The epistemic force of beauty in laws of nature does not reduce to simplicity

Some people think that simplicity of laws of nature is a guide to truth, and some think beauty of laws of nature is. One might ask: Is the beauty of laws of nature a guide that goes beyond simplicity? Are there times when one could make epistemic decisions about the laws of nature on the basis of beauty where simplicity wouldn’t do the job?

I think so. Here is one case. Suppose we live in a Newtonian universe, and we are discovering fundamental forces. The first one has an inverse cube law. The second has an inverse cube law. These two laws account for most phenomena, but a few don’t fit. Scientists think there is a third fundamental force. For the third force law, we have two proposals that fit the data: an inverse square law and a slightly more complicated inverse cube law. It is, I think, quite reasonable to go for an inverse cube law by induction over the laws.

There is something indeed beautiful about the idea that the same power law applies to all the forces of nature. But if we just go with simplicity, we will go for an inverse square law. However, going for the inverse cube law seems clearly reasonable, and it is what beauty suggests—but not simplicity.

Here is another thought. Sometimes a fundamental law has some particularly lovely mathematical implications. For instance, a conservative force law is connected in a lovely way with a potential. But it need not be the case that a conservative force law is simpler than a non-conservative alternative. (It is true that a conservative force is the gradient of a potential. If the potential can be particularly simply expressed, this makes it easier to express the conservative force law. But we can have a case where the potential is harder to express than the force itself.)

From theism to something like Christianity

The Gospel message—the account of the infinite and perfect God becoming one of us in order to suffer and die in atonement of our sins—is immensely beautiful. Even abstracting from the truth of the message, it is more beautiful than the beauties of nature around us. Suppose, now, that God exists and the Gospel message is false. Then a human (or demonic) falsehood has exceeded the beauty of God’s created nature around us. That does not seem plausible. Thus, it is likely that:

1. If God exists, the Gospel message is true.

Furthermore, it seems unlikely that God would allow us to come up with a falsehood about what he has done where the content of that falsehood exceeds in beauty and goodness what God has in fact done. If so, then:

2. If God exists, something at least as beautiful and good as the Gospel message is true.

A small disability

On the mere difference view of disability, one isn’t worse off for being disabled as such, though one is worse off due to ableist arrangements in society. A standard observation is that the mere difference view doesn’t work for really big disabilities.

In this post, I want to argue that it doesn’t work for some really tiny disabilities. For instance, about 3-5% of the population without any other brain damage exhibits “musical anhedonia”, an inability to find pleasure in music. I haven’t been diagnosed, but I seem to have something like this condition. With the occasional exception, music is something I either screen out or a minor annoyance. Occasionally I find myself with an emotional response, but I also don’t like having my emotions pulled on by something I don’t understand. When I play a video game, one of the first things I do is turn off all music. If I could easily run TV through a filter that removed music, I would (at least if watching alone). (Maybe movies as well, though I might feel bad about disturbing the artistic integrity of the director.)

On the basis of testimony, however, I know that music can embody immense aesthetic goods which cannot be found in any other medium. I am missing out on these goods. My missing out on them is not a function of ableist assumptions. After all, if the world were structured in accordance with musical anhedonia, there would be no music in it, and I would still miss out on the aesthetic goods of music—it’s just that everybody else would miss out on them as well, which is no benefit to me. I suppose in a world like that more effort would be put into other art forms. The money spent on music in movies might be spent on better editing, say. In church, perhaps, better poetic recitations would be created in place of hymns. However, more poetry and better editing wouldn’t compensate for the loss of music, since having music in addition to other art forms makes for a much greater diversity of art.

Furthermore, presumably, parallel to music anhedonia there are other anhedonias. If to compensate for musical anhedonia we replace music with poetic recitations, then those who have poetic anhedonia (I don’t know if that is a real or a hypothetical condition; I would be surprised, though, if no one suffered from it; I myself don’t appreciate sound-based poetry much, though I do appreciate meaning-based poetry, like Biblical Hebrew poetry or Solzhenitsyn’s “prose poems”) but don’t have musical anhedonia are worse off.

