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THE LAW OF NON CONTRADICTION AND MATHS

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Page 1
Logic and Mathematics
Stephen G. Simpson
Department of Mathematics
Pennsylvania State University
www.math.psu.edu/simpson/
April 30, 1999
This article is an overview of logic and the philosophy of mathematics. It is
intended for the general reader. It has appeared in the volume The Examined
Life: A Treasury of Western Philosophy, edited by Stanley Rosen and published
by the Book-of-the-Month Club.
Contents
1 Logic
2
1.1 Aristotelean logic . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.1 Subjects and predicates . . . . . . . . . . . . . . . . . . .
3
1.1.2 Syllogisms . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2 The predicate calculus . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1 Predicates and individuals . . . . . . . . . . . . . . . . . .
5
1.2.2 Formulas and logical operators . . . . . . . . . . . . . . .
6
1.2.3 Logical validity and logical consequence . . . . . . . . . .
7
1.2.4 The completeness theorem . . . . . . . . . . . . . . . . . .
9
1.2.5 Formal theories . . . . . . . . . . . . . . . . . . . . . . . .
9
2 Foundations of mathematics
10
2.1 The geometry of Euclid . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Formal theories for mathematics . . . . . . . . . . . . . . . . . . 12
2.2.1 A formal theory for geometry . . . . . . . . . . . . . . . . 12
2.2.2 A formal theory for arithmetic . . . . . . . . . . . . . . . 13
2.2.3 A formal theory of sets . . . . . . . . . . . . . . . . . . . 15
3 Philosophy of mathematics
17
3.1 Plato and Aristotle . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The 20th century . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 The future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References
20
1
1 Logic
Logic is the science of formal principles of reasoning or correct inference. Histor-
ically, logic originated with the ancient Greek philosopher Aristotle. Logic was
further developed and systematized by the Stoics and by the medieval scholastic
philosophers. In the late 19th and 20th centuries, logic saw explosive growth,
which has continued up to the present.
One may ask whether logic is part of philosophy or independent of it. Ac-
cording to Bochenski [2, §10B], this issue is nowhere explicitly raised in the
writings of Aristotle. However, Aristotle did go to great pains to formulate the
basic concepts of logic (terms, premises, syllogisms, etc.) in a neutral way, in-
dependent of any particular philosophical orientation. Thus Aristotle seems to
have viewed logic not as part of philosophy but rather as a tool or instrument
1
to be used by philosophers and scientists alike. This attitude about logic is
in agreement with the modern view, according to which the predicate calculus
(see 1.2 below) is a general method or framework not only for philosophical
reasoning but also for reasoning about any subject matter whatsoever.
Logic is the science of correct reasoning. What then is reasoning? According
to Aristotle [13, Topics, 100a25], reasoning is any argument in which certain
assumptions or premises are laid down and then something other than these
necessarily follows. Thus logic is the science of necessary inference. However,
when logic is applied to specific subject matter, it is important to note that
not all logical inference constitutes a scientifically valid demonstration. This is
because a piece of formally correct reasoning is not scientifically valid unless it
is based on a true and primary starting point. Furthermore, any decisions about
what is true and primary do not pertain to logic but rather to the specific subject
matter under consideration. In this way we limit the scope of logic, maintaining
a sharp distinction between logic and the other sciences. All reasoning, both
scientific and non-scientific, must take place within the logical framework, but
it is only a framework, nothing more. This is what is meant by saying that logic
is a formal science.
For example, consider the following inference:
Some real estate will increase in value.
Anything that will increase in value is a good investment.
Therefore, some real estate is a good investment.
This inference is logically correct, because the conclusion “some real estate is
a good investment” necessarily follows once we accept the premises “some real
estate will increase in value” and “anything that will increase in value is a
good investment”. Yet this same inference may not be a demonstration of its
conclusion, because one or both of the premises may be faulty. Thus logic can
help us to clarify our reasoning, but it can only go so far. The real issue in this
particular inference is ultimately one of finance and economics, not logic.
1
The Greek word for instrument is organon. The collection of Aristotle’s logical writings
is known as the Organon.
2
We shall now briefly indicate the basics of Aristotelean logic.
1.1 Aristotelean logic
Aristotle’s collection of logical treatises is known as the Organon. Of these
treatises, the Prior Analytics contains the most systematic discussion of formal
logic. In addition to the Organon, the Metaphysics
2
also contains relevant
material. See Aristotle [13] and Ross [19].
1.1.1 Subjects and predicates
Aristotelean logic begins with the familiar grammatical distinction between sub-
ject and predicate. A subject is typically an individual entity, for instance a man
3
or a house or a city. It may also be a class of entities, for instance all men. A
predicate is a property or attribute or mode of existence which a given subject
may or may not possess. For example, an individual man (the subject) may or
may not be skillful (the predicate), and all men (the subject) may or may not
be brothers (the predicate).
The fundamental principles of predication are:
1. Identity. Everything is what it is and acts accordingly. In symbols:
A is A.
For example, an acorn will grow into an oak tree and nothing else.
2. Non-contradiction. It is impossible for a thing both to be and not to be. A
given predicate cannot both belong and not belong to a given subject in a
given respect at a given time. Contradictions do not exist. Symbolically:
A and non-A cannot both be the case.
For example, an honest man cannot also be a thief.
3. Either-or. Everything must either be or not be. A given predicate either
belongs or does not belong to a given subject in a given respect at a given
time. Symbolically:
Either A or non-A.
For example, a society must be either free or not free.
2
The Metaphysics is Aristotle’s treatise on the science of existence, i.e., being as such. It
includes a detailed analysis of the various ways in which a thing can be said to be.
3
We use man in the traditional sense, equivalent to “human being”. There is no intention
to exclude persons of the female gender.
3
These principles have exercised a powerful influence on subsequent thinkers.
For example, the 20th century intellectual Ayn Rand titled the three main
divisions of her best-selling philosophical novel Atlas Shrugged
4
[18] after the
three principles above, in tribute to Aristotle.
