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Mathematics (Greek, ‘what should be learned’) is an area of thought basic to the whole of modern civilization. Without the power of numbers, finance, commerce, science, engineering and technology would all collapse, and we would be faced with a return to barter economies.
Despite the vital importance of mathematics, it is difficult to provide a short definition of it. This is to a certain extent because people\'s ideas of the nature of mathematics, its content and its aims, have changed throughout history, partly because the contexts in which it has been used have become more and more diverse (as have the methods used in order to keep pace with it), partly because the methods of mathematics change drastically between school and undergraduate levels of education, and partly because there is no longer, in the 20th century, a consensus among mathematicians and philosophers as to the meaning and nature of mathematics.
Despite the diversity of the subject, there are unifying themes within it as taught and researched today. The most important of these is the idea of proof. Mathematics at its purest consists of the drawing of conclusions from clearly defined assumptions purely through the use of logical thought. Thus mathematics is closely related to logic, though most mathematicians today would deny the claim of logicism that their subject is merely a branch of logic.
The difficulty is with the assumptions. These are limited only by consistency, that they should not allow the proof of contradictory statements; however, they may contradict the assumptions made by other mathematicians (the prime example of this being Euclidean and non-Euclidean geometry). They could, and often are, drawn from scientific observation, but mathematics is not a science, because the exploding of some or all of its hypotheses by new observations does not mean that mathematical research cannot be carried out (on the consequences of the assumptions that are no longer tenable in the real world). They can be creations purely of the mind, interesting ideas that seem to be productive of study, but equally, this does not make mathematics a branch of philosophy either.
It is impossible to know when mathematics began in the human past. Cultures all over the world independently seem to have developed counting systems of varying complexity, from Australian aborigines with words for ‘one’, ‘two’ and ‘many’ to the sophisticated Arabic number system which is used throughout the business and scientific world today. The crucial point that marks the emergence of mathematics from mere counting is demonstrated even in the simple aboriginal system mentioned above that of abstraction.
At the beginning of mathematics stand the numbers. They have formed the basis of a vast number of mathematical systems, and have been subjected to intense scrutiny by philosophers of mathematics (who wanted to know the answers to such questions as ‘Why is 2 + 2 = 4?’ and ‘What is a number anyway?’). They are still not very clearly understood, and many fundamental questions remain unanswered in the field of number theory, the study of numbers. This is also one of the areas of mathematics which impinges most closely on the everyday world; every time we spend money, for instance, or catch a train from the correct platform we practise our expertise in the field of numbers. However, arithmetic gives many people problems—for example, many people are unable to add up a shopping bill or balance a chequebook. It presents a problem educationally which no one seems to have satisfactorily answered; the best way to ensure that school-leavers have the necessary mathematical knowledge to function in today\'s world is still a matter of great debate in educational circles.
The idea that zero was a number sparked off a revolution in mathematics. It led the Indian mathematicians who first realized it to come up with a new system of notation for numbers, on which today\'s Arabic system with place values is based. This made the operations of arithmetic vastly easier; from the province of the most advanced minds of each generation in the Middle Ages, arithmetic became the (almost) universal property it is today. From the point of view of mathematics, the importance of the greater ease of calculation was the way that investigation of properties of numbers became more possible, leading to the study of algebra. There are today two branches of algebra, abstract algebra and linear algebra. Linear algebra concerns itself with the solution of linear equations (see linearity). Abstract algebra is the study of properties abstracted from the properties of numbers and arithmetic (the existence of operations mimicking some of the properties of addition and multiplication, for example). Both have become vast fields of study in their own right, and have affected almost every other branch of mathematics.
The other main division of pure mathematics in ancient times was that of geometry. This was considered the most philosophically satisfying branch of mathematics by the ancient Greeks; and, indeed, until the end of the Middle Ages, the word ‘mathematics’ meant geometry. Geometry is the study of the relationship of spatial objects, built up of (idealized) points and lines, to one another. It is an abstraction of the calculations used in building and the measuring of land.
