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Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics
This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition of mathematic fundamentals. Rather than a system of theorems with completely developed proofs or examples of applications, readers will encounter a coherent presentation of mathematical ideas that begins with the natural numbers and basic laws of arithmetic and progresses to the problems of the real-number continuum and concepts of the calculus.
Contents include examinations of the various types of numbers and a criticism of the extension of numbers; arithmetic, geometry, and the rigorous construction of the theory of integers; the rational numbers, the foundation of the arithmetic of natural numbers, and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; operating with sequences and differential quotient; remarkable curves; real numbers and ultrareal numbers; and complex and hypercomplex numbers.
In issues of mathematical philosophy, the author explores basic theoretical differences that have been a source of debate among the most prominent scholars and on which contemporary mathematicians remain divided. "With exceptional clarity, but with no evasion of essential ideas, the author outlines the fundamental structure of mathematics." — Carl B. Boyer, Brooklyn College. 27 figures. Index.
Contents include examinations of the various types of numbers and a criticism of the extension of numbers; arithmetic, geometry, and the rigorous construction of the theory of integers; the rational numbers, the foundation of the arithmetic of natural numbers, and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; operating with sequences and differential quotient; remarkable curves; real numbers and ultrareal numbers; and complex and hypercomplex numbers.
In issues of mathematical philosophy, the author explores basic theoretical differences that have been a source of debate among the most prominent scholars and on which contemporary mathematicians remain divided. "With exceptional clarity, but with no evasion of essential ideas, the author outlines the fundamental structure of mathematics." — Carl B. Boyer, Brooklyn College. 27 figures. Index.
272 pages, Paperback
First published January 1, 1997
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163 people want to read
About the author
Friedrich Waismann
43 books11 followersFriedrich Waismann was an Austrian mathematician, physicist, and philosopher. He is best known for being a member of the Vienna Circle and one of the key theorists in logical positivism
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Displaying 1 - 5 of 5 reviews
January 3, 2023
Every year I try to read one serious science or math book over the holidays. It's hard to pick books that are at the right level. The book has to be rigorous, but it also has to be easy enough that my imperfect memories of second year calculus and linear algebra are enough for me to fully understand the subject matter. So in prior years I have had successes and failures. I greatly enjoyed Leonard Susskind's Theoretical Minimum books, but when I tried to teach myself fluid dynamics, my head was spinning, and I didn't finish the book, and I was only able to halfway understand the tensors used in general relativity. This book has none of those problems. It hit my sweet spot with interesting ideas that I hadn't previously fully appreciated explained in a way that requires no more than high school algebra.
My biggest takeaway from the book was that in some ways our fundamental ideas of numbers and arithmetic are arbitrary definitions and systems built on sand. Many basic principles of math are "just so" stories, but that's OK because we have come up with computational systems that seem to work, that put limits on the extent of the arbitrariness and that give many interesting and often useful results. I certainly never before thought of basic operations such as addition, subtraction, multiplication and division as being fundamentally different calculating systems for natural numbers, integers, rational numbers and real numbers, whose similarity and translatability across number systems needs to be proved or that the links between systems are sometimes just conventions based on the philosophy of "permanence." And I was blown away by the concept that there is a one to one correspondence between the points on the one dimensional side of square to the points in the two dimensional interior and that the same can be extrapolated to an n-dimensional hypercube whose interior points can be matched to the points on a single edge. Cool stuff.
Then I got to the discussion of ultrareal numbers, which Mr. Waismann dismisses as having little value. I'm not sure what, if any value has been found for these numbers a hundred years after this book was written, but the curt dismissal of their value felt to me like Moses not believing that the God who gave him the burning bush, the plagues, the parting of the Red Sea and the revelation at Sinai could make water come from a rock. Come on, Fred! Surely these things have some real interest. It just isn't immediately obvious. But this is a minor gripe against a book that has helped me to hone my mathematical mind and has introduced me to some new and important ideas that I had maybe heard before in passing but had not thought about on a meaningful level until working through the proofs under Mr. Waismann's guidance.
My biggest takeaway from the book was that in some ways our fundamental ideas of numbers and arithmetic are arbitrary definitions and systems built on sand. Many basic principles of math are "just so" stories, but that's OK because we have come up with computational systems that seem to work, that put limits on the extent of the arbitrariness and that give many interesting and often useful results. I certainly never before thought of basic operations such as addition, subtraction, multiplication and division as being fundamentally different calculating systems for natural numbers, integers, rational numbers and real numbers, whose similarity and translatability across number systems needs to be proved or that the links between systems are sometimes just conventions based on the philosophy of "permanence." And I was blown away by the concept that there is a one to one correspondence between the points on the one dimensional side of square to the points in the two dimensional interior and that the same can be extrapolated to an n-dimensional hypercube whose interior points can be matched to the points on a single edge. Cool stuff.
