This is not one claim, but two separate claims. First, what we call “mathematics” today emerged specifically in the fifth century BCE. Second, that this emergent mathematics consisted of four components: arithmetic (the mathematics of numbers), geometry (the mathematics of two-dimensional and three-dimensional shapes), astronomy (the mathematics of celestial motions), and harmonics (the mathematics of music).

Let us consider the two claims separately.

Fifth century emergence

The first claim rests on a number of assumptions that need to be made explicit. The first is about location. The context within which mathematics as-we-know-it emerged is implicitly taken to be the Greek-speaking world. The mathematical practices of other cultures – Mediterranean and beyond – emerged much, much earlier than the fifth century BCE. In other words, the assertion here is really that Greeks of the Classical era invented modern mathematics.

Whether that is true depends not on hard evidence, but on what we take to be the defining characteristics of mathematics. And herein lies the second implicit assumption. It presupposes that “we” agree on what “we” call mathematics today. And yet, there is no homogenous “we” whose answer to the question, “What is mathematics?”, would be in unison, like a choir singing the same note.

What is what “we” call mathematics? Is it the subject we learn at school? But children all around the world – in different parts of Europe even – learn mathematics in significantly different ways: some will be taught to privilege mental arithmetic, whereas for others calculating with an abacus will be prioritised. If we look at the maths that anyone in the UK is supposed to learn – the current GCSE syllabus – the deep roots of most of it (problem-solving on specific exercises, rather than proofs) are in fact historically to be found in Egypt and Mesopotamia, several centuries before the fifth century BCE. One clear example can be found in the Rhind papyrus (see the image at the top of this article), which dates to the Second Intermediate Period of Egypt in the second millenium BCE.

Should we call mathematics what is taught at universities? But even that tends to be subdivided into “pure” mathematics, statistics, computational mathematics, and more. Whether one of those – even “pure” mathematics – is more “mathematics” than the rest is up for debate, and thus quite far from any consensus.

However, let us assume we are referring specifically to the emergence of axiomatico-deductive proof, a demonstrative method which starts from undemonstrated premises – some of which are called axioms – and proceeds logically from the general to the particular: a process called deduction. Euclid’s Elements (early third century BCE) is probably the earliest extant fully-structured exemplification of the method, but its earliest fully-structured discussion is probably in Aristotle’s Prior Analytics and Posterior Analytics (fourth century BCE), where he provides an account of deductive demonstration using mathematical examples.

The first claim is therefore, in my view, either badly formulated, and thus neither true nor false, because it elevates what is essentially a limited perspective to a universal “we”, when in fact views about we call mathematics today are hugely diverse, or it is partially true, but chronologically off by around a century.

The four disciplines

The four disciplines – arithmetic, geometry, astronomy, and harmonics – correspond to something called the quadrivium, i.e. the more advanced stage of education in a traditional liberal arts curriculum. This claim is then true, in that these four mathematical disciplines were seen as a compact, whose study would encompass the study of mathematics, but it is also false, in that there is no incontrovertible evidence that this compact had become fully established already in the fifth century BCE. The traditional liberal arts curriculum – what was known in Greek as enkyklios paideia – seems to have emerged in the Hellenistic period, and thus no earlier than the third or even second century BCE.

What we do find in the fourth century BCE, is a discussion of all four mathematical disciplines in Plato’s Republic (book 7), where they are presented as an essential part of the education of the philosopher king. Whether Plato was reflecting the extant educational practice of his times, or proposing something new and unusual, is up for debate. Given however the context of the Republic as a whole - a dialogue about the good, in the course of which a set of characters imagine how they could build the ideal state, and given also Plato’s critical attitude to the mathematical practices common at his time, it seems more likely that his “curriculum” did not reflect a widespread way of teaching mathematics in the fourth, let alone the fifth, century BCE.

In fact, in a later dialogue, also on the theme of the ideal state, Plato presents only arithmetic, geometry, and astronomy as a coherent group (Laws 817e):

There still remain, for the freeborn, three branches of learning: of these the first is reckoning and arithmetic; the second is the art of measuring length and surface and solid; the third deals with the course of the stars, and how they naturally travel in relation to one another.

The following passage has however been attributed to the Pythagorean philosopher Archytas, a slightly older contemporary of Plato’s, whose work may have been known to Plato (Archytas ap Porphyr, in Ptol. Harm.):

The mathematicians seem to me to have arrived at true knowledge, and it is not surprising that they rightly conceive the nature of each individual thing; for, having reached true knowledge about the nature of the universe as a whole, they were bound to see in its true light the nature of the parts as well. Thus they have handed down to us clear knowledge about the speed of the stars, and their risings and settings, and about geometry, arithmetic and sphaeric, and, not least, about music; for these studies appear to be sisters.

Archytas is referring to the four-fold structure of mathematics as something that pre-existed him. These words however have not come down to us in one of Archytas’ own books, none of which survive. Rather, they have been transmitted by Porphyry (Ptol. Harm.), a third-century CE Neoplatonic philosopher, who premises them with the following (emphasis mine; transl. Ivor Thomas):

Let us now cite the words of Archytas the Pythagorean, whose writings are said to be mainly authentic. In his book On Mathematics right at the beginning of the argument he writes thus (…)

Maybe then the words cited above are really by Archytas, in which case it could be said there is some truth to the claim. But it cannot be considered to be universally true because not everybody – not even in the Greek world – was on board with it. On the other hand, maybe the words cited above are not really by Archytas, or perhaps, in true Pythagorean fashion, the words found in Porphyry’s source are attributing greater antiquity to an idea that was in fact more recent.

Conclusion

The first claim works off two key assumptions: that there is an agreement today about what mathematics is, and that its origins lie in the Greek speaking world of the fifth century BCE. Even on a sympathetic interpretation, neither part of this claim has much validity.

The second claim is harder to determine: it is either chronologically off or true, depending on whether you believe that: (a) Archytas is telling the truth about the antiquity of the four-fold structure of mathematics, and; (b) original fragments of Archytas’ text survived through the approximately seven centuries that separate him from Porphyry.

It is for these reasons we rate the overall claim as misleading.