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 and philosophers to this day argue about the fundamental nature of math. Do we humans invent it, or do we discover it? This is the main question all the different positions answer a little bit differently. If numbers are real, mind-independent things, then they were discovered, if not then we seem to invent them.

Positions

Platonists also sometimes called mathematical realists such as Gödel and Penrose claim that mathematical objects (numbers, sets, functions, etc.) exist independent of human thought, in a timeless, abstract realm. Just as astronomers discover planets, mathematicians discover mathematical truths. The obvious question or problem with this view is that it is unclear where and how these objects exist and how we humans access them.

Logicians claim that mathematics is reducible to pure logic, that is, all mathematical truths are logical truths in disguise. The promoter of this field of theory (Frege, Russell, Whitehead, Carnap) was to derive all of mathematics from logical axioms and rules of inference but Gödel’s incompleteness theorems showed that not all mathematical truth can be captured this way. Logicists often assumed numbers were real abstract entities, but that they could be reduced to logical notions. So Logicism is both a program (reduce math to logic) and compatible with a Platonist ontology.

Struturalists like Benacerraf, Resnik and Shapiro claim that mathematics studies structures rather than individual objects. Numbers, for example, matter only in terms of their place in a structure (like the natural number sequence). This can explain why different systems (e.g. sets, Peano arithmetic) can represent the same mathematics. But this raises a question: what are structures made of? Structuralism can be seen as a refinement of Platonism: instead of individual objects existing in an abstract realm, what exists are structures (like the natural number sequence or the real line).

Formalists like Hilbert, Kleene claim that mathematics is a game with symbols, played according to rules. Mathematical statements are just strings of symbols manipulated within formal systems. These mathematicians try to avoid metaphysical questions about existence and Being. However, one might ask why does a mere game of symbols describe the universe so well? Why is mathematics so effective in its application? Formalists undercut the meaning of mathematical objects outside of the formal system within they are defined. Formalism lost its status as a full foundation mainly because its central promise (to secure all of mathematics by purely formal means) turned out to be impossible. Gödel showed that in any sufficiently strong formal system (like arithmetic), there will be true statements that cannot be proven within the system and that a system cannot prove its own consistency (if it actually is consistent). Threfore, math is not fully capturable as a “formal game” and Hilbert’s goal of proving the consistency of mathematics within a finitary framework is impossible. This meant Hilbert’s dream was unachievable in principle. Formailsm is also counter intuitive to the mathematicians because they experience proofs as discoveries, not as arbitrary symbol games. But it’s still indispensable as a methodological backbone. Philosophers now usually combine it with other views rather than defending it on its own.

Intuitionists and constructivists like L.E.J. Brouwer, Heyting and Bishop claim that mathematics is a creation of the human mind. This ties mathematics to human mental activities but it also restricts many classical results (e.g., nonconstructive proofs are invalid) because this foundational theory rejects the law of excluded middle in infinite contexts (e.g., “either a number is prime or it isn’t” is not true unless we can prove one or the other). Intuitionism is a strict version of constructivism. Both reject non-constructive proofs and see math as human-made mental constructions. Many constructivists are intuitionists in practice, though some are looser about which constructions “count”.

Nominalists such as Ockham, Field, and Goodman suggest that mathematical objects do not exist at all, not even as abstract entities. In their view, mathematics is just a convenient shorthand for talking about the physical world or about symbol manipulations. One variant, i.e. fictionalism, claim that mathematics is like a useful story—not true, but indispensable. The other variant, that is, deflationary nominalism assume that numbers are just ways of speaking about properties of the world (e.g., “there are three apples” describes a property of apples, not an abstract “3”). Norminalism avoids mysterious abstract realms but it is hard to or even impossible to account for the objectivity and universality of mathematics.

The there are empiricists like Mill, Lakatos and Putnam (at some stage) who think that mathematics is ultimately derived from our experience with the world, much like the sciences. In this view, mathematical knowledge grows by conjecture, refutation, and revision, not by pure deduction. This makes sense of the evolving, human side of mathematics but seems at odds with the timelessness and necessity of mathematics.

Social constructivists think of mathematics as a social product, created within communities of mathematicians. Promoters of this position such as Hersh, Ernest, and partly Kuhn argue that the truth of mathematics is determined by social agreement and shared practices. This explains why mathematics changes historically and culturally but it seems to downplay the objectivity of mathematics (e.g. would \(\pi\) stop to being 3.14159… if sociaty said so?). Both empiricism or quasi-empiricism and social constructivism emphasize the human, evolving nature of math. They are compatible with one another because they downplay the idea of math as eternal, perfect truth.

