Brillouin Science And Information Theory Pdf File' title='Brillouin Science And Information Theory Pdf File' />Characteristics of Modern Mathematics Mathematical Science Technologies. What are the characteristics of mathematics, especially contemporary mathematics Ill consider five groups of characteristics Applicability and Effectiveness, Abstraction and Generality, Simplicity, Logical Derivation, Axiomatic Arrangement, Precision, Correctness, Evolution through DialecticCharacteristics of Modern Mathematics. In the article What is Mathematics, I have posited that Mathematics arises from Mans attempt to summarize the variety of empirical phenomena that he experiences, and that Mathematics advances through the expansion and generalization of these concepts, and the improvement of these models. But what are the characteristics of mathematics, especially contemporary mathematics Ill consider five groups of characteristics Applicability and Effectiveness,Abstraction and Generality,Simplicity,Logical Derivation, Axiomatic Arrangement,Precision, Correctness, Evolution through Dialectic. Though each of these characteristics presents unique pedagogical challenges and opportunities, here Ill focus on the characterisics themselves and leave the pedagogical discussion to Ebr. Brillouin Science And Information Theory Pdf File' title='Brillouin Science And Information Theory Pdf File' />Pedagogical matters are discussed in the article Teaching Mathematics in Tunic. Wide Applicability and the Effectiveness of Mathematics. General applicability is a recurring characteristic of mathematics mathematical truth turns out to be applicable in very distinct areas of application in phenomena from across the universe to across the street. Why is this What is it about mathematics and the concepts that it captures that causes this Mathematics is widely useful because the five phenomena that it studies are ubiquitous in nature and in the natural instincts of man to seek explanation, to generalize, and to attempt to improve the organization of his knowledge. As Mathematics has progressively advanced and abstracted its natural concepts, it has increased the host of subjects to which these concepts can be fruitfully applied. Fully relativistic pseudopotential abinitio calculations have been performed to investigate the high pressure phase transition, elastic and electronic properties of. Abstraction and Generality. Abstraction is the generalization of myriad particularities. It is the identification of the essence of the subject, together with a systematic organization around this essence. By appropriate generalizations, the many and varied details are organized into a more manageable framework. Work within particular areas of detail then becomes the area of specialists. Put another way, the drive to abstraction is the desire to unify diverse instances under a single conceptual framework. The online version of Handbook of Optical Constants of Solids by Edward D. Palik on ScienceDirect. Beginning with the abstraction of the number concept from the specific things being counted, mathematical advancement has repeatedly been achieved through insightful abstraction. These abstractions have simplified its topics, made the otherwise often overwhelming number of details more easily accessible, established foundations for orderly organization, allowed easier penetration of the subject and the development of more powerful methods. Simplicity Search for a Single Exposition, Complexity Dense Exposition. For the outsider looking in, it is hard to believe that simplicity is a characteristic of mathematics. Yet, for the practitioner of mathematics, simplicity is a strong part of the culture. Simplicity in what respect The mathematician desires the simplest possible single exposition. Through greater abstraction, a single exposition is possible at the price of additional terminology and machinery to allow all of the various particularities to be subsumed into the exposition at the higher level. This is significant although the mathematician may indeed have found his desired single exposition for which reason he claims also that simplicity has been achieved, the reader often bears the burden of correctly and conscientiously exploring the quite significant terrain that lies beneath the abstract language of the higher level exposition. Thus, I believe it is the mathematicians desire for a single exposition that leads to the attendant complexity of mathematics, especially in contemporary mathematics. Logical Derivation, Axiomatic Arrangement. The modern characteristics of logical derivability and axiomatic arrangement are inherited from the ancient Greek tradition of Thales and Pythagoras and are epitomized in the presentation of Geometry by Euclid The Elements. It has not always been this way. The earliest mathematics was firmly empirical, rooted in mans perception of number quantity, space configuration, time, and change transformation. But by a gradual process of experience, abstraction, and generalization, concepts developed that finally separated mathematics from an empirical science to an abstract science, culminating in the axiomatic science that it is today. It is this evolution from empirical science to axiomatic science that has established derivability as the basis for mathematics. This does not mean that there is no connection with empirical reality. Quite the contrary. But it does mean that mathematics is, today, built upon abstract concepts whose relationship with real experiences is useful but not essential. These abstractions mean that mathematical fact is now established without reference to empirical reality. It may certainly be influenced by this reality, as it often is, but it is not considered mathematical fact until it is established according to the logical requirements of modern mathematics. Why the contemporary bias for axiomatic Mathematics  Why is axiomatic mathematics so heavily favored by modern mathematics For the same reason it was favored in the time of Euclid in the presence of empirical difficulties, linguistic paradox, or conceptual subtlety, it is an anchor that clarifies more precisely the foundations and the manner of reasoning that underlie a mathematical subject area. Once the difficulties of establishing an axiomatic framework have been met, such a framework is favored because it helps ease the burden of many, complicated, inter related results, justified in various ways, and inter mixed with paradoxes, pitfalls, and impossible problems. 5 Meter Air Rifle Target Pdf Job. It is favored when new results cannot be relied upon without complicated inquiries into the chains of reasoning that justify each one. The value of axiomatic mathematics  What the axiomatic approach offers is a way to bring order to a subject area, but one which requires deciding what is fundamental and what is not, what will be set up higher as a first principle and what will be derived from it. When it is done, however, it sets a body of knowledge into a form that can readily be presented and expanded. Appealing and effective axiom systems are then developed and refined. Their existence is a mark of the maturity of a mathematical subfield. Proof within the axiomatic framework becomes the hygiene that the community of working mathematicians adopts in order to make it easier to jointly share in the work of advancing the field. Axiomatic Mathematics as Boundaries in the Wilderness  In all cases of real mathematical significance, the selection of axioms is a culminating result of intensive investigations into an entire mathematical area teeming with phenomena, and the gaining of a deep understanding that results finally in identifying a good way to separate the various phenomena that have been discovered. So, though the axioms may sound trivial, in reality, the key axioms delineate substantially different structures. In this sense, axioms are boundaries that separate structurally distinct areas from each other, and, together, from the rest of wild mathematics. For example the triangle inequality is a theorem of Euclidean geometry. But it is taken as an axiom for the study of metric spaces. By doing so, this one axiom forces much of the Euclidean isometric structure. As such, it becomes a code or litmus test for the Euclidean ness of a space. Thus, from this point of view, non axiomatic mathematics is the mathematics of discovery.