandern’s 1993 book “Dark Matter, Missing Planets & New Comets” and his later article on the "Structure of Matter". Most people remember him, if at all, for his "wild" theory about life on Mars. That's really too bad, since that was not important.

    His theory, which he called the "Meta Model" and presented in the first five chapters of the book, is what he should be remembered for. He pointed the way towards a model for universal gravity that could get physics "unstuck". And it has nothing magical or obscure about it.

    With more mundane aspects of the theory, he successfully predicted a number of generally unexpected things, like rolling boulders on asteroids and the timing of earth's encounters with comet debris.

    His was a voice that should not be overlooked and I think we should hear more about his models!

  10. Subtle point of grammar. Ceased takes the infinitive, i.e. "ceased to exist". Had you used the verb stopped you would have been correct, i.e. "stopped existing".

    1. Hi Norman,
      Thanks for the careful reading; I have fixed this.


  11. It is true that the object of study of physics is not mathematics; but when one does not have an intuition about the physical phenomenon, it is the mathematics that tells the experimenter "where" and "what" to look for. On the other hand, it seems to me that aesthetics is an intuition about general frameworks and works as a type of discrimination, an economic way of thinking; It saves us from analyzing the "absurd", "the impossible", and the less probability, this has worked so far; but who knows?

    1. Luis,
      Rationalism is not at all against aesthetics or creative intuition. It simply orders the criterion of logical coherence and of coherence before any other criterion, after the experiment or when the experiment cannot discriminate. Only rationalism can discriminate between science and pseudo-science

    2. Juan Pablo thank for your comment;
      It seems to me that what you are saying is some kind of autism; there is no such rationality independent of intuitions and emotionality; When any physical phenomenon is studied, it is placed in general frames (x, y, z, t), the fact that there is no complete physical description of those general frames tells me that they are still very intuitive; But if the root of each physical concept is reached, many are supported by intuitions and even conventions.

  12. I believe Sheldon Glashow was thinking along the lines of unifying the fermions with that 3 orthogonal electric-colour charge idea in the 1970's. I don't know why physicists haven't persued this approach. It is a simpler way to classify fermions and gluons too. But I wouldn't know, I am only a simple mathematician.

    1. John, you are exactly correct -- well done!

      I was hoping someone would know that fascinating bit of history. Glashow's cube (as I call it; it's been rediscovered multiple times in the decades since, including by me) is not well known because Glashow posed the idea only once in print, as a cube-shaped "mnemonic" for recalling the charge combinations in each of the three fermions families. It is surprising that he never published a paper about it, since he had clearly put a fair bit of thought into the idea. It may even have been an observation by one of his close peers, since Glashow never really seemed to take ownership of it.

      As to why such a rather remarkable pattern was never explored or expanded upon, I can think of at least five reasons:

      (1) Glashow's introduction of the cube practically screamed in capital letters "Do not take this seriously!" Coming from Glashow, that had huge weight. (Tribalism, anyone? AIs don't care.)

      (2) The existing standard model use of two forces color not only worked, it was... well... just way cooler than a boring 3-space generalization of Maxwell's charge displacement. SU(3) was a bit like a badge of honor, a Mystery not easily understood, and thus a source of both pride and a feeling of accomplishment. (AIs don't do this.)

      (3) Electric charge had been known literally for millennia, and so was obviously more fundamental than this grotesquely confined color-charge newcomer. (AIs don't care about history.)

      (4) The standard model of separate electric and color charges gives correct results, so why rewrite it? Such an effort might well take decades, yet would presumably give the same results. (AIs don't worry about rework. Also, simpler models often turn out to be more experimentally predictive.)

      (5) Burrowing even deeper: the neutrino and antineutrino both have zero charge, and so are degenerate -- have the same location -- in Glashow's unified charge 3-space. You have to go to W particle bridge vectors between the two stacked Glashow cubes -- creating a 4-cube really -- to resolve this degeneracy.

    2. @Terry Bollinger
      It is quite remarkable. I see the task of the philosopher of science as that of a "semantic compiler" which ideally leads him to ask questions that we do not ask, to be surprised where on is not surprised (or more). Which, like AI according to you, frees us from our habits of thought and prejudices. Including aesthetic prejudices, those that result from what is contingent in the history of a science, etc.
      In France, we have the great figure of Bachelard who spoke of epistemological obstacles, evidences and / or categories of common sense that the mediation of science leads to overcome. But I believe that there exists in science a category of "non-bachelardian" obstacles: "learned obstacles", those that a science builds itself, and that it will therefore find it harder to get around. IMO that is the first reason Fundamental Physics Lost Its Way.

    3. Jean-Paul,

      Thank you for thought-provoking response and insight, new to me, that the AI goal of isolating analytics from the inadvertent influences of the mundane is the shared objective sought by many philosophers for centuries. That is a humbling and powerful lesson, and I accept it gratefully.