There is also the view hold by e.g. George Lakoff and Rafael Núnez that mathematics arises from human cognition and the way our brains metaphorically extend embodied experience. (e.g. “two apples” -> abstract “2”). This can explain the origins of mathematics in human thought and language but it seems to leaves unclear why mathematics describes physics so uncannily well. The cognitive-metaphor approach sits well with constructivist or social accounts, since all highlight the human origin of mathematics. It doesn’t claim math is eternal and abstract—just that it arises from how humans conceptualize and share patterns.

Pragmatists like Dewey, Quine and Putnam (he shifted his position between realism and pragmatism) wanna departure from these question and claim that mathematics is what works—a toolbox for reasoning, modeling, and solving problems. Its value lies in usefulness, not metaphysical truth. This captures the applied power of math but it intentionally just sidesteps the question of what math is; it’s more about what math does. Pragmatists sometimes label the question of what mathematics is as nonsensical. Pragmatism is flexible: it says math is valuable insofar as it works. It can easily blend with formalism (“we use formal systems because they’re useful”) or with nominalism (“numbers don’t exist, but they’re handy fictions”).

There are, of course, also total oppositions. Platonism and nominalism are incompatible: either numbers exist independently, or they don’t exist at all. Also intuitionism and classical mathematics (Platonism, logicism, formalism) are incompatible: intuitionism rejects the law of excluded middle, which most other schools accept. Furthermore, formalism goes against Platonism: one says math is just symbols, the other says it describes real abstract things.

In practice many working mathematicians are informal Platonists (they talk as if math is “discovered”), but when pressed, they often lean on pragmatism (“we use whatever methods work”) or formalism (rigorous proofs are what count). Philosophers sometimes combine structuralism + constructivism (math studies structures, but they exist only through our constructions) or platonism + quasi-empiricism (there are timeless truths, but our knowledge of them grows fallibly, like in science).

Kant, Heidegger, and Wittgenstein

Kant (1724–1904) 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. In fact, Kant argued that mathematical truths are not purely analytic (like logic) but also not empirical. Instead, they are grounded in the forms of intuition—space and time—which structure human experience. For Kant, Euclidean geometry was the prime example: its truths were necessary, universal, but grounded in our a priori intuition of space. Similarily for Kant, arithmetic was grounded in our temporal intuition (counting as successive additions in time) and numbers emerge from our understanding of the difference between one and many things; an experience that requires a priori categories. Of course, the discovery of non-Euclidean geometry and relativity theory undermined Kant’s idea that Euclidean space is the necessary form of intuition, but neo-Kantian views still inspire debates about the cognitive grounding of math. His view influenced constructivist/intuitionist ideas (Brouwer admired Kant’s stress on the mind’s role).

The difficulty in grasping the stated 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). Heideggers position on the question of the nature of mathematics is oblique. It is a mode of revealing Being. He wasn’t a philosopher of math in the analytic sense, but he reflected deeply on what “mathematical” means. He tied the rise of modern science to the “mathematical projection” of nature (geometry, analytic mechanics, etc.).

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 have a box and in everyone’s box is something they call “a beetle”. However, no one is ever allowed to look into anyone elses box. They can only look into their own box—the beetle is an analogy of our personal feelings, e.g. pain. 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 “beetle”. 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 meaning is or can be private to us. We think that no one knows my personal color or pain sensation. In other words, no one can see the beetle in my box. Of course, it is true that your pain can not be felt by someone else. But the meaning of sensation words such as “pain” and “red” isn’t given by referring to some private inner introspective something—a sensation to which you alone have access to. The meaning of these words come from how the word is used in shared practices and language-games. In Wittgenstein’s 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. It cannot have a role in that teaching because we cannot get at it linguistically.

