What is Mathematics?

Introducing the requirement of mathematical philosophy in every day needs of the physical reality, the book delves into the concepts of the subject in a more logical approach. The chapters might be better appreciated if the initial note of their organization and preliminary requirement are followed as specified. The Chapter I The Natural Numbers deals with the laws of arithmetic and the principle of mathematical induction with quite a few applications, some being left for the reader to solve making the chapters more challenging yet no less gripping. Continuing in its Supplement to Chapter I The Theory of Numbers, the primes are taken up in a more abstract sense and humbling the original chapter both in volume and concept, some unsolved problems are referred as are discussed modulo arithmetic, the concept of congruence, theorem of relative primes as an extension to the theorem of prime by Fermat, the development of Pythagorean triple and the elegance of their primitive forms. Utilisation of Euclidean Algorithm for searching of common factors between integers, the concept was used more than once not only for finding the highest factor that divides set of integers, but also referring to that in detailing Euler function of relative primes and culminating in analysing Diophantine equations with pointers to their solvabilities. The Chapter II Number System of Mathematics dwells in denumerability of numbers, algebra, graphic representation of numbers by analytical geometry, complex plane and transcends to Liouville's theorem. This fascnitating chapter stresses the psychological need of mathematical evolution while acknowledging the hesitant steps that apparently is the barrier of quantifying abstract thinking. It is followed up with Supplement to Chapter II The Algebra of Sets that lays the foundation for set theory, which was touched up on the previous chapter. Mathematical tool for engineers seems to be the best way to describe Chapter III Geometrical Constructions: The Algebra of Number Fields, where the number fields and constructions are introduced to appreciate the fundamentals of geometrical constructions. The problem of Appolonius is stated with its proofs by different perspective, the unsolvability of various Greek problems within their domain of constraints are logically demonstrated, inversion with applications to various problems are elaborated. However, the most interesting part of the chapter seems to be Mascheroni Construction which required me to consult http://mathafou.free.fr/themes_en/compas.html to solve a proposed exercise. The prefaces by the authors with their subsequent revision by Richar Courant and Ian Stewart indicates that some chapters were later appended as logic was refined as it branched. The toughest is Chapter IV Projective Geometry. Axiomatic and Non-Eucildean Geometries that required visits to stackexchange.com, files.eric.ed.gov, nabla.hr, amsi.org.au to understand how to approach certain problems suggested. The chapter will give a brand new insight to geometry itself. The apparently less complicated Chapter V Topology is about a different geometrical aspect that deliberates on Euler characteristics of general surfaces including that with holes as well as special surfaces like Moebius strip with its unique features. Not only does the chapter contains geometrical revelation like that for the Jordan curve but also how the fundamental theorem of algebra has a topological perspective is given as the caveat. Chapter VI Functions and Limits are a revelation on the topics. The topics have been presented with the abstract elegance that several never think of while utilizing the concepts for solving mathematical intricacies. As the name Supplement to Chapter VI More Examples on Limits and Continuity obviously suggests, this helps in elaborating the preceding concepts with several examples and exercises for the readers that reveal their beauty and give them the touch of the classic. The next Chapter VII Maxima and Minima delves into the geometrical treatment of the extremum problems with illustrations of some natural phenomenon that corroborates the models with the physical laws. But its superbly beautiful aspect lies in the investigations of the existence of extremums that are quite sometimes taken for granted, which accurately portrays the Dirichlet conditions with quite a few interesting ideas. Academic twist awaits while going through Chapter VIII The Calculus that begins with integral calculus, generally reserved for the stage, following the understanding of differential calculus in conventional approaches to study this particular branch of mathematical creativity. Utilizing the basic summation method of computing definite integrals, the specific subject is elaborated with several revealing observations and inequalities. Then comes the derivative and how Fermat stimulated its necessity for obtaining extrema of functions. The derivative is formulated, analysed and symbolised. Lucid illustrations and challenges follow to compute derivatives of several algebraic expressions. Then comes the treatment for a few of the trigonometric observations. The requirement of continuity of functions for differentiability is elaborated. Applications start with derivatives applied in deriving acceleration and instantaneous velocity for moving bodies, the geometrical interpretation of 2^nd derivative, evaluation of minima and maxima among others. Then comes the techniques of differentiation and some of their very interesting applications. The notation that was used by Leibniz for differentiation represented a radical mathematical thinking, though the actual notion was only, at best, vaguely understood at the time by the thinkers in the field. With a logical note that justifies why great minds would proverbially arrive at similar results, the fundamental theorem of calculus is evaluated. The mechanical algorithm for the usual process of integration is justified and applications provided. The 1^st being Leibniz' formula for obtaining ℼ/4, followed by logarithm and exponential, with logarithm suprisingly yet logically preceding. The properties of the base of the natural logarithm are detailed and utilized followed by infinite series for logarithm. The concept of differential equations follows with applications in principles of physics, growth, vibrations and compound interest. The Supplement to Chapter VIII starts with basic principles of differentiability, integrals with its varied applications and continues with orders of magnitude, infinite series, the interesting advantage of the complex variable for their applications regarding seemingly unrelated domains, the harmonic series, the zeta function and the sine series by Euler, a beautiful discovery in the subject and the logical yet a bit approximate introduction to the prime number theorem. The prime numbers are the start to Chapter IX Recent Developments, which is a fascinating rambling on the more modern developments in this beautiful subject. Beginning with several elusive problems of prime viz., the Goldbach Conjecture and the Twin Prime Problem, the book stresses the need of novel methods for their decisive directions, which is followed up by the famous Last Theorem of Fermat that started as the unsolved mystery when the book was initially published but was demystified in 1994 and thus was inserted in fresh editions. Then, there is a discussion on Continuum Hypothesis, which may puzzle the intellectual greatly, the fashion of Set-theoretic notations through the ages, the tricky Four Colour Theorem, the proof of which literally challenged mathematical practices. The fractals with the apparently bizarre Hausdorf dimensions follow that will stretch the brain deeper towards abstract realms. The links and knots come next and then comes another abstract analysis of a mistake in a previous chapter, now logically rectified, concerning a tricky problem of mechanics having seriously simpler mathematical consequences. Then the algorithmic efficiency of computation is discussed with reference to the Problem of Steiner to search for the smallest length to bridge a network of points. The related intricacies brewed by Soap Films and Minimal Surfaces are briefed as we reach the concluding Nonstandard Analysis that immensely satisfies the wits of the readers. The book ultimately converges with Apppendix: Supplementary Remarks, Problems, And Exercises, a tutorial of varied problems to entertain and challenge with help from sites like math.stackexchange the wits of the readers ready to tackle problems on the field of arithmetic, topology, calculus, limits, geometry, algebra, functions, analytical techniques that completes this enthralling treatment.