      My interest in philosophy was recently renewed by my reading the “Approximation” chapter in Ronald Green’s Time to Tell. If there was any room between what Green was saying and what I said a few days ago here in Backreaction, I could not detect it. Green is coming from such a different analytical background that I have difficulty truly comprehending how he came to his conclusions.

      Since the emergence of very similar results from diverse starting assumptions is an excellent indicator (heuristic) both of the veracity of the co-emergent result, and of the probability that there is shared deep infrastructure behind the co-emergence, the fact that philosophy and physics can lead to such similar outcomes makes this co-result worthy of closer inspection. (Incidentally, the diverse-roots, shared-outcome heuristic is also used in the derivation of DNA trees-of-life.)

      My own views have changed reluctantly and over decades. Regarding the issue of Approximation, the largest driver has been trying to understand the relationship between the trio of fermion particle families, quantum number conservation, and the fabric of spacetime. My current view is that Platonic perfection is not just inherently non-real, but also excruciatingly counter-productive, inciting immense amount of non-productive human and computer work, such as when quantum simulators begin with the assumption that they must sum infinities of infinitesimal states to predict the future. The photographic negative of that is to make only some finite set of environmentally and spacetime-imposed superselection rules real. If this negative-image view of quantum mechanics is correct, it should have experimentally accessible consequences. A pointed example is that negative-image quantum mechanics, dark wave functions, would mean that quantum computer can never be anything more than classical wave computing done in the most cumbersome and error-prone fashion imaginable, by using tiny statistical waves that require huge levels of sampling. Since wave computing is hugely cheaper, faster, and easier using big bright lasers over cold quantum atoms, an proof that wave functions are dark functions — in effect that the computational power of MWI is non-existent because no such worlds exist — would suggest that the research investment strategy in quantum computing is in need of a major overhaul in favor of laser-based wave computing.

      I will look up Bachelard. Respecting Sabine’s desire to keep off-topic threads from overloading her review task: If you know of any good paper titles for the learned obstacles, you might try emailing them to me.

      My next post here will expand on two obstacle examples: Glashow and Mach.

    4. Glashow and Mach provide two examples of how social norms and community-shared perceptions can unduly influence even the greatest minds in physics.

      The Glashow cube example is an extraordinary bit of history, particularly since he published it at the beginning of what turned into a multi-decade effort to use increasingly large symmetry groups and particle accelerators to unify the electromagnetic and strong forces. Glashow seems to have realized there was at least the possibility of an extraordinarily simple resolution to electric-strong unification, so why did he not encourage at least a modicum of research into this a persistent and certainly relevant pattern in the particle data?

      My suspicion, nothing more, is that the Glashow cube approach to force unification was just too radical to be acceptable at that time, particularly in the face of all of the very real and extremely impressive successes particle physics was having on other fronts (the Standard Model). Glashow at least captured the idea, even if only in the form of a trivial mnemonic. Yet at the same time he used a style of presentation that safeguarded him from accusations of trying to diminish the importance of the most sacred of all forces, excepting perhaps gravity. Everyone knew, at an almost religious level of perception, that the electromagnetic force was cleaner and thus surely more “fundamental” than the new, ugly, and weirdly complicated strong force.

      (If my use of the word “religious” seems unfair in this context, please consider these two assertions: “God said, Let there be light: and there was light,” from the Hebrew scriptures; and “Matter and antimatter particles are always produced as a pair and, if they come in contact, annihilate one another, leaving behind pure energy,” from CERN’s matter-antimatter asymmetry web page. At least at a reflexive perception level, most physicists interpret “pure energy” to mean pure electromagnetic energy: light, with everything else as “details”.)

      Another example is of the damage that can occur when even brilliant minds are stubbornly resistant to self-examination is Mach. A very smart man by all accounts, Mach strangely was also one of the last prominent scientists to not believe in atoms. Thus when Mach encountered the brilliant and deeply insightful work of Boltzmann, a scientist who not only accepted atoms as a given but who went so far as to propose that it is the irreversible chaotic behavior of atoms, their thermodynamics, that defines the arrow of time, Mach responded with unrelentingly venomous career and personal attacks on Boltzmann, ultimately driving Boltzmann to suicide. All of this happened because Mach was so deeply confident in his out-of-date and flatly wrong understanding of matter that he could not even contemplate the idea of a thoughtful discourse on the subject with Boltzmann. He instead “gut reacted” to Boltzmann as a heretic, a non-believer in Mach’s deeply held preconception that science should be so perfect and Platonic that not even the unseemly granular coarseness of atoms could be allowed.

