1 1 Introduction: Aristotle's Philosophy of Mathematics Aristotle's comments about mathematics ...

171 Pages·2009·1.21 MB·English

by  Eva Cadavid

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Introduction: Aristotle’s Philosophy of Mathematics

Aristotle’s comments about mathematics have led commentators to think his view
displays naiveté, incoherence, misunderstanding of Plato, and that he accepts a
platonic ontology of mathematical objects. These incompatible interpretations result
from an apparent inconsistency in Aristotle’s view. In this dissertation I first present
that inconsistency in Aristotle’s view of mathematics. Call this ‘the puzzle.’ I then
present four possible positions open to him to avoid an inconsistency. I call these ‘the
positions.’ I argue that one of the positions lacks textual support while the text offers
some evidence for the other three positions. This leaves three viable positions. After
examining each of the three viable positions, I defend a fictionalist interpretation of
Aristotle.

The Puzzle: the Ontological Status of Mathematical Objects
There is no one Aristotelian text devoted to mathematics. Instead, Aristotle’s views
can be found in Posterior Analytics, Metaphysics, De Anima, and Physics. These
texts show that Aristotle believed first that we could have mathematical knowledge
and second that mathematics is an ideal science. These two points are not disputed by
the interpreters I discuss. Aristotle’s views on the ontology of mathematical objects
are less clear. He says in Metaphysics XIII.2 that mathematical objects exist neither
separately from nor in sensible objects. He concludes that they either do not exist or
have a qualified existence.
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The question of whether mathematical objects exist, and if they do, how they exist,
indicates a tension in Aristotle’s ontological views. This tension is the puzzle that
challenges commentators. The tension consists in making the following two
statements consistent. Aristotle claims that:
1) All existing things are sensible objects; and
2) There are mathematical truths.
The first statement is straightforward. The only entities that exist independently of
anything else are sensible substances. The second statement requires explanation.
Aristotle doesn’t state (2) explicitly about mathematical truths. Instead, he accepts
that there is mathematical knowledge1 but knowledge will be grounded on truths, in
this case, mathematical truths. There is an ontological basis for our concepts
according to Aristotle. Our concepts must match the real world as do our knowledge
claims. The question is whether or not there is a need for an object that directly
matches the concepts or knowledge claims made by the mathematical sciences.
Being committed to the existence of mathematical truths strongly suggests the
existence of mathematical objects. Yet Aristotle argues in Metaphysics XIII.1 that
mathematical objects, whether they exist or not, are not sensible objects. The puzzle
is that the tension between (1) and (2) appears incapable of resolution without
sacrificing consistency.

1 Aristotle claims that there are mathematical sciences and they are the most exact
sciences.
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The Positions
In order to maintain consistency and the truth of statements (1) and (2), there are four
possible positions available to Aristotle. They are the following:
(I) there are no mathematical truths;
(II) not all existing things are sensible objects;
(III) mathematical truths do not require the existence of mathematical objects
as substances; and
(IV) mathematical objects are sensible substances.
The first and second positions are the denials of (2) and (1). No Aristotle
commentator has put forth an interpretation defending the first position. The
interpretations most often found in the literature today are versions of the other three
positions. Ian Mueller chooses a position that most closely fits (II).2 Julia Annas
presents an interpretation that most closely fits (III).3 Jonathan Lear embraces a
combination of the third and fourth options.4

The Plan of the Dissertation
The dissertation starts by presenting the background to Aristotle’s philosophy of
mathematics in Plato’s views from the middle period. I do not discuss Plato’s views
in their entirety but focus on his views on mathematical knowledge and mathematical
objects. This includes his criticisms of the mathematicians as they appear in the

2Ian Mueller, “ Aristotle on Geometrical Objects,” Archiv für Geschichte der
Philosophie, 52 (1970).
3Aristotle, Metaphysics Books M and N. Trans. and notes by Julia Annas (Oxford:
Clarendon Press, 1976).
4 Jonathan Lear, “Aristotle’s Philosophy of Mathematics.” Philosophical Review, 91
(1982).
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Republic and in the Theaetetus. Mathematical knowledge, once gained, is stable. Its
objects are also stable and unchanging. The mathematician works from first principles
and proceeds to provide proofs. Plato finds this objectionable if one does not have a
way to know the first principles. Plato presents a view of knowledge that relies on
definitions and stable objects that allow for those definitions to be unshakable once
known. Chapter 1 presents an interpretation of Plato’s middle period views on
mathematics. It includes a discussion of definition, knowledge, mathematical
knowledge, and mathematical objects. Plato’s middle period views influence
Aristotle’s views on the same topics.

