THE LAW OF NON CONTRADICTION AND MATHS

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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

1.1 Aristotelean logic . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1 Subjects and predicates . . . . . . . . . . . . . . . . . . .

1.1.2 Syllogisms . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 The predicate calculus . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1 Predicates and individuals . . . . . . . . . . . . . . . . . .

1.2.2 Formulas and logical operators . . . . . . . . . . . . . . .

1.2.3 Logical validity and logical consequence . . . . . . . . . .

1.2.4 The completeness theorem . . . . . . . . . . . . . . . . . .

1.2.5 Formal theories . . . . . . . . . . . . . . . . . . . . . . . .

2 Foundations of mathematics

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

3.1 Plato and Aristotle . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 The 20th century . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 The future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

References

Page 2

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

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.

The Greek word for instrument is organon. The collection of Aristotle’s logical writings

is known as the Organon.

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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

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

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.

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.

We use man in the traditional sense, equivalent to “human being”. There is no intention

to exclude persons of the female gender.

Page 4

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

[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].

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.

Page 5

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

Page 6

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.

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

∨

∼

⊃

≡

∀

∃

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

. . . ⊃ . . .

“if . . . then . . . ”

≡

bi-implication

. . . ≡ . . .

“. . . 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.

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.

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].

This is the so-called “material implication”: Φ

⊃ Φ

is equivalent to ∼ (Φ

& ∼ Φ

This is called bi-implication because Φ

≡ Φ

is equivalent to (Φ

⊃ Φ

) & (Φ

⊃ Φ

Page 7

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:

Page 8

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

& Φ

) ⊃ Ψ

where Φ

and Φ

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

, . . . , Φ

just in case

(Φ

& · · · & Φ

) ⊃ Ψ

is logically valid. Here Φ

, . . . , Φ

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

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.

See however Bochenski [2, §16E].

Page 9

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

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

& Φ

) has appeared,

we create two new nodes carrying ∼ Φ

and ∼Φ

respectively. The thought

behind these new nodes is that the only way for ∼(Φ

& Φ

) to be the case is if

at least one of ∼ Φ

or ∼ Φ

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

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

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.

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].

An explicit contradiction is a pair of formulas of the form Ψ, ∼ Ψ.

The algorithms in question may be implemented as Turing machine programs. This

undecidability result is known as Church’s theorem. See Mendelson [15].

Page 10

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.

Page 11

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

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.

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].

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.

Our modern notion of a formal theory (see 1.2.5 above) is a variant of Aristotle’s concept

of scientific method.

Page 12

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

plane geometry.

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

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.

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.

Page 13

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

be the formal theory based on Tarski’s axioms.

Remarkably, Tarski has demonstrated that T

is complete. This means that,

for any purely geometrical

statement Ψ, either Ψ or ∼Ψ is a theorem of T

Thus we see that the axioms of T

suffice to answer all yes/no questions of

Euclidean plane geometry. Combining this with the completeness theorem of

Godel, we find that T

is decidable: there is an algorithm

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.

Our primitive predicates for arithmetic are N (“number”), A (“addition”),

M (“multiplication”), I (“identity”). The atomic formulas Nx, Axyz, Mxyz,

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.

Such algorithms have been implemented as computer programs. They are useful in

robotics and other artificial intelligence applications.

Two recent studies of formal arithmetic are Hajek/Pudlak [6] and Simpson [20].

Page 14

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

be the formal theory specified by the above primitives and axioms.

It is known that T

suffices to derive many familiar arithmetical facts. For

example, 2+2 = 4 may be expressed, awkwardly

to be sure, as (1+1)+(1+1) =

This kind of awkwardness can be alleviated by means of various devices. In particular,

Page 15

((1 + 1) + 1) + 1 or

∃x ∃y ∃z ∃w (Mxxx & Axxy & Axyz & Axzw & Ayyw)

and this formula is indeed a theorem of T

, i.e., a logical consequence of the

axioms of T

. Another theorem of T

∀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

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

is necessarily

either inconsistent

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

but also T

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.

This theory includes

both T

and T

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.

In this formula, the variables x, y, z, w play the role of 1, 2, 3, 4 respectively.

A formal theory is said to be inconsistent if its axioms logically imply an explicit contra-

diction. Such theories are of no scientific value.

A recent study of second-order arithmetic is Simpson [20].

Page 16

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

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

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

includes a theory equivalent to T

. After that, one

can follow the ideas of Descartes and Weierstrass to see that T

also includes a

theory equivalent to T

. It turns out that the rest of modern mathematics can

also be emulated within T

. 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.