    5. @Terry Bollinger
      You understand that i writed "that the AI ​​goal of isolating analytics from the inadvertent influences of the mundane is the shared objective sought by many philosophers for centuries". Uh, its just my vision of what philosophy IN sciences ought to make, and what i try to make on particulars points of modern physics.

      I don't understand most of the paragraph beginning with "My own views have changed". Is what you call "platonism" the realism in mathematic's philosophy ? Or if you mean that all nature's knowledge is approximative , never perfect , obviously i agree : we will never be able to know the ultimate reality - the thing in itself of Kant- But when we calculate an integral, it seems to me that we are simulating an infinite sum of infinitesimal elements of an object supposed to be continuous, whatever it is in reality: it's just a technique that works. And i don't see what is your "negative-image view of quantum mechanics" where wave functions are "dark functions"

      For Bachelard's idea of epistemological obstacle, see The Formation of the Scientific Mind. Clinamen, Bolton, 2002. Among its examples, the old idea of "phlogiston" as the substance of fire.

      For the Glashow's cube, excuse my ignorance. Is it linked with GUT (which has been rules out by the no decay of proton) ?

      For "pure energy viewed as light" : de Broglie looked photon as a particle with a extremelly small proper mass. Yes, "pure energy" is not the dual of "pure mass" . In FQT, all is excitation of quantic void.

      For Mach, his prejudice was against "Metaphysics" , that meaned for him assumption upon unobservables realities, even with observables effets. In his time : atoms, Newton's absolute space to explain fictive strengths (bucket experience) - so he proposed his own relational theory of inertia. What guided him was fierce positivism, but not platonism as far as I know.

  13. Judging from the popular writings of many scientists there are two issues. The one most directly related to math is a kind of mathematical Platonism (which to be blunt I suspect if nonsense even for mathematicians,) where the equations are treated as the technical or formal cause. But there are no more efficient or material causes than there are final causes.

    The other issue is a commitment to the notion that science is not about describing reality, just correlating outcomes of experiments. So far as I can tell, this means progress in science is largely driven by the technology of instruments. And the mathematics of fundamental physics has a relationship to experiment as formalized mathematical proofs to applied mathematics. Since it seems to me applied mathematics is the engine even of math this seems to pose a problem.

    What's not clear to me is whether a commitment to quantum field theory/rejection of general relativity requires antirealism or the quasi-Platonic view or least if the person is to be consistent?

  14. Sabine wrote:
    " Lindley, for example, expresses at some point his frustration with a professor who explained (entirely correctly, if you ask me) that “a tensor is an object that transforms as a tensor” with a transformation law that the professor presumably previously defined."
    I find the professor's definition awful. Would it be good to define a matrix as a 2-dimensional array that satisfies some predefined transformation law? Of course not. Once you know what kind of mathematical object the matrix is supposed to represent (for instance, a linear map or a quadratic form) then the transformation laws resulting from a change of basis can be derived naturally. The same goes for tensors.

    1. Pascal,

      What you say is just wrong. You cannot derive the properties of a physical theory (coordinate invariance) from math alone. How a tensor transforms is a matter of definition, and this definition is there because it agrees with observation. You are incorrectly using knowledge that you *already have* -- after Einstein sorted it out and his theory was confirmed -- to then declare you magically know the transformation behavior of a tensor. The professor's definition is, needless to say, entirely correct and in fact I do not think it can be improved on without making it simply wrong.

      "Would it be good to define a matrix as a 2-dimensional array that satisfies some predefined transformation law? "

      No, because that's not what a matrix is, it's just the wrong definition.

    2. Well, I was thinking of a tensor as an abstract mathematical object. From this point of view, the transformation laws *can* be derived from math alone, in more or less the same way as for a matrix. Of course, what cannot be derived from math alone is that these abstract mathematical objects are useful to describe physical reality.

    3. Pascal,

      "Well, I was thinking of a tensor as an abstract mathematical object. From this point of view, the transformation laws *can* be derived from math alone"

      No, you can't. The transformation law is part of the definition what it means to be a tensor. You do not derive it, you postulate it.

    4. I suspect that we may not be talking about the same thing. Let’s try to recall what are the transformation laws for matrices, that is, 2-dimensional arrays. If a matrix A is viewed as a linear map from a vector space E to itself, the transformation law under a basis change is that A is replaced by P^{-1}AP, where P is the transition matrix from one basis to another.

      Let us now interpret a symmetric matrix A as representing a quadratic form (x -> x^tAx).
      We obtain a different transformation law. Namely, under the change of variables x = Py, the new matrix of the quadratic form is P^t.A.P.

      Now, back to tensors. A tensor is a multidimensional array (of dimension 3 or more). All I’m saying is that, depending on the kind of mathematical object represented by the tensor, one can figure out transformation laws under a change of basis (like in the matrix case, but the formulas become a bit more complicated). This should be rather uncontroversial.

