Content uploaded by Liaqat Ali Khan

Author content

All content in this area was uploaded by Liaqat Ali Khan on May 18, 2015

Content may be subject to copyright.

International Journal of Mathematics and Computational Science

Vol. 1, No. 3, 2015, pp. 98-101

http://www.aiscience.org/journal/ijmcs

* Corresponding author

E-mail address: akliaqat@gmail.com

What is Mathematics - an Overview

Liaqat Ali Khan

*

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Abstract

Mathematics is based on deductive reasoning though man's first experience with mathematics was of an inductive nature. This

means that the foundation of mathematics is the study of some logical and philosophical notions. We elaborate in simple terms

that the deductive system involves four things: (1) A set of primitive undefined terms; (2) Definitions evolved from the

undefined terms; (3) Axioms or postulates; (4) Theorems and their proofs. We also include some historical remarks on the

nature of mathematics.

Keywords

Mathematics Education, Deductive Reasoning, Inductive Reasoning, Primitive Undefined Terms, Axioms, Theorem,

Direct Proof, Indirect Proof, Platonism, Formalism

Received: April 4, 2015 / Accepted: April 15, 2015 / Published online: May 5, 2015

@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license.

http://creativecommons.org/licenses/by-nc/4.0/

1. Introduction

Mathematics is not only concerned with everyday problems,

but also with using imagination, intuition and reasoning to

find new ideas and to solve puzzling problems. One method

used by mathematicians in discovering new ideas is to

perform experiments. This is called the "experimental

method" or "inductive reasoning". When a scientist takes a

large number of careful observations and from them infers

some probable results or when he repeats an experiment

many times and from these data arrives at some probable

conclusion, he is using inductive reasoning. That is to say,

from a large number of specific cases he obtains a single

general inference.

The other method is based on reasoning rather than on

experiments or observations. This is called "deductive

reasoning". When a mathematician begins with a set of

acceptable conditions, called the hypothesis and by a series

of logical implications reaches a valid conclusion, he

employs deductive reasoning. The major difference in the

two methods is implied in the two words: “probable” with

respect to inductive reasoning and `valid' relative to

deductive reasoning. For example, if we perform an

experiment successfully say a thousand times, then another

twenty successful trails would lend credence to the result, but

we have no assurance whatever that the experiment will not

fall on the very next trail. On the other hand in a deductive

system, once we accept the hypothesis, the validity of our

conclusion is inevitable provided each implication in the

reasoning process is a logical consequence of what which

proceeds it. Here "consistency" of a logical system means

that no theorem of the system contradicts another and

"validity" means that the system's rules of proof never allow

a false inference from true premises.

2. Deductive Reasoning

System

As mentioned above, mathematics is based on deductive

reasoning though man's first experience with mathematics

was of an inductive nature. The ancient Egyptians and

Babylonians developed many mathematical ideas through

observation and experimentation and made use of this

mathematics in their daily life. Then the Greeks became

International Journal of Mathematics and Computational Science Vol. 1, No. 3, 2015, pp. 98-101 99

interested in philosophy and logic and placed a great

emphasis on reasoning. For example, in Geometry, the

axiomatic development was first developed by them from

500 to 300 BC, and was described in detail by Euclid around

300 BC. They accepted a few most basic mathematical

assumptions and used them to prove deductively most of the

geometric facts we know today. Our high school geometry is

an excellent example of a deductive system. Recall that in the

study of geometry, we began with a set of undefined terms,

such as point, line etc. We then made some definitions, for

example, those of angle, parallel lines, perpendicular lines,

triangle etc. Next, we listed a number of statements

concerning these undefined and defined terms which we

accepted to be true without proof; these assumptions we

called, axioms or postulates. Finally, we were able to prove a

considerable number of propositions or theorems by

deductive reasoning.

In summary, we observe that the study of foundations of

mathematics involves an abstract deductive system consisting

of:

1. A set of primitive undefined terms;

2. Definitions evolved from the undefined terms;

3. Axioms or postulates;

4. Theorems and their proofs.

We now discuss each of them as follow.

UNDEFINED TERMS: To build a mathematical system

based on logic, the mathematician begins by using some

words to express their ideas, such as `number' or a `point'.