1.1.2 Syllogisms
According to Aristotelean logic, the basic unit of reasoning is the syllogism. For
example, the real estate inference which was presented above is a syllogism. It
is of the form
Some A is B.
All B is C.
Therefore, some A is C.
Here A denotes real estate, B denotes increase in value, and C denotes a good
investment. Just as in the case of this example, every syllogism consists of two
premises and one conclusion. Each of the premises and the conclusion is of one
of four types:
universal affirmative:
All A is B.
universal negative:
No A is B.
particular affirmative: Some A is B.
particular negative:
Some A is not B.
The letters A, B, C are known as terms. Every syllogism contains three terms.
The two premises always share a common term which does not appear in the
conclusion. This is known as the middle term. In our real estate example, the
middle term is B, i.e., that which increases in value.
In order to classify the various types of syllogisms, one must take account of
certain symmetries. In particular, “no A is B” and “no B is A” are equivalent,
as are “some A is B” and “some B is A”. Furthermore, the order of the two
premises in a syllogism does not matter. Allowing for these symmetries, we
can enumerate a total of 126 possible syllogistic forms. Of these 126, only 11
represent correct inferences. For example, the form
all A is B, all B is C, therefore all A is C
represents a correct inference, while
all A is B, all C is B, therefore some A is C
does not.
The classification of syllogisms leads to a rather complex theory. Medieval
thinkers perfected it and developed ingenious mnemonics to aid in distinguishing
the correct forms from the incorrect ones. This culminated in the famous pons
asinorum (“bridge of asses”), an intricate diagram which illustrates all of the
syllogistic forms by means of a contrast between the good and the pleasurable.
See Bochenski [2, §24H, §32F].
4
A survey conducted for the Book-of-the-Month Club and the Library of Congress in 1991
found that Atlas Shrugged is the most influential book in the United States of America, second
only to the Bible. See http://www.lcweb.loc.gov/loc/cfbook/bklists.html.
4
1.2 The predicate calculus
In 1879 the German philosopher Gottlob Frege published a remarkable treatise,
the Begriffsschrift (“concept script”) [22]. This brilliant monograph is the origin
of modern logical theory. However, Frege’s account was defective in several
respects, and notationally awkward to boot. Instead of Frege’s system, we shall
present a streamlined system known as first-order logic or the predicate calculus.
The predicate calculus dates from the 1910’s and 1920’s. It is basic for all
subsequent logical research. It is a very general system of logic which accurately
expresses a huge variety of assertions and modes of reasoning. We shall see that
it is much more flexible than the Aristotelean syllogistic.
1.2.1 Predicates and individuals
In the predicate calculus, the subject/predicate distinction is drawn somewhat
differently from the way it is drawn in Aristotelean logic. The main point here
is that, in the predicate calculus, a subject is always an individual entity, never
a class of entities. For example, an individual man can be treated as a subject,
but the class of all men must be treated as a predicate. Since a subject in
the predicate calculus is always an individual entity, it is usual to speak of
individuals rather than subjects. We shall follow this customary practice.
The predicate calculus makes heavy use of symbolic notation. Lower-case
letters a, b, c, . . . , x, y, z, . . . are used to denote individuals. Upper-case letters
M, N, P, Q, R, . . . are used to denote predicates. Simple assertions may be
formed by juxtaposing a predicate with an individual.
For example, if M is the predicate “to be a man” and a is the individual
“Socrates”, then Ma denotes the assertion “Socrates is a man”. The symbol a
is called an argument of M. The predicate M may be applied to any individual,
and that individual is then an argument of M. If b is the individual “New
York”, then Mb asserts, falsely, that New York is a man. In general, if x is any
individual whatsoever, then Mx is the assertion that x is a man. This assertion
may or may not be true, depending on what x is. The expression Mx is called
an atomic formula of the predicate calculus.
Some predicates require more than one argument. For example, if B is the
predicate “bigger than”, then Bxy denotes the assertion “x is bigger than y”.
Thus B requires two arguments, and Bxy is an atomic formula. If we try to
use B with only one argument, we obtain something like Bx, i.e., “x is bigger
than”. This is not an atomic formula or any other kind of assertion. It is only
a meaningless combination of symbols. In analogy with English grammar, we
could say that Bxy is like a grammatically correct sentence, while Bx is merely
a sentence fragment. Such fragments play no role in the predicate calculus.
Let us now go into more detail about the role of individuals in the predicate
calculus. We have already said that lower-case letters denote individuals. We
now divide the lower-case letters into two groups: a, b, c, . .. near the beginning
of the alphabet, and x, y, z, . . . near the end of the alphabet. We insist on an
important grammatical or logical distinction between these two groups. Letters
5
of the first group are known as individual constants or simply constants. As
in the above examples, we think of them as denoting specific individuals, such
as Socrates or New York. Letters of the second group are known as individ-
ual variables or simply variables. For example, x is a variable. We think of
x as denoting not a specific individual but rather an arbitrary or unspecified
individual.
5
1.2.2 Formulas and logical operators
We have already mentioned two kinds of symbols: lower-case letters for individ-
uals (constants and variables), and upper-case letters for predicates. In addition
to these, the predicate calculus employs seven special symbols known as logical
operators
6
:
&
∨
∼
⊃
≡
∀
∃
The names and meanings of the logical operators are given by
symbol
name
usage
meaning
&
conjunction
. . . & . . .
“both . . . and . . . ”
∨
disjunction
. . . ∨ . . .
“either . . . or . . . (or both)”
∼
negation
∼ . . .
“it is not the case that . . . ”
⊃
implication
7
. . . ⊃ . . .
“if . . . then . . . ”
≡
bi-implication
8
. . . ≡ . . .
“. . . if and only if . . . ”
∀
universal quantifier
∀x . . .
“for all x , . . . ”
∃
existential quantifier
∃x . . .
“there exists x such that .. . ”
Here x is any variable.
A formula is a meaningful expression built up from atomic formulas by
repeated application of the logical operators. In the above table, an ellipsis
mark . . . stands for a formula within a larger formula.
For example, suppose we have a predicate M meaning “is a man”, another
predicate T meaning “is a truck”, and another predicate D meaning “drives”.
