Adventures in formalism (Q2904378)

Adventures in formalism is a modern investigation into the role, form, development, effect, and appreciation of formalism in mathematics. The preface briefly explores various meanings of the word `formalism' and introduces three particular types of formalism that are explored via examples throughout the book. Type I formalism stands for the process of the manipulation of symbols fueled by `reasoning by analogy', where one is willingly blind to the meaninglessness of the manipulations (e.g., early appearance of imaginary numbers). Type II formalism stands for the transition from an intuitive notion to a precise concept (e.g., the formulation of limit). Type III formalism stands for the modern axiomatic approach whereby one describes what something is by what one can do with it rather than by how to build it.NEWLINENEWLINEThe adventures recounted span numerous areas of mathematics scattered across several millennia with the bulk of the journey taking place in the past four centuries. The exposition includes all of the mathematical details needed to fully understand each and every example considered with the exception of a handful of cases. An essential part of the exposition are the lively descriptions and analyses of the reactions of (both prominent and less prominent) mathematicians to various developments.NEWLINENEWLINEWhile portions of the book will be accessible to readers with no mathematical training, the book will be particularly appealing to those with some university level mathematical background. To the expert mathematician the book offers a very broad view of the development of modern ideas, as well as a wide selection of historical case studies, interesting in their own right, that can effectively be used as supplements when teaching analysis or algebra courses. To the advanced student/beginning researcher the book will demonstrate that the birth of some of the most fundamental aspects that we take for granted today was fraught with difficulties and was usually met with opposition even by the greatest minds in the history of mathematics. To the beginning or intermediate student the book offers more than a glimpse as to what mathematics is all about.NEWLINENEWLINEThe first adventure is a tour of the intricacies of infinite series. Type I formalism is explored in depth through many examples of various formal computations with infinite series performed by prominent mathematicians in the early days of the calculus in a section named Preposterously absurd flights of fancy. Different type II formalisms are discussed with all the mathematical detail laid out (with particular attention to Bolzano's axiomatization of summation). Finally, a type III formalism is presented by taking an axiomatic approach to generalized sums. This chapter represents the most accessible portion of the book, demanding very little mathematical sophistication on the part of the reader while presenting a very well-rounded historical account exemplifying all of the subtleties and mathematical and psychological effects of formalism.NEWLINENEWLINEChapter II is a plethora of classic examples of formalism. There is first a short discussion of various attempts to deal with the concept of infinity (including Galileo's famous and less famous writings on the subject). The major source of examples given are from analysis and include, with plenty of mathematical detail and sophistication, Taylor series, the Euler-Maclaurin summation formula, and the Dirac \(\delta\)-function.NEWLINENEWLINEChapter III is a thorough adventure exploring many niches in the history of the process of extending number systems and potential alternatives to now common approaches. Starting with sections on the negative numbers and the rational numbers, there is then a very detailed section on fields of fractions. The classical passage to the real numbers then occupies two sections exploring various approaches including those of Dedeking and Weierstrass. Following is a treatment of nonstandard analysis via the hyperreals, giving an exact account of the contributions of Schmieden and Laugwitz, and Robinson. Next, general metric space completion is discussed and the chapter closes with a very interesting account of monsters -- counter intuitive examples allowed by the formalism. In particular, the existence of a nowhere differentiable continuous function is discussed with the trajectory of the problem clearly marked, from Bolzano's unpublished work to Weierstrass's function.NEWLINENEWLINEChapter IV starts off with a discussion of the philosophy and utility of axiomatization exploring where axioms come from, the notion of truth, convenience, system refinement, pure formalism, and the consequences of axioms. In that context the basic notions of logic (model theory) are introduced and the completeness theorem and its consequences are discussed. This chapter is mathematically less self-contained than the others. Some notions of definability are discussed and second-order logic is briefly recounted. The chapter finishes with numerous very pleasing essay questions.NEWLINENEWLINEThe final chapter, The crisis of intuition, is largely a philosophical discussion containing many quotations, mainly from the Hilbert vs. Brouwer debate of the early 20th century.NEWLINENEWLINETo conclude, the contents of the book justify its title as it will take the reader through almost uncountably many adventures in the realms of pure, applied, and the history of mathematics. The work is designed to appeal to readers of varying degrees of mathematical expertise and is written in such a way that it is easy to tune-in to almost any portion of it that interests the reader. It is written with a lot of care for detail and presentation, choosing a narration style that makes the reader feel as if the author is right there next to her, smiling.