Then I got to the discussion of ultrareal numbers, which Mr. Waismann dismisses as having little value. I'm not sure what, if any value has been found for these numbers a hundred years after this book was written, but the curt dismissal of their value felt to me like Moses not believing that the God who gave him the burning bush, the plagues, the parting of the Red Sea and the revelation at Sinai could make water come from a rock. Come on, Fred! Surely these things have some real interest. It just isn't immediately obvious. But this is a minor gripe against a book that has helped me to hone my mathematical mind and has introduced me to some new and important ideas that I had maybe heard before in passing but had not thought about on a meaningful level until working through the proofs under Mr. Waismann's guidance.
May 22, 2013
Even though I took Real Analysis two years ago, this book helped me take a step back and see the big picture. And see the value of analysis, in a way that I hadn't, really, when I was taking the class. I think this would be a great book for a motivated general reader, and an excellent book for someone who has taken--or is taking--a basic proofs or "transition" course.
Another reviewer mentioned that the proofs are pretty dry with little hand-holding. The typical reader tends to feel this way about pretty much any rigorous discussion of these topics. In this particular book it seemed to me that proofs and demonstrations were actually "narrated" quite fluidly and helpfully.
Another reviewer mentioned that the proofs are pretty dry with little hand-holding. The typical reader tends to feel this way about pretty much any rigorous discussion of these topics. In this particular book it seemed to me that proofs and demonstrations were actually "narrated" quite fluidly and helpfully.
September 15, 2022
If not as a great researcher, the author distinguishes himself not only as a teacher but as one up-to-date in the latest trends in the mathematical world of his time, the interwar years when, under the impulse of Emmy Noether in Göttingen, algebra was being recast into the abstract guise into whose recondite mysteries generations of students have been inducted ever since. The textbook presently under review, Einführung in das mathematische Denken: Die Begriffsbildung der modernen Mathematik (1936, reprinted in 1970), product of Friedrich Waismann’s pedagogical efforts during the early part of his career, stands out from other literature of its type for the educated public in being rigorous throughout and thoroughly modern in its perspective.
Quick overview of contents: the different kinds of number, the difference between arithmetic and geometry, the foundation of everything on whole numbers, elementary arithmetic, induction, formalism versus logicism, limit processes, sequences and series, real numbers (Cantor versus Dedekind), complex numbers and a concluding reflection on invention versus discovery [Erfinden oder Entdecken?]. The theme of the book revolves around the different kinds of numbers and the passage from one to another (from the natural numbers to the integers, from integers to rationals, from rationals to reals etc.). A number system is characterized by axioms; when we extend our concept of number in order to be able to solve previously unsolvable problems, something new is necessarily introduced but on the other hand a lot of the old structure may be preserved. This is known to historians as Hankel’s principle of permanence [Prinzip der Permanenz der Rechenregeln, p. 21ff]. For instance, after a section on the application of the principle of complete induction to division of one natural number by another (producing possibly a repeating decimal expansion):
Nun beginnt die Lage zu klären. Der ‘Sprung vom Endlichen ins Unendliche’ ist in Wirklichkeit der Übergang zu einem neuen Kalkül, der keine logische Folge des alten ist, in keiner Weise aus ihm abgeleitet werden kann, sich aber in bestimmter Weise an ihn anlehnt. [p. 68]
There are no hard and fast rules as to what new axioms to postulate other than the ultimate test: does the extension thereby won lead to a productive realm of mathematics? For as Waismann explains, what gives meaning to a mathematical term is its application [Verwendung, pp. 164-165]. John von Neumann once remarked that one never understands mathematics, but just gets used to it: is this celebrated quip entirely just? Waismann is an operationalist; by repeatedly operating with a novel kind of quantity we become familiar with it. For it seems that our minds can comprehend what they do – but then, if we become sufficiently acquainted with how the idea works in practice, doesn’t this mean that we do understand it somehow? In the ordinary course of affairs, discursively that is; what von Neumann alludes to is the circumstance that our faculty of intellectual intuition, to the extent that we have it at all, is feeble.
Waismann explains everything in sufficient detail with plenty of formulae and anticipates possible questions, misunderstandings or objections on the part of the reader. A typical move is the introduction of the rational numbers as equivalence classes of ordered pairs of natural numbers, subject to certain rules for performing arithmetical operations. He gives complete derivations of some simple results (e.g. commutativity of addition or existence of a multiplicative inverse for non-zero rationals), enough to give a flavor of elementary research in the field but normally shies away from adverting to any difficult theorems – so a careless reader could gain a misleading impression that research in mathematics is easier than it really is, for what stands behind the formulation of successful new concepts and axiom schemes is an intuition honed on the patient study of many hard examples. This concern aside, perusing this work is like accompanying the author on a pleasant journey for a while and getting to see what happens behind closed doors when a circle of mathematicians invents something hitherto unconceived of.
