For some, this question may suggest an evasive adventure by the meanings of words; a mere semantic reverie. There still those for whom the actions of “discovering” and “inventing” are equivalent, and thus the discussion becomes irrelevant. However, considering mathematics in one way or another, it changes its status before other sciences and society and has direct implications on its teaching.
The difference between discovery and invention
When talking about discovering “something”, it is admitted that such a “thing” has an existence a priori. An example of this was the discovery of America in the 16th century. The continent has always existed, it wasn’t created by Europeans. For them, what happened was a process of discovery. Similarly, we can consider that the original inhabitants also discovered the existence of these newcomers to their lands.
And an invention? In this case, the “thing” invented did not exist at all before whoever invented it. For example, the magnetic compass. The Earth’s magnetism preexists to any human action. However, the creation of an artifact that uses this phenomenon to help locate it on the surface of the planet is an invention!
Bringing this discussion to mathematics scope, we will resume our initial question: discovery or invention? If mathematical entities such as numbers and geometric figures were discovered, this means that they preexisted in some way. If, on the contrary, they were invented, it means that their existence began as the end of process of invention.
In the following we will present some constitutive elements of these two perspectives.
Mathematics as discovery
The origins of this perspective are diluted along its own historical construction. However, we can identify strong traits that ground such a perspective in classical Greek thought, particularly in Plato and Aristotle.
Plato (left) and Aristotle (right) in detail of the work “School of Athens”, by Raphael.
According to Plato, ideas (eidos) would exist ‘in and of’ themselves, independent the actions of any subjects. Thus, ideas would inhabit a dimension of their own, of purity and immobility called the intelligible world. Such ideas would manifest themselves in the world which we live in, the sensitive world.
The access to the intelligible world would be reserved for philosophers, holders of reason and true knowledge. Nevertheless, it would be in the intelligible world that mathematics entities would inhabit. In this way they would exist in a full and autonomous way, endowed with perfection. Thus, knowing mathematics would mean the ownership of true science and knowledge.
In what way, then, would people have access to mathematics? Plato presents an possible answer his dialogues ‘Meno’ and ‘Phaedo’: the theory of reminiscences (anamnesis).
Reminiscence means recollection, or remembrance. Since, for Plato, souls would inhabit the intelligible world before materializing in the sensitive world, accessing the first would mean an exercise in recollection of the soul itself. In Meno, Socrates (as Platonic character) proposes to demonstrate the theory to the other character, Meno.
Socrates asks Meno to choose one of his servants and questions him at length on an classic Geometry problem known as ‘the duplication of square’. Throughout his conversation with the servant, Socrates problematizes the answers given and creates new questions leading the servant’s reasoning and putting him in doubt or in apparently insoluble situations (aporia).
Socrates: — You realize, Meno, what point he has reached in his recollection. At first he did not know what the basic line of the eight-foot area square was; even now he does not yet know, but then he thought he knew. He answered confidently, as if he knew, and he did not think he was at a loss, but now he thinks he is at a loss; and so, although he does not know, neither does he think he knows. […] Look, then, at how he will emerge from his perplexity while searching together with me. I will do nothing but ask questions, not instruct. Watch whether you find me instructing and explaining instead of asking for his opinion. […] So these opinions were in him all along, were they not? […] These opinions have so far just been stirred up, as in a dream, but if he were repeatedly asked these sorts of questions in various ways, you know that in the end his knowledge about these things would be as perfect as anyone’s. […] If he always had it, he would always have known. If he acquired it, he cannot have done so in his present life.
Once mathematical entities would preexist in intelligible world and would be accessed from recollection of a previous life, this perspective places mathematics as discovery — i.e. Mathematical truths are therefore discovered, not invented. Perhaps not a discovery in the same sense previously exemplified. It isn’t a search for something external, such as a continent or a natural phenomenon. On the opposite, it’s a search inside the soul itself.
According to Michael Foucault, such a conception coated the mathematics with ideality and occurred throughout history, being questioned only to be repeated and purified. This metaphysical view of mathematics has spread its influence till modern days. Nevertheless, this conception has also been criticized by a number of intellectuals, mathematicians and scientists. The main arguments of this criticism are founded on the conception that mathematics would be a human invention.
Mathematics as invention
For Platonism, mathematical entities were eternal, i.e., they had no beginning. However, to contest such an idea also means to discuss a possible origin for mathematics.
Some historians support the origins of mathematics in the observation of nature and the surrounding world by early humans. According to this approach, the human ability to recognize and compare sizes, shapes and quantities constitutes the core of mathematical knowledge.
Another approach points out, through the effects of labor. This is not antagonistic to the first approach. We understand that both complement each other. As hominids began to walk erectly, their hands were free to make tools, construct artifacts. Thus, notions such as symmetry, measures, proportionality and counting gradually developed according to the development of the labor itself. Added to this approaches, there are also signs that link the development of mathematics to religion, ritualistic doing, and even in games and toys.
In this sense, objects seen in nature are never exactly geometry forms, for example. There are no perfect straight lines, circles, squares in nature. The creativity, inventiveness and analytical spirit of human beings have created ideal models of these natural forms. An exercise of transformation from concrete to abstract. And even more, by using such abstract models to unravel the mysteries of nature and solve everyday problems, the human being resigns this knowledge by using it in the real "concrete" world.
South American Indigenous people identified the equilateral triangle as an optimal solution for the creation of an apparatus for sifting manioc flour (Gerdes, 2013, p. 49).
In this sense, Mathematics isn't a collection of objects and concepts endowed with an a priori existence. It's a invention of the humankind in response of its observation and action on the world.
References & Further reading
Foucault, Michel. (1982). The archaeology of knowledge: and the discourse on language. Vintage.
Gerdes, Paulus. (2013). Ethnogeometry: awakening of geometrical thought in early culture. Lulu.
Plato. (2002). Meno.
Courant & Robbins (1996). What is mathematics?
Hersch, Reuben (1997). What is Mathematics, really?