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What does ars mathematica or mathematicus mean?

From Augustine's commentary to the Genesis we have the following statement:

Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium, maxime dicentes vera, cavendi sunt, ne consortio daemoniorum animam deceptam, pacto quodam societatis irretiant.

Or from the Codex Iustinianus:

Artem geometriae discere atque exerceri publice intersit. Ars autem mathematica damnabilis interdicta est

Surely we don't assume that mathematics is so bad.

If I didn't know anything else, from the context I would assume numerology is meant here.

But mathematici / ars mathematica is translated as astrologers/astrology.

And Thomas Aquinas did understand it that way, as he uses the quote from Augustine to support his views on astrology in the Summa Theologiae:

Et ideo astrologi in multis vera praenuntiant, et praecipue in communibus eventibus, qui dependent ex multitudine. Alio modo, propter Daemones se immiscentes. Unde Augustinus dicit, "Fatendum est, quando a mathematicis vera dicuntur, instinctu quodam occultissimo dici, quem nescientes humanae mentes patiuntur. [...]" Unde concludit, "quapropter bono Christiano sive mathematici, sive quilibet impie divinantium, et maxime dicentes vera, cavendi sunt, ne consortio Daemoniorum animam deceptam pacto quodam societatis irretiant."

Still, I find this puzzling. In the quote from the Codex Iustinianus, why was ars mathematica contrasted to ars geometriae?

It is often claimed that ars geometriae didn't mean geometry but mathematics in general. But that still doesn't make so much sense, too.

At some point much later, in modern Latin, mathematicus seems (from my cursory knowledge) to always just mean mathematical, for example in Newton's Philosophiae Naturalis Principia.

Thomas Aquinas again, as an example for medieval Latin, though he understood Augustine's use of mathematicus as meaning astrological himself uses mathematicus as just meaning mathematical. In De Substantiis Separatis he writes:

Unde Plato duo genera rerum a sensibilibus abstracta ponebat: scilicet mathematica, et universalia, quae species sive ideas nominabat. Inter quae tamen haec differentia videbatur: quod in mathematicis apprehendere possumus plura unius speciei, puta duas lineas aequales, vel duos triangulos aequilateros et aequales; quod in speciebus omnino esse non potest, sed homo in universali acceptus, secundum speciem est unus tantum.

So did ars mathematica also mean mathematics in classical (and late) Latin? Or did it only mean astrology?

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    This is an interesting set of questions, but it feels a bit too broad. Asking all those questions in one question makes this difficult to answer. Is your last paragraph your main question? Consider separating your question into several questions or emphasizing what is your main question you actually want answered. – Joonas Ilmavirta Jun 13 '16 at 12:24
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Here is the Lewis and Short definition of mathematica:

mathēmaticus, a, um, adj., = μαθηματικο:ς, of or belonging to mathematics, mathematical (class.). Adj.: mathematica nota, Vitr. 1, 1: artes, Plin. 30, 1, 1, § 2: cogitatio, Macr. Somn. Scip. 2, 2: disciplinae, i. e. geometry, arithmetic, astronomy, music, geography, optics, Gell. 1, 9, 6.
— Subst. Mathēmaticus, i, m. A mathematician, Cic. de Or 1, 3, 10; id. Ac. 2, 36, 116; id. Tusc. 1, 2, 5; Sen. Ep. 88, 26.
— An astrologer (post-Aug.): mathematici, genus hominum potentibus infidum, sperantibus fallax, quod in civitate nostra et vetabitur semper et retinebitur, Tac. H. 1, 22: nota mathematicis genesis tua, Juv. 14, 248; Tert. Apol. 43: qui de salute principis ... mathematicos consulit, cum eo qui responderit, capite punitur, Paul. Sent. 5, 21, 3.
— Mathēmatica, ae, f. Mathematics, Sen. Ep. 88, 23; v. l. ma-thēmaticē ( = μαθηματική, sc. τέχνη).
— Astrology: addictus mathematicae, persuasionisque plenus, cuncta fato agi, Suet. Tib. 69.

This definition tells us that a mathematicus could mean astrologer OR mathematician. The difference between the occupations (in terms of the word used) became more apparent only after the reign of Augustus. This definition also states that mathematica meant mathematics. It makes sense that it could also mean astrology seeing as mathematicus has the duel meaning, and this is backed up by a definition found here.

So, to answer your question, it appears that mathematica had the duel meaning in Classical Latin. Pre-Augustus the word primarily meant astrology, but post-Augustus other words were used instead (such as astrologia, see below), with mathematics becoming another common definition (It is important to note however that it still retained its astrologer meaning). Augustine and Justinian were both writing in Late Latin, the immediate descendant of Classical Latin, so it may have been more interchangeable for them as compared to Thomas Aquinas, who would have written in a mix between Medieval and Renaissance Latin, thus giving more time for the meanings to diverge.

Geometria would have exclusively meant geometry at the same time mathematica had duel meaning. This makes sense because geometry was far more valuable to the Romans (architecture, road building, etc.) than number operations, so it is logical that there would be a difference. Geometry to the Romans could have meant mathematics in general because what we think of as "mathematics" (algebra, calculus, etc.) either did not exist or was not very advanced, so it would have been lopped into the broader category of geometry.


Definition of astrologia:

astrologia, ae, f., = ἀστρολογία, knowledge of the stars, astronomy (class. for the later astronomia, while astrologia was used to designate astrology exclusively first in late Lat., Hier. adv. Pelag. 1, 8; cf. Isid. Orig. 8, 9), Cic. Div. 2, 42, 87 sqq.; id. de Or. 1, 16, 69; id. Off. 1, 6, 19: astrologiam Atlas Libyae filius, ut alii Aegyptii, ut alii Assyrii invenerunt, Plin. 7, 56, 57, § 203; also a work upon astronomy: occasum matutinum vergiliarum Hesiodus, nam hujus quoque nomine exstat astrologia, tradidit fieri, id. 18, 25, 57, § 213.

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    "Pre-Augustus the word primarily meant astrology, but post-Augustus other words were used instead, with mathematics becoming the primary definition". I think the citation in Lewis and Short says exactly the opposite. – fdb Jun 13 '16 at 16:06
  • @fdb I apologize about that. I guess I misinterpreted the definition. Thanks for bringing that to my attention! I'll change that... – Sam K Jun 13 '16 at 16:11
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Many of the definitions and classical quotations in Forcellini make mathematica mean astrology. However, to illustrate the modern sense, Forcellini quotes from Seneca, Ep. 88, including this passage, where mathematice is rendered as "the science of numbers" in this English translation:

Philosophia nil ab alio petit, totum opus a solo excitat: mathematice, ut ita dicam, superficiaria est, in alieno aedificat; aliena accipit principia, quorum beneficio ad ulteriora perveniat.

Now philosophy asks no favours from any other source; it builds everything on its own soil; but the science of numbers is, so to speak, a structure built on another man's land – it builds on everything on alien soil; it accepts first principles, and by their favour arrives at further conclusions.

Elsewhere in the same letter, Seneca seems to use geometria as the science of measurement, and mathematica for astronomy (not astrology).

Forcellini quotes the following sentence from Macrobius, Somn. Scip., apparently providing a Late Latin usage of mathematica in its modern sense:

Corpora mathematicæ cogitationi tantum subjicienda, non sensui.

but I was not able to find this sentence in the original! I did find two occurrences of mathematica in that work, which, at least by my beginner's level of Latin, appear to have the modern sense.

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