What is Mathematicalia?

Why such a name for a website? It all depends on the meaning of the Latin word mathematicalia, which simply refers to ‘mathematical things’ or ‘things that relate to mathematics’. This is precisely what one can find within this webpage: materials, information, and news concerning the premodern mathematical thought.

To better understand what mathematicalia means to a premodern thinker, we might have a closer look at a few occurrences of this Latin term in the history of philosophy. A good starting point, though, is its Greek ancestor, τὰ μαθηματικά.

Aristotle’s μαθηματικά

In book 13 of Metaphysics, Aristotle wonders whether mathematical entities (τὰ μαθηματικά) exist or not; if they exist, whether they are in sensible things or separated; and whether they simply exist or they exist in a special way.

The first of these dilemmas – do mathematical entities exist in sensible things? – had been already solved in book 3, when discussing the issue of the existence of ‘intermediate beings’ between the sensible reality (i.e., the world we experience) and the eidetic world (i.e., the Platonic ideas). Mathematical entities are not to be found in sensible things, as such assumption would indeed lead to absurd conclusions: two solid bodies (the empiric one and the ‘intermediate’ geometric one) would exist in the same place; mathematical entities conceived as properties of things in motion would be also in motion; physical bodies, which are divisible, would not be as such, for they would contain mathematical entities, which in turn are indivisible. However, mathematical entities are not even separated, that is, existing independently from physical bodies, for also this stance would generate unreasonable ramifications. We would have, for instance, a pointless proliferation of geometric entities: besides the material solid, there would also be the separate geometric solid (S1); beside the separate geometric solid (S1) we would also have the geometric solid (S2), surface (s2), line (L2), and point (P2), and so on. Which of these entities would then represent the specific object of geometry? Moreover, if mathematical entities would be separated, then also the axioms on which mathematical profs rely would be separated from both sensible things and mathematical entities. Such axioms would be of mathematical nature, but at the same time different from mathematical entities, i.e., they would still be some kind of mathematical substances which would not correspond to any of the mathematical objects such as lines, points, numbers, etc. These are just a few of the many arguments offered by Aristotle – we won’t dwell on all of them here!

So how do mathematical entities exist according to Aristotle’s account? Mathematical entities – including their universal propositions, notions, and proofs – are about sensible things, considering them as magnitudes and numbers. The mathematician abstracts these mathematical properties, isolating them from all other mathematically irrelevant aspects of the sensible thing, such as the movement, the color, etc. The mathematician studies what is sensible, but not in as much as it is sensible; rather, he/she considers the mathematical properties (surfaces, lines, numbers, etc) as if they were separated. Tὰ μαθηματικά do not exist independently from sensible things, rather, they are ‘isolated’ from these by mathematicians – they have a separate existence only in their minds. The specific way of knowing mathematical entities is therefore the abstraction.

Mathematicalia in the Middle Ages

Aristotle’s view on mathematical entities was the most popular in the Middle Ages, at least especially after the rediscovery of his philosophical works (e.g., On the Soul, Physics, Metaphysics, and other treatises on natural philosophy) thanks to the translations from Arabic and Greek into Latin. (This is not totally true, as also Boethius, who lived in the 6th century, defended the same Aristotelian account together with a more Neopythagorean one. Click here to read the full story.

Medieval occurrences of the term mathematicalia mirror some fundamental aspects of Aristotle’s conception of mathematics. Two cases from the 13th and 14th century will do: one text by Albert the Great and a passage from a supposed commentary by John Buridan.

The indefatigable champion of Aristotelian philosophy, Albert the Great (d. 1280), comments on a passage from Peter Lombard’s Sentences, clarifying what are the ‘hard things’ that Peter Lombard is referring to:

In another 14th-century commentary on Aristotle’s De anima, we find the same account of mathematicalia. The anonymous commentator was inspired and influenced by John Buridan (d. 1361 ca.), one of the most influential Masters of Arts at the University of Paris – so that this text was initially ascribed to Buridan himself and referred to as prima lectura. Besides the querelle about its authorship (see here), this text is revealing of the spreading of the Aristotelian account of mathematical objects within the university context. Here’s the passage concerning our mathematicalia:

Moreover, young people can learn mathematics very quickly and successfully, which is not the case for the other parts of philosophy, such as metaphysics, physics, and ethics:

Clerici licet rudissimi mathematicalia possunt scire, quamvis ad alias scientias non valeant attingere.