Here M and T are predicates which require only one argument apiece. The
predicate D requires two arguments: the driver, and the vehicle being driven.
5
The idea of using letters such as x and y as variables is of great value. Historically,
the creators of the predicate calculus borrowed this idea from the mathematical discipline
known as algebra. Recall that algebra is a kind of generalized arithmetic. In algebra there
are constants, i.e., specific quantities such as 2, the square root of 10, etc., but there are
also variables such as x, y, etc. The key idea of algebra is that a variable x represents an
unspecified or unknown quantity. It always stands for some quantity, but it may stand for any
quantity. The use of variables makes algebra much more powerful than arithmetic. Variables
help us to express and solve equations such as 2x + 3y = 11 involving one or more unknown
quantities. Variables can also be used to express arithmetical laws such as x + y = y + x.
6
The first five logical operators ( & , ∨, ∼ , ⊃, ≡) are equivalent to so-called “Boolean logic
gates” of electrical engineering. Formulas built from them may be viewed as representations
of the binary switching circuits that control the operation of modern digital computers. See
Mendelson [14, 15].
7
This is the so-called “material implication”: Φ
1
⊃ Φ
2
is equivalent to ∼ (Φ
1
& ∼ Φ
2
).
8
This is called bi-implication because Φ
1
≡ Φ
2
is equivalent to (Φ
1
⊃ Φ
2
) & (Φ
2
⊃ Φ
1
).
6
Thus Mx, Ty, and Dxy are atomic formulas meaning “x is a man”, “y is
a truck”, and “x drives y”, respectively. A typical formula built from these
atomic formulas is
∀x (Mx ⊃ ∃y (Ty & Dxy))
which we can translate as “for all x, if x is a man then there exists y such that
y is a truck and x drives y”. In other words,
Every man drives at least one truck.
Similarly, the formula
∀y (Ty ⊃ ∃x (Mx & Dxy))
translates to
Every truck is driven by at least one man.
In writing formulas, we often use parentheses as punctuation marks to in-
dicate grouping and thereby remove ambiguity. If parentheses were not used,
one could construe the formula ∼ Ty & Dxy in two logically inequivalent ways:
as (∼ Ty) & Dxy (“y is not a truck, and x drives y”), or as ∼(Ty & Dxy) (“y
is not a truck that x drives”). The parentheses allow us to choose the meaning
that we intend.
The predicate calculus is very rich in expressive power. For example, the four
Aristotelean premise types discussed in 1.1.2 can easily be rendered as formulas
of the predicate calculus. Letting A and B be predicates which require one
argument apiece, we have
universal affirmative
all A is B
∀x (Ax ⊃ Bx)
universal negative
no A is B
∀x (Ax ⊃ ∼ Bx)
particular affirmative some A is B
∃x (Ax & Bx)
particular negative
some A is not B
∃x (Ax & ∼ Bx)
In the second line of this table, the universal negative “no A is B” could have
been rendered equivalently as ∼∃x (Ax & Bx), or as ∀x (Bx ⊃ ∼Ax).
The above table may tend to gloss over a subtle but philosophically signifi-
cant difference between Aristotelean logic and the predicate calculus. Namely,
where Aristotelean logic views A as a subject and B as a predicate, the predi-
cate calculus views both A and B as predicates. This is typical of the different
perspectives involved. Aristotelean logic emphasizes the universal essences of
subjects or entities, while the predicate calculus elevates predicates to a position
of supreme importance.
1.2.3 Logical validity and logical consequence
A formula of the predicate calculus is said to be logically valid if it is necessarily
always true, regardless of the specific predicates and individuals involved. For
example, the three fundamental principles of Aristotelean logic (see 1.1.1 above)
correspond to formulas as follows:
7
Identity:
∀x (Ax ≡ Ax).
Non-contradiction: ∼∃x (Ax & ∼ Ax).
Either-or:
∀x (Ax ∨ ∼Ax).
These formulas are logically valid, because they are “necessarily” or “automat-
ically” or “formally” true, no matter what predicate may be denoted by the
symbol A.
The predicate calculus concept of logical validity subsumes the Aristotelean
syllogism. Each syllogism corresponds to a logically valid implication
(Φ
1
& Φ
2
) ⊃ Ψ
where Φ
1
and Φ
2
are formulas expressing the two premises and Ψ expresses the
conclusion. For example, the syllogism
some A is B, all B is C, therefore some A is C
has a predicate calculus rendition
((∃x (Ax & Bx)) & (∀x (Bx ⊃ Cx))) ⊃ (∃x (Ax & Cx))
and this formula is logically valid.
More generally, a formula Ψ is said to be a logical consequence of a set of
formulas Φ
1
, . . . , Φ
n
just in case
(Φ
1
& · · · & Φ
n
) ⊃ Ψ
is logically valid. Here Φ
1
, . . . , Φ
n
are premises and Ψ is a conclusion. This
is similar to the Aristotelean syllogism, but it is of wider applicability, because
the premises and the conclusion can be more complex. As an example, the 19th
century logician Augustus DeMorgan noted
9
that the inference
all horses are animals,
therefore, the head of a horse is the head of an animal
is beyond the reach of Aristotelean logic. Yet this same inference may be para-
phrased as “if all horses are animals, then for all x, if x is the head of some
horse then x is the head of some animal”, and this corresponds to a logically
valid formula
(∀y (Hy ⊃ Ay)) ⊃ (∀x ((∃y (Rxy & Hy)) ⊃ (∃y (Rxy & Ay))))
of the predicate calculus. Here H, A, R denote “is a horse”, “is an animal”,
“is the head of”, respectively. Thus DeMorgan’s conclusion is indeed a logical
consequence of his premise.
9
See however Bochenski [2, §16E].
8
1.2.4 The completeness theorem
Formulas of the predicate calculus can be exceedingly complicated. How then
can we distinguish the formulas that are logically valid from the formulas that
are not logically valid? It turns out that there is an algorithm
10
for recognizing
logically valid formulas. We shall now sketch this algorithm.
