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The Society for Ancient Greek Philosophy Newsletter

1983

Aristotle's Philosophical Principles of Mathematics

Hippocrates George Apostle Grinnell College

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Recommended Citation Apostle, Hippocrates George, "Aristotle's Philosophical Principles of Mathematics" (1983). The Society for Ancient Greek Philosophy Newsletter. 5. https://orb.binghamton.edu/sagp/5

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ARISTOTLE'S PHILOSOFHICAL PRINCIPLES OF MATHEMATICS 1-J i ppooret fes G<'lZ>rae Arpos-fle. ''(;Jzsi-fl'"fl/ 11 IC/ g3 The definition of mathematics as the science which investigates the

properties of quantities was first conceived by the ancient Greeks and was

formulated later both philosophically and scientifically by Aristotle in his

various works. Later mathematicians accepted this definition tacitly or by

habit, for philosophical problems about mathematics as a whole were not much of

a concern t0 them nor was the field of mathematics far. advanced in research to

suggest alternative definitions. Early last century Gauss reaffirmed this

definition in his treatise, � Foundations £!. Mathematics.

From the latter part of last century until today, however, there has been

a tendency away from this definition and in the direction of what was thought

to be a better and more general definition; and in the opinion of most modern

mathematicians and philosophers the old definition is too limited to cover modern

mathematical research. The introduction of the so-called "non-Euclidean geometries"

and of transfinite numbers, too, contributed somewhat to this tendency; for, it

was t11ought, if the parallel postulate did not possess the absolute truth which was once attributed to it, it �ould have only hypothetical truth, if any truth at all, and today we find many if not most mathematicians and even philosophers taking the position that mathematics is not interested in the truth or falsity of its principles but on�y in consistent sets of postulates and the deduction of theorems from those postulates.

As a consequence of such thinking there arose a number of new definitions which give the appearance of being general enough to include all actual and perhaps possible mathematical research. Peirce regarded mathematics �s the science which draws necessary conclusions, Russell identified mathematics with logic, Hilbert emphasized the symbolic nature of mathematics, and others posited such wide concepts for mathematical objects as order, intuition, relations, and the like.

Now a fair criticism and evaluation of the old definition presupposes an 2

understanding of the terms in that definition and the principles according to

which the formulation was made. Unfortunately, however, the critics failed

on both counts; for they knew neither the meaning of those terms nor the

principles according to which the formulation was made.

First, I shall present Aristotle's definition of mathematics and the

principles which he uses in formulating it; second, I shall discuss some a./501 definitions which have been given lately and their difficulties; and t:hil

I shall show that Aristotle's definition best fits modern mathematical research.

DEFINITION: Mathematics is the science which investigates

generically, specifically, and analogically the properties

of quantities and whatever belongs to quantities.

Aristotle did not give expressly this definition, but it can be gathered from

what he says in his various works. Whether he wrote a work on mathematics or

not is not known. Now the key terms in this definition are "science," "quantity,"

"generically," "specifically," "analogically," "belonging," and "property,"

1''irst, let us turn to the term "science". It has two senses for Aristotle.

Its main and narrow sense is: necessary knowledge of what exists through its

cause. The other sense inc�udes the principles and the logical proofs of theorems

from the principles. So the definition of science in this sense would be:

universal knowledge of principles and demonstrations of properties from those

principles under one genus of existing things or under one aim; and the definition

of a theoremiould be: a demonstrated statement which signifies an attribute

as belonging to a subject through the cause. In the case of mathematics, quantity is the genus.

Now the principles in a science are four in kind. They are (1) the indefinable concepts, (2) the definitions, (3) the hypotheses, and (4) the axioms. The premises come from the definitions and the hypotheses; and as for the axioms, they are not premises but what some moderns call "directive" or

"regulatory" principles which are used to de:nonstrate c •nclusions from premises. 3

Second, the meaning of the term "quantity" is clear to those who have read

carefully Aristotle's Categories and Book Delta of the Metaphysics. 'Quantity'

is a category, and its two immediate species are 'number' and 'magnitude'• ,and

by "number" Aristotle means what nowadays call "a natural number which is greater

than 211, Bertrand Russell, whose ignorance of the history of mathematics

"' surp�sses even his ignorance of Aristotle's logic, chooses to reject the ancient

definition in his Introduction to Mathematical Philosophy. I quote:

"It used to be said that mathematics is the science of 'quantity'.

'Quantity' is a vague word, but for the sake of argument we may

replace it by the word •number'. The statement that mathematics

is the science of number would be untrue in two different ways.

On the one hand, there are recognized branches of mathematics

which have nothing to do with number - all geometry that does

not use coordinates or measurement, for example; projective

geometry and descriptive geometry, down to the point at which

coordinates are introduced, does not have to do with number, or

II even with quantity in the sense of greater and � •••

In this passage, Mr. Russell considers the word "quantity" as vague; and this shows that he did not do his homework. Again, he replaces the word "quantity" by the word "number" and chen tries to refute the ancient definition; and this is like setting up a straw man and then knocking him down. Again, the expression

"mathematics is the science of quantity" was given as a definition and not as a fact, but truth and falsity do not apply to definitions; yet he regards the exfression as being untrue, that is, as false. Again, he is unaware of the fact that the relations of greater and less apply even to magnitudes in geometry which does not use coordinates, as in the case of plane geometry. There are other errors, but these should be enough for a logician.

