Concrete and abstract numbers

Question

Recently I heard about the concepts of concrete number = numerus numeratus and abstract number = numerus numerans. See here. These seem to be mathematical-philosophical specialist terms of medevial (European) origin with a "scholastic sound".

In

Smith, D.E. (1953). History of Mathematics. Vol. II. Dover. pp. 11–12. ISBN 0-486-20430-8.

one finds

Question: Is this distinction still in use in contempory mathematics, at least at some places? Or is it a premodern concept which has completely disappeared?

A search for "concrete number" gives 54 results in this forum, but I do not have the impression that its use has a philosophical background. Similarly "abstract number" gives 15 results.

Update:

Here are some sources which prove the origin in the medieval (European) world of ideas:

1. Joh. Micrealius, Lexicon Philosophicum (1661)

Numerus est compositarum unitatum aggregatio. Vocatur alias multitudo & quantitas discreta.
Numerus physicis alius est numerans, alius numeratus.
Numerus numerans seu formalis est, quem anima apprehendit abstractum ab omnia materia. Dicitur etiam matbematicus.
Numerus numeratus et materialis est, cujus unitates sunt res. Sic Aristoteles definivit tempus per numerum, puta numeratum, quia tempus est numerus motus per prius et posterius. Dicitur etiam physicus.

2. Time and Eternity in Augustine (and in Medieval Scholasticism)

In short, using scholastic terminology, time is the numerus numeratus motus, but something is numerus numeratus, only if there is a numerans, although that which is the numerus numeratus can exist even if no numerans exists, but then it is not numerus numeratus, but numerus numerabilis.

3. Thomas de Aquino, Commentaria, Libros Physicorum, IV, Lectio 17

Uno modo id quod numeratur actu, vel quod est numerabile, ut puta cum dicimus decem homines aut decem equos; qui dicitur numerus 'numeratus', quia est numerus applicatus rebus numeratis. Alio modo dicitur numerus 'quo numeramus', idest ipse numerus absolute acceptus, ut duo, tria, quatuor.

IMO, the text is a little bit misleading. Ancient Greek mathematics deals with arithmos that can correspond to "abstract" number. But obviously Greeks used numbers to count: sheeps, coins, soldiers... — Mauro ALLEGRANZA, Jan 07 '21 at 13:28
What bis correct is to say that for Ancient Greek mathematics the only kind of "numbers" was the natural numbers. See e.g. I.Grattan-Guinness, Numbers, Magnitudes, Ratios, and Proportions in Euclid's Elements (1996) — Mauro ALLEGRANZA, Jan 07 '21 at 13:30
See also the post regarding the Pythagorean assumption that all segments are commensurable — Mauro ALLEGRANZA, Jan 07 '21 at 13:35
@MauroALLEGRANZA I haven't heard about these concepts before. But it seems that ancient Greek mathematics didn't use them; they belong to the medieval world of ideas. See also http://www.xlatangente.it/page.php?pid=3174 . — Paul Frost, Jan 07 '21 at 14:38
The distinction seems to correspond to one in physics, between dimensionless quantities and those requiring specification of the unit. — John Bentin, Jan 07 '21 at 14:42
The History of Science and Mathematics StackExchange might've been a better place for this question. — Blue, Jan 10 '21 at 01:03
In the Elements (Euclid) numbers are often reasoned with abstractly as line segments. Without what we would call an integer or real number attached, they are referred to and treated as, numbers. Sometimes these segments are divisible into multiples of a unit, with unit presented as a 'common notion' (i.e. axiom). Otherwise the segment is considered incommensurable, but still a number: What we would consider irrational. I don't know the history of your terms, but to me it maps to this distinction between commensurable and incommensurable, literally, rational and irrational. — , Jan 10 '21 at 02:34

mucciolo · Accepted Answer · 2021-01-11T01:09:12.597

Past personal studies of mine of the mathematical foundations of dimensional analysis lead me to believe that the following answers are plausible ones.

Is it a premodern concept which has completely disappeared?
Quite the opposite. These concepts continued to be developed until they gave rise to a branch of mathematical physics known today as dimensional analysis. The concept of an abstract number is nowadays refereed as a dimensionless quantity and the concept of concrete number as dimensional quantity.

Is this distinction still in use in contemporary mathematics, at least at some places?
Yes. Whenever in mathematical modeling we feel the presence of dimensional physical quantities in a determined model is making its analysis harder, in the light of Buckingham $\pi$ theorem we might try to remove them through a technique known as nondimensionalization. Roughly, we try to algebraically combine dimensional quantities (concrete numbers) into dimensionless quantities (abstract numbers) and then rewrite the model in terms of these dimensionless quantities. A proof of Buckingham's $\pi$ theorem as well as the mathematical foundations of dimensional analysis can be found at Jan-David Hardtke (2019). On Buckingham's Π-Theorem. arXiv:1912.08744 [math-ph].

An example of nondimensionalization in the branch of the qualitative analysis of differential equations can be found at section 1.2 of J.D. Murray (2011). Mathematical Biology: I. An Introduction, Third Edition., and in the branch of statistics can be found at Pearson's correlation coefficient which is essentially the nondimensionalization of the covariance between two random variables.

In case you don't get any answer you find helpful then I suggest emailing Jan-David Hardtke whose e-mail can be found at his above linked article.

Thank you for your answer. I am not completely convinced that the old concept of abstract number is the same as that of a dimensionless quantity, but certainly the concept of concrete number corresponds to that of a dimensional quantity. In my opinion an abstract number is something like a Platonic universal, but perhaps I am wrong. For example, the fine-structure constant is an inherently dimensionless quantity, but I would not regard it as an abstract number. — Paul Frost, Jan 16 '21 at 10:38
You are welcome, Paul. Unfortunately this is as far as I am able to help you, nevertheless I reiterate the possibility of emailing your thoughts to a "dimensional analyst" or perhaps a math history researcher. I am sure that at least one of them will make himself available to help you advance your investigation. — mucciolo, Jan 16 '21 at 12:41

Concrete and abstract numbers

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