Is Mathematics discovered or invented?

The question of whether mathematics is discovered or invented has been a topic of philosophical debate for centuries. There are different perspectives on this issue, and both viewpoints have their proponents. Let's explore both sides of the argument:

Mathematics as Discovered:

1. Platonism: This philosophical view asserts that mathematical entities and concepts exist independently of human thought or invention. According to Platonism, mathematicians discover mathematical truths and concepts, much like explorers discovering physical phenomena. These mathematical truths are considered to be part of an abstract, objective reality.
2. Existence Beyond Humanity: Proponents of this view argue that mathematical truths and relationships are universal and exist independently of human existence. They are "out there" waiting to be uncovered by mathematicians. Theorems and mathematical structures are discovered, not invented.
3. Consistency with the Physical World: Many argue that the remarkable consistency of mathematics with the physical world suggests that mathematics is a discovery. Mathematical concepts and equations often provide highly accurate descriptions of natural phenomena.

Mathematics as Invented:

1. Formalism: The formalist perspective sees mathematics as a human invention, created by mathematicians to organize and describe patterns and relationships. In this view, mathematics is a product of human creativity and language.
2. Use of Axiomatic Systems: Mathematicians often work within axiomatic systems, starting with a set of axioms and deriving theorems from those axioms. This process of theorem proving is seen as a human invention.
3. Variability Across Cultures: Different cultures have developed their own mathematical systems, suggesting that mathematics is culturally constructed rather than universally discovered.
4. Evolution and Development: Mathematics has evolved and developed over time. New mathematical concepts and structures have emerged as human understanding has advanced. This suggests that mathematics is not static but is shaped by human creativity and invention.

To venture into these uncharted territories, let's dive into a part of the book 'Complex Analysis' of Ian Stewart and David Tall where they discuss this topic in Chapter 0. We will transcribe this part, trying to be as faithful to the text as possible. (Here we will start the transcription):

Students trying to understand new concepts are in a similar position to the pioneers who first investigated them. At any stage in our education, we build not just on our current knowledge, but on a variety of beliefs and intuitions that are often vague and may not be consciously recognized. As a trivial example, children familiar with counting numbers may find it hard to adapt their thinking to negative numbers, or rational numbers. When faced with questions like ‘what is 3 minus 7?’ or ‘what is 3 divided by 7’, intuition based solely on whole numbers leads to the answer ‘can’t be done’. That makes it hard to understand ?4 or 3/7. In fact, these is not really trivial examples, because the world’s top mathematicians, centuries ago, were just as confused by the question ‘what is the square root of minus one?’ Even their terminology – ‘imaginary’ – reveals how puzzled they were. Intuitively they considered numbers to be ‘real’ – not in the sense we now use to distinguish real from complex, but as direct representations of real measurements. The new objects behaved like numbers in many ways, but they seemed not to correspond directly to reality.

In such circumstances, it can be tempting to discard existing intuition completely. But it is more sensible to adapt the intuition to fit the new circumstances. It is much easier to do arithmetic with negative numbers or fractions if you remember how to do it with whole numbers; it is much easier to do algebra with complex numbers if you bear in mind how to do it with real numbers. So, the trick is to sort out which aspects of existing intuition remain valid, and which need to be refined into a broader kind of understanding.

One way to approach this issue is to take seriously a question that is often asked but seldom answered satisfactorily: is mathematics discovered or invented? One answer is to dismiss the question and agree that neither word is entirely appropriate; moreover, they are not mutually exclusive. Most discoveries have elements of invention, most inventions have elements of discovery. Galileo would not have discovered the moons of Jupiter without the invention of the telescope. The telescope could not have been invented without discovering that sand could be melted to make glass.

But leaving such quibbles aside, we can make a rough distinction between discovery, which is finding something that is already there but has not hitherto been noticed, and invention, which is a creative act that brings into being something that has not previously existed. There is a case to be made that in this sense, mathematicians invent new concepts but then discover their properties. For example, complex integration is all about ‘paths’ in the complex plane. Intuitively, a path is a line drawn by moving the hand so that the pencil remains in contact with the paper – no jumps. We might choose to formalise this notion as a continuous curve – the image of a continuous map from a real interval to the complex plane. We might be interested in how the pencil point moves along this curve, which requires the map itself, not just its image. Sometimes we might wish the path to be smooth – to have a well-defined tangent.

As it happens, we need all of these notions. Intuitively, they are all based on the same mental image. Formally, they are all very different. They have different definitions, different meanings, and different properties. A smooth path always has a meaningful length, for instance; a continuous path may not. The definitions we settle on in this book fit conveniently into the standard ideas of analysis, but they are not built into the fabric of the universe. We chose them, and by so doing we invent concepts such as ‘path’, 'curve’, and ‘smooth’.

On the other hand, once a concept has been invented, we cannot invent its properties. When we also invent the concept ‘length’, we discover that every smooth path has finite length. We cannot ‘invent’ a theorem that the length of a smooth path can be infinite. If we weaken ‘smooth’ to ‘continuous’, however, we can discover that infinite lengths are possible; indeed, ‘length’ need not have a sensible meaning at all. In short: invention opens up new mathematical territory, but exploring it leads to discoveries. We may not know what things are present in the territory, but we do not get to choose them.

