Mathematics, music, and effing the ineffable

Some people say: “Mathematics and science can never describe the beauty of a sunset, or the feeling I get from listening to a beautiful piece of music!”

But they are wrong.

It all depends on what your mean by the word describe. How many ways are there to describe something?

You can describe something using a list of traits.  Specifically, the beauty of some object (call it object X) can be described by listing the traits or characteristics it has that contribute to it being beautiful. So, you can imagine a long list (or what mathematicians call a “set”) of possible beauty-causing attributes,

B = {b1, b2, b3, b4, ..., bn}

and you can describe the beauty of object X (let’s say, a gorgeous awe-inspiring sunset in the Bahamas) with the subset of B containing only those elements of B that are present in X. We can call this BX where in set notation, where


Now, the list in the set BY (where Y is, say, Bach’s Brandenberg Concerto #3) will be different than the list in set BX, because although each is “beautiful” they have different characteristics that make them beautiful.

We can take this one step further, though!

Since different traits might contribute different amounts to the beauty of any given object, we might actually want to describe the beauty of object X using a vector of coefficients βi for each trait bi in the set B.  The more any given bi contributes to the beauty of an object, the higher that object’s βi will be.

Obviously, if X doesn’t have trait bi then βi in that vector BX would be equal to zero.


“No, no, no.. wait! Hang on a moment! That isn’t what I mean at all!”

What do you mean, then?

“I mean a description that captures how it makes me feel… the sheer magnitude of the wonder and awe!”

Well, if “wonder” is assigned to beauty descriptor variable b34, for example, then the magnitude of wonder would be coefficient β34

“No, not just the magnitude as a number, but a description that conveys the feeling.”

Now this is where it gets interesting.


Do you believe in the ineffable?

All languages (and I include in this both English and mathematics, both prose and Morse Code, both poetry and the binary language of moisture evaporators) operate on the following assumption: there is some kind of transformation process through which your mental state can be encoded into some set of symbols which then can be decoded by some other person’s mind to produce a mental state that somehow resembles the mental state you started with.

your mental state → some set of symbols → another person’s mental state

The other person’s mental state might not be exactly the same as the mental state you tried to encode with your language, but it has to have at least some key characteristics in common for the communication to be deemed “successful”. (Figuring out how one can tell if a communication has actually been successful by this criterion is a whole other complicated discussion that we will sidestep for now.)

If you say “math can’t convey the feeling of a sunset” then you are saying: it is impossible for me to choose a set of mathematical symbols that will get your feelings to be the same as (or “close enough to”) my feelings about the sunset.

But why?

Why would you think that is true?

Suppose I told you that I have been using beauty trait vector B my entire life, and I know the exact value of  β34 (and all the other coefficients as well) every single time I’ve experienced any kind of sunset. I have that vast reservoir of personal experiences to draw in, and a strong mental association between those experiences and each of the symbols in that vector.

Because I’m so familiar with these variables — the words of this language — that list of numbers { β1... βn } it just like any list of words in any other language: they are each associated with personal experiences, feelings, senses, and memories. And when I hear you say, for example, that the sunset last night had b34 > 0.98, that will be just as emotionally meaningful to me as hearing any other description that includes soaring poetic words such as “radiant” or “glowing” or “glorious” or “breathtaking”.

The variables are simply another kind of language. If you and I both understand the meanings of the words (variables), then you can pick those variables that are most closely matched with your experience, and when I hear you tell me those variables and their values, I will be able to call up those same experiences from my own memories.

Now, there are those who say that no language can convey the feeling of a beautiful sunset, or the feeling that a piece of music gives you. They believe such feelings to be ineffable: indescribable, or unable to be put into words.

But that is bullshit.


Either nothing is effable, or everything is.

There is no such thing as a “complete description” of something. There can’t be: a description by its very definition identifies a finite set of characteristics or relationships that characterize a thing, using a finite list of description terms. It selects some features as the things that matter, and disregards others as background or unimportant.

How can I describe the screwdriver in the toolbox in my garage?

I can describe it in terms of its shape, and the materials out of which it is made; I can describe its possible uses, both those intended and unintended by its maker and its owner; I can describe its history, and how that particular screwdriver came into being, where I bought it, and the projects in which it has been involved; I can describe its physical location, as well as how it relates to other types of tools, both physically in my toolbox and in the conceptual framework of “toolness”; I can describe the ways that its temperature changes during different times of the day; I can even describe the impact its existence would have on the emotions and economics of isolated tribal cultures, were they to come into contact with it.

All of these are valid “descriptions” of my screwdriver, and none of them is complete. Even added together, they would not form a complete description. How could it? Someone will surely come along and find a new set of associations, meanings, or consequences in the future. An alien race with sensory systems unknown to any human will have a sensory experience of the screwdriver that is literally (neurologically) impossible for humans to have… but without a description of that experience, the description of the screwdriver cannot be (in the ultimate, objective sense) complete.

Does that mean even screwdrivers are ineffable?

Wouldn’t that mean that pretty much everything is ineffable?

Eff that.  That bar is too high.

To me, it makes more sense to say that we can describe a sunset just as well as we can describe anything else… and that mathematics can describe aesthetics just as well as any other language… and leave it at that.

It’s up to you to decide how “well” you think that really is.

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