Why Does Some Science Work So Good? Part 1: According to Hermann Weyl


Weyl Composite 02Most people do not appreciate that some forms of science are better—much, much better—than others. What makes for good science? We consider some quotes from the famous mathematician and physicists Hermann Weyl that shed light on this issue.

I am now embarking on a deeper inquiry of ideas I expressed in What Is Science? It might seem as if I am getting off the tracks of my normal metaphysics of linking Western and Eastern thought because we will be going into Western thinking at a deeper level. But a lighter seasoning of Eastern thought will be peppered throughout, and it will all be brought back around in the end, so just hang in there!

In What Is Science? I posited that science is a weak form of samadhi. To translate this into more familiar terms: what we call science is a type of image in the mind that weakly resonates with, or captures a facet of the essence (artha) of some aspect of nature.

In general this takes the forms of mathematical expressions that capture the dynamics of some aspect of nature. Hence, the intimate link with the Hindu idea of gunas. It is interesting to contrast this viewpoint to the one expressed by Hermann Weyl. Weyl’s view reinforces the “weak samadhi” view and fleshes in some essential details. By considering Weyl’s views it helps us appreciate what it means to “resonate” with some aspect of nature, and why this causes some sciences to be better than others.

For example, physics is much more successful than say, biology. Where biology is just a mass of details held together tenuously by the “theory” of evolution, physics is a series of interconnected models of reality that resolve some of the deepest metaphysical issues to have plagued the mind of man since…well…since man had a mind to plague. For example, modern physics has much to say about the natures of time, space, energy, matter, force, and other issues that have a long history in metaphysics, stemming back to the ancient Greeks some 2500 years ago and the ancient Indians over 5000 years ago.

Further, scientific success can be gauged in terms of the technological applications it spawns. Quantum mechanics, the theory behind atoms, has allowed us to have massive control over material substances. This has allowed the creation of cell phones, computers, MRI machines, and lasers, to name only a few technologies.

On the other hand, what have we gotten from, say, the human genome project, in which all the genes in human chromosomes have been identified? Granted there have been some inroads towards applications. For example, this information allows insight into why some people better respond to cancer drugs than other people (because they have different genes. Why the different genes make a difference is not necessarily known). This leads to a concept of “personalized medicine” where drug treatments can be individually tailored for a patient based on their genes. At present this applies to a very limited set of cases and is not a wide-spread procedure.

But in general, if we were to put the technological achievements to stem from physics on a weighing scale against those achieved from biology then there would be no contest. The results from physics dwarf those of biology, period.

Why? I think Weyl gets to the heart of the matter. Let’s see what he had to say about this issue.

Biographical Stuff
First though: Who is Hermann Weyl and why should we care what he has to say? Weyl was one of the most important mathematicians and theoretical scientist of the 20th century. He was also respectful of philosophy and did not hesitate to draw on it in expositions of his work. He is less well-known to the general public than say Einstein or Feynman, but his work is of the same first-rate importance as these more famous people.

The physicists Freeman Dyson wrote in Weyl’s obituary in the journal Nature in 1956:

“Among all the mathematicians who began their working lives in the twentieth century, Hermann Weyl was the one who made major contributions in the greatest number of different fields. He alone could stand comparison with the last great universal mathematician of the nineteenth century, Hilbert and Poincaré.”

For anyone who knows about math, comparing Weyl to Hilbert and Poincaré says it all. Let’s now jump into the main topic at hand.

Weyl On Science
We focus on two Weyl quotes. The first is from The Open World essay. This is taken from page 54 of the book Mind and Nature Selected, Writings on Philosophy, Mathematics and Physics edited by Peter Pesic. Note the following contains a quote within a quote.

“The transformation of metaphysical questions of cause into the scientific questions of law is taught by all great scientists. The discovery of the law of falling bodies is the first important example…It is more important to investigate the law according to which the acceleration varies. Again Newton says:

‘I have not yet been able to determine from the phenomena the cause of these properties of gravitation….It is sufficient that gravitation exists, that is acts according to the laws we have formulated…’

Dynamics, according to the doctrines of d’Almbert and Lagrange, requires no laws which extend to the causes of physical phenomena and to the essence of such causes; it is closed in itself as a representation of the regularities of the phenomena.”