In order to recognize that a formula Φ is logically valid, it suffices to construct
what is known as a proof tree for Φ, or equivalently a refutation tree for ∼Φ.
This is a tree which carries ∼Φ at the root. Each node of the tree carries a
formula. The growth of the tree is guided by the meaning of the logical operators
appearing in Φ. New nodes are added to the tree depending on what nodes have
already appeared. For example, if a node carrying ∼(Φ
1
& Φ
2
) has appeared,
we create two new nodes carrying ∼ Φ
1
and ∼Φ
2
respectively. The thought
behind these new nodes is that the only way for ∼(Φ
1
& Φ
2
) to be the case is if
at least one of ∼ Φ
1
or ∼ Φ
2
is the case. Similarly, if a node carrying ∼∀x Ψ has
already appeared, we create a new node carrying ∼Ψ , where Ψ is the result of
substituting a new constant a for the variable x throughout the formula Ψ. The
idea here is that the only way for the universal statement ∀x Ψ to be false is if Ψ
is false for some particular x. Since a is a new constant, Ψ is a formula which
may be considered as the most general false instance of Ψ. Corresponding to
each of the seven logical operators, there are prescribed procedures for adding
new nodes to the tree. We apply these procedures repeatedly until they cannot
be applied any more. If explicit contradictions
11
are discovered along each and
every branch of the tree, then we have a refutation tree for ∼ Φ. Thus ∼Φ is
seen to be logically impossible. In other words, Φ is logically valid.
The adequacy of proof trees for recognizing logically valid formulas is a major
insight of 20th century logic. It is a variant of the famous completeness theorem,
first proved in 1930 by the great logician Kurt Godel [5, 22].
On the other hand, the class of logically valid formulas is known to be
extremely complicated. Indeed, this class is undecidable: there is no algorithm
12
which accepts as input an arbitrary formula Φ and outputs “yes” if Φ is logically
valid and “no” if Φ is not logically valid. In this sense, the concept of logical
validity is too general and too intractable to be analyzed thoroughly. There will
never be a predicate calculus analog of the pons asinorum.
1.2.5 Formal theories
The predicate calculus is a very general and flexible framework for reasoning. By
choosing appropriate predicates, one can reason about any subject whatsoever.
These considerations lead to the notion of a formal theory.
10
The details of this algorithm are explained in modern logic textbooks. Variants of it have
been programmed to run on digital computers. They form the basis of a system of computer
logic. See Fitting [4].
11
An explicit contradiction is a pair of formulas of the form Ψ, ∼ Ψ.
12
The algorithms in question may be implemented as Turing machine programs. This
undecidability result is known as Church’s theorem. See Mendelson [15].
9
In order to specify a formal theory, one first chooses a small collection of
predicates which are regarded as basic for a given field of study. These predicates
are the primitives of the theory. They delimit the scope of the theory. Other
predicates must be defined in terms of the primitives. Using them, one writes
down certain formulas which are regarded as basic or self-evident within the
given field of study. These formulas are the axioms of the theory. It is crucial
to make all of our underlying assumptions explicit as axioms. Once this has been
done, a theorem is any formula which is a logical consequence of the axioms. A
formal theory is this structure of primitives, axioms, and theorems.
As a frivolous example, we could envision a theory of cars, trucks, and
drivers. We would begin with some primitives such as C (“is a car”), T (“is a
truck”), D (“drives”), M (“is a man”), etc. We could then write down certain
obvious or self-evident axioms such as ∀x (Mx ⊃ ∼ Cx) (“no man is a car”),
∀x ((∃y Dxy) ⊃ Mx) (“every driver is a man”), etc. Then, within the con-
straints imposed by the axioms, we could investigate the logical consequence
relationships among various non-obvious assertions, such as
∼∃x (Mx & ∃y (Dxy & Cy) & ∃z (Dxz & Tz))
(“nobody drives both a car and a truck”). Additional predicates V (“is a vehi-
cle”) and P (“is a driver”) can be defined in terms of the primitives. The defining
axioms for V and P would be ∀y (V y ≡ (Cy ∨ Ty)) and ∀x (Px ≡ ∃y Dxy), re-
spectively. In this fashion, we could attempt to codify all available knowledge
about vehicles and drivers.
More seriously, one could try to write down formal theories corresponding
to various scientific disciplines, such as mechanics or statistics or law. In this
way one could hope to analyze the logical structure of the respective disciplines.
The process of codifying a scientific discipline by means of primitives and
axioms in the predicate calculus is known as formalization. The key issue here
is the choice of primitives and axioms. They cannot be chosen arbitrarily. The
scientist who chooses them must exercise a certain aesthetic touch. They must
be small in number; they must be basic and self-evident; and they must account
for the largest possible number of other concepts and facts.
To date, this kind of formal theory-building has been convincingly carried
out in only a few cases. A survey is in Tarski [21]. The most notable successes
have been in mathematics.
2 Foundations of mathematics
Mathematics is the science of quantity. Traditionally there were two branches
of mathematics, arithmetic and geometry, dealing with two kinds of quantities:
numbers and shapes. Modern mathematics is richer and deals with a wider
variety of objects, but arithmetic and geometry are still of central importance.
Foundations of mathematics is the study of the most basic concepts and
logical structure of mathematics, with an eye to the unity of human knowledge.
10
Among the most basic mathematical concepts are: number, shape, set, function,
algorithm, mathematical axiom, mathematical definition, mathematical proof.
The reader may reasonably ask why mathematics appears at all in this vol-
ume. Isn’t mathematics too narrow a subject? Isn’t the philosophy of mathe-
matics of rather specialized interest, all the more so in comparison to the broad
humanistic issues of philosophy proper, issues such as the good, the true, and
the beautiful?
There are three reasons for discussing mathematics in a volume on general
philosophy:
1. Mathematics has always played a special role in scientific thought. The
abstract nature of mathematical objects presents philosophical challenges
that are unusual and unique.
2. Foundations of mathematics is a subject that has always exhibited an
unusually high level of technical sophistication. For this reason, many
thinkers have conjectured that foundations of mathematics can serve as a
model or pattern for foundations of other sciences.