Third, let us turn to the term "property". It means an attribute which is not the essence of a subject but belongs to that subject and to no other subject. The reason why the definition of mathematics uses the' term "property" CVkw31† \<† wA3† w5eU†
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to generic or specific properties. For example, the number system has analogical

properties; for here the unit taken as a principle may be any magnitude, such

as a line or a surface or a volume or an angle, and what we call "the origin"

or "zero" serves as an initial principle if a magnitude is considered without

direction, but as an initial principle of direction if a magnitude is considered

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with directions. /\

The principles above with other details are sufficient to set up analytic

geometry in three spacial dimensions, for there are three such dimensions,

each of which has two directions relative to a principle of direction, and

what is now called the "origin" is what Aristotle would call the initial principle

� of direction for all six directions. Fur ther, alalytic geometry would be v

considered by Aristotle as a mixed science, with magnitude as the subject but

with divisible numbers as attributes, for the unit in this geometry, unlike

the unit in the theory of natural numbers, is infinitely divisible and not

indivisible. Calculus would be a brance of mathematics whose subject is studied

analogically, but we are omitting the de1ai.ls here.

As for Russell's assertion that quantity does not apply to projective and

descriptive geometries, it is false, as we have indicated, for both those

geometries have magnitudes as subjects, and magnitudes are qu1µ1tities as

already stated. Topology, too, would come under geometry for the same reason,

and so would algebra; but the foundations of these have to be laid down not in

the hypothetical manner in which it is d·ne nowadays, but in a manner in which

the existence principles are introduced. And this brings us to the problems

of existence in mathematics.

As we have stated at the start, mathematicians nowadays are inclined to

bypass the problem whether the postulates laid down are ture or not, as long

as all the postulates as a set are consistent - so to say. But they assume

truth, which signifies existence, even if they do not mention it. For a set

of postulates is said to be consistent if from them and all the demonstrable " which follow it is impossible to choose two statements which are contradictories. ;

But why avoid contradictories? The only reasonable answer is that mathematicians

are anxious to keep the law of contradiction, and in keeping it, they believe

that it is true, otherwise there would be no point in insisting on consistency.

In other words, they believe in the truth of the law of contradiction, which

is one of their postulates and a postulate without w!Uch they cannot proceed.

But if they so believe, it can be shown that they have to believe in other

truths which they use. So why believe in the truths of some postulates and omit their belief in others.

There is another difficulty. What is the point of deducing theorems if

we are not concerned with their truth? It is said that science seeks truth,

and if truth is omitted, deduction loses its dignity and worth which is usually attributed to it. Are we then investigating things or playing games? Paying mathematicians to play games is certainly strange. If, on the other hand, we attend to mathematical results rather than to what is said about them, we find overwhelming evidence that tnose results are truly applicable and are therefore true insofar as they are applicable. In other words, two and two make four, and there is no doubt about it; and the universal use of arithmetic confirms without doubt the accuracy and truth of arithmetic. Again, the use of geometry, too, is not regarded as faulty, otherwise there would have been a change in its

"'" postulates and rulers and compasses and the like. A.

A theoretical problem arose last century, however, when the so-called

"non-Euclidean geometries" were put forward as alternatives to Euclidean geometry; for doubt arose as to the absolute truth of the Euclidean postulates, especially the parallel postulate. Do the non-Euclidean geometries contradict Euclidean geometry? In words, they do; for the parallel poatulate is different in each case in such a way that each of them contradicts the other two. It is agreed, however, that each of these geometries is consistent within itself, and further, that either all of them are consistent or all of them are inconsistent because 7

transformations exist which transform each of them into the others. Consequently,

it appears that either all are true, or all are false whether partly or wholly.

Now it is easy to define a straight line so as to mean not what Euclid

meant by it but something else and then show that straight lines always meet

in a plane or that many straight lines can be drawn through an external point

parallel to a given line, and this is exactly the situation. But doing so is

like calling a man "a cat" and then proving that many American cats have

received the Nobel Prize. First, all models offered of non-Euclidean straight

lines are either arcs of great circles on a sphere or circular arcs within a

plane circle and perpendicular to that circle or S':me other kinds of lines

which are not straight in Euclid's sense. Second, because of the transformations

stated above, the theorems of Euclidean geometry are exactly the same as those

of each of the other geometries, except for words, end this shows, as Poincare

has shown in his Science and Mypothesis, that, for example, the Riemannean

triangle has the same properties as the Euclidean spherical triangle. This

indicates th":t Riemann equivocates when he uses the term "straight line" since what he means or should mean is an arc of a great circle. Third, let us consider the projective plane and its postulates. The term "straight line" here is used as a genus of the so-called "straight lines" in each of the different geometries, and its properties are those which are common to the so-called "straight lin�s" in each of those geometries. So this is a further equivocation of the term

"straight line." Moreover, a postulate is added to the postulates of the projective plane to give the Euclidean plane, and another and different postulate is added likewise to give each of the other geometries. The upshot of ail this is the following: (1), the method of studying properties in geometry by beginning with the projective plane is sound and even approved by Aristo�e as being better than that of Euclid, since to begin with the more universal is to be more scientific because this method is better by nature, and (2), this study can be done within Euclidean geometry, and (3) there will be clarity in meaning and the *5k;Z6<† `A† ;fyH~`7+wH_Z† HA† Hw† Hk† R*9<† dN+H[† wE+w† wE;† w;hU†
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