What do you think?
Rate this book
Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability
The intellectual and human story of a mathematical proof that transformed our ideas about mathematics. In 1824 a young Norwegian named Niels Henrik Abel proved conclusively that algebraic equations of the fifth order are not solvable in radicals. In this book Peter Pesic shows what an important event this was in the history of thought. He also presents it as a remarkable human story. Abel was twenty-one when he self-published his proof, and he died five years later, poor and depressed, just before the proof started to receive wide acclaim. Abel's attempts to reach out to the mathematical elite of the day had been spurned, and he was unable to find a position that would allow him to work in peace and marry his fiancé. But Pesic's story begins long before Abel and continues to the present day, for Abel's proof changed how we think about mathematics and its relation to the "real" world. Starting with the Greeks, who invented the idea of mathematical proof, Pesic shows how mathematics found its sources in the real world (the shapes of things, the accounting needs of merchants) and then reached beyond those sources toward something more universal. The Pythagoreans' attempts to deal with irrational numbers foreshadowed the slow emergence of abstract mathematics. Pesic focuses on the contested development of algebra—which even Newton resisted—and the gradual acceptance of the usefulness and perhaps even beauty of abstractions that seem to invoke realities with dimensions outside human experience. Pesic tells this story as a history of ideas, with mathematical details incorporated in boxes. The book also includes a new annotated translation of Abel's original proof.
222 pages, Paperback
First published May 1, 2003
4 people are currently reading
115 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 13 of 13 reviews
December 17, 2025
Für uns gibt es kein Ignorabimus, und meiner Meinung nach auch für die Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heiße im Gegenteil unsere Losung: Wir müssen wissen, Wir werden wissen.
………………………………………………………………………………........
David Hilbert (1862-1943)
This famous cri de coeur, exclaimed by the elderly statesman of the mathematical world at the summit of his métier as the closing lines of a radio address to the German public in 1930, reflects the inexperienced mentality of the adolescent. Flush with elation over his new-found powers of reasoning, one deems oneself invincible, supposing that any problem, be it ever so difficult, will yield to solution with sufficient exertion. Most everyone confronts soon enough the limits of his mettle, but the failure of an individual says nothing about mathematical truth, per se. The ethos of the field dictates trust in the eventuality that someone with enough genius must come along, sooner or later. And so it reliably proved to be, down through the centuries running into the turn of the modern era and the high-water mark of the French Enlightenment.
Thus, Niels Henrik Abel’s (1802-1829) demonstration that there does not exist a solution expressible as a rational function in radicals for the roots of a general quintic polynomial signalizes a watershed in the history of mathematics. For ‘what Abel found is indeed surprising and strangely beautiful’ [p. 3], in the words of the author of the present book, Peter Pesic’s Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability (MIT Press, 2003). At just 153 octavo pages disregarding appendices, a quick read which covers the history of algebra in general, not just Abel’s contribution, starting with the ancient Greeks and culminating in the mathematical world of the 1820’s, then dominated by Paris and Berlin, where, as a promising young upstart from provincial Norway our hero barely managed to secure a foothold in the profession: all the distinguished establishment figures whom Abel called upon dismissed him off-hand, failing to see the value or potential of his work, but the hapless young man did have the good fortune to seal an acquaintance with the amateur and publisher August Crelle, who was impressed enough to be willing to publish several papers of Abel’s in his Journal für die reine und angewandte Mathematik (universally known simply as Crelle’s Journal), then a leading venue for breaking mathematical discoveries.
The down-to-earth exposition is clear and always interesting, if not penned in the eloquent style of a distinguished philologist. A nice feature is the provision of quite informative inset boxes giving worked examples at some length, so the reader can see for himself what is going on in the discussion in the main text. Pesic thinks very much like a trained modern mathematician who is interested in old results from the point of view of what they say about our current understanding of the mathematics itself, not as a historian who cares about the contemporary context. Thus, his book’s genre belongs more to popular science than to the (professional) history of mathematics.