The greatest achievement of Greek mathematics was the creation of the paradigm of an axiomatic system by Euclid (3rd century BCE): a small collection of self-evident rules and definitions from which a vast amount of knowledge can be derived in steps which are apparent to any reasonably intelligent and sufficiently well-trained reader. Euclid axiomatized the study of geometry and derived a large number of results from his seven axioms in his book the Elements, one of the great classics of mathematics. Various criticisms can today be made of Euclid, and there are a few mistakes in his proofs (most of which were not discovered for well over 2,000 years). His work nevertheless stands as the bench mark against which all subsequent mathematical thought must be measured.
The major problem with Euclid is that one of his axioms, the parallel axiom, was considered questionable. The reason for this is that it involved the possibility of the extension of a given line infinitely. The parallel axiom states that ‘Given a line and a point, there is a unique line through the point which does not meet the first line, no matter how far it is extended’. (An easier way to say this is ‘Parallel lines do not meet.’) How do we know that this holds outside the galaxy, for example? (Euclid himself appears to have had doubts about this axiom; he proved as many results as possible before he first used it.) During the 17th and 18th centuries, many mathematicians attempted to prove the parallel axiom from the other axioms, but their efforts proved fruitless. One method was to assume a contrary axiom instead such as ‘Given a line and a point, there are two lines through the point which do not meet the first line, no matter how far they are extended’ and use it to deduce an absurdity, a contradiction of one of the other axioms, thus showing that the contrary axiom could not be true: this is an important mathematical method known as proof by contradiction. Independent attempts to do this in the 19th century led mathematicians to believe that it was not in fact possible to do so, that the system of geometry with such an axiom was perfectly self-consistent. This discovery led to perhaps the most important mathematical development for many centuries, that of non-Euclidean geometries in which parallel lines might meet. These seemed to be just an academic curiousity, until Einstein\'s theory of relativity, which relies on non-Euclidean geometry, was verified by physical observation. It is, Einstein showed, only locally that the universe appears to follow a Euclidean pattern.
The two major branches of mathematics, pure mathematics and applied mathematics, are complementary parts of a whole, rather than divisions, although mathematicians view themselves as ‘pure’ or ‘applied’ and sometimes almost despise those in the other part of the subject. Neither could exist without the other. Without the impetus and inspiration provided by physical problems, pure mathematics would wither and die. Without the work done in pure mathematics, often in areas that no one expected to have application, applied mathematicians would be left in the dark. In only one important instance historically has an applied mathematician created a new body of mathematics; this was the development of the infinitesimal calculus by Isaac Newton (1642 - 1727) for application to the problem of determining planetary motion. Newton\'s work also led to an influential school of philosophical thought, which still affects the way that many people think today: the idea of determinism, which states that the path of the universe is predetermined and can be known in advance.
During the 19th century, the accuracy of many parts of mathematics was called into question. Some parts of mathematics, notably calculus and probability, did not seem to rest on sure foundations. Mathematicians began to demand more from a proof than that it was apparently convincing to the intuition. They began to search for greater rigour. This meant that every step had to be explicitly stated (or at least be capable of being backed up by a chain of such statements); every supposition had to be laid bare, so that it could be scrutinized by others. The very process of deduction was studied in greater depth, a study which led to the development of symbolic logic by , Gottlob Frege (1848 - 1925), which became the accepted basis and fundamental language of mathematics. Simultaneously with this development came the study of set theory by , Georg Cantor (1845 - 1918), which became the basic building blocks of every mathematical theory. Cantor defined what mathematics would talk about, in however disguised a form; Frege defined how it should be talked about.
This new concern with the fundamental basics of mathematics came about because of the discovery of non-Euclidean geometry. To philosophers before the 19th century, mathematics represented absolute truth, the very material of thought at its purest, and the system of Euclidean geometry was held up as the supreme example of a priori truth (one that follows without the need for assumptions to be made). This view of mathematics was called Platonism. However, the acceptance by mathematicians of a body of work contradictory to Euclidean geometry destroyed this whole comforting notion. There were three main responses to this, new ways of defining the meaning of mathematics.
