It is necessary to point out that due to the subject
complexity, our efforts will be to explain basic characteristics of both

Mathematics and Chess which have been posed by well-known Mathematicians
and Chess players. Accordingly, we are not interested in exposing facts,
for example, from the Theory of Knowledge, Psychology, Epistemology or
going further into the technical and sophisticated aspects of Chess.

To begin with, let us examine some qualities of Mathernatics.

People having poor experience in Mathematics believe that knowing how to add, subtract, multiply or divide enables them to say that they could master Mathematics. Others possessing some skill in performing quick calculations think they are "Mathematicians". In both cases, they indicate they do not know about the meaning of Mathematics:

- Mathematics as an expression of the human mind reflects the active
will, the contemplative reason, and the desire for aesthetic perfection.
Its basic elements are logic and intuition, analysis and construction,
generality and individuality. Though different traditions may emphasize
different aspects, it is only the interplay of these antithetic forces
and the struggle for their synthesis that constitute the life, usefulness,
and supreme value of mathematical science. (Courant& Robbins, 1941).

- The first major step which the Greeks made was to insist that Mathematics
must deal with abstract concepts... On the basis of elementary abstractions,
mathematics creates others which are even more remote from anything real.
Negative numbers, equations involving unknowns, formulas, and other concepts
we shall encounter are abstractions built upon abstractions. Fortunately,
every abstraction is ultimately derived from, and therefore understandable
in terms of, intuitively meaningful objects or phenomena. The mind does
play its part in the creation of mathematical concepts, but the mind does
not function independently of the outside world. Indeed the mathematician
who treats concepts that have no physically real or intuitive origins is
almost surely talking nonsense .

Another common method of reasoning is induction. People use this method of reasoning every day. Inductive reasoning is in fact the method must commonly used in experimentation. An experimentation is generally performed many times, and if the same result is obtained each time, the experimenter concludes that the result will always follow. The essence of induction is that one observes repeated occurrences of the same phenomenon and concludes that the phenomenon will always occur.

There is still a third method of reasoning, called deduction. Let us consider an example. If we accept as basic facts that honest people return found money and that John is honest, we may conclude unquestionably that John will return money that he finds. In deductive reasoning we start with certain statements, called premises, and assert a conclusion which is a necessary or inescapable consequence of the premises.

All three methods of reasoning, analogy, induction, and deduction, and other methods, are commonly employed. There is one essential difference, however, between deduction on the one hand and all other methods of reasoning on the other. Where as the conclusion drawn by analogy or induction has only a probability of being correct, the conclusion drawn by deduction necessarily holds. Despite the usefulness and advantages of induction and analogy, mathematics does not rely upon these methods to establish its conclusions. All mathematical proofs must be deductive.

Each proof is a chain of deductive arguments, each of which has its premises and conclusion.

Finally, we point out that Mathematics must not be considered only as a system of conclusions drawn from premises or postulates. Mathematicians must also discover what to prove and how to go about establishing proofs. These processes are also part of Mathematics and they are not deductive:

- In the search for a method of proof, as in finding what to prove, the
mathematician must use audacious imagination, insight, and creative ability.
His mind must see possible lines of attack where others would not. In the
domains of algebra, calculus, and advanced analysis especially, the first-rate
mathematician depends upon the kind of inspiration that we usually associate
with the creation of music, literature, or art.

A Chess game is a war between two medieval Kingdoms.
In medieval times, when Kingdoms were small, absolute monarchies, if the
King was imprisoned or captured the war was over. So it is in the game
of Chess. The game is finished when one of the Kings is captured. It may
here be noted that Chess is not necessarily a game of elimination but rather
a game of tactics. However, elimination of the opponent's pieces plays
an important part since by so weakening or wearning down your opponent
the end is hastened. A general definition is given by Mason: "Chess is
a process of thought conditionated and limited by the Institutes and Rules
of the Game. The judgments of thought are certified or visibly expressed
upon the chessboard in movements of various forces".^{4} (Mason,
1946)

The invention of Chess had been credited to the Persians, the Chinese, Arabs, Jews, Greeks, Romans, Babylonians, Scythians, Egyptians, Hindus, Irish and the Welsh. Although the precise origin has been lost in obscurity, it continues to excite the speculation of men of learning at one end of dilettantes at the other. Careful research has called it an "ancient" game; the foolhardy are quite ready to underwrite exact dates. Other characteristics are pointed out by Mason (1946).

- But there is a mischievious imagination abroad that it is a difficult
game. It takes time. Its intricacies and profundities are not rightly within
mastery of the average human intellect. This, in a sense, is true enough,
else Chess would not be Chess. That it cannot be all known and mastered
by anybody is truly its chiefest, crowning merit. It is an instrument all
may play, no two precisely alike, and yet everyone his best. Too much time
may be devoted to it. Chess is a science as well as an art. In its exercise
the tendency is to premature mechanical facility, rather than to a clear
perception of principles; though upon this, of course, all true and lasting
faculty necessarily depends.

In the first place, topnotch Chess requires visual imagery. Before you make a contemplated move, you have to visualize how the board will look after you make it, and then how it will be changed by your opponent's response, and how it will look after you meet another possible answer. You also need patience and restraint.

The quick thinker is often a fool. You need a good memory too. Memory has two components: ability to retain, and ability to recall. The chessplayer needs both. Finally, Chess calls for a certain kind of "reasoning". This reasoning consists of joining together the above elements in order to give an appropiate response to any move. This, then, is the "putty" which holds the "blocks" together. The "blocks" are memory, patience and imagery. The putty is associative reasoning. In daily life you use some of these processes, but you also use other intellectual techniques. For instance, inductive reasoning is not much used in chess, but it pays dividends in business and professional life.