Audience of the book: anyone comfortable with elementary mathematics and eager to acquire the most advanced modern point of view on the subject, without rigmarole. Even someone who has studied at the graduate level may profit from Waismann’s lucid explanations. He excels at allaying fears and motivating the constructions by which present-day pure mathematicians nowadays go about their business, which could well appear strange and abstruse to an amateur who wishes to learn about how the professionals really do things – one won’t get anything like this in other popular literature!
Quick overview of contents: the different kinds of number, the difference between arithmetic and geometry, the foundation of everything on whole numbers, elementary arithmetic, induction, formalism versus logicism, limit processes, sequences and series, real numbers (Cantor versus Dedekind), complex numbers and a concluding reflection on invention versus discovery [Erfinden oder Entdecken?]. The theme of the book revolves around the different kinds of numbers and the passage from one to another (from the natural numbers to the integers, from integers to rationals, from rationals to reals etc.). A number system is characterized by axioms; when we extend our concept of number in order to be able to solve previously unsolvable problems, something new is necessarily introduced but on the other hand a lot of the old structure may be preserved. This is known to historians as Hankel’s principle of permanence [Prinzip der Permanenz der Rechenregeln, p. 21ff]. For instance, after a section on the application of the principle of complete induction to division of one natural number by another (producing possibly a repeating decimal expansion):
Nun beginnt die Lage zu klären. Der ‘Sprung vom Endlichen ins Unendliche’ ist in Wirklichkeit der Übergang zu einem neuen Kalkül, der keine logische Folge des alten ist, in keiner Weise aus ihm abgeleitet werden kann, sich aber in bestimmter Weise an ihn anlehnt. [p. 68]
There are no hard and fast rules as to what new axioms to postulate other than the ultimate test: does the extension thereby won lead to a productive realm of mathematics? For as Waismann explains, what gives meaning to a mathematical term is its application [Verwendung, pp. 164-165]. John von Neumann once remarked that one never understands mathematics, but just gets used to it: is this celebrated quip entirely just? Waismann is an operationalist; by repeatedly operating with a novel kind of quantity we become familiar with it. For it seems that our minds can comprehend what they do – but then, if we become sufficiently acquainted with how the idea works in practice, doesn’t this mean that we do understand it somehow? In the ordinary course of affairs, discursively that is; what von Neumann alludes to is the circumstance that our faculty of intellectual intuition, to the extent that we have it at all, is feeble.
Waismann explains everything in sufficient detail with plenty of formulae and anticipates possible questions, misunderstandings or objections on the part of the reader. A typical move is the introduction of the rational numbers as equivalence classes of ordered pairs of natural numbers, subject to certain rules for performing arithmetical operations. He gives complete derivations of some simple results (e.g. commutativity of addition or existence of a multiplicative inverse for non-zero rationals), enough to give a flavor of elementary research in the field but normally shies away from adverting to any difficult theorems – so a careless reader could gain a misleading impression that research in mathematics is easier than it really is, for what stands behind the formulation of successful new concepts and axiom schemes is an intuition honed on the patient study of many hard examples. This concern aside, perusing this work is like accompanying the author on a pleasant journey for a while and getting to see what happens behind closed doors when a circle of mathematicians invents something hitherto unconceived of.
Audience of the book: anyone comfortable with elementary mathematics and eager to acquire the most advanced modern point of view on the subject, without rigmarole. Even someone who has studied at the graduate level may profit from Waismann’s lucid explanations. He excels at allaying fears and motivating the constructions by which present-day pure mathematicians nowadays go about their business, which could well appear strange and abstruse to an amateur who wishes to learn about how the professionals really do things – one won’t get anything like this in other popular literature!
December 6, 2012
Pokes around at the foundations and structure of math and the various types of numbers (natural, integers, rationals, reals, complex). In between many long pages of proofs, in which he demonstrates how those various number systems can be constructed with a minimum of assumptions, the author touches on subjects such as the origin of symbolic math language (in Babylon, where phonetic and ideographic spoken/written languages came together, opening up a world of possibilities), and different philosophical views on the meaning of math and numbers; e.g., apparently math can never be completely and consistently wrapped up into a single neat and tidy system, though that was a goal for many mathematicians for a while, and the author also argues the view that numbers have no objective existence outside the human mind/experience (in keeping with his logical positivist philosophical views). Many famous mathematicians throughout history are referenced. Really the book is mostly dry proofs with little hand-holding, but good for a deeper look into math.
April 5, 2015
Just realized that thinking is quite different from being able to do some calculations. Honestly, this book was a straight punch in my fragile knowledge; there are numbers and definitions all over the place. Time to unlearn stuff and build something more solid. We don't even know what a number is; this is quite exciting since everything is possible.
Displaying 1 - 5 of 5 reviews