[adrian_b]

This kind of sloppy writing is too frequently encountered and it is annoying: "Archimedes didn’t think of 22/7, 223/71, or pi as numbers; to him, they were ratios of magnitudes".

This has nothing to do with "thinking". This is just about (incorrect) language translation and mathematical terminology.

In Ancient Greek and Latin and in any other old languages, the word "number" designated the result of a counting operation, i.e. it corresponds to what in modern mathematical terminology is named "natural number", so it must be translated as such in a mathematical context.

In Ancient Greek and Latin, the word "magnitude" (or "measure") designated the result of a measurement operation, like the measurement of a length with a standard ruler or the measurement of a weight with weighing scales, i.e. it corresponds to what in modern mathematical terminology is named "real number", so it must be translated as such in a mathematical context.

There is no difference between Archimedes and a modern mathematician, both distinguish natural numbers and real numbers, but since they speak different languages, they use different words for these 2 concepts. Whenever you read an ancient mathematical text, "number" must be understood as "natural number", while "magnitude" or "measure" must be understood as "real number".

The names do not matter, this is just a problem of language translation (which is done usually incorrectly for texts with mathematical terms, like also for texts with specialized terms from physics, chemistry, biology or mineralogy, because the translators have little knowledge about those domains).

The important difference between ancient mathematics and modern mathematics is that the ancients did not have a method of construction of the real numbers from natural numbers, which has been conceived only in the 19th century, by the time of Cantor, Dedekind etc. The ancients also did not have the method of abstract algebra, where you prove something for a set of axioms of a given form (e.g. of group, of ring, of field etc.) and then the proof is valid for any set where that set of axioms is satisfied. Abstract algebra is also a creation of the 19th century.

Because of that, the ancients had 2 independent sets of axioms for natural numbers and for real numbers, even if many of them were identical in form. The consequence of this was that many theorems and demonstrations had to be duplicated for natural numbers and for real numbers, because something proven for one of them could not be applied directly to the other.

[whiteandnerdy]

I think there's a sense in which moderns feel that the reals (or at least the rationals) are a natural category: that 5 and 0.3 are the same "kind of thing". Mathematicians talk about the distinction between different classes of numbers, but to most people they're all numbers.

Do you think that sense was shared by the ancients, or do you think that the linguistic distinction mirrored a stronger mental separation between the two? It sounds like it might have done if they had to do the work of duplicating proofs from one to the other. Did they have a single word to describe the shared concept?

[adrian_b]

To the ancients, as already clearly expressed by Aristotle, natural numbers and real numbers were both quantities. Quantities were classified in discrete quantities, of which natural numbers are an example and continuous quantities, of which real numbers are an example. This classification remains completely valid today.

The ancient mathematicians and philosophers were using the word "quantity" in most contexts where modern people use the word "number", i.e. when the word is applied to different kinds of "numbers", not just to natural numbers.

There is no difference in thinking between ancients and moderns, it is just a difference in the words that happen to be used.

Both the similarities and the differences between natural numbers and real numbers are well entrenched in most natural human languages since many millennia ago, before any scientific theory of quantities, numbers and magnitudes, as exemplified by the similarities and differences between questions like "How many ... do you have?" and "How much ... do you have?".

Actually I consider the ancient usage of the words as more sound than the modern usage. There appears little justification for the modern usage of the word "number" instead of the previous usage of "quantity", except that "number" is a shorter word than "quantity", so the change in terminology is just due to laziness, not to any theoretical reason. However what has been gained by saying "number" instead of "quantity" when the wider sense is intended, has been lost due to the requirement for qualifying "number" as "natural", "real", "integer" etc., when the narrower meaning is intended.

Etymologically, "number" is the result of counting, which real numbers and many other kinds of "numbers" that correspond to continuous quantities are not.

[areyousure]

Suppose we designate a unit line segment, and from that we pick out a line segment of (what we now call) length 2 and separately a rectangle of area 3. To "an ancient" (like Aristotle), does it make sense to add the length and the area to get 5? If you then drew a line segment of length 5, would they say it has the same magnitude?

(My understanding is "no" to both questions.)

[adrian_b]

Today, as always, it is a serious mistake to add a length with an area.

Nowadays, everyone who attempts to do such things should understand that the value of a physical quantity is the product between a scalar (i.e. real number) with a unit of measurement.

The addition of distinct physical quantities is impossible, because the addition between distinct units of measurement is undefined (because there is no useful definition).

However, if you detach the scalars from the complete values of some physical quantities, those are elements of the field of the real numbers, so you can do any operations with them, not only addition, but you can compute arbitrary functions, like transcendental functions, e.g. logarithms (which you cannot compute using as argument a complete physical quantity, including a unit of measurement).

In the ancient Greek mathematics, the notions of physical quantity and measurement were not well formalized, even if they had some intuitive understanding, so they would not normally try to perform invalid operations with physical quantities. The concepts of physical quantity and measurement became well understood only in the 19th century, e.g. in the works of people like Weber and Maxwell.

Nevertheless, in the ancient Greek texts there are also examples of numbers that were detached from some geometric quantities and then used in unrelated operations where their origin was no longer taken into account. A simple example is the use in various arithmetic problems of several kinds of "geometric" numbers, e.g. square numbers, triangular numbers, cubic numbers or pyramidal numbers, i.e. numbers that were computed using formulae for areas or volumes, but which were used in applications were there were no areas or volumes involved.

Another example is the use of mechanical devices for the computation of some irrational or transcendental functions, which could be used for solving some famous problems like the trisection of an angle, the quadrature of a circle or the doubling of a cube. In such mechanical devices there were e.g. some lengths that were numerically equal with areas or volumes in the related problems. (While such mechanical devices, which were examples of analog computers, have been built and they have solved the problems, such solutions were considered cheating, because those problems had been formulated with the restriction of using only a straightedge and a compass for their solution, which is now known to be impossible.)

The essence of analog computing, whence the name "analog" comes, is that you have in the analog computer some physical quantities that are numerically equal (using some arbitrary units) with some physical quantities in the problem that must be solved by the computer, even if the nature of the physical quantities is very different. Analog computing is an important example of detaching the real number value from the complete physical quantity, and as I have said, there are examples of simple analog computers, dedicated to the computation of some irrational or transcendental functions, already since the Greek Antiquity.

[johnisgood]

Right, I read Aristotle's metaphysics, the terms used refer to concepts for what we use a different word, today.

So what you said is very important to take into consideration.