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The Story of Proof: Logic and the History of Mathematics
How the concept of proof has enabled the creation of mathematical knowledge
The Story of Proof investigates the evolution of the concept of proof―one of the most significant and defining features of mathematical thought―through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.
Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved.
Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field’s power and progress.
The Story of Proof investigates the evolution of the concept of proof―one of the most significant and defining features of mathematical thought―through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.
Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved.
Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field’s power and progress.
456 pages, Hardcover
Published November 15, 2022
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357 people want to read
About the author
John Stillwell
51 books60 followersJohn Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University
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February 11, 2024
Nos mathematici sumus isti veri poetae,
Sed quod fingimus nos et probare decet.
Poets in truth are we in mathematics,
But our creations also must be proved.
[Leopold Kronecker, Ueber den Zahlbegriff (1887)]
To be a mathematician by vocation and not just by profession, one ought always to keep a store of reflections on the nature of the discipline itself in the back of his mind and ponder them on occasion. These will nourish his productivity at a deep level, indirectly and unconsciously even if not directly made a conscious theme by which to guide everyday practice. For, as is the case with every endeavor of the mind, the quality of the final result is a function of the care one cultivates in getting there (the degree of one’s dedication to correctness, that is). Therefore, we must welcome it when another work issues from the pen of a gifted expositor of the history of mathematics, such as we have here with John Stillwell in his recently published The Story of Proof: Logic and the History of Mathematics (Princeton University Press, 2022).
The demand for proof, more than anything else has been distinctive of mathematics ever since the ancient Greeks and must be viewed as a foundation stone of western civilization. Nevertheless, proof itself is hardly a timeless Platonic ideal and methods of proof have evolved through the course of mathematical experience. Geometry, as practiced by Euclid, appeals to spatial intuition and invites a constructive procedure of solution of its problems, as was indeed the rule all the way to Kant. Algebra, on the other hand, is more computational. A result may derived by following a sequence of steps in which the elementary operations are applied to the algebraic expression at hand until the desired result pops out (indeed, the very meaning of the word algebra derives from the Arabic for al-jabr, which designates the ‘reunion of broken parts’). After the development of algebraic notation in the sixteenth century and of infinitesimal methods in the seventeenth, the power of the technique becomes all the more apparent. Thus, one of the themes in the present work is to trace these developments all the way to the victory of logical formalism around the turn of the twentieth century. Still, as Stillwell notes, even though mathematicians now reason in an all but exclusively logical mode, spatial intuition continues to play a role in guiding deliberation, behind the scenes as it were. This too he wishes to exemplify.
Another theme will be the role of abstraction. For what drives advances in every science is conceptual understanding. Recognize what is really at issue in a problem, abstract it and formulate a clear definition of the corresponding concept, and one will be well on the way to a solution. As Stillwell remarks in the preface, the web of concepts wielded by a contemporary mathematician can be no less intricate than the chain of logical steps that must be fashioned in order to derive a proof of a theorem. Thus, a good deal more space in the present work will be dedicated to the careful explanation and definition of concepts than is usual in popular histories of mathematics.
A glance at the table of contents reveals an extensiveness of coverage, with topics in sixteen chapters ranging over Euclidean and projective geometry, algebra, algebraic geometry, calculus, number theory, the fundamental theorem of algebra, non-Euclidean geometry, topology and lastly analysis in its modern guise as it was initiated during the nineteenth century. Then chapters eleven to sixteen round out the story with an extended look at the arithmetization of the continuum, set theory, axiomatics, logic and computation and incompleteness (concluding with proof theory). In each chapter, one will find an overview of selected results – meaning that Stillwell typically supplies detailed definitions of basic concepts and a sketch of the respective field highlighting a few key theorems, accompanied by incisive comments on the motivations that must have actuated their original discoverers. The text is complemented by plenty of illustrative figures and diagrams. This reviewer, accustomed to old-fashioned black-and-white printing, finds the color coding of section headings to be rather a bit annoying; it seems superfluous, anyway (he will admit that color printing does come in helpful to comprehending many of the figures). A paragraph or two of remarks at the end of every chapter contain valuable high-level reflections, something at which Stillwell excels in this and other works of his.
