What is Mathematicalia?

Why this name? 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. In doing this, we might also notice the special status of mathematical objects with regard to both epistemology and ontology. A good starting point 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:

Scanduntur autem ardua triplici gradu: quaedam enim accedunt ad animam nostram sensu, imaginatione, et intellectu: quaedam imaginatione, et intellectu tantum: quaedam autem intellectu tantum. Prima sunt alta tantum, secunda altiora, tertia altissima. […] et separatis secundum rationem a motu et materia, sed tamen per esse conjunctis, et haec sunt mathematicalia.The hard things <we can deal with> can be divided according to three levels <of difficulty>: (1) those which reach our soul via the senses, the imagination, and the intellect; (2) those which reach our soul via the imagination and intellect only; (3) and those which enter our soul only via the intellect. The first ones are simply difficult, the second ones are more difficult, the third ones are the most difficult. [The second ones] are separated from movement and matter via the reason, but they are nevertheless joined to these in their being, and these are the objects of mathematics.

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:

Etiam mathematicalia non sunt per se sensibilia, cum sint abstracta a motu et materia sensibili; sed numerus, magnitudo et figura, quae ponuntur sensibilia communia, sunt entia mathematicalia.Even the mathematical objects are not sensible things in themselves, as they are abstracted from movement and sensible matter; but the number, the magnitude, and the geometric figure, which are conceived as being common to sensible things, are indeed mathematical beings.

From these two brief examples we can draw that mathematicalia are something deeply linked to the material world and that they also exist independently from sensible reality thanks to our reasoning. This means that they have a special status with regard to both ontology and epistemology.

As Albert remarks, they are indeed something hard to deal with. This opens up another nuance of the word mathematicalia: the set of notions and concepts on which different mathematical disciplines are built. According to the 13th-century polymaths Roger Bacon (Opus maius IV), learning mathematics is easy to clerics, who are not however fitted for other sciences:

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

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:

pueri melius et citius addiscunt mathematicalia […] et etiam per experientiam scimus, quod pueri melius addiscunt et capiunt mathematicalia, quam alias partes philosophiae.