**by Enrique Diaz G.**

Our purpose for writting this article is to attempt to answer the question: Is there any relationship between thinking mathematically and thinking in the game of Chess? In other words, must a person possessing an active mind in Mathematics become necessarily a good Chess player have skills in Mathematics?

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 Mathematics.

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).

Even though at the beginning this definition seems difficult to understand, it is the best approximation to comprehend the whole sense of mathematics.

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 .^{2} (Kline, 1962).

After this brief glance at the meaning of Mathematics, let us seethe most commonly methods used in this science. According to Kline (1962), the major method of obtaining knowledge is reasoning, and within the domain of reasoning there are several forms. One can reason by analogy, which consists of finding a similar situation or circumstance and to argue that what was true for the similar case should be true of the one in question. Of course, one must be able to find a similar situation and one must take the chance that the differences do not matter.

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.^{3} (Kline, 1962).

Let us consider now the game of Chess showing some of its characteristics and trying to find out any special method of reasoning that Chess players could use. First of all, we are not going to explain the game as accurately as in a Chess book. Instead, we will describe the game in a rather general form.

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 wearing down your opponent the end is hastened. A general definition is given by Mason: “Chess is a process of thought conditioned 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 mischievous 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.

Now, after these rough explanations about Chess, let us see what attributes a person must possess in order to become a good Chess player. In other words, what is the pattern of intellectual skills that makes one man a good chess player while the other remains a duffer?

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 chess player 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 appropriate 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 chess player; 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 finally 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 chess player. 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.^{5} (Binet, 1946).

Also, we have Binet’s thinking about this matter:

Conversely, mathematicians have after been interested in Chess. However, few famous mathematicians have been first-rate chess players … 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 common 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.^{6} (Binet, 1966).

Now, let us consider a good Chess player, for example, the so-called, chess master. Could he become a good mathematician also? One categorical, answer is expressed by Horowitz and Rothenberg. ,

As strange as it may seen, the chess player’s skill may have no relationship whatever to any other facet of his personality or activity. The common belief that expert chess players are good mathematicians is fiction. On the other hand, good mathematicians may turn out to be good chess players … One conclusion and one only is a safe one: Expert Chess-players are able to play Chess expertly.^{7} (Horowitz & Rothenberg, 1963).

Again Poincare points out that:

…, but, however extraordinary he (a chess player) 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).

Another interesting point of view concerning this point is set up by Abrahams:

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 formulate 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.^{8} (Abrahams, 1951).

To summarize then, we can say that up to now there is not any valuable reason to support the theory that a Chess player must possess abilities related to Mathematics. Lastly, we will indicate some ideas about Chess as a mental process.

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 uselessness” … The great Chess masters, like the great poets, the great composers, the great artists, 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 idiosyncrasies. Through seeing a clever maneuver, 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).

**REFERENCIAS**

Abrahams, Gerald. (195 1) The Chess Mind. London.

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

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