Indeed, as an historian as well as teacher, Stillwell has evidently had plenty of time over a long career to digest a wide range of material spanning the entire tradition, and draws on his erudition in the considered selection of subjects to cover. For instance, in the early chapter on the ancient Greeks, he treats not only the number-theoretical proof of the incommensurability of the diagonal to the sides of a square – with which everyone these days will be familiar, but also another method of proof that the Greeks hit upon and which may have impressed them even more: namely, the so-called anthyphaeresis = how the Euclidean algorithm for finding the greatest common denominator of two natural numbers, when applied to irrational numbers, never terminates. Stillwell comments on how a high level of proof technique obtained in ancient times, yet however, the art of algebra was missing – what gave European mathematicians their edge during the early modern period. Stillwell himself does not comment on this aspect, but we ourselves may remark on how al-Kindī writing in the ninth century, engaged in founding what would become the great tradition of medieval Islamic mathematics leading to the discovery of and elaboration of algebra, declares his aim to be nothing less than to complete what the ancients began (cf. our review of al-Khwārizmī’s The Beginnings of Algebra, here).
A few highlights: a thorough treatment of the epochal developments in algebra and geometry during the nineteenth century in chapters four and seven through ten. Replete with the usual panoply of diagrams, but as always, with Stillwell we get a little added value in the form of his reflections on the significance of what is being done, as for instance in the following passage:
Non-Euclidean geometry put an end to the idea that axioms are universal truths from which other truths may be derived by logic. Rather, axioms are truths about certain domains called models of the axioms, and their logical consequences are true in those models. Later, algebra made this idea its modus operandi: take some axioms known to hold in certain domains, such as the group axioms, and study their consequences. In effect, axioms are used to define a certain kind of structure, and their consequences are theorems about that kind of structure. However, algebra differs from non-Euclidean geometry in having obvious models. There are simple, finite models of the axioms for groups, rings and fields, so the consistency of these axioms was never in doubt. For non-Euclidean geometry, the existence of models seemed to defy intuition. [p. 226]
In line with his theme of logic and computation, Stillwell devotes the last third of this volume to an in-depth exploration of the state of the art as it emerged through the twentieth century, when, it could be said, the topic came into its own at the forefront of research in response to the so-called crisis in the foundations of set theory. The starting point is, of course, the pioneering work of Georg Cantor on set theory. As usual, Stillwell points out why it matters:
In the 1870’s Cantor made a series of groundbreaking discoveries about infinite sets that forced a complete rethink of the ancient dichotomy between potential and actual infinity. First, Cantor (1894) showed that the set R is uncountable, which means that there is no way to finesse R as a potential infinity. [p. 291]
At the hands above all of Richard Dedekind and Karl Weierstrass, Cantor’s set theory becomes an effective tool in real analysis. In chapter eleven, Stillwell tells the whole story of its foundation, leading up to his observation that
The program of arithmetization was essentially completed when Borel (1898) observed that each continuous function may be encoded by a real number. [p. 286]
Chapter twelve contains a clear exposition of some elements of set theory, involving in particular, transfinite ordinals. A final section on inaccessibility illuminates the meaning of the replacement axiom:
This result of Zermelo was stunning, because one expects the axioms of set theory to be consistent, and hence to have a model, which should be an inaccessible set. Thus we expect inaccessible sets to exist, and it is a shock not to be able to prove it. [p. 311]
Chapter thirteen follows with the formulation of axioms for numbers, geometry and sets. To someone accustomed to informal reasoning, it has to be pointed out how tricky natural language can be. Only when subjected to close analysis do its subtleties come to light. Here is an instance in point to anyone who might be inclined to view Peano’s axioms of arithmetic as prima facie obvious:
We say that the Peano axioms are categorical, meaning that all their models are essentially the same. However, this conclusion depends on the induction axiom being second order, that is, a statement about sets X of numbers. [p. 319]