3. The philosophy of mathematics has served as a highly articulated test-
bed where mathematicians and philosophers alike can explore how various
general philosophical doctrines play out
13
in a specific scientific context.
The purpose of this section is to indicate the role of logic in the foundations
of mathematics. We begin with a few remarks on the geometry of Euclid. We
then describe some modern formal theories for mathematics.
2.1 The geometry of Euclid
Above the gateway to Plato’s academy appeared a famous inscription:
Let no one who is ignorant of geometry enter here.
In this way Plato indicated his high opinion of geometry. According to Heath
[9, page 284], Plato regarded geometry as “the first essential in the training of
philosophers”, because of its abstract character. See also Plato [17, Republic,
527B].
In the Posterior Analytics [13], Aristotle laid down the basics of the scientific
method.
14
The essence of the method is to organize a field of knowledge logically
by means of primitive concepts, axioms, postulates, definitions, and theorems.
The majority of Aristotle’s examples of this method are drawn from arithmetic
and geometry [1, 7, 9].
13
For example, philosophical intrinsicism may play out as mathematical Platonism. Philo-
sophical subjectivism may play out as mathematical constructivism. Nominalism may play
out as formalism.
14
Our modern notion of a formal theory (see 1.2.5 above) is a variant of Aristotle’s concept
of scientific method.
11
The methodological ideas of Aristotle decisively influenced the structure and
organization of Euclid’s monumental treatise on geometry, the Elements [8]. Eu-
clid begins with 21 definitions, five postulates, and five common notions. After
that, the rest of the Elements are an elaborate deductive structure consisting
of hundreds of propositions. Each proposition is justified by its own demon-
stration. The demonstrations are in the form of chains of syllogisms. In each
syllogism, the premises are identified as coming from among the definitions,
postulates, common notions, and previously demonstrated propositions. For
example, in Book I of the Elements, the demonstration of Proposition 16 (“in
any triangle, if one of the sides be produced, the exterior angle is greater than ei-
ther of the interior and opposite angles”) is a chain of syllogisms with Postulate
2, Common Notion 5, and Propositions 3, 4 and 15 (“if two straight lines cut
one another, they make the vertical angles equal to one another”) occurring as
premises. It is true that the syllogisms of Euclid do not always conform strictly
to Aristotelean templates. However, the standards of rigor are very high, and
Aristotle’s influence is readily apparent.
The logic of Aristotle and the geometry of Euclid are universally recognized
as towering scientific achievements of ancient Greece.
2.2 Formal theories for mathematics
2.2.1 A formal theory for geometry
With the advent of calculus in the 17th and 18th centuries, mathematics de-
veloped very rapidly and with little attention to logical foundations. Euclid’s
geometry was still regarded as a model of logical rigor, a shining example of
what a well-organized scientific discipline ideally ought to look like. But the
prolific Enlightenment mathematicians such as Leonhard Euler showed almost
no interest in trying to place calculus on a similarly firm foundation. Only in the
last half of the 19th century did scientists begin to deal with this foundational
problem in earnest. The resulting crisis had far-reaching consequences. Even
Euclid’s geometry itself came under critical scrutiny. Geometers such as Moritz
Pasch discovered what they regarded as gaps or inaccuracies in the Elements.
Great mathematicians such as David Hilbert entered the fray.
An outcome of all this foundational activity was a thorough reworking of
geometry, this time as a collection of formal theories within the predicate calcu-
lus. Decisive insights were obtained by Alfred Tarski. We shall sketch Tarski’s
formal theory for Euclidean
15
plane geometry.
16
As his primitive predicates, Tarski takes P (“point”), B (“between”), D
(“distance”), I (“identity”). The atomic formulas Px, Bxyz, Dxyuv, and Ixy
mean “x is a point”, “y lies between x and z”, “the distance from x to y is equal
to the distance from u to v”, and “x is identical to y”, respectively. Geometrical
15
Here “Euclidean geometry” refers to the familiar geometry in which the angles of a triangle
sum to 180 degrees, as distinct from the “non-Euclidean” (i.e., hyperbolic) geometry developed
by Bolyai and Lobachevsky in the 19th century.
16
Tarski also showed how to handle non-Euclidean plane geometry, as well as Euclidean and
non-Euclidean geometries of higher dimension, in a similar fashion.
12
objects other than points, such as line segments, angles, triangles, circles, etc.,
are handled by means of the primitives. For example, the circle with center x
and radius uv consists of all points y such that Dxyuv holds.
In geometry, two points x and y are considered identical if the distance
between them is zero. Tarski expresses this by means of an axiom
∀x ∀y ∀z (Dxyzz ⊃ Ixy).
Another axiom
∀w ∀x ∀y ∀z ((Bwxy & Bwyz) ⊃ Bxyz)
expresses the fact that, given any four points, if the second is between the first
and the third, and if the third is between the first and the fourth, then the third
is between the second and the fourth. A noteworthy axiom is
∀x ∀y ∀z ∀u ∀v ((Dxuxv & Dyuyv & Dzuzv & ∼Iuv) ⊃ (Bxyz ∨ Bxzy ∨ Byxz))
which says: any three points x, y, z equidistant from two distinct points u, v must
be collinear. This axiom is typical of two-dimensional (i.e., plane) geometry and
does not apply to geometries of dimension greater than two.
Altogether Tarski presents twelve axioms, plus an additional collection of
axioms expressing the idea that a line is continuous. The full statement of
Tarski’s axioms for Euclidean plane geometry is given at [10, pages 19–20]. Let
T
g
be the formal theory based on Tarski’s axioms.
Remarkably, Tarski has demonstrated that T
g
is complete. This means that,
for any purely geometrical
17
statement Ψ, either Ψ or ∼Ψ is a theorem of T
g
.
Thus we see that the axioms of T
g
suffice to answer all yes/no questions of
Euclidean plane geometry. Combining this with the completeness theorem of
Godel, we find that T
g
is decidable: there is an algorithm
18
which accepts as
input an arbitrary statement of plane Euclidean geometry, and outputs “true”
if the statement is true, and “false” if it is false. This is a triumph of modern
foundational research.