Next, on page 117 of the same book, from the essay Mind and Nature Weyl says:

“In physics we do not a posteriori describe what actually occurs in analogy to the classification of the plants that actually exist on earth, but instead we apply an a priori construction of the possible, into which the actual is embedded on the basis of the values and attributes indirectly determined by reactions…But construction a priori must be joined with experience and analysis of experience by experiments. “

To me, the first quote is more comprehendible. The second quote seems a little ‘blah blah blah” and requires some translation. So let’s dissect what is being said here.

Dissecting the Quotes
Allow me to restate in my own words what Weyl is saying in each quote:

Quote 1: Scientific theories in general are not concerned with the cause of a phenomenon but instead concerned with how a phenomenon behaves.

Quote 2: Scientific theories do not describe specific scenarios. They describe all possible scenarios. Then, only a small subset of scenarios described by the theory applies to real phenomena that actually exist in nature.

What these quotes mean is best illustrated by example.

Quote 1 Example
To illustrate quote 1, let’s consider Newton’s law of gravitation. It is normally presented as:

figure6This equation says that the force between two bodies with mass equals the product of the mass divided by the square of the distance between them. This is a vector form of Newton’s equation. We can’t forget that it is really a differential equation that looks like this:

newton ODE This is because F = ma, where a is acceleration, and acceleration is the second derivative of the position. So technically, Newton’s gravity law is a 2nd order differential equation. An example of how to solve this differential equation to get real life answers is given here. We stress that Newton’s law is a differential equation because this is an example of dynamics (e.g. of describing a pattern of gunas).

The first thing to notice is that the equation does not define what gravity is. It just shows how it behaves with respect to two masses M and m (or m1 and m2). As Newton admits in the quote above “I have not yet been able to determine from the phenomena the cause of these properties of gravitation”. In fact, Newton never determined the cause of gravity.

This overall situation does not change even when we consider Einstein’s idea of gravity. Instead of being an invisible force that reaches through space and binds two masses together, gravity in Einstein’s relativity comes from the bending of space-time by the two masses. For a nice intuitive understanding of Einstein’s concept of gravity see this video.

Einstein’s equations are much, much more complicated that Newton’s, but the same general idea holds. We still don’t know what gravity is. Why does mass bend space-time? Nobody knows. Einstein’s equations do not explain why mass bends space-time. Einstein’s equations are preferred over Newton’s because they calculate how masses move through space-time better than Newton’s equations.

So, as Weyl says:

“Dynamics, according to the doctrines of d’Almbert and Lagrange, requires no laws which extend to the causes of physical phenomena and to the essence of such causes; it is closed in itself as a representation of the regularities of the phenomena.”

This is a really important thing about the successful sciences: they do not try to determine causes; they simply describe how things behave.

We can isolate two key points in this quote. (1) He specifically refers to dynamics. Dynamics means “how things change with time”. How a system changes with time is the basic “how” that is captured by successful science (not always, but in most instances). (2) He says “it is closed in itself as a representation of the regularities of the phenomena”. What does this mean?

It means that the description of how something works is couched in mathematical terms. The “closed representation” refers to a mathematical description. Why math? That is what the 2nd quote is all about.

Quote Number Two
Let’s repeat the idea:

“In physics we do not a posteriori describe what actually occurs in analogy to the classification of the plants that actually exist on earth, but instead we apply an a priori construction of the possible, into which the actual is embedded on the basis of the values and attributes indirectly determined by reactions…But construction a priori must be joined with experience and analysis of experience by experiments. “

Let’s translate a little more. He says that physics does not try to describe what exists after the fact (a posteriori). His example of botany is apt; plants already exist. A biologist seeks to describe (e.g. classify) what already exists after the fact.

Not so in physics. Instead, one makes up some general idea (a “construction”) before any given fact (“a priori”). This construction will define a limited, closed sphere of applicability (ala quote 1). In this closed sphere are all of the possible scenarios within that domain of applicability. Out of all the possible scenarios, only some of them will apply to what already exists (“into which the actual is embedded”).