Now, let us consider a mathematician with all his capacity to think abstract concepts; with all his methods of reasoning, that is, reason by analogy, induction, and deduction. Will he become a good Chess player? One of the greatest mathematicians, Henri Poincare, denies this possibility:

- In the same way I should be but a poor chessplayer; I would perceive
that by a certain play I should expose myself to a certain danger; I would
pass in review several other plays, rejecting them for other reasons, and
then finallyl should make the move first examined, having meantime forgotten
the danger I had foreseen. In a word, my memory is not bad, but it would
be insufficient to make me a good chessplayer. Why them does it not fail
me in a difficult piece of mathematical? Evidently because it is guided
by the general march of the reasoning.

- Conversely, mathematicians have after been interested in Chess. However,
few famous mathematicians have been first-rate chessplayers ... I will
readily admit that a similarity exists between chess and mathematics, especially
between chess and mental arithmetic, without, however, ascribing to them
identical mental operations. Chess and Mathematics follow parallel lines.
In other words, the two types of study have a comnion direction; they presuppose
the same taste for complex mental operations which are both abstract and
precise; and they both require a strong dose of patience and concentration.

- As strange as it may seen, the chessplayer's skill may have no relationship
whatever to any other facet of his personality or activity. The common
belief that expert chessplayers are good mathematicians is fiction. On
the other hand, good mathematicians may tum out to be good chessplayers
... One conclusion and one only is a safe one: Expert Chess-players are
able to play Chess expertly.

- ..., but, however extraordinary he (a chessplayer) may be, he will
never prepare more than a finite number of moves; if he applies his faculties
to arithmetic, he will not be able to perceive its general truths by a
single direct intuition; to arrive at the smallest theorem he can not dispense
with the aid of reasoning by recurrence, for this is an instrument which
enables us to pass from the finite to the infinite, (Poincare, 1946).

- The Chess process, being intuitive, Is not mathematical in the normally
accepted sense of that term. The fact that the Chess player is controlled
by rules makes him comparable to the user of a language with a grammar
rather than to those who explicitly use rules and formulaete deductively.
The Chess player is sometimes in a position to be aided by learning and
memory. But essentially each Chess act is a fresh application of mind to
data. Than which nothing is less mathematical or less inferential.

Why has Chess remained the world's most popular game for fifteen centuries? Some authorities attribute the game's fascination to its mimicry of war and all the other struggles of "real life' , others see Chess as a convenient escape from reality. Some have found in Chess an admirable schooling for the mind; others would agree with Ernest Cassirer that "what Chess has in common with science and fine art is its utter uselesness" ... The great Chessmasters, like the great poets, the great composers, the great artistis, the great mathematicians, the great mystics, have the faculty of immersing themselves in some creative process with a concentration, a finality, that is beyond most of us... Chess concepts, like mathematical concepts, depend on formal relations, and therefore exist forever, independent of the capacity of this or that human brain to grasp them.

Now nobody, according to Abrahams (1951), has succeeded in explaining, in casual terms, how the mind apprehends in the first place, or why it falls to apprehend, whether in Chess or in any department of mental activity. The working of the mind is a fact common to intelligent human beings, and Chess has no exclusive claim of vision; for an element of vision or intuition, however slight, is involved in any mental process which is distinguishable form reflex action. But Chess is important because in it the functions of the mind are relatively clear and the mental process is less assisted than inmost other activities by positive rules. Within limits set by the material (the pieces, the board, and the matrix of paths available to pieces on the board) the mind is moving freely. Its scope is the possibility of the material, limited only by the degree of vision available to the player. Its methods, whatever they are, do not resemble the mechanical use of formula, which is the essence of mathematics. The appearance of simplicity that characterizes effective mental action is as deceptive in Chess as it is in any other department of science or art. Imagination traces its own paths and develops idiosyncracies. Through seeing a clever manoeuvre, an improving Chess player may find himself quicker at apprehending an analogous idea; and, more remarkably, quicker at apprehending a different clever possibility in a different setting.

Where Chess differs from many other activities is in that, in Chess, the mind is "influenced" by notions and ideas that it has appreciated, rather than "stocked" with them, or guided by them as one is guided by a signpost.

As to Chess ability, at the present stage of psychology,
the nature of imagination remains obscure. Therefore, it is impossible
to speak about special faculties for Chess, or even to establish any cognate
relationship between skill at Chess and other abilities. Certainly, famous
Chess masters have excelled in other, and various activities - from the
music of Philidor and the Shakespearian researches of Staunton to the medicine
of Tarrash and the engineering of Vidmar. Nor is there evidence of the
transmission of Chess skill, innate or acquired. Why some persons are good
at Chess, and others bad at it, is more mysterious than anything on the
Chess board. "Chess can never reach its height by following in the path
of science ... Let us, therefore, make a new effort and with the help of
our imagination turn the struggle of technique into a battle of ideas"
( Jose Raoul Capablanca).

- Abrahams, Gerald. (195 1)

Binet, Alfred. (1966). __Mnemonic virtuosity__. New York.

Courant, Richard and Herbert Robbins. (194 1). __What is Mathematics__?.
New York.

Horowitz, I.A. and P.L. Rothenberg. (1963). __Personality of Chess__.
New York.

Kline, Morris. (1962). __Mathematics. A Cultural Approach__.

Mason, James. (1946). __The Principles of Chess in Theory and Practice__.
Philadelphia.

Poincare, Henri. (1946). __The Foundations of Science__. Lancaster,