Clearly, it is a major departure to require second-order logic! But what really occupies Stillwell’s attention is the so-called arithmetization of syntax that takes place during the interwar decades of the 1920’s and 1930’s. Here is why, it seems, Stillwell considers it important:
Gödel’s arithmetization of syntax shows ZF minus infinity is equivalent to Peano arithmetic, in other words, set theory = arithmetic + infinity! [p. 327]
A simple point: if logic is to be reducible to computation as Leibniz wants, it would exclude the predicate calculus [p. 347]. For the remainder of the present work, the author dilates on the Banach-Tarksi paradox, Turing machines, the word problem for semigroups, Gentzen’s consistency proof for Peano arithmetic, Goodstein’s theorem and reverse mathematics (which latter subject seems to be close to Stillwell’s heart and about which he has written his earlier Reverse Mathematics: Proofs from the Inside Out, Princeton University Press, 2018). The last item to be discussed is the striking weak König lemma, an extremely powerful result to judge by the wide scope of its immediate consequences, such as the completeness of predicate logic (any valid formula is provable) and invariance of dimension (there is no continuous bijection from R^n into R^m for n≠m).
In his concluding paragraph, Stillwell’s reasons for thinking proof theory interesting tend rather to the technical. An overall comment – while rich in examples and apposite commentary, the level of the text turns out to be somewhat lighter than expected: hard-going intellectual history in the proper sense does not happen to be Stillwell’s metier. Thus, read the present work for fine-grained minutiae on the evolution of the idea of proof and its attendant techniques, but one will have to seek elsewhere for a fitting appreciation of the intellectual significance of proof in the tradition behind our modern civilization, characterized as it is by the prominent place ascribed to science and technology, as opposed to other conceivable modes of knowledge (such as divination, prophecy, authority, convention, opinion, genial intuition and so forth). A little too much of a whirlwind tour, though the connoisseur will welcome a good number interesting observations along the way.
Sed quod fingimus nos et probare decet.
Poets in truth are we in mathematics,
But our creations also must be proved.
[Leopold Kronecker, Ueber den Zahlbegriff (1887)]
To be a mathematician by vocation and not just by profession, one ought always to keep a store of reflections on the nature of the discipline itself in the back of his mind and ponder them on occasion. These will nourish his productivity at a deep level, indirectly and unconsciously even if not directly made a conscious theme by which to guide everyday practice. For, as is the case with every endeavor of the mind, the quality of the final result is a function of the care one cultivates in getting there (the degree of one’s dedication to correctness, that is). Therefore, we must welcome it when another work issues from the pen of a gifted expositor of the history of mathematics, such as we have here with John Stillwell in his recently published The Story of Proof: Logic and the History of Mathematics (Princeton University Press, 2022).
The demand for proof, more than anything else has been distinctive of mathematics ever since the ancient Greeks and must be viewed as a foundation stone of western civilization. Nevertheless, proof itself is hardly a timeless Platonic ideal and methods of proof have evolved through the course of mathematical experience. Geometry, as practiced by Euclid, appeals to spatial intuition and invites a constructive procedure of solution of its problems, as was indeed the rule all the way to Kant. Algebra, on the other hand, is more computational. A result may derived by following a sequence of steps in which the elementary operations are applied to the algebraic expression at hand until the desired result pops out (indeed, the very meaning of the word algebra derives from the Arabic for al-jabr, which designates the ‘reunion of broken parts’). After the development of algebraic notation in the sixteenth century and of infinitesimal methods in the seventeenth, the power of the technique becomes all the more apparent. Thus, one of the themes in the present work is to trace these developments all the way to the victory of logical formalism around the turn of the twentieth century. Still, as Stillwell notes, even though mathematicians now reason in an all but exclusively logical mode, spatial intuition continues to play a role in guiding deliberation, behind the scenes as it were. This too he wishes to exemplify.