2.2.2 A formal theory for arithmetic
By arithmetic we mean elementary school arithmetic, i.e., the study of the
positive whole numbers 1, 2, 3, .. . along with the familiar operations of addition
(+) and multiplication (×). This part of mathematics is obviously fundamental,
yet it turns out to be surprisingly complicated. Below we write down some of
the axioms which go into a formal theory of arithmetic.
19
Our primitive predicates for arithmetic are N (“number”), A (“addition”),
M (“multiplication”), I (“identity”). The atomic formulas Nx, Axyz, Mxyz,
17
This means that all occurrences of variables x within the formula Ψ are within subformulas
of the form ∀x (Px ⊃ . . .) or ∃x (Px & . . .). Thus we are restricting attention to the realm of
geometry and excluding everything else.
18
Such algorithms have been implemented as computer programs. They are useful in
robotics and other artificial intelligence applications.
19
Two recent studies of formal arithmetic are Hajek/Pudlak [6] and Simpson [20].
13
Ixy mean “x is a number”, “x+y = z”, “x×y = z”, “x = y”, respectively. Our
axioms will use the predicates N, A, M, I to assert that for any given numbers
x and y, the numbers x + y and x × y always exist and are unique. We shall
also have axioms expressing some well known arithmetical laws:
substitution laws:
if x = y and x is a number then y is a number, etc.
commutative laws: x + y = y + x and x × y = y × x.
associative laws:
(x + y) + z = x + (y + z) and (x × y) × z = x × (y × z).
distributive law:
x × (y + z) = (x × y) + (x × z).
comparison law:
x = y if and only if, for some z, x + z = y or x = y + z.
unit law:
x × 1 = x.
Our formal axioms for arithmetic are as follows.
substitution laws:
∀x Ixx
∀x ∀y (Ixy ≡ Iyx)
∀x ∀y ∀z ((Ixy & Iyz) ⊃ Ixz)
∀x ∀y (Ixy ⊃ (Nx ≡ Ny))
existence and uniqueness of x + y :
∀x ∀y ∀z ∀u ∀v ∀w ((Ixu & Iyv & Izw) ⊃ (Axyz ≡ Auvw))
∀x ∀y ∀z (Axyz ⊃ (Nx & Ny & Nz))
∀x ∀y ((Nx & Ny) ⊃ ∃w ∀z (Iwz ≡ Axyz))
existence and uniqueness of x × y :
∀x ∀y ∀z ∀u ∀v ∀w ((Ixu & Iyv & Izw) ⊃ (Mxyz ≡ Muvw))
∀x ∀y ∀z (Mxyz ⊃ (Nx & Ny & Nz))
∀x ∀y ((Nx & Ny) ⊃ ∃w ∀z (Iwz ≡ Mxyz))
commutative laws:
∀x ∀y ∃z (Axyz & Ayxz)
∀x ∀y ∃z (Mxyz & Myxz)
associative laws:
∀x ∀y ∀z ∃u ∃v ∃w (Axyu & Auzw & Ayzv & Axvw)
∀x ∀y ∀z ∃u ∃v ∃w (Mxyu & Muzw & Myzv & Mxvw)
distributive law:
∀x ∀y ∀z ∃t ∃u ∃v ∃w (Ayzt & Mxtw & Mxyu & Mxzv & Auvw)
comparison law:
∀x ∀y ((Nx & Ny) ⊃ (Ixy ≡ ∼ ∃z (Axzy ∨ Ayzx)))
unit law:
∃z (Nz & (∼∃x ∃y Axyz) & ∀w (Nw ⊃ Mwzw))
Let T
a
be the formal theory specified by the above primitives and axioms.
It is known that T
a
suffices to derive many familiar arithmetical facts. For
example, 2+2 = 4 may be expressed, awkwardly
20
to be sure, as (1+1)+(1+1) =
20
This kind of awkwardness can be alleviated by means of various devices. In particular,
14
((1 + 1) + 1) + 1 or
∃x ∃y ∃z ∃w (Mxxx & Axxy & Axyz & Axzw & Ayyw)
21
and this formula is indeed a theorem of T
a
, i.e., a logical consequence of the
axioms of T
a
. Another theorem of T
a
is
∀x ∀y ∀z ∀w (((Axzw & Ayzw) ∨ (Mxzw & Myzw)) ⊃ Ixy)
expressing a familiar cancellation law: if either x + z = y + z or x × z = y × z,
then x = y.
On the other hand, the axioms of T
a
are by no means exhaustive. They
can be supplemented with other axioms expressing the so-called mathematical
induction or least number principle: if there exists a number having some well-
defined property, then among all numbers having the property there is a smallest
one. The resulting formal theory is remarkably powerful, in the sense that its
theorems include virtually all known arithmetical facts. But it is not so powerful
as one might wish. Indeed, any formal theory which includes T
a
is necessarily
either inconsistent
22
or incomplete. Thus there is no hope of writing down
enough axioms or developing an algorithm to decide all arithmetical facts. This
is a variant of the famous 1931 incompleteness theorem of Godel [5, 22]. There
are several methods of coping with the incompleteness phenomenon, and this
constitutes a currently active area of research in foundations of mathematics.
The contrast between the completeness of formal geometry and the incom-
pleteness of formal arithmetic is striking. Both sides of this dichotomy are of
evident philosophical interest.
2.2.3 A formal theory of sets
One of the aims of modern logical research is to devise a single formal theory
which will unify all of mathematics. Such a theory will necessarily be subject
to the Godel incompleteness phenomenon, because it will incorporate not only
T
g
but also T
a
.
One approach to a unified mathematics is to straightforwardly embed arith-
metic into geometry, by identifying whole numbers with evenly spaced points
on a line. This idea was familiar to the ancient Greeks. Another approach is
to explain geometry in terms of arithmetic and algebra, by means of coordinate
systems, like latitude and longitude on a map. This idea goes back to the 17th
century mathematician and philosopher Rene Descartes and the 19th century
mathematician Karl Weierstrass. Both approaches give rise to essentially the
same formal theory, known as second-order arithmetic.
