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Aristotle on Number Theory

Reason the question came up in the Categories:

Does Aristotle view counting as an activity, and not as a number line?

Does Aristotle view numbers as sets, groups of objects?

Plato would say hes not talking about number here, since it is not about ideal numbers having

independent existence. Is Aristotle orienting number to objects?

What are the characteristics of number evident fromCategories 4b20 - 6a35?

Number is (from ): separate, having-been-distinguished,

having-been-removed-across-the-frontier.

"The parts of a number have no common boundary at which they join together," e.g. two '5's of 10are not joined at any boundary.

The parts do not have towards each other, or are situated somewhere but have a type of

order (): 1 is counted before 2, 2 is counted before 3, etc. (cf below: knowledge of the

number line.)

As a quantity, it does not admit of more or less: one three is not more three than another three, nor

more threethan five.

Called unequal and equal.

Part of the difficulty of determining whether he is talking about abstract numbers or

things-as-quantified is the phrase . It could be translated as five or five things. Is he talking

about mathematical objects? Ideal numbers? Physical objects? None of these? Physical objects certainly

hold some manner of position () towards each other - Aristotle denies that number does so he cant be

talking about physical objects quaphysical objects. Aristotle seems to say that numbers as mathematical

objects has an ordering (), so is his theory of number about non-ideal mathematical objects?1

In Physics 219b5-9, Aristotle distinguishes 2 senses of number: (1)what is counted or countable

[physical referent]and (2)what we count with [mathematical object].(Annas, 97)

(1)seems to mean some element of a things, or a group of things, existence, inhering in the objects

themselves and enabling it to be counted.

1Can we make an assumption that there is a unified theory of number in Aristotle? Annas thinks so: All these passages

show a consistent attitude to the question of the existence of numbers. (Annas, 99)

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One element of a things existence is its unity - its one-ness (which Aquinas inQuestiones de

Veritatetakes to be a transcendental). But Aristotle says that Unity is not itself a number:

We should generally say by any number except unity but a unit here is excluded from the meaning of number.

According to Aristotle, the unit is not a number: cf.Metaph. N. I. 1088a4-8: The term one means that it is a

measure of some multitude, and number that it is a measured multitude, or a multitude of measures. It is only

reasonable therefore that one is not a number, for neither is a measure measures, but the measure and the one

are the beginning, i.e. the unit if the beginning of number but not a number. (Heath p.84)

To count, we need to pick something out as our unit. This unity we pick is what numbers measure. One is

the measure of number. (Annas, 99) Counting is then the notation of unified beings.

One, unity, and unit all indicate the same element: a measure picked out by the mind to

measure a multitude. We focus on a particular unity and make that the measure by which we count: If we

have 10 sheep and 10 men, we have two species of the genus 10. Counting then, is the measuring of

ones and all numbers are derivative from counting.2

In sense (1) then, numbers are measures of the unities of a group of objects . Numbers are

derived from counting and are identical with measure and measuring: Aristotle is quite happy to

interchange number and counting with measure and measuring. (Annas 98) Since counting is - in

every instance - concerning with a unit, and every unit is a definite thing, then counting will always be of

specific things: When we have 10 dogs and 10 sheep we have different units and therefore different tens3

(because they are tens of different kinds of thing) but we still have the same number. (Annas 107)

Therefore definitenumbers have species, depending upon what you are counting.

Exempli gratia: 1000 is the measure of the body of students. 1000 is impossible to be had apart

from counting the number of students who go here. 1000 is not a thing itself, but a measure of the unit

we have chosen: student, which is itself recognizable by the fact that the substance man unifies each

student. Therefore our genus is 1000 and our species is students.

Is there any difference between numbers and counting? Yes, since numbers are accidents of things

and counting is an activity of the mind. Numbers, though derived from counting, can be considered apart

and their various aspects may be considered (such as equal and unequal). Their ontological status is

questionable, since Aristotle does not want to grant them independent status as numbers and since they are

2 Measure is that by which quantity is known and quantity quaquantity is known either by a one or by a number, and all

number is known by a one. 1052b20 (Ross translation)

A number is several ones or a certain quantity of them. Hence number must stop at the indivisible two and three are

derivative words, and so is every other number. (Phys. III. 7.207a33-b34, cf. Heath p110)3Numbers are derived from one, and one is - in every kind - a definite thing: In a sense, unity means the same as being.

[...] To be one is just to be a particular thing. (Metaphysic s1054a10-19)

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differentiae. Which brings us to sense (2).

(2) seems to mean the numbers themselves - quantity quaquantity [mathematical objects].