Plato’s influence is noticeable in Aristotle’s high regard for mathematics. He agrees
that knowledge is of the universal and in the case of mathematical sciences, it always
holds. Aristotle’s views on mathematical knowledge and mathematical objects are
also very much influenced by Plato’s middle period views. Aristotle offers an
account of scientific knowledge that requires experience and establishes knowledge
of the first principles of a science through induction. The first principles can then be
taken for granted and the scientist proceeds to offer demonstrations like the
mathematicians do in Plato’s Republic, Meno, and Theaetetus. Plato places
mathematical objects in a privileged place because they are on the intelligible side of
the divided line. Yet they are also lower in clarity and intelligibility than the
philosophical objects. Aristotle places mathematics among the privileged sciences
alongside physics and first philosophy. It is one of the theoretical sciences. It also
stands between physics and first philosophy. It is more exact than the physical
sciences and its objects are unchanging. Plato has the objects of mathematics as
separate from the sensible world. Aristotle does not think the objects of mathematics
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exist separately from the sense perceptible, but he thinks that they are separable in
thought. He thinks that the mathematician does not say anything false or does
anything improper by considering the objects of mathematics in isolation from matter.
Chapter 2 presents Aristotle’s views on mathematical sciences and how they differ
from other sciences. It also offers an account of knowledge, the role of first
principles, definition, and a preliminary account of mathematical objects. This
chapter also raises the OSMO puzzle about the ontological status of mathematical
objects and the four possible ways of responding. It acknowledges that there is
overlap among positions.

After pointing out the connections between Plato and Aristotle, I present a
preliminary account of Aristotle’s view on the mathematical sciences and their
relation to the natural sciences. This includes a discussion of knowledge as well as
objects. Definition plays a role in knowledge since the mathematician, like any
scientist, studies the relations between universals, and the universals include
definitions that capture the essential properties. For an account of Aristotle’s
philosophy of mathematics, it is important to understand his methodology, including
abstraction, and ontology. Chapter 2 discusses Aristotle’s method and explains the
puzzle that motivates this dissertation. Chapter 2 then presents the four positions in
response to the puzzle. The remaining chapters of the dissertation concern solutions
to the puzzle.

The puzzle structures chapters 3-5. Chapter 3 focuses on the first two positions: (I)
there are no mathematical truths; and (II) not all existing things are sensible objects.
The first position is dismissed quickly because there is no textual evidence in support
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of it. The second position is not so easily dismissed. The implication of (II) is that
mathematical objects are independent, abstract objects that have the ontological status
of substances. Aristotle argues against Plato for having such a position, yet his own
views can be interpreted as implying that there are mathematical objects that are not
sense perceptible and yet function as subjects. Ian Mueller presents an interpretation
of Aristotle’s philosophy of mathematics that separates geometrical objects into
fundamental and secondary objects. The secondary objects are connected to sensible
objects only through extension. Mueller’s account, although trying to avoid position
(II) has the implication that secondary objects are abstract objects removed from
sense perceptibles.

Chapter 4 focuses on positions (III) and (IV). I show that Lear’s interpretation is
representative of the fourth position but also overlaps with the third position. Position
(III) states that mathematical truths do not require the existence of mathematical
objects as substances. One way of understanding this position is to consider
fictionalist theories in contemporary philosophy of mathematics, which is sketched
below. Position (IV) says that mathematical objects are sensible substances.
According to him, Aristotle’s view is that mathematical objects are sense perceptible
objects that can be fictionally considered as separate from sense perceptible matter.

Chapter 5 critically engages Julia Annas’ interpretation as representative of
nominalism and compatible with fictionalism. This chapter gives the foundation for a
fictionalist interpretation of Aristotle’s philosophy of mathematics. It shares some
common elements with other interpretations, for instance Mueller’s, Lear’s, and
Annas’, but it differs from them by meeting the following eight criteria: (1) Be
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consistent with the text; (2) Maintain Aristotle’s rejection of Platonic forms; (3)
Explain the connection between mathematical objects and sensible objects, and why
mathematical truths apply; (4) Explains the privilege of mathematics over other
sciences; (5) Explain Aristotle’s expression of ‘existence in a way’; (6) Account for
both types of abstraction, ‘éfa€resiw’ (aphaireisis) and ‘√’ (hei); (7) Give an account
of both geometry and arithmetic; and (8) Give an account of intelligible matter. The
other three interpretations meet most of the criteria but not all of them.