What does this mean? Again, let’s look at what Newton did and what he did not do. Newton did not try to make a theory that explains the relationship between the Sun and the Earth and the other planets known at the time (which went up to Saturn I believe). Newton did not try to a posteriori describe the known planets. Kepler did. This is a big difference between them. Kepler tried to explain the specific orbits observed at that time in the solar system. Kepler did what a biologist would have tried to do: invent a theory that accounts for only the known planets at the time.

Newton took a different approach. He constructed a closed system of representation. He made a theory that said: “masses are attracted to each other by a force called gravity”. But even this was not enough. He didn’t use words. He said: F = GmM/r2. This formula is the “closed system of representation”. It describes all possible interactions between two masses m and M. It is completely generic. It describes no specific occurrence of gravitational interaction. Instead, in the particular form written, it describes every possible gravitational interaction between any two bodies, m and M.

This is a key idea. Successful science does not focus on specific examples. It seeks to express some general principle which will apply to an entire class of phenomena.

One may consider the theory to be an accounting device. It accounts for all possible scenarios. Then there is a procedure for extracting out specific instances from all of the possible ones. In the case of Newton’s law, specific cases are defined by assigning numbers to m, M, and r, (r is the distance between m and M).  In such a way, Newton could apply his theory to model the solar system. But there are infinite combinations of masses and distances that could be plugged into the equation. However, most of these combinations will represent nothing that actually exists.

So, in this sense, a successful scientific theory can either be construed as a horn o’ plenty, or as a tremendous case of overkill.

Math is Necessary
This “overkill” feature of a successful scientific theory is a consequence of using mathematics. When Weyl speaks of “the possible” he generally refers to mathematical objects called “manifolds”. The term “manifold” is the mathematical way to say “a set of mathematical objects”. It is the manifold that Wyle calls “the possible”.

Calling a scientific theory a “law” is over-dramatizing. It is an equation, or system of equations. Equations nowadays are thought of as formally similar to computer programs. Equations are machines or programs that transform inputs to outputs. The manifolds that define the scope of the inputs and outputs of equations which constitute a theory are “the possible”.

There is another aspect to Weyl’s statement when he says “…the actual is embedded on the basis of the values and attributes…” This is also a mathematical thing. When we use Newton’s law, we do not input the Earth and the Moon into the equations, whatever that is supposed to mean. No, we abstract the Earth and the Moon to three numbers: M (mass of the Earth), m (mass of the moon) and r, the distance between their centers. These are the “values and attributes” used to embed the actual into the possible.

So, the scientific theory per se is completely abstract. Elsewhere Weyl says explicitly that it is just a system of symbols, specifically of a mathematical type. These he calls “free constructions” meaning we are free to make up any theory we can imagine, yet that is constrained by the internal logic of math. We would not generally invent a system where 1 = 2, which is just nonsense.

Weyl says time and again that math is the border between absolute freedom and absolute bondage. We are absolutely bound by the self-evident constraints of logic and number, but within these bounds we are free to construct whatever we want. No one today can deny that this has emerged and grown into the most fertile and productive domains of human endeavor.

The Other Side of the Coin: Experience
Finally he says: “But construction a priori must be joined with experience and analysis of experience by experiments.” So while we are free (within the confines of math and logic) to construct any theory we wish, in the end, it must be held up next to experience. Otherwise, it is just a piece of art; perhaps beautiful as a piece of mathematics, but the construction is not science.

This too is not straight-forward. Again, Weyl used Newton as an example. What is this “mass” Newton spoke of? What is the meaning of m and M in the gravity equation? It is not just the volume of “stuff” in our hands, nor what we observe via our eyes (e.g. a star for example). It is not even the weight of a thing; how much something weighs decreases the farther one moves away from the Earth.

