Thursday, July 14, 2016

Idealism And Modern Science: Space And Time

Kant, again

So, another post on High German Idealism. Before, I'd been pretty kind and polite, even being careful to point out good parts in Hegel. But today, I'd like to point out a major error in the philosophy of Kant, Schopenhauer and others, one that makes much of their exposition wrong as a matter of strict fact. This major error has to do with the division between the underlying "noumenal world"/World As Will and the phenomenal world of experience/World As Representation. Kant and Schopenhauer believe that spatial and temporal order of the world is part of human experience, but not the world in itself - this is simply completely wrong. That the world of experience is 3+1 dimensional is a deep fact about the underlying physical world in itself. Further, human perception does not, in fact, take place in a mostly "geometric" manner, by which Kant would mean specifically Euclidean geometry. This is not a minor flaw, but appreciating it requires far more technical apparatus than Kant and Schopenhauer had access to, even giving them the benefit of deep insight through dim appreciations. I'm going to go about this exposition quickly but carefully. First, I will again recapitulate the core Kantian argument so that it will be understood that Kant meant these terms literally. I'll give an example of a seemingly objective property of an object that is actually not mind-independent. After that, I will give a brief description of the correct modern understanding of these facts.

Kant began his career as a serious natural philosopher in an initially typical 18th century mold. He studied physics with great intensity and soon was setting very difficult physical problems for himself to solve. He had been a minor player in the debate over whether energy or momentum is conserved - unfortunately only publishing after it was understood that both were. This book, though confused by Cartesian concepts, also contains many important insights - such as the correct general explanation of inverse square laws. Kant appreciated that conservation of momentum implied that a collapsing cloud of particles would force the system to rotate and flatten out. Eventually, he argued, the cloud will condense into a star and planets. Kant used this to explain the two dimensionality of the solar system, and went much further than that. He argued that the solar system itself was part of a much larger scale condensation, what we now call the Milky Way Galaxy. At the time, this was a novel and innovative hypothesis. It turned out to be impressively correct. Such arguments shows that Kant was familiar with conservation laws and how they can be used to give powerful qualitative arguments. When Kant was 30, he won a prestigious prize for a demonstration that resistance to tides causes the rotation of planets to slow. This implicitly involves energy considerations and demonstrated that the solar system could not be infinitely old (an open question before this).

Kant's peaceful potential life as an eminent but minor Prussian physicist was ruined one day when he happened to read a book by David Hume, the greatest of all philosophers. Kant's research had convinced him completely of the correctness of Newtonian mechanics - classical mechanics to me and you. But Hume had a devastating and novel argument in favor of skepticism of what is called "induction" - essentially learning. Induction obviously cannot be justified by empirical knowledge, since learning from observation requires learning. No particular induction can be justified on general logical principles - since such an induction would be a general deduction. There are some truths - such as conservation of energy - that are either true or not about particular systems, we have to learn whether they are. So this kind of reasoning is not enough. Finally (and this was Hume's addition to the skeptical argument) induction cannot be justified inductively - that's a vicious circle!

Hume forced Kant to see how delicately his hypotheses leaned on conservation laws that he (and everyone at that time) barely understood and were certainly not necessarily true. In particular, Hume forced Kant to question whether we could learn the laws of physics. Kant spent years - decades - attempting to carefully develop both the system of the world and our knowledge of it into a coherent whole even given Hume's critique. In 1781, he hastily published a massive tome containing all of his work, after a health scare convinced him he would die unpublished if he did not.

Kant's system is not easily explained, partly because it involves such a complex mingling and careful separation of the world as it is and our perception of it. But it goes something like this. The world in itself consists of innumerable particles. The particles move around by exchanging energy and momentum with each other, according to specific - Newtonian - laws. But we cannot see the naked system of the world - instead we see a coarse grained, psychological, language drenched, enculturated, clothed system. In modern physics, we represent the whole system by a vector, and the motion of the particles are given by the so-called "Hamiltonian" of the system. This gives the fine-grained reality of the system, but the world of experience with fluids, pressures, etc. "exists" only as an approximation at a coarse grained higher level.


The whole world of experience may be coarse-grained, but that doesn't make it "subjective", in the usual sense. Purely physical systems interact on a coarse grained level. The usual thermodynamic functions are minimal statistics of systems, so that any reasonable description of the world must include some of them. They are "forced moves" in Dennett's words.