Why, one might well wonder at this juncture, is Abel’s finding of unsolvability more significant than the irrationality of √2? Though the latter is usually portrayed as a disaster in popular accounts, Pesic acknowledges that the extant sources do not permit us to say with confidence that the ancient Greeks experienced it as such, rather than welcome it as a mysterious property of the world. In any case, the irrationality of √2 failed to occasion a change in the concept of number for the Greeks; they always consistently maintained that numbers must be whole integers, which can be considered in proportions. Say, 3:2, which we moderns think of as the rational number 3/2. As it happens, the Greek manner of viewing things lies closer to the technical definition in modern algebra of a field of fractions over a ring in an integral domain. What is more, it can be elaborated into Eudoxus’ elegant theory of continuous magnitudes which hews fairly close (allowing for the anachronism) to Dedekind’s theory of real numbers based on his cuts.
Hence, the balance of evidence indicates that the Greeks did not view the existence of what we would call irrational magnitudes as posing a crisis to their concept of number. Being mainly interested in geometrical demonstrations and possessing at best rudiments of what later became known as algebra, they would not have perceived, say, the irrationality of √2 as meaning the unsolvability in rationals of the quadratic equation x² – 2 = 0. For they would have no doubt that √2 exists as the hypotenuse of a right triangle of unit sides.
Therefore, Abel and his contemporaries found themselves faced with an essentially unanticipated phenomenon:
Viète and Descartes were optimistic about being able to solve polynomial equations of any degree : Descartes recommends reduction to lower degree, for instance, using tricks involving conic sections, but although his method is not general, he supposes something like this should always be possible in principle [p. 58]….The quintic’s intractability was considered a challenge that simply awaited the right assault, and it was widely believed to be solvable until almost 1800. [p. 67]
Carl Friedrich Gauss (1777-1855) was the first to suspect otherwise.
In chapters five through eight, Pesic unfolds his main undertaking, to describe the proximate route by which Abel was able to reach his epochal conclusion that the general quintic is intractible (which does not exclude the possibility that certain special quintics might be solvable; indeed, some are). The major antecedents are due to Joseph-Louis Lagrange (1736-1813) and Paolo Ruffini (1765-1822). For Lagrange had the ingenious idea to contemplate the effect of permuting the roots among themselves, and came up with his so-called resolvent which gets to the heart of the matter. Ruffini realized the relevance of Lagrange’s resolvent to the question of solvability of polynomial equations of higher than fourth degree, but gave a defective proof.
Nevertheless, Ruffini’s work, transmitted through the medium of Augustin-Louis Cauchy (1789-1857), supplied Abel with his starting point. By illustrating Abel’s method under the simpler case of the quadratic equation, Pesic motivates the reasoning behind Abel’s 1824 paper, announcing unsolvability of the quintic. From this formulation it becomes evident why solvability is related to symmetry under permutation of the roots, a subject followed up upon very nicely in chapters seven and eight, by means of ample diagrammatical illustrations. Here we meet another Romantic hero, Évariste Galois (1811-1832), who in a remarkable fit of inspiration completes the program Abel began. Here is how Pesic explains the inner connection between the two gifted mathematicians:
Galois’ work agrees entirely with Abel’s results. What is new in Galois is a turn toward abstraction in an essentially modern way, leading to a complete understanding of solvability, which Abel lacked. [pp. 108-109]
If the reader is alert, perhaps he will hit upon the same question that occurred to this reviewer in the course of Pesic’s exposition: if radicals do not suffice to express the general solution to polynomials of degree five or higher, maybe one could attain the desired formulae upon enlarging the collection of functions one is willing to entertain beyond nested radicals? The answer turns out to be, yes:
Now we can complete the odyssey of the quintic. It turns out that their solutions can be identified with the so-called elliptic modular functions (defined as sums of powers times trigonometric functions). These functions are based on the ‘theta functions’ that Abel and Jacobi began to study in 1827, though it was not until 1858 that Charles Hermite, Leopold Kronecker and Fernando Brioschi independently showed their relation to the quintic. By 1870, Jordan had proved that any polynomial equation, of whatever degree, could be solved by using these generalized functions. [p. 146]
But in one of those paradoxes of which the history of mathematics is full, these startling results have been pretty well bypassed and overlooked, owing to a shift in emphasis by practicing mathematicians. Chapter nine, The Order of Things, traces the rise of a self-conscious discipline of algebra in the aftermath of Abel and Galois. Progress towards the modern view, self-evident to everyone today, was halting. Pesic dwells on some of the stations along the way, such as, preeminently, the Irish mathematical physicist William Hamilton’s discovery in 1833 of the noncommutative algebra of quaternions, which, due to its connection to the rotations of 3d Euclidean space, lies at the inception of the vector analysis so familiar to generations of physicists ever since.