No, even mass is an indirect concept. It is something that is defined and measured operationally, and the meaning only exists in the context of the theory. To add a third quote relevant to this point (same book, Man and the Foundations of Science essay, pg. 181):

“In order to measure the mass of a body one must therefore characterize how it reacts with other bodies. Inertial mass is a concealed character; we cannot perceive it directly as we can its color. Moreover, the determination of mass is possible only on the basis of a law of nature to which that notion is bound.” (His emphasis)

Later (pg. 183) he says:

“Thus we had better not commit ourselves to any definition and rather develop the theory as a symbolic construction with unexplained symbols and only at the end indicate in which way certain derived quantities may be checked by observation. The theory then becomes a connected system that only as a whole may be confronted with experience”

Now we come to appreciate that experience only touches a theory at a few points. There is much more “meat” to the theory. It is a “connected system”. We might as well consider the elements of the theory as pure abstractions than to even try to map them into our direct perceptions and experience.

Then, within the theory, “derived” relationships provide the concrete points of contact between the mind and perception. These points of contact between the theory in the mind, and the perceptions of the senses provide the empirical link between theory and experiment that is the essence of science.

To Wyle, perception stands on one side of an unbridgeable divide between our direct conscious experience of perception (which he calls “intuition”) and the intellectual construct which is the theory. We will discuss Wyle’s particular brand of philosophical dualism next time.

Let’s summarize the main points discussed above:

  1. There is good science. It has the form of not trying to describe causes, but to describe the behaviors of things, the dynamics of things.
  1. The good science is grounded in a mathematical framework. The math framework is a “free construction” of the human intellect bound within the confines of mathematical necessity (by which is meant 1 + 1 necessarily equals 2, not 6).
  1. The math framework contains every possibility that could ever occur (“the possible”) within the semantic confines of the framework. Only a subset of these possibilities will be realized in real natural systems (”embedding of the actual in the possible”).
  1. The links with experience and perception are indirect. The terms of the theory (variables, parameters) make sense only in the framework of the theory. There is some restricted domain of our actual experience that can be quantified in terms of the theory.

I am pleased as Punch that Weyl’s ideas are consonant with the philosophy of science I expounded in What Is Science? (e.g. that science is a weak form of samadhi). When writing WIS, I was unaware of Weyl’s ideas.  As seen above, he expresses quite clearly the nature of the dynamic “resonance” between ideas in the mind (the “free constructions of the possible”) and perceptions in our experience.

The key thing to emphasize is that they are not direct one-to-one mappings. The qualities of our perceptions must be abandoned to get at the deeper truths of nature. The deeper truths “wear” or disguise themselves as the qualities of perception. But what is being perceived, the appearances (Kant’s “phenomena”) are only indicators pointing to something deeper. This something deeper is the dynamics, the gunas; the patterns of movement and relationship that constitute the “skeleton”, if you will, of our experience.

Next time, we will get into the fact that Wyle, like Wilder Penfield, was a scientific dualist. Wyle clearly demarcated between our direct subjective experience and the objective world. We will see that his ideas too point to the solutions obtained in yoga thousands of years ago.

Go to Part 2 of my discussion of Hermann Wyle’s ideas.

7 thoughts on “Why Does Some Science Work So Good? Part 1: According to Hermann Weyl

  1. There will always be success in science for that which can largely be reduced to the material and mechanical as some things can. No-one denies that. Problems arise when that approach is applied to everything.

    • Hi Thanks for replying! It is nice of you to stop by and offer a comment. Here is a quote from Weyl (pg 185) that nicely addresses your comment: “The mechanistic world picture is dead, beyond restoration“. My next post will go into his views along these lines. But this is a big deal coming from Wyle. Also, this was said by him in 1949. So, it is not the idea of things being reduced to material and mechanical.

      There is no reduction going on in this picture. Nothing is being explained in terms of some basic unit. Instead, things are understood as being dynamical systems, as being patterns of change.

      And, if you take yoga at face value, in terms of its ideas of the gunas and Prakriti, then yes, the approach can be applied to “everything”, where “everything” means “maya”. Of course consciousness itself will ever defy being caged by mere intellectual understanding, will defy being grasped by any level or function of the mind. But the illusions that fill consciousness, which are just patterns of movement, these are able to be captured in the terms Wyle is expressing. Which is why I feel his thoughts are important enough to write about.

      Again, thanks for popping in!