In general, the lack of an ability to appreciate a major artistic modality is surely a loss in one’s life. It need not be a major loss: one can compensate by enjoying other modalities. But it is a loss.

In the case of a more major disability, there can be personal compensations from the intrinsic challenges arising from the disability. But really tiny disabilities need not generate much in the way of such meaningful compensations.

Here’s another argument that musical anhedonia isn’t a mere difference. Suppose that Alice is a normal human being who would be fully able to get pleasure from music. But Alice belongs to a group unjustly discriminated against, and a part of this discrimination is that whenever Alice is in earshot, all music is turned off. As a result, Alice has never enjoyed music. It is clear that Alice was harmed by this. And the bulk of the harm was that she did not have the aesthetic experience of enjoying music—which is precisely the harm that the person with music anhedonia has.

Objection 1: Granted, musical anhedonia is not a mere difference. But it is also not a disability because it does not significantly impact life.

Response 1.1: But music is one of the great cultural accomplishments of the human species.

Response 1.2: Moreover, transpose my argument to a hypothetical society where it is difficult to get by without enjoying music, a society where, for instance, most social interactions involve explicit sharing in the pleasure of music. In that society, musical anhedonia may make one an outcast. It would be a disability. But it would still make one lose out on one of the great forms of art, and hence would still be a really bad thing, rather than a mere difference.

Objection 2: There is a philosophical and a spiritual benefit to me from my musical anhedonia, and it’s not minor. The spiritual benefit is that I look forward to being able to really enjoy music in heaven in a way in which I probably wouldn’t if I already enjoyed it significantly. The philosophical benefit is that music provides me with a nice model of an aesthetic modality that is beyond one’s grasp. Normally, “things beyond one’s grasp” are hard to talk about! But in the case of music, I can lean on the testimony of others, and thus talk about this art form that is beyond my grasp. And this, in turn, provides me with a reason to think that there are likely other goods beyond our current ken, perhaps even goods that we will enjoy in heaven (back to the spiritual). Furthermore, music provides me with a conclusive argument against emotivist theories of beauty. For I think music is beautiful, but I do not have the relevant aesthetic emotional reaction to it. My belief that music is beautiful is largely based on testimony.

Response 2: These kinds of compensating benefits help the mere difference view. Even if one were able to get tenure on the strength of a book on the philosophy of disease inspired by getting a bad case of Covid, the bad case of Covid would be bad and not a mere difference. The mere difference view is about something more intrinsic to the condition.

Beauty and simplicity in equations

Often, the kind of beauty that scientists, and especially physicists, look for in the equations that describe nature is taken to have simplicity as a primary component.

While simplicity is important, I wonder if we shouldn’t be careful not to overestimate its role. Consider two theories about some fundamental force F between particles with parameters α₁ and α₂ and distance r between them:

1. F = 0.8846583561447518148493143571151840833168115852975428057361124296α₁α₂/r²

2. F = 0.88465835614475181484931435711518α₁α₂/r^{2 + 2−64}.

In both theories, the constants up front are meant to be exact and (I suppose) have no significantly more economical expression. By standard measures of simplicity where simplicity is understood in terms of the brevity of expression, (2) is a much simpler theory. But my intuition is that unless there is some special story about the significance of the 2 + 2⁻⁶⁴ exponent, (1) is the preferable theory.

Why? I think it’s because of the beauty in the exponent 2 in (1) as opposed to the nasty 2 + 2⁻⁶⁴ exponent in (2). And while the constant in (2) is simpler by about 106 bits, that additional simplicity does not make for significantly greater beauty.

A curious but common art form

A curious art form that blends nature with artifice is endemic in our culture, and likely a cultural universal. Many people modify their bodies (e.g., muscle building, hair-styling, etc.) and then combine them in harmonious ways with other physical objects attached to the body, such as paint, clothing, jewelry, etc., deliberately to create a work of art that is a hybrid of a living thing and typically (but not always) non-living accessories.