23
This theory includes
both T
a
and T
g
and is adequate for the bulk of modern mathematics. Thus the
standard mathematical notation such as x+2 can be incorporated into the predicate calculus.
21
In this formula, the variables x, y, z, w play the role of 1, 2, 3, 4 respectively.
22
A formal theory is said to be inconsistent if its axioms logically imply an explicit contra-
diction. Such theories are of no scientific value.
23
A recent study of second-order arithmetic is Simpson [20].
15
decision about whether to make geometry more fundamental than arithmetic or
vice versa seems to be mostly a matter of taste.
A very different approach to a unified mathematics is via set theory. This
is a peculiarly 20th century approach. It is based on one very simple-looking
concept: sets. Remarkably, this one concept leads directly to a vast structure
which encompasses all of modern mathematics.
A set is a collection of objects called the elements of the set. We sometimes
use informal notations such as x = {y, z, . . .} to indicate that x is a set consisting
of elements y, z, . .. . The number of elements in a set can be arbitrarily large
or even infinite. A basic principle of set theory is that a set is determined
by its elements. Thus two sets are identical if and only if they have the same
elements. This principle is known as extensionality. For example, the set {a, b, c}
is considered to be the same set as {c, a, b} because the elements are the same,
even though written in a different order.
Much of the complexity of set theory arises from the fact that sets may
be elements of other sets. For instance, the set {a, b} is an element of the set
{{a, b}, c} and this is distinct from the set {a, b, c}.
For a formal theory of sets, we use three primitives: S (“set”), I (“identity”),
E (“element”). The atomic formulas Sx, Ixy, Exy mean “x is a set”, “x is
identical to y”, “x is an element of y”, respectively. One of the ground rules of
set theory is that only sets may have elements. This is expressed as an axiom
∀w ∀x (Ewx ⊃ Sx). In addition there is an axiom of extensionality
∀x ∀y ((Sx & Sy) ⊃ (Ixy ≡ ∀w (Ewx ≡ Ewy)))
and an axiom ∃x (Sx & ∼∃w Ewx) expressing the existence of the empty set,
i.e. a set { } having no elements. A list of all the axioms of set theory may be
found in textbooks [11, 15]. Let T
s
be the formal theory of sets based on these
axioms.
The set theory approach to arithmetic is in terms of the non-negative whole
numbers 0, 1, 2, 3, . . .. These numbers are identified with specific sets. Namely,
we identify 0 with the empty set { }, 1 with {{ }}, 2 with {{ }, {{ }}}, 3 with
{{ }, {{ }}, {{ }, {{}}}}, etc. In general, we identify the number n with the set
of smaller numbers {0, 1, . . ., n − 1}. Among the axioms of T
s
is an axiom of
infinity asserting the existence of the infinite set ω = {0, 1, 2, 3, . . .}. One can
use the set ω to show that T
s
includes a theory equivalent to T
a
. After that, one
can follow the ideas of Descartes and Weierstrass to see that T
s
also includes a
theory equivalent to T
g
. It turns out that the rest of modern mathematics can
also be emulated within T
s
. This includes an elaborate theory of infinite sets
which are much larger than ω.
The set-theoretical approach to arithmetic and geometry is admittedly some-
what artificial. However, the idea of basing all of mathematics on one simple
concept, sets, has exerted a powerful attraction.
24
The implications of this idea
are not yet fully understood and are a topic of current research.
24
The idea of set-theoretical foundations gave rise to the “new math” pedagogy of the
1960’s. For a lively discussion, see Kline [12].
16
3 Philosophy of mathematics
In this section we indicate some issues and trends in the philosophy of mathe-
matics.
3.1 Plato and Aristotle
The objects that are studied in mathematics tend to be somewhat abstract
and remote from everyday perceptual experience. Therefore, the existence and
nature of mathematical objects present special philosophical challenges. For
example, is a geometrical square different from a square floor tile? If so, then
where is the geometrical square? Is it on the floor, in our minds, or somewhere
else? And what about sets? Is a set of 52 cards something other than the cards
themselves?
The ancient Greek philosophers took such questions very seriously. Indeed,
many of their general philosophical discussions were carried on with extensive
reference to geometry and arithmetic. Plato seemed to insist that mathematical
objects, like the Platonic forms or essences, must be perfectly abstract and have
a separate, non-material kind of existence. Aristotle [1, 7, 13, 19] dissected and
refuted this view in books M and N of the Metaphysics. According to Aristotle,
the geometrical square is a significant aspect of the square floor tile, but it can
only be understood by discarding other irrelevant aspects such as the exact
measurements, the tiling material, etc. Clearly these questions provide much
food for philosophical analysis and debate.
3.2 The 20th century
In the 20th century, the advent of the predicate calculus and the digital com-
puter profoundly affected our view of mathematics. The discovery that all of
mathematics can be codified in formal theories created a huge stir. One expres-
sion of this excitement was the rise of an extreme philosophical doctrine known
as formalism.
25
According to formalism, mathematics is only a formal game, concerned solely
with algorithmic manipulation of symbols. Under this view, the symbols of the
predicate calculus do not denote predicates or anything else. They are merely
marks on paper, or bits and bytes in the memory of a computer. Therefore,
mathematics cannot claim to be any sort of knowledge of mathematical objects.
Indeed, mathematical objects do not exist at all, and the profound questions
debated by Plato and Aristotle become moot. Mathematics is nothing but a
kind of blind calculation.
The formalist doctrine fits well with certain modern trends in computer
science, e.g., artificial intelligence. However, formalism has proved inadequate as
an integrated philosophy of mathematics, because it fails to account for human
mathematical understanding, not to mention the spectacular applications of
mathematics in fields such as physics and engineering.
25
See for example Curry [3].