The dissertation concludes by developing the fictionalist account of mathematical
objects suggested by Annas’ interpretation but not fully developed in her writings.
Chapter 5 argues that Physics II.2 and Metaphysics XIII.3 support the interpretation
that there are two types of fictionalism in Aristotle’s view. The process of abstraction
for the mathematical sciences allows for the mathematicians to treat the objects of
mathematics as separate from sense perceptibles. This process is engaging in
fictionalism by making as if the objects are separate. The other type of fictionalism is
about the pure objects of mathematics. When the mathematicians study the objects of
mathematics as if they are independent substances, they are engaged in object
fictionalism, the view that mathematical objects do not exist but are fictionally
posited by mathematicians. However, the fictional mathematical objects are closely
connected to particular, sense perceptible objects and the way they are. The fiction
depends on reality.
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Chapter 1: Plato’s Legacy

Introduction
Any study of Aristotle’s philosophy of mathematics should begin with a clear
discussion of Plato’s views on mathematics. The focus for this chapter and chapter 2
will be the influence Plato’s middle period had on two issues in Aristotle’s
philosophy of mathematics: (a) the ontological status of mathematical objects and (b)
the methodology employed in mathematics.5 Plato explicitly states what the
ontological status of mathematical objects is, and he critically considers the
methodology of the mathematicians. These two issues are also the focus of
Aristotle’s views although his position differs from Plato’s. For Plato, mathematical
objects have independent existence, they are completely separable from sense
perceptible objects, and they are more real than perceptible objects. In his criticism of
mathematics, Plato both acknowledges the importance of mathematical knowledge
and highlights the strengths and weaknesses of the mathematicians’ methodology.
Aristotle explicitly rejects the claim that mathematical objects are completely
separable and more real than sense perceptible objects instead saying they only have
qualified existence.6 However, he agrees with Plato that there are mathematical

5 The ‘methodology employed in mathematics’ refers to the approach used by
mathematicians to gain knowledge.
6Qualified existence is Aristotle’s answer to the tension between believing that:
1) All existing things are sensible objects; and
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truths that we can come to know. Furthermore, his theory about scientific knowledge,
specifically about mathematics, incorporates and explains what Plato presents as the
strengths and weaknesses of the mathematical sciences.

This chapter will focus on the Meno, Republic, and Theaetetus as representative of
Plato’s middle period views on mathematics and the Forms. I take the chronological
order of these dialogues to be the Meno (a transitional dialogue from the early to
middle period) the Republic, and the Theaetetus (a late middle period dialogue).7 The
Meno investigates the definition of virtue and whether it is teachable. It also presents
the Theory of Recollection, the Method of Hypothesis, a theory of knowledge and
uses a number of mathematical examples. The Republic investigates the definition of
justice and presents theories on political systems, psychology, education, metaphysics
and knowledge. Mathematics is extensively discussed in relation to several of these
theories. The Theaetetus investigates the definition of knowledge and presents
arguments against Protagoras and relativism. The Theaetetus also offers an
examination of judgments. The interlocutors in this dialogue are mathematicians,8

2) There are mathematical truths.
The puzzle raised by the tension between (1) and (2) will be presented in chapter 2
and resolved in chapters 3-5.
7 I follow Gregory Vlastos in this assignation. The only deviation is that I take
Republic I to be also written in the middle period as C.D.C. Reeve persuasively
argues in C.D.C. Reeve, Philosopher-Kings: The Argument of Plato’s Republic,
(Indianapolis: Hackett Publishing Company, 1988): spec. pg. 3-42.
8 Theodorus was a Greek mathematician and philosopher. Theaetetus was a Greek
geometer.
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and many of the examples used are mathematical. The mathematical views that
Aristotle attributes to Plato closely resemble the views in these three dialogues.
Plato influences Aristotle’s views knowledge and ontology. Specifically Plato’s
views on definition, knowledge, mathematical knowledge, and mathematical objects
explain Aristotle’s own focus on definition and its connection to knowledge, the
universal aspect of knowledge for Aristotle, and his views on mathematical
knowledge and objects. This chapter will present Plato’s views stressing the topics
rather than the Platonic texts. The discussion will proceed by considering
mathematical objects, knowledge, definition, and accounts.

This chapter presents Plato’s views on mathematical objects, mathematical
knowledge, and what he finds faulty with mathematical investigations as a context for
Aristotle’s position on mathematical objects. Section 1.1 presents an overview of
Plato’s views on the ontological status of mathematical objects focusing primarily on
the Republic. Section 1.2 lays the groundwork for the distinction between
philosophical and mathematical knowledge. To do so, I examine what Plato says
about knowledge, opinion, definition, and proper accounts. Section 1.3 presents
Plato’s criticism of the mathematicians and their approach to knowledge. These
criticisms explore the distinctions between philosophical and mathematical
knowledge in Plato.

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