The first is logicism, which seized on the work of Frege and Cantor to assert that mathematics was basically a branch of the study of logic. The second was formalism, which said that mathematics was all about the manipulation of symbols under formal rules—for example, saying that 2 + 2 = 4 means that it is always possible to replace the symbols 2 + 2 by the symbol 4. The third school was that of intuitionism, which said that the basis of mathematics was ultimately on thought and the intuition (and therefore rejected Cantor\'s work on the concept of infinity, for example). All these schools of thought have today been discredited to varying degrees—formalism and its goals by Gödel\'s incompleteness theorem, formalism and logicism by the feeling that they are trivializing the study of mathematics, and intuitionism under its own weight (as proofs became far harder with intuitionistic limitations on what could be said) and because most of the results obtained by intuitionistic means were only confirmations of results earlier obtained by mainstream mathematicians.
The result is that today there is no real consensus about the fundamental nature of mathematics. The subject, however, still goes ahead, even without a philosophical background, because of its uses and because of its internal elegance and the satisfaction it brings to those who study it.
In the 20th century, the role of mathematics in understanding the universe has become more and more apparent. Until the late 19th century, Newton\'s mathematics ruled supreme in physics. The universe ran deterministically, following the rules that Newton had laid down. Towards the end of the century, it became more and more apparent that something was lacking. This lack was filled by two physical theories which relied on more recent developments in mathematics: Einstein\'s theory of relativity (which relied on non-Euclidean geometry) and Planck\'s Quantum mechanics (which relied on statistics) and was directly opposed to a deterministic view of the universe. In quantum mechanics, particles (or waves) are viewed in a totally non-deterministic manner; their position, velocity, mass or whatever is merely statistical.
The first reason that mathematics is important is that of application (see applicability of mathematics). This is the way in which mathematics is used by scientists. First, scientists extract what they consider to be the fundamental nature of the problem they are studying in a mathematic form (the process of abstraction). they then use mathematics to manipulate these statements to deduce a conclusion. This is then translated back into the predicted results of an experiment (the process of application), and they will then perform the experiment and check the results obtained against those predicted by the mathematics.
Mathematics has proved so successful in its applications that if the results of the experiment do not tally with what the scientists expect, they will not conclude that mathematics cannot be applied to their problem. They will rather conclude that their measurements are incorrect or that their original abstraction failed to take into account some important factor, or contained an unwarranted assumption, or that their reasoning was flawed. The applications that have been made have proved almost endless. Not only those sciences in which exact measurements can be made (such as branches of physics or engineering) have benefited from the use of mathematics, but also those where measurements cannot be exact (such as sociology or quantum mechanics), where the techniques of statistics and stochastics have proved invaluable. These vast successes have led to the application of mathematics to areas which at first seem far from obvious: music, fine art and literature, for example, have been produced and analyzed according to mathematical models; the methods of logic have been applied to linguistics; major philosophical world-views (determinism for example) have been inspired by mathematics. Advances in technology are daily being made which depend on mathematics in crucial ways, the most obvious being that of the computer. But mathematics is not only the underpinning of scientific thought. There are many areas studied in mathematics with no thought of their applicability to the ‘real world’; pure mathematics is the study of mathematical objects for their own sake. Many areas of mathematics seem today to bear no possible relation to the real world, particularly some fields of set theory and logic. It must be remembered, though, that this was also true of non-Euclidean geometry until Einstein, or of symbolic logic until Turing. Techniques from pure mathematics have always illuminated applied mathematics. Pure mathematics is the study of objects of human invention, of concepts with no existence necessary outside the human mind; it comprises the most complex imaginative structure ever built by humankind. It can be viewed—indeed, is viewed by many—as an intellectual game, a superior version of chess in which players are limited only by their imagination, but it is also a game which continues to shape the scientific view of the universe around us. SMcL
Further reading P.J. Davies and , R. Hersh, The Mathematical Experience; , P. Hoffman, Archimedes\' Revenge: the Joys and Perils of Mathematics; , M. Kline, Mathematics in Western Culture; , I. Stewart, The Problems of Mathematics. |
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