      Best wishes,


  2. PeterJ

    Hang on Don. You write ‘”physics is a series of interconnected models of reality that resolve some of the deepest metaphysical issues to have plagued the mind of man since…”

    This is the first thing you’ve written with which I’d take issue in a major way. It appears to be just plain wrong. In my view it would be a matter of definition that physics cannot solve metaphysical problems. Evidence please. 🙂

    • Hi Peter!
      Uh oh! I’ve been called out by a professional metaphysician! 🙂

      I guess I have lately been swept away by Weyl’s take on things. Perhaps I should have started the sentence you quoted with “According to Weyl..” Which is the fact that I am aping him here, at least with regard the “series of interconnected models” part of the sentence. The remainder I inferred from what he said. It is Weyl’s take on space, time and causality via Relativity, and on mass and fields, via quantum mechanics, I am getting at here. There is no question, again upon reading Weyl, that Relativity tosses out many notions of space and time that are prior to it, with radical implications for causality. We all hear the idea of a “light cone”, but I, for one, never really appreciated how radical a notion it is until reading Weyl.

      As regards the metaphysics part, maybe your expertise can help me here. I am still gestating part 2 of this post, which is about Weyl’s metaphysics. He often refers to the models of physics as “the transcendental” and he is using this term specifically in reference to Kant. As you know, Kant said we can never know with any justifiable certainty the transcendental “thing in itself”. So, it is intriguing that Weyl uses this term to refer to physics models. There are two issues: (1) I am still trying to decode what he means by this term, and (2) I want to explore the implications of what he means.

      As far as (1) goes, I think he is following Kant insofar as he is implying about physics models something to the effect: we cannot directly perceive the transcendental, but we can make guesses about what it is, and express those guesses as mathematical theories (which are then subject to a consistency check against direct experience, e.g. experimentation). Again, a key idea here is “field of possibles”. The models are massive overkill in terms of describing specific instances, but, in some sense, reflect the essence of the object being considered (space-time, mass, etc). I think this qualifies as metaphysics. What do you think?

      Further, I am surprised by how his thinking dovetails with my idea of science as weak samadhi, and the link between gunas and dynamics. So, this opens a non-Western dimension to the discussion. I think you might agree that yoga gives access to direct experience of things that are considered “metaphysical” in Western thought? If we combine Weyl’s thinking with my view of science and yoga, metaphysics becomes a whole different beast. Something that is not only accessible to logic and thought, but to experience too.

      I don’t know. What do you think? Do you think this line of thought is way off base? As usual, you know I value your ideas, Peter, and am always grateful for your input.



      • PeterJ

        Oh boy, Half a dozen cans of worms at the same time. I see what you’re saying, I think, but would disagree on the details. Mostly this may be (as usual) about the use of words.

        I don’t know what Weyl had in mind, but I wonder if by ‘transcendental’ he meant ‘meta-scientific’. The scientific method is formal when it deals with data collection and analysis, but not for theory creation. Other than that a theory cannot contradict the data it seems to be transcendental to the scientific process, or to the data. It is an interpretation. In the same way, when speaking of Nagarjuna’s worldview in relation to his proof of it, I would call his proof logical and metaphysical, but his worldview would be ‘transcendental’ since it is an interpretation of a logical result. At least one other interpretation of his result (or theory of why the logic comes out as it does) is possible, and logic cannot decide between them. Weyl seems use the term in this sort of way. Perhaps the fact that we cannot prove that a scientific theory is true would make it ‘transcendental’ in this sense. Or maybe not. This is thinking out loud.

        I would say that when we discuss metaphysics we are doing metaphysic and not physics. ‘Fundamental physics’ would be an oxymoron. Only metaphysics deals with the fundamental and this is written into its definition. The idea that physics might solve a metaphysical problem would be incoherent. It would be the most common silly idea in physics in my opinion.

        It was only that one claim about metaphysics that bothered me. Weyl was a good metaphysician and I don’t see why we should be crediting physics for his conclusions. I suspect that most physicists believe that nobody has ever solved a metaphysical problem in the whole of history, so crediting them with solving any seems a bit generous. Also, it makes a bit of a mockery of the dictionary.

        I’m not at all sure that I agree with Kant’s use of ‘transcendental’, He assumes a ‘thing-in-itself’ and never recovers.

        The points about the samadhi, gunas and dynamics is over my head at the moment but I’m working on it. Fascinating, and it seems to make some sense, but I’m not even up to speed on the gunas.