    5. Pascal,

      "A tensor is a multidimensional array (of dimension 3 or more)."

      A tensor is a multidimensional array with a certain transformation behavior under basis change. It is simple enough to write down multidimensional arrays that do not transform like tensors. I'm sure if you think about it for a moment examples will come to your mind.

    6. Sabine, may I appeal to wikipedia’s authority here? In the article on tensor products, a tensor is defined as an element of the tensor product of vector spaces. In order to write down the corresponding array, we need to fix bases of the vector spaces. If the bases are changed, we can compute the elements of the new array by expressing the elements of the old basis in the new basis. This transformation is certainly not presupposed, it is derived from the familiar rules of linear algebra.

      But we are warned in this article that the term tensor “refers to many other related concepts as well”.
      So let’s look at these concepts at wikipedia’s main article on tensors. They present the concept of tensors from several points of view. As a physicist I suppose that you are most familiar with the approach given in their section on tensors “as multidimensional arrays” (where by the way they give the example of the transformation law for a matrix like I did in my previous comments). In that section they define tensors as multidimensional arrays endowed (the horror!) with certain transformation law. This point of view is attributed to Ricci. I very much prefer the somewhat more abstract approaches in the next 2 sections (based on multilinear maps and tensor products) because it looks much less arbitrary. You will see that no transformation laws is postulated in these 2 sections, and they explain the correspondence with the physicists’ point of view. In particular, indices are covariant and contravariant depending on whether you take the tensor product with a vector space or V or its dual V^*.

    7. Pascal,

      Needless to say, there are many different ways to formulate the same definition. And they all agree that the definition of a tensor includes the transformation behavior. If you define a tensor as a multilinear map on a space which transformation behavior you know, instead of an array with a certain transformation behavior, you have shoved the transformation behavior of the tensor in the requirement that it is a multilinear map over a space with elements whose transformation behavior you know and the requirement that this definition is coordinate-independent.

      Look, let me put this differently. If I give you a 4x4 array of physical properties that you can measure in any reference frame that you want, how do you decide whether it is a 2-tensor? Answer: You look whether it has the right transformation behavior.

    8. > how do you decide whether it is a 2-tensor?
      > Answer: You look whether it has the right
      > transformation behavior.

      That may be how you decide that you want to model the physical properties as a tensor, but it isn't the way a mathematician would define what a tensor is.

    9. As we already noted, there are many different ways to write down the same definition. And both Lindley and are are talking about physics, not math.

    10. Pascal and David Marcus,

      Are you acquainted with how tensors are defined in MTW?

      They define tensors as a "machine" (i.e., a function or transformation) that takes as inputs a finite number of vectors and/or 1-forms (i.e., covectors), is linear in each input, and produces a number as an output. This is basically the second Wikipedia definition, though I think MTW makes it more concrete.

      Perhaps you guys find that more "mathematical"?

      I myself did find MTW's explanation easier to grasp than the traditional definition that Sabine is appealing to.

      The two definitions are, in fact, trivially equivalent, and to use tensors effectively, you really need to be comfortable with both definitions and the fact that they are indeed equivalent.


    11. I am not familiar with MTW but yes, what you're saying looks very much like the Wikipedia definition based on multilinear maps.

    12. Pascal,

      "MTW" is the classic text on GR, Misner, Thorne, and Wheeler's Gravitation, usually just called MTW.

      For those interested in books on these topics, here is a good annotated list of books on GR and differential geometry: I own a number of them and generally agree with the comments on the list. Two additional books I would have put on the list are John Wheeler's pop-sci (but serious) A Journey into Gravity and Spacetime and the second edition of Adler, Bazin, and Schiffer's Introduction to General Relativity.

      (I think you yourself actually sent me to Adler, Bazin, and Schiffer a while back for Buchdahl's theorem, so I know you yourself are familiar with many of these: thanks, by the way, for that reference -- it was very useful.)

      Full disclosure: I studied under Kip Thorne and Dick Feynman, both doctoral students of John Wheeler. Wheeler was a bit of a "mad genius" in twentieth-century American physics: his contributions ranged from nuclear physics to General Relativity to the foundations of quantum mechanics.

      I had a chance to attend a guest lecture he gave, but, as a lowly undergrad, was too shy to chat with him (I wandered in to the lecture hall by accident in mid-afternoon, when Wheeler was drawing some diagrams on the board for his talk -- a perfect chance to meet the guy, but I was too nervous).

      All the best,


    13. ""MTW" is the classic text on GR, Misner, Thorne, and Wheeler's Gravitation, usually just called MTW."
      OK, I had missed that reference! I did google "MTW" and "tensor", and ended up instead on slides on optimal transportation and a conjecture by Cédric Villani...

