Mathematical Foundation#
Mathematics and music, the most sharply contrasted fields or scientific activity, are yet so related as to reveal the secret connection binding together all the activities of our mind. – Hermann von Helmholtz
What is Mathematics?#
Mathematicians to this day argue about the fundamental nature of math. Do we humans invent it, or do we discover it? Platonists and realists claim that numbers are real, mind-independent things, implying that they were discovered. Conceptualists say that numbers exist but are just mind-dependent ideas; nominalists say that numbers do not exist. The latter two positions would imply that numbers are invented.
Kant suggested that the prerequisite of our experiences is a pure concept of understanding (Verstand). A Kantian category is a characteristic of the appearance of any object in general, before it has been experienced (a priori). Kant believed that without it, cognition is impossible because we would not be able to find any structures within the data. In his view, neither pure reason nor pure experience suffice; we need both. He thought that this is an a priori truth that we can only understand by using exclusively a priori methods, i.e., pure reason. From that perspective, numbers can not come from pure experience which is a truth we understand by pure reason. Numbers emerge from our understanding of the difference between one and many things; an experience that requires a priori categories.
The difficulty in grasping that question is that we are thrown into math, as Heidegger would say. Teachers tell us what it is and how it works. We use it before we reflect on it or understand it; it is ready-to-hand (zuhanden). Math, similar to language, is like a virus or parasite that circulates through the human mind and reproduces itself like an autopoietic system. It is closed and gets irritated by new observations such that it extends or adapts itself (through us).
The late Wittgenstein argues that the meaning of words is determined by their use. This seems like a trivial observation but his claim was in fact very radical because it rejects the view that words refer to some essences. Furthermore, it offers arguments agains the view that language can describe reality accurately. The use of a word changes over time. Words disappear and new words are created. For Wittgenstein, the fact that language is ambigious wasn’t a downside but a feature—a view which is uncontroversial today. Reality is a mess and language is flexibel enough not to accurately descrite the mess but to navigate through it.
Take for example the word game. Many games share common characteristics and properties but whenever you think there is a good definition that fits all games, we can find counter examples. It is quite difficult to work out the essence of games. So what is a game? A game is what we call a game and we call those things and activities ‘game’ because it is (socially) useful to do so. It helps us navigate through the mess. At some point in time, someone called foodball a ‘game’ and that description kept being used because it was useful and made sense at the time and place. Our language is deeply connected to social practices. There would be no words, no language, and no thinking without the social.
Wittgenstein brings up the example of a beatle in a box. Imagine a group of people each box in everyone’s box is they call a beetle. However, no one is ever allowed to in anyone elses box. They can only look into their own box. However, they can talk meaningful about beetles even though they don’t know what others have in their boxes or indeed if there is anything in them at all. The thing in each person’s box is irrelevant to the shared public meaning of the term ‘beetles’. The words meaning is determined by the use people make of the term and not the private something hidden away from view. Whatever is or isn’t in each person’s box is irrelevant. It cancels out. Wittgenstein uses this analogy to go agains the assumption that experiences are private to us. We think that no one knows my personal color or pain experiences. In other words, no one can see the beetle in my box. But the meaning of sensation words such as pain and color isn’t given by referring to some private inner introspective something—a sensation to which you alone have access to. In his view there cannot be more to the public meaning of our language than we are capable of teaching each other. The private something—the beetle—may or may not exist and may or may not be different from other beetles, cannot have a role in that teaching because we cannot get at it.
If this is true, then the same has to be the case for mathematical terms such as numbers. With this view, the word two does not point to some essence or abstract idea that ‘lives’ in Plato’s world of ideas. Instead it is a result of social practices. It is a term born out of social practice to communicate such that it helps us to navigate the mess. This, of course, tells us nothing about the question of the ontological existence of the number two. But Wittgenstein would probably say, that to ask for such a thing is impossible. The only thing we have is experience, social practice and language which we constructed over time not to describe reality but to navigate through it.
I partly agree with Kant, even though I think he highly overrates reason over all the other bodily processes and thinks that there is still something absolut and pure to establish and ground. I don’t think we need some eternal Platonic space of perfect ideas that is hidden somewhere. Such meta-concepts are shortcuts for something we do not understand yet or something we can deal with by describing it in this way. The Platonic worldview is mystical, religious, absolute and in some sense too certain. It introduces something extra, something godly that we do not need. And it devalues life as something to overcome instead of something to embrace. Call me impulsive, but after I read Nietzsche, Heidegger, and Wittgenstein, I despise the Platonic worldview more and more.
Platon believed that these abstractions are more accurate than what we perceive. He thought that the perception of a chair is an obscuration of the perfect idea that shines through it. Thereby contradictiong Wittgenstein by establishing an essence of the chair. I think it is the other way around, and instead of perfect and imperfect, I use the terms concrete and abstract. The chair is concrete, but our perception of it has nothing to do with it with the exception that it enables us to deal with it. Our perception is the first abstraction born out of Kant’s understanding—we are able to perceive it as one thing. The second-order abstraction of it, e.g., using the word chair, is a silent agreement of a set of perceptions we call chair. For a number, e.g. the number one, this is similar.
It is no surprise that the Greeks had troubles with defining negative numbers because of a lack of experience. Since they focused mostly on geometry, negative numbers were never an issue. The invention of money made negative numbers a necessity thus they were invented and discovered. I think this is an argument for Kant’s assumption because if negative numbers came from within, there is no reason why the Greeks had trouble with finding them.