These words are undefined and are sometimes called

`primitive terms'. These words usually have some meaning

because of experience we have had with them. It may seem

strange that in mathematics, a field with which precision and

accuracy are commonly associated, we do not (and cannot)

`start from scratch' but find it necessary to begin with a set of

undefined terms. Why do we not start with precise

definitions? An attempt to define any of the fundamental

undefined terms, such as point, set, number or element

demonstrate that we are soon led to what is referred to as

‘circular reasoning’. For example, let us try to define `point'.

What is a `point'? A point is a position at which something

exists. But what is meant by position? The location of an

object, naturally. But what does location of an object imply: a

point. So we are back where we started. In a like manner, an

attempt to define any of the other undefined terms of

mathematics would also result in circular reasoning. Hence, it

should now be clear that the use of primitive terms is

indispensable because they serve as the foundation upon

which the system rests.

For obvious reasons, the primitive terms, a mathematician

chooses must be simple in form and as small in number as

possible. They usually appeal to the intuition, more or less,

but it is important to distinguish the intuitive ideas behind

them and the part they play in the theory. It is not completely

true to say that a primitive term has no formal meaning. It

may have content because of the logical position we put in it.

DEFINITIONS: A definition, Bertrand Russell says, is a

declaration that a certain newly introduced term or

combination of terms is to mean the same as a certain other

combination of terms, of which the meaning is already

known. It assigns a meaning to a term by means of primitive

terms and terms already defined.

It is to be observed that although we employ definitions, yet

"definitions" does not appear among our primitive ideas

because, strictly speaking, the definitions are no part of our

subject. Practically, of course, if we introduce no definitions,

our formulae would very soon become so lengthy, as to be

unimaginable.

In spite of the fact that definitions are theoretically

superfluous, it is nevertheless true, that they often convey

more important information than is contained in the

proposition in which they are used. Definitions clarify and

simplify expressions. We need to define our terms so that we

can use short names for complex ideas. Also definitions

contain an analysis of a common idea and can, therefore,

classify, that we wish to single out quadrilaterals with

opposite parallel sides. We may do this by means of a

definition: "a parallelogram is a quadrilateral whose opposite

sides are parallel". If we assume in this definition that

`quadrilateral', `opposite sides' and `parallel' have been

defined previously, then what we have done is to define the

class of parallelograms.

AXIOMS AND POSTULATES: At the start of every

mathematical theory (such as Real Numbers System, Group

Theory, Topology, Quantum Mechanics), some kinds of

foundations are needed. For this purpose, a set of

independent fundamental statements is asserted. These

assertions are called axioms and postulates. Both the axioms

and the postulates have their roots in antiquity. To quote

Aristotle, "Every demonstrative science must start from

indemonstrable principles. Otherwise, the steps of

demonstration would be endless". Both the axioms and the

postulates presumably are principles, so clearly true that we

accept them without a corresponding proof. In Euclid's time

(300 BC), axioms referred specifically to an assumption in

geometry. Today the distinction is disregarded and both terms

are used interchangeably.

The axioms of a mathematical theory are usually stipulated at

the beginning of the theory, immediately after announcing

the primitive terms. These terms are the bricks with which we

100 Liaqat Ali Khan: What is Mathematics - an Overview

build up these axioms. The axioms may contain such

statements as: "Things equal to the same thing are equal to

one another". "Every line is a set of points". They are

necessary if we are to avoid an infinite regression which

would certainly result if we only accepted what we could

prove. Once the axioms have been chosen, we become more

severe about the subsequent propositions.

THEOREMS AND THEIR PROOFS: A `theorem' is a

statement whose truth is established by formal proof. The

bulk of any branch of mathematics consists of the collection

of theorems that pertain to that particular area. Much of the

beauty of mathematics lies in the sequential development of

the subject through the proofs of its theorems. A `proof' is a

chain of reasoning that succeeded in establishing a

conclusion by showing that it follows logically from

premises that already are known to be true. In proving a

theorem, we may use our undefined and defined terms, and

our axioms and of course any theorem we prove, the more

knowledge we have at our disposal to prove additional

theorems. In any mathematical theory, to prove the first

theorem, A (say), the only arguments that can be used are the

axioms. And to prove a second theorem, B (say), we may use

the axioms and Theorem A and similarly for the subsequent

theorems. Hence we state the principle: "a proof

demonstrates the validity of a proposition using an argument

based entirely on the axioms and the previously established

theorems".