A large proportion of our population engages in this art form on a daily basis, but I don’t know a good name for the works of this art in the languages I know. We have two English words that come close, “fashion” and “cosmetics”, but both are specific to aspects of the art rather than the work as a whole. We might try to explain this odd lack by saying that there is a sense in which the work is the person (for, after all, normally when we imagine a person, we imagine them accoutred). We might call the art form "anthropocosmetique", using the archaic spelling to hearken back etymologically to earlier English uses (evoking shades of Bulwer's use of "cosmetique" and his specific contexts) that may be closer to the Greek roots, but also emphasizing the human component in the work.

An interesting feature of the works of anthropocosmetique art is their diachronic character. They are often created for a specific occasion—a day, a party, a liturgical celebration—and disassembled into their constituents afterwards, typically without any feeling that one has destroyed something of great value in disassembly.

At the same time, some people engage in a larger art form, one spread over multiple occasions, consisting of sequences of the anthropocosmetique art, with similarities and differences from occasion to occasion in the particular works of each occasion being aesthetically relevant perhaps in something like the way that themes and variations are the warp and weft (not respectively) of music.

Sometimes there is a melding between the anthropocosmetique and other art forms, especially performance arts like dance.

And another curious fact is that many of the most famous works of fine art are actually meta-art: they are themselves portrayals of the works of the anthropocosmetique art.

Meaning and beauty

1. Only intelligent beings and things produced by them have objective meaning.
2. Something that is objectively meaningless is not objectively beautiful.
3. The earth is objectively beautiful.
4. The earth is not intelligent.
5. So, the earth is produced by an intelligent being.

God and Beauty

Here is my talk from the Cracow philosophy of religion conference in September:

Two beauties

In a number of cases of beauty, beauty is doubled up: there is the beauty in an abstract state of affairs and there is the beauty in that state of affairs being real, or at least real to an approximation. For instance, the mathematics of Relativity Theory is beautiful in itself. But that it is true (or even approximately true) is also beautiful.

This shows an interesting aspect of superiority that painting and sculpture have over the writing of novels. The novelist discovers a beautiful (in a very broad sense of the word, far broader than the “pretty”) abstract state of affairs, and then conveys it to us. But the painter and sculptor additionally doubles the beauty by making something real an instantiation of it, and it is by making that instantiation real that they convey it to us. The playwright is somewhere in between: the beautiful state of affairs is made approximately real by a play.

The above sounds really Platonic. But we can also read it in an Aristotelian way, if we understand the abstract states of affairs as potentialities. The painter, sculptor and novelist all discover a beautiful potentiality. The painter and sculptor brings that potentiality to actuality. The novelist merely points it out to us.

Taste and cross-cultural encounters

After visiting the British Museum yesterday, I find it rather hard to take seriously the argument for the relativity of beauty from the diversity of taste. It seems clear that just as C. S. Lewis has argued for a moral core cutting across cultures, one can argue that there is an aesthetic core across cultures.

There is, however, an interesting apparent difference between the diversity of taste and the diversity of morals. I think a cross-cultural encounter involving a difference of taste regarding the best cultural artifacts—by each culture’s own standards—should typically lead to a broadening of taste. But a cross-cultural moral encounter should not typically lead to a broadening of morals. Very often, it should lead to a narrowing of morals: for instance, one culture learning from the other that slavery sex is wrong.

Why this difference? I think it may come from a difference in quantifiers.

As Aquinas already noted (in a somewhat different way), to be morally good, an action has to be good or neutral with respect to every relevant dimension of moral evaluation. If it is good with respect to courage and kindness and generosity, but it is bad with respect to justice (Robin Hood?), then the action is plain wrong. Thus as new dimensions of moral evaluation are discovered, as can happen in cross-cultural encounter, we get a narrowing of the actions that we classify as morally good.

On the other hand, for an item to be beautiful, it only needs to be beautiful with respect to some relevant dimensions of beauty. A musical performance is still beautiful on the whole even if the orchestra is dressed in dirty rags, and a painting can be beautiful even if it reeks of oil. Thus as we discover new dimensions of beauty, we get a broadening of the pieces that we classify as beautiful.