This does not mean we have to experience negative numbers to find them. Einstein came up with his theory by virtually thinking about it. Back then, there was no way to experience it. But I would argue that he had lived through everything to make his leap of understanding. He had learned a good fundament, lived in a certain time and had a brilliant mind. Similar to thinking about pink elephants, he knew \(A\) and \(B\) and was able to connect both. In other words, he had the imagination to come up with a concept that was useful enough that it rendered Newton’s theory ‘wrong’ or maybe to use a better term inaccurate.
Only because we have never experienced a perfect circle does not mean we can not find a description of one. I have never experienced a pink elephant, but I can perfectly picture one. Does this mean there has to be another perfect world where the idea of a pink elephant lives? The world of ideas cripples the fascination with the infinite complexity of what there is. It trys to bind the infinite to the finite, Kierkegaard might say. Wittgenstein would probalby say that the word “existing” in the language game of mathematics is completely different from its meaning in any other language games. He would argue that Plato confused different language games, i.e., the mistake is a missuse of language.
We have never experienced a perfect circle. But we found a model called circle that models circle-like objects that we perceived. Such an object does not exist, like an apple right in front of me but it is a good abstraction of circle-like things.
The abstraction is an objectified version that we often silently agree on because our experiences endorse it. In that sense it is not really objective, rather we call things objective we all agree on. The language game of mathematics is one most people agree on because it turned out so amazingly useful. Because mathematics is unreasonable useful in describing nature we are tempted to believe that there has to be traces of an objective truth. I belive this. But at the time of writing I also believe that we cannot prove this and we can never know with certainty. Our language and we as creatures are finite. Therefore, grasping the infinite is impossible. Reality will always be a mystery. However, we can tinker with it and see where it goes.
To conclude, in Wittgensteinian view, in the beginning mathematics was invented according to social practices: the differentiation of one thing from another thing; the handling of multiply things which all named by the same word; the exchange of goods and services. It required a close observation of nature and there might be traces of an objective truth in it but we can never know. Therefore, I claim that mathematics is invented but this invention is closely bound to observations of nature and therefore it is also discovered.
Do we need Mathematics?#
Do you ever saw a cat jumping over a fence? It is an elegant and precise act. So how can we neglect the cat’s intuitive understanding of gravity? In a way the cat has a better grasp on it than we do. Similarily, as musicians, we are perfectly able to create great sounds and tones without knowing what complex numbers or the Fourier transform is. We do not really care about sample rates or the Nyquist sample theorem. In fact, we might even have a better understanding of it without knowing the term and how it is defined by a mathematical formula.
Mathematics is a kind of technology, that is, a way of seeing. However, as it turned out, it is an especially effective way of seeing. Mathematics helps us to find patterns not only in our constructed world but within imaginary objects. Much like art, it can set up a new world. It can enable the differentiation between form and medium which leads to our appreciate of the physical, as well as formal world. A purely mathematical understanding might undermine the multiplicity of an artwork but the beauty of mathematics, which lies in its sense-making potential, can bring us to new hights. Making sense of what we do and what we experience and to act on behalf of it is a way of life and a way to create art and mathematics offers us a large ocean of senes.
You probably know about these artificial intelligence driven image generators. They are capable of generating beautiful images guided by a simple textual description. The user types in her imagination of a landscape, and the artificial intelligence generates different versions of it. One can even specify certain artists to achieve a particular style. After the images are rendered, the user selects her favorites and continues. It is fascinating, but at the same time I wonder if this process is satisfactory for the user. Is she really producing art? I would argue that the result belongs not only to her but is a product of second-order observation. It is more like a cultural production of the Zeitgeist of computation than the work attributed to anyone or any group. The experience of generating artificial images is shallow because the effort, the struggle, and the social and cultural context is missing. Furthermore, the logic of a loose coupled medium seems to be broken because the artist can not color each pixel as she desires. The outcome is contingent but probably not contingent enough? Do we still ask: Why did the algorithm produced this outcome over any other?
Whenever we use tools that hide specific details, we limit ourselves to the interface of those tools. Furthermore, our artwork will be a product of those tools. That can be a good thing! The sound generated by playing the piano is, in part, the product of the piano. But the artist has mastered it; he has a certain degree of control over the instrument. In the example above, the artist has very little control; she is highly dependent on the suggestions of the machine. She is guiding the machine but can no longer take over control and we as the audience know this and we know that the artist knew when she created the artwork. There is just not enough control which translate to creative freedom—the confrontation of choice—that makes the creation of a contingent alternative version of reality (the artwork) possible. I think this property is required to catch attention and to prompt question about by making the difference between medium and form visible.
Therefore, control is important and that is why I like SuperCollider: it offers a lot of control; it does not hide too much of the concrete. Of course, there is always a tradeoff between the abstract and the concrete. We are never entirely concrete because some abstraction will always separate us from the real thing: the digital machine.
The question is: how close do we want to get? How much control do we want to have? How much effort do we want to put in? I like to think that the artwork can not be separated from the process that created it. It should tell a story; it should be imbued with emotions, passion, and struggle.
The mathematical foundation of signal processing can help us to get closer to the concrete and to get even more control over the creative process. It can help us not only to understand but experience the process. Coming back to my introduction of the book: technical tools open as well as close spaces of possibilities. They turn the ship such that the horizon of the possible changes and they have agency over us, they call us to act and to deal with the world in a certain way. Math in that sense can also conceal aspects of artistic craftsmanship.