Julia Annas takes Aristotles account in Physics and inMetaphysics I to be decidedly anti-Platonic in

motive. Number, Aristotle wants to show, is inseparable from the being of definite objects. It is not out

there apart and above the objects. In counting, the objects quantity is not compared to an independent

number-being. Insofar as they have independent existence from the mind, Aristotle nails down numbers

to the countability of physical objects. The matter of mathematical objects is the , intelligible

matter as distinct from sensible matter. (Heath 224)

Interestingly, because they are derived from counting, numbers cannot have infinitely great

magnitude. Whatever [a numbers] size potentially, that size it can be actually hence, since there is no

sensible magnitude that is infinite, it is not possible to have an excess over every determinate magnitude if

it were, there would have to be something greater than the universe. This implies that all number - actual4

numbers - are known by abstraction from physical objects. Numbers are, for Aristotle, inseparable5

from the real existence of the thing from which it is derived: Aristotle goes on to say that he holds, as

before stated, and that it is obvious, that the objects of mathematics are not separable from sensible

things. (Heath 220)

Numbers above one, considered as what we count with, are the result of the human activity

of counting. This seems evident from Aristotles reluctance to posit an infinitely great number. Annas

writes, in describing the implications of calling time a number,

[Aristotles] point is that since time is a kind of number it has the sort of existence appropriate to

a number: that is, it has no existence independently of activities of counting. The clear implication

of this is that (as with number) it is misleading to say that time exists if this is taken to imply that

the existence of time is independent of that of human activity - in this case the activity of

time-keeping. (Annas 101)

4The possible bisections of a magnitude are infinite in number this infinite is potential, not actual, but you can always

assume a number (of such bisections) exceeding any assigned number. But this number is not separable from the process ofbisection, and its infinity is not a stationary one but it is in process of coming to be, like time and the number of time. With

magnitudes the contrary is the case for the continuous magnitude is divisible ad infinitum, but in the direction of increase

there is no infinite. Whatever its size potentially, that size it can be actually hence, since there is no sensible magnitude that

is infinite, it is not possible to have an excess over every determinate magnitude if it were, there would have to be

something greater than the universe.

(Phys. III. 7.207a33-b34, cf. Heath p110)5Even the definition of an odd number sounds physical. In the Topicshe speaks of a contemporary definition - which he

rejects - of an odd number as a number having a middle. Unclear in what way odd could mean having a middle

except as a physical object. (Heath p.91)

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All numbers above one are derivative of one, and - it seems to me - have only res cogitansstatus (a mental

idea that we have in organizing the world). The meaning of statements about numbers is exhausted by6

statements in which no such apparent reference [to number] occurs, which involve reference only to

counting and measuring. (Annas, 99) Number is not counting. Counting is the measuring (and therefore

creation/abstraction) of number based on unity in being. (Annas 103) It seems that Aristotle must - if he

were to systematically write on number - place numbers ontology in thought. But thought is not in one of

his categories so where does it belong?

Which leaves us with a question: where/how do we get our knowledge of the number sequence?

Numbers are derivative of multitudes of unities and dont have real existence outside of counting, but it is

veryunclear how we get our idea of the number sequence. Aristotle would presumably call it by

abstraction, but how can the number sequence be abstracted from a group of physical objects? How do we

know that 2 is after 1, and 3 after 2? Where does this come from in Aristotle?

Say that number is a res cogitans. It remains that number is itself objective (and not arbitrary) in

some degree. Yes, we can have different base systems (binary vs. Hindu-Arabic ten-base-system) but there

is a consistent order to it. No culture counts randomly.This is what I take Aristotle to mean by a

certain order ( ) if not the certain position towards other parts ( ). One comes after

another, but it doesnt lieanywhere, so that it cant have position. The ontological status of the number

sequence is unclear.

Aristotle thinks that number is discrete(i.e. parts without a common boundary, discontinuous,

definite). [For not one of the parts of number is a common boundary. (Cat 4b25)] Numbers, as discrete

quantities, have an order (), but not position () towards each other.

But, at any rate, concerning number, one cant observe its parts holding some position

towards each other or pick out where it lies. Nor can you pick out which of the parts join together

with others. (Cat 5a23-26)

Similarly with a number also, in that one is counted before two and two before three in this

way they may have a certain order, but you would certainly not find position. (Akrill)

A number line - insofar as it shows the - is accurate, but insofar as it seems to show that numbersborder each other is a misrepresentation. As a number sequence, it is accurate.

What then would Aristotle do with decimals?

6I am unsure as to whether Aristotle would have had the concept of res cogitans. It is certainly anachronistic to read that

back into Aristotle (it is Latin, after all), but it seems that it is the correct category for where Aristotle would place it had he

had access to this concept.

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