But very little of our representations of the world are forced! In fact, perception is highly dependent on language, culture and conditioning. The most famous example - first pointed out by John Locke - is color. To an English speaker, it seems to be an objective fact that the sky on a hot cloudless day is blue and the sea on that same day is also blue. But if I were to mention this to Homer, he would be shocked! How could the sun bright sky and the wine dark sea be "the same color"? The answer is that in my culture - the culture of English speaking people - we learned to divide up the spectrum of light in ways that some are called blue and others not.

 This is a somewhat dishonest way of living, color is not so simple, color perception even less than that.  To our culture - you and I are English speakers after all - black is white. It's plain to see that the dark blue sea is the same color as the bright blue sky. If we merely apply the same argument to grey, we see that black and white are obviously the same color.

How do we survive zebra crossings then? In spoken language, our culture simply partitions sufficiently dark greys into black and sufficiently light ones into white. It's hardly less arbitrary than most of life. Our non-linguistic experience of color that motivates most of our actions may be different. Generally, we try to keep life or death situations away from subtle color gradations.

Of course, much more than just color is part of the World As Representation that isn't grounded in the underlying physical world (Schopenhauer's World As Will) in a unique way. Psychologists study physical perception in the form of affordances. Beyond that there are complex social systems that include languages, governments, markets and all that goes with them (such as philosophy).

Emmy Noether

So, if so much of the world is ungrounded in the huge vector and the Hamiltonian rules that make up The Dang-An-Sich, what makes me so sure that space, time or spacetime is part of it? In order to understand this, you have to use tools far more modern than anything to which Kant had access. Kant understood that the rules of The Dang-An-Sich conserved certain global properties such as energy and momentum. This was not an easy thing to figure out, and he had to do it for himself. But in order to understand space and The Dang-An-Sich, one must understand the connection between conserved properties and symmetry. This could not have been done before group theory, it could not have been done before Lie groups and algebras, it could only have been done by someone who understood them both. The person who did so was Emmy Noether, and this alone would have made her one of the most important persons in mathematical physics. The fact that the theory of groups was entirely absent from physics before her makes her probably the most important person in the history of mathematical physics. Kant appreciated that conservation of momentum was a fact about physical systems, what he did not and could not have known that this equivalent to the existence of an symmetry operator on the laws of physics - on the Hamiltonian. This can be strengthened by Wigner's Theorem - not only must every conservation law give a differential symmetry, but the symmetry must take a very special form. To say that these theorems is the very foundation of modern physics would be to understate how central they are.

Let's look in particular at conservation of momentum, which suffices to give the philosophical flavor. It arises from the following symmetry: if every particle was moved in a way that keeps all the relative distances the same, then the relative motions of each particle wouldn't change. This symmetry operator defines the three dimensions of space. This is a fact about the Hamiltonian, a fact about The Dang-An-Sich and therefore has nothing to do with perception. It is not even a coarse grained fact, but applies on the microscopic level. Perception may take advantage of this organization, though it only does less than one might think. Actual perception is a lot more edge detection and topological relations, Euclidean geometric representation (with it's angles etc.) is learned.

The above argument has many slight alterations important to physics but not philosophy. The symmetry operators that defines actual physical space are called the Poincare Group and they give rise to geometry which is relativistic - not Euclidean. But these alterations, constrained as they are by Noether's and Wigner's Theorem, cannot alter the simple fact is that spacetime is part of the organization of the world in itself and the Kantian/Schopenhauerian thesis that it is not is simply incorrect.


  1. with schopenhauer's voice: where did your math come from?

    i mean, common, don't be a slacker, at least include some cognition-based reasoning.
    or will you refute some philosophy based on banach–tarski next?

    1. Thank you for your comment!

      Yes, that's a very good point, Schopenhauer was much better on the psychological and cognitive side than Kant. Unfortunately, he based his philosophy of space and time on Kant's incorrect one. Space and time are, contra Kant and Schopenhauer, part of the underlying World As (universal) Will and not part of the individual human world as experience. Our visual World As Representation is not a manifold, even though underlying reality may be. James Gibson recapitulates many of Schopenhauer's insights without Schopenhauer's incorrect assumptions. This deserves more than the tossed aside sentences I gave it.

