Development of Modern Mathematics

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To select points on the time scale of human events and to designate the intervals thereby demarcated as "era" is always an arbitrary act. Inevitably, it is influenced by the point of view, the experience, and the purposes of the selector. In particular, any designation of a specific date as marking the commencement of the "modern era" in math would be affected by the math interests of the designator. Even though one were a "universalist" in maths, possessing wide knowledge of all its branches, his judgment in this  regard might not find acceptance.

            Probably one of the last mathematicians who could be termed a universalist was the late John von Neumann who expressed the view that "the calculus was the first  achievement of modern maths" - a judgment not many would be likely to accept. Certainly, the two centuries following the formulations of the calculus by Newton and Leibnitz constituted a period of high productivity in math research. But it was a period characterized by the rough methods of the pioneer and seemingly imbued with the philosophy (attribute to d"Alembert), "Forge ahead, and faith will come to you".

            Not until the nineteenth century do we find the inception of those features that appear to distinguish modern maths from its progenitors of the preceding two centuries. In the works of Cauchy and other early-nineteenth-century mathematicians the attention paid to greater rigour   points to a change in the philosophy underlying math research. And not until the latter part of the nineteenth century do we find clearly exhibited those characteristics that appear to distinguish modern math from earlier forms of the science. It would seem reasonable, therefore, to say the by "modern maths" is meant the maths of, roughly, the 19th century. More important than dates, however, are the general characteristics of modern maths.

General Characteristics of Math in the Modern Era

            A New Point of View. A prominent feature of modern math - one that serves to distinguish it from the maths of earlier eras - is a point of view,  a new conception of the nature of maths. From this point of view, maths is seen not as a description of an external world of reality, not merely as tool for studying such a world, but rather as a science in its own right. No longer simply a servant of the natural sciences, maths has achieved a status seemingly independent of the natural sciences while still lending aid to the latter. Having moved from its earlier dependence n natural phenomena for new ideas from within itself.

            The achievement of such a status was not, of course, the result of determination on the part of particular individuals to declare the independence of maths from natural phenomena. Rather it was the result of a natural evolution one can detect as long ago as Babylonian times, when meager attempts as what we would call "number theory" were made by the temple scribes. When sufficiently developed, maths breeds a fascination its devotees find irresistible. No, doubt, during the Sumerian-Babylonian era the needs of a fast-developing society, requiring quicker aids to computation, Iay at the root of the development of multiplication tables, tables of reciprocals, and the like. But inevitably certain curious aspects of numbers, when so manipulated and tabulated, evinced themselves and inculcated in the observer a desire to know "why".

            The "why" that probably had most to do with development of the new viewpoint in modern math was a question related to the parallel axiom of Euclid's geometry. As is well known, the question whether the addition of this axiom to Euclid's other postulates was necessary stimulated a great deal of research from the third century B.C. t the time of its solution in the non-Euclidean geometries of the nineteenth century.

            Parallel with these developments were the generalizations of number being invented by the algebraists. Prominent among the proponents of such generalizations were Hamilton and Grassmann, who realized that the axiomatic method could be applied just as effectively to algebra as to geometry. Much as Lobahevsky and Bolyai, in order to found their non-Euclidean geometry, substituted for the parallel axiom a statement contradicting it, so Hamilton finally decided to defy the commutative "low" of algebra for multiplication. Certainly, such work contributed to the new point of view especially when it became apparent that the new types of non-commutative systems had applications in the physical sciences.

            By the end of the nineteenth century the arbitrary character of any system of  algebra or geometry, whether it was a classical form or one of the newly invented types, was becoming evident to the math world. And when classical geometry and algebra lost their previously assumed property of "necessity", this loss could not help but create the new point of view. In maths the practical result was the flowering of the modern form of the axiomatic method, which has subsequently proved so important for research and is certainly to be regarded as one of the major achievements of the nineteenth century.

            A Higher Level of Abstraction. A natural concomitant of the new form of the axiomatic methods was the greater abstraction that characterizes modern math. An important feature of axiomatic method in its modern form is its treatment of basic terms as undefined; thus, in geometry, instead of trying to give a spurious definition of "point" as in Euclid, one usually attributes to the term no meaning except what may be implied by the axioms. This attributes to the term no meaning except what may be implied by the axioms. This attitude was, perhaps, a result of  (1) the emergence of models in both algebra and geometry in which the basic terms no longer had their usual intuitive meaning - as, for , instance, geometries in which the "point" were objects totally unlike the classical conception - and (2) the development of the rigorous math (as opposed to the dictionary) type of definition, which forced the realization that continual definition could lead only to endless regression or to a vicious circle;  one cannot give explicit definition of every basic concept of an axiomatic system. An important by product of this treatment is that one can interpret the undefined terms in any concrete way he pleased so long as the axioms turn out to be true in the interpretation;  one then all theorems of the theory at his disposal in the new interpretation.

            One may protest that abstractness is not peculiar to modern maths. Certainly, the geometry of Euclid was a grand abstraction from physical space. But the type of abstraction to be found in modern maths is of an even higher order. Modern maths is more concerned with a conceptual world that is at least one stage removed from concepts that are derived directly from sense perceptions; the objects, relation, and operations with which it deals are already themselves abstractions. It is this kind of abstractness that characterizes modern maths.  Greater Rigour; "Self-suffiiency" . Another aspect of modern maths, helping to distinguish it from the maths of earlier eras, is the greater demand for rig our in the demonstrations of proofs for theorems and the validation of proposed  theories. Unquestionably, the development of the modern axiomatic method contributed to this since the method made more precise the assumptions underlying a proof and allowed less play to vague intuition (except as an aid to discovery). But the contributions of the nineteenth century to such topics as the theory of limits, the real number system, foundations of geometry, the extensions of the notion of "number" in algebra, and symbolic logic made possible a precision that was formerly unattainable. It seemed no longer necessary to appeal to "common sense" or to notions generally accepted but not well defined. And this helped tremendously to foster in the math community of the late nineteenth century a feeling of self-sufficiency, - for maths came to be independent of the physical world for its theoretical justification. It had laid down its own foundations.