The implications of this idea

are not yet fully understood and are a topic of current research.

The idea of set-theoretical foundations gave rise to the “new math” pedagogy of the

1960’s. For a lively discussion, see Kline [12].

Page 17

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.

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.

See for example Curry [3].

Page 18

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

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.

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

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?

See the essays of Brouwer and Kolmogorov in [22].

Page 19

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

, would be rigorously justified by reference to the finite. This is

Hilbert’s program of finitistic reductionism.

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

is not finitistically reducible,

there are other formal theories which are finitistically reducible, in the precise

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

, which is physically meaningful and makes absolutely no reference to actual infinity.

Page 20

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.

References

[1] Hippocrates G. Apostle. Aristotle’s Philosophy of Mathematics. University

of Chicago Press, 1952. X + 228 pages.

[2] I. M. Bochenski. A History of Formal Logic. Chelsea Publishing Company,

New York, 2nd edition, 1970. Translated and edited by Ivo Thomas, XXII

+ 567 pages.

[3] Haskell B. Curry. Outlines of a Formalist Philosophy of Mathematics. Stud-

ies in Logic and the Foundations of Mathematics. North-Holland, 1951. VII

+ 75 pages.

[4] Melvin Fitting. First-Order Logic and Automated Theorem Proving. Grad-

uate Texts in Computer Science. Springer-Verlag New York Inc., 2nd edi-

tion, 1996. XVI + 326 pages.

[5] Kurt Godel. Collected Works. Oxford University Press, 1986–1995. Volume

I, XVIII + 474 pages, 1986, Volume II, XVI + 407 pages, 1990, Volume

III, XX + 532 pages, 1995.

[6] Petr Hajek and Pavel Pudlak. Metamathematics of First-Order Arithmetic.

Perspectives in Mathematical Logic. Springer-Verlag, 1993. XIV + 460

pages.

[7] Thomas Heath. Mathematics in Aristotle. Clarendon Press, Oxford, 1949.

XIV + 291 pages.

[8] Thomas Heath. The Thirteen Books of Euclid’s Elements. Dover Publica-

tions, Inc., New York, 2nd revised edition, 1956. Three volumes.

[9] Thomas Heath. A History of Greek Mathematics. Dover Publications, Inc.,

New York, 1981. Volume I, XV + 446 pages, Volume II, XI + 586 pages.

[10] Leon Henkin, Patrick Suppes, and Alfred Tarski, editors. The Axiomatic

Method, With Special Reference to Geometry and Physics, Studies in Logic

and the Foundations of Mathematics. North-Holland, 1959. XI + 488 pages.

[11] Thomas J. Jech. Set Theory. Pure and Applied Mathematics. Academic

Press, New York, 1978. XI + 621 pages.

Page 21

[12] Morris Kline. Why Johnny Can’t Add. St. Martin’s Press, 1973. 173 pages.

[13] Richard McKeon, editor. The Basic Works of Aristotle. Random House,

1941. XXXIX + 1487 pages.

[14] Elliott Mendelson. Boolean Algebra and Switching Circuits. Schaum’s Out-

line Series. 1970. 213 pages.

[15] Elliott Mendelson. Introduction to Mathematical Logic. Wadsworth, 3rd

edition, 1987. IX + 341 pages.

[16] Leonard Peikoff. Objectivism: The Philosophy of Ayn Rand. Dutton, New

York, 1991. XV + 493 pages.

[17] Plato. Dialogues. Oxford, Clarendon Press, 4th edition, 1964. Translated

by Benjamin Jowett.

[18] Ayn Rand. Atlas Shrugged. Random House, New York, 1957. V + 1168

pages.

[19] David Ross. Aristotle. Barnes and Noble, 5th revised edition, 1964. XIV

+ 300 pages.

[20] Stephen G. Simpson. Subsystems of Second Order Arithmetic. Perspectives

in Mathematical Logic. Springer-Verlag, 1999. XIV + 445 pages.

[21] Alfred Tarski. Introduction to Logic and to the Methodology of Deductive

Sciences. Oxford University Press, New York, 4th edition, 1994. XXII +

229 pages.

[22] Jean van Heijenoort, editor. From Frege to Godel: A Source Book in

Mathematical Logic, 1879–1931. Harvard University Press, 1967. XII +

660 pages.

THE LAW OF NON CONTRADICTION AND MATHS

Wednesday, September 26, 2007

STRING THEORY

PARALLEL UNIVERSE AND NON CONTRADICTION

Tuesday, September 25, 2007

MATHS AND THE LAW OF NON CONTRADICTION

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