Another theme will be the role of abstraction. For what drives advances in every science is conceptual understanding. Recognize what is really at issue in a problem, abstract it and formulate a clear definition of the corresponding concept, and one will be well on the way to a solution. As Stillwell remarks in the preface, the web of concepts wielded by a contemporary mathematician can be no less intricate than the chain of logical steps that must be fashioned in order to derive a proof of a theorem. Thus, a good deal more space in the present work will be dedicated to the careful explanation and definition of concepts than is usual in popular histories of mathematics.
A glance at the table of contents reveals an extensiveness of coverage, with topics in sixteen chapters ranging over Euclidean and projective geometry, algebra, algebraic geometry, calculus, number theory, the fundamental theorem of algebra, non-Euclidean geometry, topology and lastly analysis in its modern guise as it was initiated during the nineteenth century. Then chapters eleven to sixteen round out the story with an extended look at the arithmetization of the continuum, set theory, axiomatics, logic and computation and incompleteness (concluding with proof theory). In each chapter, one will find an overview of selected results – meaning that Stillwell typically supplies detailed definitions of basic concepts and a sketch of the respective field highlighting a few key theorems, accompanied by incisive comments on the motivations that must have actuated their original discoverers. The text is complemented by plenty of illustrative figures and diagrams. This reviewer, accustomed to old-fashioned black-and-white printing, finds the color coding of section headings to be rather a bit annoying; it seems superfluous, anyway (he will admit that color printing does come in helpful to comprehending many of the figures). A paragraph or two of remarks at the end of every chapter contain valuable high-level reflections, something at which Stillwell excels in this and other works of his.
Indeed, as an historian as well as teacher, Stillwell has evidently had plenty of time over a long career to digest a wide range of material spanning the entire tradition, and draws on his erudition in the considered selection of subjects to cover. For instance, in the early chapter on the ancient Greeks, he treats not only the number-theoretical proof of the incommensurability of the diagonal to the sides of a square – with which everyone these days will be familiar, but also another method of proof that the Greeks hit upon and which may have impressed them even more: namely, the so-called anthyphaeresis = how the Euclidean algorithm for finding the greatest common denominator of two natural numbers, when applied to irrational numbers, never terminates. Stillwell comments on how a high level of proof technique obtained in ancient times, yet however, the art of algebra was missing – what gave European mathematicians their edge during the early modern period. Stillwell himself does not comment on this aspect, but we ourselves may remark on how al-Kindī writing in the ninth century, engaged in founding what would become the great tradition of medieval Islamic mathematics leading to the discovery of and elaboration of algebra, declares his aim to be nothing less than to complete what the ancients began (cf. our review of al-Khwārizmī’s The Beginnings of Algebra, here).