17
By way of reaction against formalism, several alternative doctrines have been
advocated. One of these is constructivism, the idea that mathematical knowl-
edge can be obtained by means of a series of purely mental constructions. Under
this view, mathematical objects exist solely in the mind of the mathematician,
so mathematical knowledge is absolutely certain. However, the status of math-
ematics vis a vis the external world becomes doubtful. An extreme version of
constructivism is so solipsistic that it does not even allow for the possibility of
mathematical communication from one mind to another.
An additional disturbing feature of constructivism is that it entails rejection
of the basic laws of logic. To see how this comes about, consider some specific
mathematical problem or question
26
of a yes/no nature, for which the answer is
currently unknown. (Mathematics abounds with such questions, and the Godel
incompleteness phenomenon suggests that such questions will always exist.)
Express the “yes” answer as a formula Ψ and the “no” answer as the negated
formula ∼Ψ. Since the answer is unknown, neither Ψ nor ∼ Ψ is in the mind of
the mathematician. Therefore, according to the constructivists, the disjunction
Ψ∨∼Ψ is not a legitimate mathematical assumption. Thus Aristotle’s either-or
principle (see 1.1.1 and 1.2.3 above) must be abandoned.
27
Constructivism has the merit of allowing human beings to possess mathe-
matical knowledge. However, the constructivist rejection of the external world
and of Aristotelean logic are highly unpalatable to most mathematicians and
mathematically oriented scientists. For this reason, constructivism remains a
fringe movement on the 20th century mathematical landscape.
Another 20th century philosophical doctrine has arisen from set-theoretical
foundations. The reliance on infinite sets suggests many perplexing questions.
What do such sets correspond to in reality? Where are they, and how can the
human mind grasp them? In order to boldly answer these questions, and as a
reaction against formalism, many researchers in axiomatic set theory have sub-
scribed to what is known as set-theoretical Platonism. According to this variant
of the Platonic doctrine, infinite sets exist in a non-material, purely mathemat-
ical realm. By extending our intuitive understanding of this realm, we will be
able to cope with chaos issuing from the Godel incompleteness phenomenon.
The most prominent and frequently cited authority for this kind of Platonism
is Godel himself [5].
There is a good fit between set-theoretical Platonism and certain aspects of
20th century mathematical practice. However, as a philosophical doctrine, set-
theoretical Platonism leaves much to be desired. Many of Aristotle’s objections
to the Platonic forms are still cogent. There are serious questions about how a
theory of infinite sets can be applicable to a finite world.
We have mentioned three competing 20th century doctrines: formalism, con-
structivism, set-theoretical Platonism. None of these doctrines are philosoph-
ically satisfactory, and they do not provide much guidance for mathematically
oriented scientists and other users of mathematics. As a result, late 20th century
26
For example, we could consider the following difficult question of Goldbach. Can every
even number greater than 2 be expressed as the sum of two prime numbers?
27
See the essays of Brouwer and Kolmogorov in [22].
18
mathematicians have developed a split view, a kind of Kantian schizophrenia,
which is usually described as “Platonism on weekdays, formalism on weekends”.
In other words, they accept the existence of infinite sets as a working hypothesis
in their mathematical research, but when it comes to philosophical speculation,
they retreat to a formalist stance. Thus they have given up hope of an integrated
view which accounts for both mathematical knowledge and the applicability of
mathematics to physical reality. In this respect, the philosophy of mathematics
is in a sorry state.
3.3 The future
From the Renaissance through the 20th century, Aristotle’s ideas about the
nature of mathematical objects have been neglected and ignored. Now the
time seems ripe for a renovation of the philosophy of mathematics, based on
Aristotelean and neo-Aristotelean [16] ideas and bolstered by the techniques of
modern logic, including the predicate calculus.
The great mathematician David Hilbert anticipated such a renovation in his
1925 essay, On the Infinite [22]. Hilbert was aware that, according to modern
physics, the physical universe is finite. Yet infinite sets were playing an increas-
ingly large role in the mathematics of the day. Hilbert therefore recognized that
the most vulnerable chink in the armor of mathematics was the infinite. In order
to defend what he called “the honor of human understanding”, Hilbert proposed
to develop a new foundation of mathematics, in which formal theories of infinite
sets, such as T
s
, would be rigorously justified by reference to the finite. This is
Hilbert’s program of finitistic reductionism.
28
Although Hilbert did not cite Aristotle, we can imagine that Hilbert would
have profited from an examination of Aristotle’s distinction between actual and
potential infinity. An actual infinity is something like an infinite set regarded
as a completed totality. A potential infinity is more like a finite but indefinitely
long, unending series of events. According to Aristotle, actual infinities cannot
exist, but potential infinities exist in nature and are manifested to us in various
ways, for instance the indefinite cycle of the seasons, or the indefinite divisibility
of a piece of gold.
In any case, it turned out that Hilbert had stated his program in too sweeping
a fashion. The wholesale finitistic reduction which Hilbert desired cannot be
carried out. This follows from Godel’s incompleteness theorem [5, 22]. The
remarkable results obtained by Godel in 1931 caused the philosophical ideas of
Hilbert’s 1925 essay to fall into disrepute. Hilbert’s grand foundational program
appeared to be dead, broken beyond hope of repair.
The last 20 years have seen a revival of Hilbert’s program. Recent founda-
tional research [20] has revealed that, although T
s
is not finitistically reducible,
there are other formal theories which are finitistically reducible, in the precise
28
Hilbert is often inaccurately described as a formalist. The details of Hilbert’s program
will not be presented here, but see [20, 22]. Roughly speaking, a formal theory is said to be
finitistically reducible if it can be embedded into some very restricted formal theory such as
T
a
, which is physically meaningful and makes absolutely no reference to actual infinity.
19
sense envisioned by Hilbert. Moreover, these other formal theories turn out to
be adequate for a very large portion of mathematics. They do not encompass
actual infinities such as ω, but they do include the main results of arithmetic
and geometry and allied disciplines.
This new research has not yet had an impact on the philosophy of mathe-
matics or on mathematical practice. Philosophers and mathematicians are free
to choose which directions to pursue and which techniques to emphasize. Only
time will reveal the future evolution of the philosophy of mathematics.
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20
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21
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