    14. Baez' reading list includes Gregory Naber, "Spacetime and Singularities: An Introduction" (without description). I have studied that book, as it is very good preparation for Hawking and Ellis. A relevant reminder from Bishop and Goldberg: "Tensors generally admit several interpretations in addition to being multilinear function with values in R. For tensors arising in is rarely the case that the multilinear function interpretation is the most meaningful in a physical or geometric sense." (page 79, Tensor Analysis on Manifolds). Bishop and Goldberg, as MTW write, "is the number one reference." (page 196, Gravitation).

    15. Gary Alan,

      Thanks for the suggestion: Hawking and Ellis is one of those books (another is PCT, Spin and Statistics, and All That) that I have owned forever and that I can make sense of an individual page. But it's never quite "come together" for me. Perhaps the book you point to will help.

      You quoted:
      >" is rarely the case that the multilinear function interpretation is the most meaningful in a physical or geometric sense."

      Yeah, it's the problem of math books that start out "A vector space is a set V and two operations + and x such that..."

      Perfectly logical and perfectly correct... but also perfectly impenetrable and perfectly useless unless you already know what vectors are!

      I believe tensors started out in the theory of elasticity (stress and strain and all that) and are probably fairly intuitive there. But for people who start blind with any of the mathy definitions, yeah, it's rough.

      I'm working on a monograph explaining GR through the Schwarzschild solution without any use of tensors: it takes off from Wheeler's emphasis on tidal forces -- you can do an awful lot in GR without the high-powered mathematical apparatus.

      Of course, you cannot fully master GR without tensors, but I think starting out with the more physical approach may provide more motivation for the tensors in the end.

      I think this might have been Wheeler's initial idea for MTW, but MTW tries a bit too hard to be all things to all people. I think of the book I am working on as sort of the natural follow-on to Wheeler's A Journey into Gravity and Spacetime

      I think Wheeler himself thought of his and Taylor's Exploring Black Holes: Introduction to General Relativity as the natural follow-on, so maybe I am aiming to be volume 3!

      Alas, I do not think I write as well as John Wheeler, but I'm giving it a shot.


    16. An idiosyncratic physics monograph is Brillouin's "Tensors in Mechanics and Elasticity" (1964), available electronically at Brillouin says: "The study of matrices has some impressive analogies with the tensor calculus but also some essential differences." (page 44). Many years ago I enjoyed the Schaum's Outline Series, working my way through Kay's Tensor Calculus. Amusingly, Kay writes: " While physicists readily recognize the importance and utility of tensors, many mathematicians do not." (1988). I wondered if that was ever so.

    17. These days, tensors have become all the rage in fields such as signal processing, statistics and machine learning. And they were long ago recognized as important in computational complexity theory thanks in particular to the pioneering work of Volker Strassen. So tensors have definitely made it out of physics!

    18. @PhysicistDave

      He might have called them a 'machine' but he named a whole chapter after differential forms - which are just those anti-symmetric tensors.

      Personally speaking, whenever someone calls something a machine in this kind of context it generally means that they are encouraging you to ignore the full formalism and focus on using it. That is learn by use rather than learn by understanding - which might seem a bit wrong-headed in a discipline which is all about understanding, but given that so many things we learn, we learn by use then I think it's about right given that physics is about physical things and so - things we use.

      By the way, tensors have moved on since MTW were writing. There are such things as tensor and multilinear categories used in TQFTs ...

    19. Mozibur wrote to me:
      >He might have called them a 'machine' but he named a whole chapter after differential forms - which are just those anti-symmetric tensors.

      I'm afraid you are sorta missing the boat here!

      The most important tensor in GR is not anti-symmetric -- I'm talking about the metric tensor g, of course.

    20. @Physicist Dave

      The metric tensor is just the generalisation of the inner product in Euclidean space. It's an obvious generalisation to make once you have the notion of a manifold.

      Actually, according to Chern, Riemann orginally went much further, and defined what is now called Finsler geometry, that is defining a geometry via an arbitrary norm on the tangent manifold. Not all norms are induced by inner products.

      As the inner product is symmetric then obviously the metric is also symmetric.

      What I find interesting is that mathematicians focus on anti-symmetric tensors, that is differential forms. But since arbitrary tensors can be decomposed into anti-symmetric and symmetric tensors then shouldn't there also a calculus of symmetric tensors? And what distinguishes the anti-symmetric part from symmetric tensors.

      Well, the anti-symmetric tensors are topological in the way that the symmetric tensors are not. It turns out that you can characterise the topology of a manifold, it's singular cohomology, through anti-symmetric tensors, which form the smooth cohomology, that is the de Rham cohomology.