Sometimes – in fact, very often – we discover that our inventions have features that we neither expected nor intended them to have. We discover, perhaps to our dismay, that the image of a smooth path can have a right-angled corner. We did not expect that: a corner does not feel ‘smooth’. But its possibility is a direct consequence of the definition we invented.

When this kind of thing happens, we have two choices. Accept the surprises as the price for having a nice, tidy definition; or rule them out by changing the definition – inventing a more comfortable alternative. In practice we often do both, by giving the alternative a different name. Here we could (and do) define a ‘regular path’ to be a smooth path γ : [a, b] → ? for which γ'(t)≠0 whenever t ∈ [a, b]. Now the image cannot have a sharp corner. On the other hand, every theorem about regular paths must now take account of the consequences of that extra condition. We also have to remember that some theorems may be valid for regular paths but not for smooth paths, and so on.

As we move from intuitive ideas to formal ones, we also refine our intuition so that it matches the formal theory better. Formal calculations start to make sense, not just as strings of symbols that follow from previous strings, but as meaningful statements that agree with our new intuitions. From this point of view, the history of complex analysis is the story of intuition co-evolving with an increasingly formal approach. This suggests that mathematicians lost interest in the meaning of complex numbers when they incorporated them into their intuitive assumptions and beliefs. With the apparent conflicts resolved by these refined intuitions, they were free to push the subject forward, no longer worried that it did not make logical sense.

When a mathematical area ‘settles down’ into a mature theory, there is a broad consensus that certain concepts provide the most convenient route through the material. These concepts then become standard – things like ‘continuous’, ‘connected’, and so on. They get taught in lecture courses and printed in books. We may start to feel that the standard definitions are the only reasonable ones. Even so, we are always free to work with different concepts if that seems sensible, or even to modify definitions while retaining the same name – though that can be dangerous. Today’s concept of continuity is quite different from what it was in the time of Euler, but we use the same word; we just bear in mind that it now has a specific technical meaning. A historian reading Euler would need to be on their guard.

It is also worth remarking that many mathematical concepts seem more natural to us than others. Counting numbers are very natural (we even call them the ‘natural numbers’). The number i was baffling for centuries (and was called ‘imaginary’ as a result). Our culture, our society, and even our senses, predispose us towards certain concepts. Euclid’s points and lines correspond to early stages of the processing of images sent from the retina to the visual cortex. Newton’s concept of acceleration being related to an applied force reflects the way our ears sense accelerations and make us ‘feel’ a push –a force.

It then becomes easy to imagine that mathematics somehow already exists in a realm outside the natural world. Even if humans invented numbers, in retrospect they seem such a natural idea that surely, they were just hanging around waiting to be invented. If so, that is more like discovery. This view is often called Platonism: the idea that mathematical concepts already exist in some ideal form in some kind of world outside the physical universe, and mathematicians merely discover how these ideal forms work. The contrary view is that mathematics is a shared human construct, but that construct is by no means arbitrary, because every new invention is made in the context of existing knowledge, and every new discovery must be logically valid.

A major theme of this book is that many apparently puzzling aspects of complex analysis can be made more intuitive by paying attention to the geometry of the complex plane (in a broad sense, including its topology). This brings one of the human brain’s most powerful abilities, visual intuition, into play. For this reason, we draw a lot of pictures. However, a picture, and our visual intuition, can be misleading unless we examine the unstated assumptions that they involve. By doing so, we can refine out intuition and make it more reliable. For this reason, we do not just introduce important definitions and then deduce theorems that refer to them. We try to relate those definitions to intuition, to make the proofs easier to understand. Then we exhibit some of the positive results that arise, to convince you that the new concept is worth considering. And then . . . we show you that sometimes the formally defined concept does not behave the way intuition might suggest. Sometimes it turns out to be useful to strengthen the definition so that it matches intuition more closely. Sometimes we refine our intuition so that it matches the formal definition. Sometimes we can even do both, in which case we have to make some careful but useful distinctions.

The historical events sketched earlier in this chapter offer many examples of this process. The square root of minus one went from being a puzzling idea that seemed to have no meaning to one of the most important concepts in the whole of mathematics. Along the way, mathematicians’ intuition for ‘number’ underwent a revolution. We can now to some extent short-circuit the historical debates – what were hang-ups then need not be hang-ups now – but when a new idea puzzles us and doesn’t seem to make sense until we finally sort it out, it is helpful to remember that the mathematical pioneers often experienced exactly the same feelings, for much the same reasons. (Here finish the transcription)

In reality, the answer to the question may not be an either/or proposition. It's possible that there are elements of both discovery and invention in mathematics. The process of exploring and proving mathematical theorems may involve both discovering preexisting truths and inventing new mathematical concepts and relationships.

Ultimately, the view one adopts may depend on their philosophical perspective and the specific aspect of mathematics being considered. The debate itself is a testament to the richness and complexity of the field of mathematics.

You may share your thoughts in comments. Eager to know them! Until the next article...