Three places for beauty in representational art

There seem to be three senses in which beauty can be found in a piece of representational art:

1. The piece represents something as beautiful.

2. The piece in and of itself is beautiful.

3. The task of representing is performed beautifully.

One can have any one of the three without the others. For instance, the one-line poem “The kitty was pretty” satisfies 1 but fails 2 and 3. Though, to be precise, I think sense 1 is not a real case of something being beautiful, but only of something being represented as beautiful. The kitty could be ugly and yet described as pretty.

I think 3 is particularly interesting. It opens up the way for works of art that are in themselves not beautiful and that do not represent beauty, but which do a beautiful job of representing their objects (Sartwell says that Picasso’s Guernica may be beautiful; I think my aspect 3 of the beauty of representational art may explain this). Note that “beautiful” here does not merely mean “accurate”, as the case of my one-line poem shows, since that poem may represent the beauty of a cat with perfect accuracy, but there is very little of the beautiful about how it accomplishes this.

Fundamental bearers of aesthetic properties

I am finding myself frustrated trying to figure out whether the fundamental bearers of aesthetic properties are mental states or things out in the world. When I think about the fact that there does not seem to be any significant difference between the beauty of music that one actually listens to with one’s ears versus “music” that is directly piped to the auditory center of the brain, that makes me think that the fundamental bearers of aesthetic properties are mental states.

But on the other hand, when I think about the beauty of character exhibited by a Mother Teresa, I find it hard to think that it is my mental states—say, my thoughts about Mother Teresa—that bear the fundamental aesthetic properties. If I thought that it was my mental states that are the bearers of aesthetic properties, then I would think that a fictional Mother Teresa is just as beautiful as a real one. But it seems to me that a part of the beauty of the real Mother Teresa is that she is real.

Perhaps the fundamental bearers of aesthetic properties vary. For music and film, perhaps, the fundamental bearers are mental states: the experiences one paradigmatically has when listening and viewing (but which one could also have by direct brain input). For the characters of real people, perhaps, the fundamental bearers are the people themselves or their characters. For the characters of fictional people, perhaps, the fundamental bearers are mentally constituted (in the mind of the author or that of the audience or both).

Maybe the beauty of a real person is a different thing from the beauty of a fictional character. This kind of makes sense. For we might imagine an author who creates a beautiful work of literature portraying a nasty person: the nasty person qua fictional character is beautiful, but would have been ugly in real life, perhaps.

But I hate views on which we have such a pluralism of fundamental bearers of a property.

Musical beauty and virtual music

We have beautiful music at home on a hard drive. But wait: the arrangement of magnetic dipoles on a disc is not musically beautiful! So it seems inaccurate to say that there is music on the hard drive. Rather, the computer, hard drive, speakers and the orientations of magnetic dipoles jointly form a device that can produce the sound of beautiful music on demand.

One day, however, I expect many people will have direct brain-computer interfaces. When they “listen to music”, no sounds will be emitted (other than the quiet hum of computer cooling fans, say). Yet I do not think this will significantly change anything of aesthetic significance. Thus, the production of musical sounds seems accidental to the enjoyment of music. Indeed, we can imagine a world where neither composers nor performers nor audiences produce or consume any relevant sounds.

Perhaps, then, we should say that what is of aesthetic significance about my computer, with its arrangements of magnetic dipoles, is that it is a device that can produce musical experiences.

But where does the musical beauty lie? Is it that the computer (or the arrangement of magnetic dipoles on its drive) is musically beautiful? That seems wrong: it seems to be the wrong kind of thing to be musically beautiful. Is it the musical experiences that are musically beautiful? But that seems wrong, too. After all, a musical performance—of the ordinary, audible sort—can be musically beautiful, and yet it too gives rise to a musical experience, and surely we don’t want to say that there are two things that are musically beautiful there.

Perhaps a Platonic answer works well here: Maybe it is some Platonic entities that are trulymusically musically beautiful, and sometimes their beauty is experienced in and through an audible performance and sometimes directly in the brain?

Another possibility I am drawn to is that there is a property that isn’t exactly beauty, call it beauty^*, which is had by the musical experiences in the mind. And it is this property that is the aesthetically valuable one.

And of course what goes for musical beauty goes for visual beauty, etc.