Kinds of Proofs: There are two kinds of proof: direct proof

and indirect proof. Most theorems have the form "a statement

p implies another statement q". To demonstrate such a

statement we proceed with an assumption usually called the

hypothesis in the following ways:

Assert p (i.e. suppose p is given). From this we construct a

demonstration that ends with the statement q.

This program makes up what we called a "direct proof".

The `indirect proof', also called "proof by contradiction"

(reductio ad absurdum, in Latin), depends essentially on the

notion of negation. This idea can be stated in the following

form:

"To prove the Theorem A indirectly, we affirm its negation.

From this we construct an argument that concludes with the

negation of a result already known to be true. This is a

legitimate proof of Theorem A".

Truth of Assertions: We have spoken of the truth of certain

assertions. What does the word `truth' mean in this context?

A proposition is true if it can be proved by means of the

axioms and theorems proved previously. Notice that a

theorem may be true in one theory but false in another; it all

depends on the initial axioms. In "plane geometry", the

statement that the sum of the angles of a triangle is two right

angles is true, but it is no longer true in "Riemannian

geometry". In "classical mechanics" mass is indestructible,

but in "quantum theory", a mass can be destroyed and

replaced by energy.

3. Some Remarks and Quotes

A. The above described Deductive System is also called

Formalism. In fact, there are two dominant schools of

thought about the nature of Mathematics: one is the

Platonist or Realist (deriving from Plato) and the other is

Formalist. The Platonists believe that mathematical objects

exist independent of us and inhabit a world of their own.

They are not invented by us but rather discovered.

Formalists on the other hand believe that there are no such

things as mathematical objects. Mathematics consists of

definitions, axioms and theorems invented by

mathematicians and have no meaning in themselves

except that which we ascribe to them. This school of

thought was introduced by David Hilbert in 1921.

During the 1920's shock waves had run through the

science of physics, because of Heisenberg's Uncertainty

Principle (introduced first by the German physicist Werner

Heisenberg in 1927). This principle states that you can

never simultaneously know the exact position and the

exact speed of an object. In 1931, a 25 year old Austrian

mathematician Kurt Gödel shocked the worlds of

mathematics and philosophy by showing that there are

mathematical truths which simply cannot be proved.

Kurt Gödel (1906-1978) was regarded as a brilliant

mathematician and perhaps the greatest logician since

Aristotle. His famous “incompleteness theorem” was a

fundamental result about axiomatic systems, showing that

in any axiomatic mathematical system, there are

propositions that cannot be proved or disproved within the

axioms of the system. In particular the consistency of the

axioms cannot be proved. This ended a hundred years of

attempts to establish axioms which would put the whole of

mathematics on an axiomatic basis. These included some

major attempts by several logicians and mathematicians of

that time (such as Germans' Richard Dedekind (1831-

1916), Georg Cantor (1845-1918), Friedrich Frege (1848-

1925), David Hilbert (1862-1943), Ernst Zermelo (1871-

1953), Italian's Giuseppe Peano (1858-1932), ), Dutch

L.E.J. Brouwer (1881-1966), British Bertrand Russell

(1872-1970)).

B. Gödel's results did not destroy the fundamental idea of

formalism, but it did demonstrate that any system would

have to be more comprehensive than that envisaged by

International Journal of Mathematics and Computational Science Vol. 1, No. 3, 2015, pp. 98-101 101

Hilbert and others. In fact, these results were a landmark

in 20th-century mathematics, showing that mathematics is

not a finished object, as had been believed. It also implies

that a computer can never be programmed to answer all

mathematical questions. Among physicists, Gödel is

known as the man who proved that time travel to the past

was possible under Einstein's equations.

C. Mathematics may be defined as the subject in which we

never know what we are talking about, nor whether what

we are saying is true – (Bertrand Russell, Mysticism and

Logic (1917) Ch. 4).

D. "Obvious" is the most dangerous word in mathematics

(E.T. Bell, 1883-1960).