        In passing – thanks for some very enjoyable discussions.

      • Hi Peter.

        Yes, thank you too for the very stimulating discussion. It is much appreciated. Yes, I think he means “meta-scientific” but also I get the sense he is really referring to Kant to and talking about us trying to tackle realities that transcend our ability to perceive them directly or know then completely. In What is Science? I make the statement that no one in the 20th century tried to incorporate Kant, but I am beginning to think I was simply wrong, and Weyl actually did tackle the issue of the “transcendental” that Kant raised. I think what Weyl is saying is we can make guesses about what is outside of our mind.

        As you say, theory creation is not at all formal. Weyl would use the word “intuition” to describe the source of theory creation, which he seems to use as a synonym for the totality of our direct conscious experience (thinking, perceiving, cognizing, etc. all of it together).

        These “guesses” must take the form of mathematical theories. As such both math and mathematical science are alike in this regard. Math provides a “realm of the possible” for making guesses about what is outside of our minds (in an Kantian sense). Then, these guesses need to dovetail with some aspect of our “intuition” (e.g. direct experience). This is the empirical part. He is emphatic that there is no a priori method to guess what and where the math model will match with experience, only that it ultimately must to be scientific.

        He does not distinguish theory and data as you seem to be doing. They are not independent in any sense. The “data” is inherent in the theory. Like in the above post where the notion of mass is defined within the theory. It is not an observable at all. Only velocity can be directly observed. The changes in velocity (momentum) can also be observed. Then mass can be calculated from the theory. We have no “intuition” (direct experience) of mass. He goes to some length to disabuse the reader of the idea that there is somehow a direct correlation between sensory qualities and the idea of mass. They are vastly different according to him.

        So, I think it is a different angle from what you said above. He is not seeking any kind of justification. He only seems to be explaining his conception of what constitutes science. To him, a theory is true only insofar as their is some aspect of intuition/direct experience that can be calculated from the theory. But it is always tentative as the extension from Newton to Einstein showed. Newtons equations are a subset of Einsteins, but the interpretations, the math theories, are radically different. And there are more points of contact with “intuition” using Einsteins framework than using Newton’s, and that is why it is “better”.

        But in all these cases, what he seems to be saying is that all we can know are the models in our minds and their seeming correspondence with some aspect of direct experience. I think this is his answer to Kant’s dilemma. We can never know directly the transcendental outside of the mind, but we can get indirect glimpses of facets of it via this process of math model construction/correspondence with events within our minds and consciousness.

        Overall, I think it is pretty clever. Great actually. Outstanding. It’s the best I read. And I am particularly disposed to it as it gets to the gunas/samadhi issue too.

        As to the link with metaphysics, I think I take your point. Weyl is certainly unique amongst physicists and mathematicians for his expertise in mixing math, science and philosophy. Very influenced by Plato and Leibniz he was. So no, physics isn’t solving metaphysical problems at face value. I think, like Kant, Weyl is offering a method to reconcile our inability to ever solve a metaphysical problem with 100% certainty with the fact that we can sometimes solve specific, little problems, like motion, or heat, or light.

        Finally, about Kant. He is a complex character. I don’t know the full scope of his thinking, only mostly what I have read others say about it. It might be a little harsh to say he “assumes” the thing in itself. I am more generous and like to think he deduced it. Either way, it is not like deducing the consequence that 100 + 10 = 110 from the fact that 1 + 1 = 2. It is more just a way to state the paradox of consciousness trying to understand what consciousness is. I think Kant’s is a useful way to express this paradox and is productive as it opens things up for creative activity, like what Weyl does.

        The gunas/dynamics connection is important. Its in What is Science, but I won’t get into it here.

        Again, Peter, thanks so much for the fun, stimulating discussion! Best, Don.

      • PeterJ

        That all makes sense. The Kant thing I’ll stick to my guns since although he would say that he deduced the thing-in-itself it is actually not a necessary logical conclusion but a choice of solution. I also am not a Kant expert, however, so may be missing something. I would also disagree with Weyl that we cannot solve a metaphysical problem with certainty. But these quibbles are beside the point.

        Au revoir

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