Natural law: Between objectivism and subjectivism

Aristotelian natural law approaches provide an attractive middle road between objectivist and subjectivist answers to various normative questions: the answers to the questions are relative to the kind of entity that they concern, but not to the particular particular entity.

For instance, a natural law approach to aesthetics would not make the claim that there is one objective beauty for humans, klingons, vulcans and angels. But it would make the absolutist claim that there is one beauty for Alice, Bob, Carl and Davita, as long as they are all humans. The natural lawyer aestheticist could take a subjectivist’s accounts of beauty in terms, of say, disinterested pleasure, but give it a species relative normative twist: the beautiful to members of kind K (say, humans or klingons) is what should give members of kind K disinterested pleasure. The human who fails to find that pleasure in a Monet painting suffers from a defect, but a klingon might suffer from a defect if she found pleasure in the Monet.

Beauty and goodness

While listening to a really interesting talk on beauty in Aquinas,I was struck by the plausibility of the following idea (perhaps not Aquinas'): The good is what one properly desires to be instantiated; the beautiful is what one properly desires to behold. So the distinction between them is in how we answer Diotima's question about desire (or eros): what do we want to do with the object of desire?

Everything is beautiful

Consider something visually ugly, say one of my school painting projects. The colors are poorly chosen and the lines don’t do a good job representing what it’s meant to represent. (I am not being modest.)

But now suppose we live in an infinite universe or a multiverse, so that every possible intelligent species is realized. It is very likely that there will be some intelligent species whose electromagnetic spectral receptivities are such that the colors in the lines look gorgeous to it, and harmonize in a wonderful abstract way with the shape of the lines. This is, of course, a chance matter—I wasn’t making the painting for that mode of visual receptivity. Let’s say that the species is the xyllians. We can still say that what I made is an ugly work of art, but it is also a part of the natural world, and considered as a part of the natural world it is visually_x (i.e., as seen with the electromagnetic reception apparatus of xyllians) beautiful while being visually_h (i.e., as seen with human electromagnetic reception apparatus) beautiful.

Moreover, it is irrelevant whether the xyllians and humans exist. Whether they exist or not, my painting is visually_x beautiful and visually_h ugly. All that’s needed is that the xyllians and humans could exist. Thus, my painting really is both beautiful and ugly, even if we are the only intelligent species. And it is just as objectively beautiful as it is objectively ugly. I wasn’t supposing that the xyllians misperceive: just that they have a different pattern of spectral receptivities. We can suppose that xyllian visual perception is just as accurate in reflecting the world, including my unhappy artistic productions, as ours is.

This means that an argument from particular beauty for the existence of God must be run cautiously. Sure, sunsets and goldfish are beautiful. But so is any child’s scrawl, and quite likely any physical object is beautiful with respect to some possible sensory apparatus. Particular instances of beauty are easy to find and should not surprise us. What could surprise us, however, is:

1. That the particular sensorily beautiful things around us—such as sunsets and goldfish—are in fact beautiful with respect to the sensory apparatus of the intelligent species that dwells near them.

We might also attempt to mount arguments from beauty to God on the basis of these remarkable facts:

2. That there is such a property as (objective) beauty at all.

3. That we are able to perceive beauty.

4. That we enjoy beauty.

5. That we are able to make correct judgments of beauty.

And bracketing the question of arguing for the existence of God on the basis of beauty, the realization that all material things are beautiful should lead us to glorify God. For while I said that it’s chance that my poor attempts at painting are visually_x beautiful, that’s only so loosely speaking. God is omnirational, and that the paintings are visually_x beautiful is a redeeming quality that surely God did not fail to intend.

The variety of beauty

A crucial part of Diotima’s ladder is the progress from sensible beauty to the non-sensible beauty of mind, law and mathematics. From time to time I’m struck by how very strange it is that such very different things as paintings, faces, poems, minds and theorems have beauty in common.

If one has a view of beauty as that which gives a certain “aesthetic pleasure”, it’s easy to explain this: it is not that surprising that different inputs could give rise to the same kind of pleasure. But that view of beauty is false. (We would not make my preschool scribbles more beautiful than Monet’s mature paintings by brainwashing people into taking more aesthetic pleasure in the former than in the latter.)