      In fact, Heydari, Boroojerdian & Peyghan have developed an analogue of the differential forms that they call the symmetric calculus. They define symmetric analogues of the Lie bracket, the Lie derivative and the differential which they call - surprise - the symmetric bracket, the symmetric Lie derivative and the symmetric differential.

      Moreover, they prove that the symmetric differential of a 1-form on a Riemannian manifold vanishes iff the form is a Killing cofield.

      Of course Killing cofields/fields form the Lie algebra of the group of isometries of the Riemannian manifold.

      This just goes to show that whilst the anti-symmetric calculus is about topology, the symmetric calculus is about geometry - aka the metric.

      By the way, how did Wheeler come across to you? According to Susskind when he first walked into his office he looked like a waspish conservative, which given Susskinds politics, put him off immediately. (He was originally going to walk into his office looking like a plumber - not unusual - since at the time he was working as a plumber). But he was just mesmerised by Wheeler talking about physics ...

    21. Mozibur said: "What I find interesting is that mathematicians focus on anti-symmetric tensors, that is differential forms."
      Symmetric tensors and homogeneous polynomials are essentially the same mathematical objects (same relation as between a quadratic form and a symmetric matrix). So it would be only a slight exaggeration to say that all of algebraic geometry is about symmetric tensors...

    22. @Pascal:

      They are used for different purposes so they are actually different.

      We use numbers to measure both mass and length, but no-one would mistake a million kilograms for a million metres.

    23. The reason mathematicians prefer the conceptual definition of a tensor is that it has higher explanatory power. It explains why tensors transform the way they do.

  15. I think the problem may not be that mathematical, though I suspect we are getting some razzle dazzle from that. After all, Einstein appealed to Riemannian geometry to fix relativity with gravitation, and that was 60+ year old mathematics. By our standards things such as Thurston’s theorems, Bott periodicity and the rest would be of such a date. Much of the efforts with advanced mathematics are not that different from what Einstein did, at least as far as advancement of mathematics.

    The problems I think have one to do with scale of physics, where to get an exciton of quantum gravitation requires enormous energy and that these matters are also becoming cosmological. It might almost be compared to Leucippus, Democritus, and Lucretius who advanced the idea of atomic theory in an age where there was no way to test the idea. They were right in the end of course, but they lacked the tools to make it work.

    We might have to throw in the towel at some point. Velenkin showed that an inflationary manifold is not geodesic or b-complete into the past. This means there is then maybe some deeper origin to that spacetime, which of course is the mother of pocket worlds. So, the mother may itself by a daughter of something deeper! We may have some issues with getting any data on that, even if we get inflationary cosmology supported firmly by data. We are just little critters running around a mote orbiting one of about a mole of stars in … ; the extremes are boggling. If all possible explanations require further explanations, or all we have are model-theories all the way down, turtles all the way down, then at some point we are bound to lose contact. We may be suffering from the beginning of that.

    However, I think a problem is the schemes of appealing to ever larger gauge groups may be wrong. This approach in a funny way requires ever more degrees of freedom, roots, weights and quantum states or particles, when maybe we need to go the other way. These large groups might play some role, such as a large number of particle-states being the result of some degeneracy breaking etc, but on some level we may be barking up the wrong tree.

    1. Lawrence Crowell wrote:
      >These large groups might play some role, such as a large number of particle-states being the result of some degeneracy breaking etc, but on some level we may be barking up the wrong tree.

      Are you suggesting that, to coin a phrase, we might be "lost in math"?

      Lawrence also wrote:
      >We might have to throw in the towel at some point. Velenkin showed that an inflationary manifold is not geodesic or b-complete into the past. This means there is then maybe some deeper origin to that spacetime, which of course is the mother of pocket worlds.

      Is this the same point that Guth makes towards the end of his book when he says that inflation cannot extend infinitely far into the past? Do you know of any reference that explains how and why this is true at a level we ordinary physicists (i.e., who are not experts on inflation) can grasp?


      All the best,


    2. I generally do not read that many popularization of physics or science. My non-physics and math reading tends to be with classic literature. Vilenkin did write not long ago a book titled something like Many World From One that one can look up. The last popularization of physics I bought and read was Sabine's book.

      The past incompleteness of inflation can be seen fairly easily, The line element or metric

      ds^2 = dt^2 - a(t)dΣ^2

      for dΣ^2 the line element of space gives the FLRW Hamitonian constraint

      ℌ = 0 = (å/a)^2 – H^2 - k

      for H = 8πGρ/3c^2 = Λ/3 the Hubble parameter. This can be written as the integral

      ∫da/a = ∫Hdt,

      with k = 0 and this can be integrated. The difficult with assuming this can be extended to an arbitrary time into the past is that the left-hand side is ln(a), that diverges for a → 0 and this corresponds to t → -∞ in a fairly nasty way. The Hubble parameter H is 1/√Area and time is a parameter such that (Ht) → entropy and with the Bekenstein bound S_ent = A/4ℓ^2 and this Planck length ℓ is a regularization cut off. This means the time as an endpoint.