A few highlights: a thorough treatment of the epochal developments in algebra and geometry during the nineteenth century in chapters four and seven through ten. Replete with the usual panoply of diagrams, but as always, with Stillwell we get a little added value in the form of his reflections on the significance of what is being done, as for instance in the following passage:
Non-Euclidean geometry put an end to the idea that axioms are universal truths from which other truths may be derived by logic. Rather, axioms are truths about certain domains called models of the axioms, and their logical consequences are true in those models. Later, algebra made this idea its modus operandi: take some axioms known to hold in certain domains, such as the group axioms, and study their consequences. In effect, axioms are used to define a certain kind of structure, and their consequences are theorems about that kind of structure. However, algebra differs from non-Euclidean geometry in having obvious models. There are simple, finite models of the axioms for groups, rings and fields, so the consistency of these axioms was never in doubt. For non-Euclidean geometry, the existence of models seemed to defy intuition. [p. 226]
In line with his theme of logic and computation, Stillwell devotes the last third of this volume to an in-depth exploration of the state of the art as it emerged through the twentieth century, when, it could be said, the topic came into its own at the forefront of research in response to the so-called crisis in the foundations of set theory. The starting point is, of course, the pioneering work of Georg Cantor on set theory. As usual, Stillwell points out why it matters:
In the 1870’s Cantor made a series of groundbreaking discoveries about infinite sets that forced a complete rethink of the ancient dichotomy between potential and actual infinity. First, Cantor (1894) showed that the set R is uncountable, which means that there is no way to finesse R as a potential infinity. [p. 291]
At the hands above all of Richard Dedekind and Karl Weierstrass, Cantor’s set theory becomes an effective tool in real analysis. In chapter eleven, Stillwell tells the whole story of its foundation, leading up to his observation that
The program of arithmetization was essentially completed when Borel (1898) observed that each continuous function may be encoded by a real number. [p. 286]
Chapter twelve contains a clear exposition of some elements of set theory, involving in particular, transfinite ordinals. A final section on inaccessibility illuminates the meaning of the replacement axiom:
This result of Zermelo was stunning, because one expects the axioms of set theory to be consistent, and hence to have a model, which should be an inaccessible set. Thus we expect inaccessible sets to exist, and it is a shock not to be able to prove it. [p. 311]
Chapter thirteen follows with the formulation of axioms for numbers, geometry and sets. To someone accustomed to informal reasoning, it has to be pointed out how tricky natural language can be. Only when subjected to close analysis do its subtleties come to light. Here is an instance in point to anyone who might be inclined to view Peano’s axioms of arithmetic as prima facie obvious:
We say that the Peano axioms are categorical, meaning that all their models are essentially the same. However, this conclusion depends on the induction axiom being second order, that is, a statement about sets X of numbers. [p. 319]
Clearly, it is a major departure to require second-order logic! But what really occupies Stillwell’s attention is the so-called arithmetization of syntax that takes place during the interwar decades of the 1920’s and 1930’s. Here is why, it seems, Stillwell considers it important:
Gödel’s arithmetization of syntax shows ZF minus infinity is equivalent to Peano arithmetic, in other words, set theory = arithmetic + infinity! [p. 327]
A simple point: if logic is to be reducible to computation as Leibniz wants, it would exclude the predicate calculus [p. 347]. For the remainder of the present work, the author dilates on the Banach-Tarksi paradox, Turing machines, the word problem for semigroups, Gentzen’s consistency proof for Peano arithmetic, Goodstein’s theorem and reverse mathematics (which latter subject seems to be close to Stillwell’s heart and about which he has written his earlier Reverse Mathematics: Proofs from the Inside Out, Princeton University Press, 2018). The last item to be discussed is the striking weak König lemma, an extremely powerful result to judge by the wide scope of its immediate consequences, such as the completeness of predicate logic (any valid formula is provable) and invariance of dimension (there is no continuous bijection from R^n into R^m for n≠m).
In his concluding paragraph, Stillwell’s reasons for thinking proof theory interesting tend rather to the technical. An overall comment – while rich in examples and apposite commentary, the level of the text turns out to be somewhat lighter than expected: hard-going intellectual history in the proper sense does not happen to be Stillwell’s metier. Thus, read the present work for fine-grained minutiae on the evolution of the idea of proof and its attendant techniques, but one will have to seek elsewhere for a fitting appreciation of the intellectual significance of proof in the tradition behind our modern civilization, characterized as it is by the prominent place ascribed to science and technology, as opposed to other conceivable modes of knowledge (such as divination, prophecy, authority, convention, opinion, genial intuition and so forth). A little too much of a whirlwind tour, though the connoisseur will welcome a good number interesting observations along the way.
May 26, 2025
I am a great admirer of Stillwell’s writing, and this book does not disappoint. Ranging broadly and authoritatively over the history of mathematics, he takes the reader into those places where proofs have been innovative and have played a critical role.
David M. Bressoud
Displaying 1 - 2 of 2 reviews