Plato’s famous explanation is that all these different things participate in the same form. But that leaves mysterious why it is that a painting that exhibits a certain harmonious play of colors and a theorem that is illuminating and unifying in a certain way both end up necessarily participating in the form of beauty. There needs to be a connection between the configurations that give rise to beauty and the participation in the form of beauty. The historical Plato seems to have thought that there was a common mathematical structure in all these configurations, but this seems quite implausible given the great variability of them.

Perhaps a theistic explanation can make some progress. All beauty is a participation in God. But God is infinitely beyond all else, so this participation is from an infinite distance, and it is not so surprising that the infinite richness of God can be participated in in infinitely many different ways.

The difficulty with this explanation is that beauty is not the only property that’s a participation in God. Every positive property is a participation in God. And some positive properties—say, knowledge—are much more unified than beauty. Perhaps it helps, though, to have the medieval view that beauty, goodness and being are all in some sense interchangeable. So perhaps every participation in God constitutes beauty, and so the great variety of participations in God gives rise to the great variety of types of beauty.

Aesthetic reasoning about necessary truths

We prefer more elegant theories to uglier ones. Why should we think this preference leads to truth?

This is a classic question in the philosophy of science. But I want to raise the question in connection with philosophical theories about fundamental metaphysics, fundamental ethics, philosophy of mathematics and other areas where our interest is necessary rather than contingent truth. Why should we think that the realm of necessity has the kind of aesthetic properties that would make more beautiful theories more likely to be true?

Here are two stories. The first story is that we are so constructed that we tend to find beauty in those philosophical theories that are true. It is difficult to explain why there would be such a coincidence if we are the product of naturalistic evolution, since it is unlikely that such a connection played a role in the survival of our species tens of thousands of years ago. If God exists, we can give an explanation: God gave us aesthetic preferences that guide us to truth.

The second story is that fundamental necessary reality is itself innately beautiful, and beautiful theories exhibit the beauty of their subject matter. And we recognize this beauty. It is puzzling, though: Why should fundamental necessary reality be beautiful? The best explanation of that which I can think of is again theistic: God is beauty itself, and all necessary truths are grounded in God.

Of course, one might simply reject the claim that our aesthetic preferences between theories lead to truth. But I think that would be the end of much of philosophy.

I think that in the order of knowing, aesthetics and ethics come first or close to first.

Spirograph

My three-year-old has been enjoying the vintage Spirograph set we have. I thought it would be fun to make a javascript/svg version. Code is public domain (just View Source in your browser). What can we deduce about humankind from the fact that we appreciate patterns like these?

Stationary circle size:
Mobile circle size:
Fractional distance of pen from center of mobile circle:
Rotate around outside of stationary circle:

The elegance of fundamental physics

Plenty of the mathematics in science is ugly. But the mathematics in fundamental physics tends to be beautiful. It could be that in the correct fundamental physics it won't be beautiful. I wonder if we have reason to think it will be. The fact that in our current fundamental physics theories there is mathematical beauty doesn't seem to say much, because our current fundamental physics is probably false.

Still, I hope that the book of the world is written by God in such a way that the mathematical elegance of approximately true theories points to the mathematical elegance of the true theory in physics. I wonder, though, if an atheist could have any reason to have such a hope.

The magical

Magical things happen. The glorious glow of the sunset, the elegant glide of the turkey vulture or the delight of conversation with friends. Such experiences of the magical are what gives life its zest.

To experience these things as magical is to experience the events as going over and beyond the merely natural. Thus, if naturalism is true, such experiences are all deceptive. And that makes naturalism a very dour doctrine indeed.

Yet even if naturalism is false, how can these experiences be of events going beyond the merely natural? A sunset is, after all, just light refracted in the atmosphere as the part of the earth on which one stands turns away from the sun. So there is something more going on than the physics describes. This something more could be intrinsic or relational or both. Perhaps the sunset reflects something much deeper beyond it. Or perhaps there is more in the very sunset than the physics describes.