      Another way to see this is that Hdt defines the motion of particles, or galaxies on a large scale, and this motion is parameterized by proper time. Momentum of a particle is given by p^2 = g_{ab}P^aP^b with P = dU/ds. This gives the energy E = √{p^2 + m^2} and so

      ∫Hdt = ∫mda/[√{(m^2 + p^2)}a] = ln[(E + m)/p] = ½ln[(γ + 1)/(γ – 1)]

      which diverges in the past or where the Lorentz gamma factors approach unity. This tells us that the time → -∞ where say the relative velocity between two particles would be zero is a divergence. This in effect corrected with the regularization above as seen in the Planck length ℓ.

      We may still in one sense have time extended t → -∞, in de Sitter spacetime with a ~ cosh(t√(Λ/3)) where the bottleneck is the Planck scale. In this setting we still have a block on classical extension to the infinite past, but where there is some condition with quantum gravitation.

      The pocket world we observe emerged from this inflationary manifold. It is either some Swiss cheese hole in the space, or maybe a spatial manifold that popped off this dS manifold. The result of Vilenkin tells us this inflationary manifold, at least as can be described with classical or semi-classical physics has a finite time into the past, where through this bottle neck is a quantum gravitational description.

      The actual impact on the so called multiverse I could go into maybe, but space and time here is too limited. A bit like extending inflation into the past.

  16. Hi Bee, I notice an unusual turn of phrase in the third paragraph from the end of this post. The common phrase is, "means to an end", rather than, "end to a means".
    Thanks for all you do.

  17. All cosmological "observational studies" are not controllable laboratory experiments. The postulated theoretical implications strongly influence the experimental interpretations. Obviously in our solar system there are neither neutron stars, gamma ray bursts nor black holes (or “anomalies” that can be interpreted as such). For comparison: The next star "from our point of view" with 4.24 light years is Proxima Centauri. A list postulated "closest to the earth" black hole candidates can be found at with a "shortest" distance of 2800 light years. Object and distance information refer to the "view of the ΛCDM model".

    The human observation period is extremely short compared to the periods of time in which cosmic movements took place and played out. To substantiate assumptions with the data from the human observation period is "far-fetched" to put it casually. Virtually all current supposedly empirical measurements are (big bang) theory laden. Postulated time spans, distances, and energy densities are subjective-theoretic. The entire present physical view is based on the paradigm of "physical space-time".

    Bringing aspects of the theory of general relativity closer to the interested reader can only be very incomplete due to the degree of difficulty of the necessary mathematics. Sabine Hossenfelder published »Screams for Explanation: Finetuning and Naturalness in the Foundations of Physics« 2018 In this scientific paper S.Hossenfelder “critically analyze the rationale of arguments from finetuning and naturalness in particle physics and cosmology. Some other numerological coincidences are also discussed.”... In addition to book readings I recommend to read this paper especially because she “transforms” math into “easy to understand” words.

  18. Dr. Hossenfelder,

    Thank you for the book review. I had another book in mind to start reading, but this one is now next in the Que. I truly love physics and I believe it is the key to understanding our universe. But as an author I have read once said, we keep looking for that key under the street light rather than in the darkness where it was lost.

    Having said this may I be so bold as to go off topic and throw out something to ponder, a rhetorical question, Are dark matter and dark energy symmetric? One pulls while the other pushes. I know there is nothing that relates these two things, other than maybe symmetry.... No answer to the question, just kind of wondering if other think about things like this.

    1. Some authors have suggested that what we call "dark matter" and "dark energy" have the same cause, namely, negative-mass matter.

  19. Sabine,
    Thanks for getting into book reviews. Could you group all your reviews and book recommendations into one place? Perhaps something like Sabine's Library?

  20. David Lindley makes an error on page 157, he writes: "...10^500 is ten with 500 zeroes after it." Well, it is really a one with 500 zeroes after it (10^1 is ten, which denotes a one followed by a single zero, 10^2 is a one followed by two zeroes...100, 10^3 is a thousand, which is a one followed by three zeroes, etc.). Additionally, he writes: "Many theorists...continue to believe that Supersymmetry must be a fact of Nature." (page 155). It would have been nice for Lindley to interject a footnote with reference citations for that claim. The researchers whose work I am familiar with never make that claim: "must be a fact." They all realize it is so-far only theoretical. Needless to say, speaking purely from history--to those who feel Supersymmetry is a "waste of time," it helps to return to the origins of the topic and there read why it is that Supersymmetry appears to possess a modicum of sense. Also, see chapter 27 of Baulieu "From Classical to Quantum Fields" or,
    on a nontechnical level, Kane's chapter 28 in "Modern Elementary Particle Physics" (1993 edition).