      The math is motivated by uncontroversial physical principles common to Kant, Schopenhauer, modern physics, etc. Metaphysical principles such as conservation of momentum or energy give rise to operators that must have a specific form - they must be a linear, unitary (or anti-linear and anti-unitary) Lie group. These operators are the laws of physics, including the structure of space and time. But there are infinitely many such groups. As Hume, Kant, Schopenhauer, etc. note correctly, we cannot know these metaphysical principles - and therefore the correct linear, unitary Lie group - with certainty. For instance, a priori, we can't know whether the operators commute or not, whether momentum or 4-momentum is conserved, etc. Therefore, we can't know, for instance, what the real structure of space and time is. But this is enough to preclude the idea that the structure of space and time is cognitive.

      The Banach-Tarski Volume Non-Conservation Paradox is terribly important, since it shows that volume is not trivial to define. It is possible that there is a philosophy out there that depends essentially on volume being conserved by rigid motions. Such a philosophy would be incorrect (or, at least, very limited). I'm not aware of any such philosophy though.

    2. schopenhauer had worldview as space, time AND causality.
      the time and space is mere order of perceptions - just succesion and position - and there was no connection between them.
      the way you use "spacetime" includes all three, together with causality, and hence word "worldview" is more correct for it.
      i dont think he was THAT pessimistic to deny the existance of worldview?
      (worldview is a wrong word here, but i dont remember what he had instead of it.)

      together to theese three he also had "concepts", which were humans-only specific, connecting everything else, resulting in language and math.
      and the point of denying was not that there's no such things as time or space or causality in the world, but that you can't tell them apart.
      the "lie group" is a concept. it's structure assuming space, positions. it's operations assuming time, change, succesion.
      you can tell "there's something", but you loose every time saying which is which.

    3. Thank you for your continued interest! Schopnehauer and Kant simply miscatergorized space and time.

      Schopenhauer's objective world was called the "World As Will". In my previous posts on this subject, I called it The Dang-An-Sich. Schopenhauer, following Kant, did _not_ believe the World As Will had spatiotemporal relations. He connected this to Platonic Froms, which are obviously not spatiotemporal.

      Schopenhauer's phrase for our experience, what you call our "worldview", can be translated as the "World As Representation". According to Schopenhauer (and Kant), the World As Representation is our subjective, spatiotemporal experience of the objective, non-spatiotemporal reality. In rare moments of philosophical reflection or artistic ecstasy, we can be conscious of the World As Will. Music is particularly valued by Schopenhauer because it is non-spatial, so that musical ecstasy is becoming part of the composer/perfomer's pure will.

      In my last post, I described how the striving of individual subsystems to retain properties divides the world into parts, with important philosophical consequences. Though I didn't bring him up, this is a basically Schopenhauerian point: the will to live is the mark of an individual. In another post, I described how Schopenhauer is correct in describing the World As Will as "worthless and empty". I'm not refuting Schopenhauer's general philosophy, only a mistaken part of it. Space and time belong to the Platonic, metaphysical, category of "will" and not the cognitive, individual category of "representation". They insisted in so many words _many times_ that the World As Will was non-spatial, acasual and beyond time! While not the fundamental aspect of their philosophy (which is why I didn't get to it until my fourth or fifth post), it is a grave mistake.

      The Lie groups do _not_ assume space, time, change or any such thing! This is a very subtle fact, but it is the whole point. The operators are results of non-spatial, non-temporal and acasual metaphysical principles that _result_ in space and time (or spacetime or more exotic geometries, depending on what unknowable metaphysical principles rule the universe). The metaphysical principles are part of Schopenhauer's World As Will. I mean this literally, I mean it how it sounds. Schopenhauer and Kant believed that the world was made up of point particles interacting with inverse square laws (they only knew about gravity and electromagnetism). This is the very first thing Schopenhauer asserts in book 2 of The World As Will And Representation. Schopenhauer correctly asserts that the laws are beyond casual explanation. Each metaphysical principle gives rise to set of operators that are equivalent to some geometry. This they did not know. The result is that the World As Will is geometric. Schopenhauer was mistaken to follow Kant in ascribing space and time to the World As Representation.

      Space and time are _real_, not cognitive. Cognitive perception is *not* geometric, just as you would predict from a corrected version of Kantian/Schopenhauerian system. Studying the history of art and art instruction, you can see clearly that geometry is always learned and never natural. This is all perfectly Schopenhauerian, with only a correction as to the right place for space.