If this is true, then the same has to be the case for mathematical terms such as numbers. With this view, the meaning of the word “two” can not be private. Instead it is a result of social practices. In his later philosophy (Philosophical Investigations [Wit03]), he rejected Platonism: there are no mathematical objects “out there”. Instead, math is a set of rule-governed activities within forms of life. It “two” is a term born out of social practice to communicate such that it helps us to navigate the mess. Proofs don’t “reveal eternal truths”; they change what counts as true by fixing the rules of use. Thus, mathematics in a Wittgensteinian view is not a body of discoveries but a collection of techniques—a human institution. Therefore, Wittgenstein anticipated social constructivism: math is a practice maintained by communities. But he also connects with formalism (math as rules), but unlike Hilbert, Wittgenstein insists that meaning comes from use, not from abstract formal systems. His influence is strongest in philosophy of mathematics education and constructivist views of math.

This, of course, tells us nothing about the question of the ontological existence of mathematical objects, e.g. the number two. But Wittgenstein would probably say, that to ask for such a thing is impossible or misguided. He would insist that to ask for the ontological status of these objects is a philosophical confusion, arising from being bewitched by grammar. “2” does not exist as an independent object but it exists in the way we use it—in counting, calculating, proving. 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. An analogous question would be to ask: “What is the ontological status of the king in chess?” A Platonist might say: “The king is a timeless abstract piece.” Wittgenstein would say: “The king exists as part of the game; it has no existence outside the practice of chess.” Likewise, “2” exists as part of the language game of arithmetic.

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. This frustrates Platonists and structuralists, who think the ontological question is real and can’t be dismissed.

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, except 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. But this meaning can shift. It has to be re-established again and again in social pracitice.

In this regard, 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. One might say: “Ok but you know what ‘pink’ and ‘elephant’ means” but the same is true for the mathematical objects from which the perfect circle emerges. 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.

\[K = \{(\cos(\phi), \sin(\phi)) : \phi \in [0;2\pi]\}\]

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 a Wittgensteinian view, 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 (which are non-arbitrary constructions) of nature (or our environment) and therefore it is also partly 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.

Digital Signal Processing

Sound is a what we call signal. A signal is any physical quantity that carries information. It is defined as any physical quantity that varies with one or more independent variables such as time (one-dimensional signal), or space (2D or 3D signal). Signals exist in several types. Of course, we are interested in the one-dimensional case and our independent variable is time \(t\). In the real world, most of signals are continuous-time or analog signals that have values continouesly at every value of time.

Furthermore, we are working on a computer—a digital device. Digital means that that the process we wanna do, such as creating or transforming a signal, has to be done in a discrete space. Therefore, to be processed by a computer, a continuous-time signal has to be first sampled in time into a discrete-time signal so that its values at a discrete set of time instants can be stored in computer memory locations. In addition, in order to be processed by logic circuits, these signal values have to be quantized into a set of discrete values. The final result is called digital signal.

So what is the field mathematics we apply is: Digital Signal Processing (DSP). The two main characters in DSP are signals and systems. A system is defined as a process whose input and output are signals. We need those to further transform the signal, for exmaple, to filter high or low frequencies. As we will see, an important class of systems is the class of linear time-invariant (LTI) systems. These systems have a remarkable property, that is, each of them can be completely chracterized by an impulse response function, and the system is defined by a convolution (also referred as filtering) operation. Thus, a linear time-invariant system is equivalent to a (linear) filter. LTI systems are classified into two types, those that have finite-duration impulse response (FIR) and those that have an infinite-duration impulse response (IIR). In practice for our musical purposes we will create filters that vary over time but the theory of LTI systems is still very useful to induce understanding.

A signal can be viewed as a vector in a vector space. Thus, linear algebra provides a powerful framework to study signals and linear systems. In particular, given a vector space, each signal can be represented (or expended) as a linear combination of elementary signals. The most important signal expansion are provided by the Fourier transform. The Fourier transforms, as with general transforms, are often used effectively to transform a problem from one domain to another domain where it is much easier to solve or analyze. The two domains of Fourier transform have physical meaning and are called time domain and frequency domain.

Sampling, or the conversion of continuous-domain real-life to discrete numers that can be processed by computers, is the essential bridge between the analog and the digital world. It is important to understand that connections between signal and systems in the real world and inside a computer. These connections are convenient to analyze in the frequency domain. Moreover, many signals and systems are specified by their frequency characteristics.

Because any linear time-invariant system can be characterized as a filter, the design of such systems boils down to the design of the associated filter. Typically, in the filter design process, we determine the coefficient of an FIR or IIR filter that closely approximates the desired frequency response specification. Together with Fourier transforms, the z-transform provides an effective tool to analyze and design filters.