E. To arrive at the simplest truth, as Newton knew and

practiced, requires years of contemplation. Not activity.

Not reasoning. Not calculating. Not busy behaviour of any

kind. Not reading. Not talking. Not making an effort. Not

thinking. Simply bearing in mind what it is one needs to

know. (George Spencer Brown, 1923-)

F. Pure mathematics is on the whole distinctly more useful

than applied. For what is useful above all is technique, and

mathematical technique is taught mainly through pure

mathematics. (G.H. Hardy, 1877-1947)

G. As one ancient stated, teaching is not a matter of pouring

knowledge from one mind into another as one pours water

from one glass into another. It is more like one candle

igniting another. Each candle burns with its own fuel.

H. For more than two thousand years some familiarity with

mathematics has been regarded as an indispensable part of

the intellectual equipment of every cultured person. Today

the traditional place of mathematics in education is in

grave danger. (Richard Courant and Herbert Robbins)

Acknowledgement

Dedicated to my teacher Professor B. A. Saleemi.

References

[1] Joseph Auslander, Review of "What is Mathematics, Really?"

by Reuben Hersh, SIAM Review, 42(1) (2000), 138-143.

[2] Briane E. Blank, Review of "The Pleasures of Counting” by T.

W. Körner, Notices Amer. Math. Soc. 45 (1998), 396-400.

[3] Douglas M. Campbell, John C. Higgins, Mathematics: People,

Problems, Results (Volume 2, Taylor & Francis, 1984).

[4] Alain Connes, A view of Mathematics (38 pages),

http://www.alainconnes.org/docs/maths.pdf

[5] J. J. O'Connor and E F Robertson, The MacTutor History of

Mathematics archive (2000), http://www-history.mcs.st-

and.ac.uk/

[6] Richard Courant and Herbert Robbins, What Is

Mathematics? : An Elementary Approach to Ideas and

Methods (1941); second edition, revised by Ian Stewart

(Oxford University Press, 1996).

[7] Underwood Dudley, What Is Mathematics For? , Notices of

AMS, Vol. 57, No. 5 (2010), 608-613.

[8] Paul Ernest, The Nature of Mathematics;

https://www.academia.edu/3188583/

THE_NATURE_OF_MATHEMATICS_AND_TEACHING.

[9] Leonard Gillman, Review of “What Is Mathematics?”, by

Richard Courant and Herbert Robbins, revised by Ian

Stewart , Amer. Math. Monthly, 105 (1998), 485-488.

[10] Arend Heyting, Intuitionistic Views on the Nature of

Mathematics, Synthese, 27(1/2), On the Foundations of

Mathematics, (May - Jun., 1974), pp. 79-91.

[11] Reuben Hersh, What is Mathematics, Really? (Oxford

University Press, 1997).

[12] T. W. Körner, The Pleasures of Counting (Cambridge

University Press, 1996).

[13] Kenneth Kunen, The Foundations of Mathematics (Studies in

Logic: Mathematical Logic and Foundations) (College

Publications, 2009).

[14] Robert H. Lewis, What is Mathematics? The Most

Misunderstood Subject,

http://www.fordham.edu/info/20603/what_math

[15] Ernst Snapper, "The three crises in mathematics: Logicism,

intuitionism and formalism", Math. Magzine, 52(4) (1979),

207-216.

[16] Ernst Snapper, What is Mathematics?, Amer. Math. Monthly,

87(7) (1979), 551-557.

[17] Ian Stewart, Concepts of Modern Mathematics (Dover

Publications, 1975).

[18] David Tall, The Transition to Formal Thinking in

Mathematics, Mathematics Education Research Journal, 20(2)

(2008), 5-24.

[19] Alan Weir, "Formalism in the Philosophy of Mathematics",

The Stanford Encyclopedia of Philosophy (Spring 2015

Edition), Edward N. Zalta (ed.), URL =

<http://plato.stanford.edu/archives/spr2015/entries/formalism-

mathematics/>.

[20] Richard B. Wells, Mathematics and Mathematical Axioms

(Ch. 23, 2006, 42 pages),

http://www.mrc.uidaho.edu/~rwells/Critical%20Philosophy%

20and%20Mind/Chapter%2023.pdf