  21. Gary,

    " The researchers whose work I am familiar with never make that claim: "must be a fact."

    Plenty of examples in my book.

  22. As late as January 2020, theorist John Ellis says this: "if you want a specific benchmark scenario for new physics at a future collider, SUSY would still be my go-to model, because you can calculate accurate predictions." (Cern Courier). He uses the terms "scenario" and "model." John Ellis also says:"People are certainly exploring new theoretical avenues, which is very healthy and, in a way, there is much more freedom for young theorists today than there might have been in the past. Personally, I would be rather reluctant at this time to propose to a PhD student a thesis that was based solely on SUSY – the people who are hiring are quite likely to want them to be not just working on SUSY and maybe even not working on SUSY at all. I would regard that as a bit unfair, but there are always fashions in theoretical physics."

  23. Tensors were the first mathematical concept I came across just before I went to university that I simply could't get my head around. I mean I could read the definition and see what it said and go through the calculations but I couldn't see what was actually happening unlike the notion of a limit, a vector or a group.

    I mean with a vector, we can see directly that there is this invariant arrow right before our eyes and that if we change our reference frame then how it looks according to that reference frame will be different. All that is immediate once understood.

    I wanted to visualise a tensor in the same way I could visualise a vector space. Not all of them, just some of them. After all, I don't need to be able to visualise 15d Euclidean space to be comfortable with working with it. Ordinary Euclidean space gives us sufficient intuition for doing so, and even upto their infinite-dimensional cousains, the Hilbert spaces (despite the subtleties).

    I don't think Lindleys description of a jelly-like substance helps. It seems to kind of cloud an already quite cloudy picture. It took me a while to understand that a tensor is simply another variant of a vector. It's a higher dimensional vector. Not the kind of ones that we're used to, the ones that live in 17d Euclidean spaces, or in Banach spaces but they are nevertheless related to them.

    A tensor, instead of having one arrow, has two or more arrows. For example, consider three arrows at a point. Then the 3d parallelogram that it defines *is* the tensor. But there is a subtlety, whereas a vector is rigid, we can rescale a tensor so long as the volume itself remains constant. That is, it has an 'internal' notion of scaling and which it has because it has more than one arrow. We can see straightaway why a vector cannot have such a notion, it has only one arrow and there is no way of rescaling it without changing the length.

    So we can fatten or squeeze the 3d parallelogram so long as its volume remains constant.

    I guess this is where Lindleys jelly analogy comes in. After all, you can squeeze jelly...

    It's easy to see from this description that the generalisation of vector addition and scalar multiplication still holds, except that there are more directions in which we can add and scale. In the example - three directions.

    Moreover, just as we have for vectors when we change the reference frame the components of the tensor change just as the tranformation formula we all know and love.

    Then instead of having this tensor all by its lonesome, we can have a field of them, just like we have vector fields. That's easy enough to visualise.

    There's actually a categorical definition which is incredibly unilluminating. It's all wrapped up in obscure mathematical jargon concerning universal properties, which to me was like a mystery wrapped inside of an enigma given that I didn't understand that particular language, and nor did it seem that those in the know were going out of their way to make it more intelligible.

  24. ...

    One day it suddenly dawned on me - as these things do - that all it was saying was that there are two ways of measuring the volume of a tensor. You can measure the length of the edges and multiply them to get a volume; or else you can just, somehow, directly measure the tensor volume. I mean, after all, you can look at a cube and kind of see its volume without going through the rigmarole of measuring its edges. In fact, more precisely, you can cut it up into smaller cubes and add its volume up that way.

    Now it turns out the former volume is *multilinear* on the edges, and the latter volume is obviously *linear* on the cubes. The universal property is that these two definitions of volume should match!

    Personally, I prefer the term characterising. Universally sounds too close the the word universe and has the wrong kind of connotations for those of us who are actually interested in the universe and rather than mathematical constructions.

    Anyway, I hope this helps someone as I wished someone had helped me with tensors and much else ...

    By the way, I've just discovered a geometric idea of how to think about spinors but I'll spare you the details until you decide to write something about them and why they are called 2-component or 4-component spinors and not something more inspiring.

    Warm wishes.

    PS. Apologies for the length ...

  25. "Notice this rent in my garment; I am at a loss to explain its presence!; I am even more puzzled by the existence of the universe".
    (Jack Vance, Tales of the Dying Earth)

    1. Have you read David Brins star-tide rising series? I read them ages ago. He wrote eco sciece-fiction long before ecology was popular or known.

    2. No. Have you got a relevant on topic qoute? I like Eco's books.



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