In the latter half of the nineteenth century, algebra came into its own. Perhaps most memorable, in retrospect, is the impetus Felix Klein imparted to the field by stressing the inner connection between algebra, in the guise of symmetry groups, and novel perspectives on the problems of geometrical space. Thus, a rich vein was opened up that would continue to be mined with great success all through the twentieth century, not least in theoretical physics with Einstein’s general theory of relativity, on the one hand, and Yang-Mills’ nonabelian gauge theory, on the other.
Pesic summarizes the prevailing situation, thus:
The historical unfolding of the notion of noncommutativity implies larger questions: what is its significance? Why should we care about it? In 1872, not long before Cayley set out group theory in full abstraction, the great mathematician Felix Klein gave a lecture at Erlangen, in which he proposed that group theory be considered the heart of mathematics, more fundamental in its import than geometry or algebra alone. He argued that each geometrical figure should be considered more in terms of its fundamental symmetry—specified by the group governing it—than by any figure or equation in itself. In Klein’s view, considering ever more general groups would be the royal road to understanding ever more complex spaces, reaching past three dimensions to manifolds of higher dimensions. The ‘Erlangen Program’ was the clarion call of a new abstract vision, going past the traditional starting points of mathematics to reveal the deepest, most general concepts underneath. It was an influential rallying cry that still resounds in contemporary mathematics. [pp. 138-139]
(The interested reader may refer to Felix Klein’s 1893 lectures given at Northwestern University in Evanston, Illinois, our review here; and to his Grundlegung der Geometrie zum Erlanger Programm, published as part of the first volume of his collected works, our review here.)
To conclude: Pesic is to be applauded for a fine piece of popular literature in the history of mathematics. Enjoy!
Let us return, in a postscript, to Hilbert: as a supreme master of contemporary algebra and number theory, he is well aware of Abel’s result and of its rich repercussions in Galois theory. Hence, for him, mathematical practice aims, among other things, to show existence resp. non-existence of a solution to a problem, along the lines originally proposed. Indeed, his renowned Nullstellensatz of 1893 boils down to proof of the existence of a finite resolution of the radical of an ideal in a polynomial ring, by non-constructive means! Therefore, he is perfectly well prepared to accept either existence or non-existence. For Hilbert, the salient point is that one can decide between the two alternatives, either way. This conviction leads him to formulate his formalist agenda, surmising that every conceivable theorem should be susceptible of proof or disproof (in elementary arithmetic, for starters). A further and natural stipulation would be consistency, viz., that no false proposition be derivable from the axioms.
It is commonly opined that this formalist program was wrecked by the young Austrian logician Kurt Gödel in 1931, with his celebrated incompleteness theorems. In a sense, yes, but not quite a refutation as it is normally taken to be. For what Gödel establishes scarcely plunges us into an abyss of unknowing; it merely indicates that the path of knowing is far subtler than anyone in modern times guessed (leaving aside Parmenides the pre-Socratic, who already saw with amazing penetration the substance of the matter, if not perhaps all the technical details, as of the early fifth century BC). The mathematician of today will be forced to recognize, contrary to naïve expectation, that consistency of arithmetic cannot be proved from within itself—a setback, indeed, but far from a terminal diagnosis! Gödel tells us, instead, that we must seek out other and stronger self-evident principles, if we wish to continue the program of complete comprehension of the logical structure of mathematical reasoning. Nowadays, questions such as these can be investigated in the burgeoning discipline known as the theory of higher cardinals, also sometimes called proof theory. To date, a surprising number of deep discoveries have been puzzled out, and there is no end to progress in sight. (See our review of John Stillwell’s The Story of Proof: Logic and the History of Mathematics, here.) Thus, the clear-eyed mathematical logician currently has no cause for despair on account of Gödel’s initially discouraging breakthrough, but should rather steel himself for further hard work and accede to what Hilbert himself declares, with unimpaired validity, in the sentence immediately preceding the passage quoted above from his 1930 radio address:
Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus gefallen.
………………………………………………………………………………........
David Hilbert (1862-1943)
This famous cri de coeur, exclaimed by the elderly statesman of the mathematical world at the summit of his métier as the closing lines of a radio address to the German public in 1930, reflects the inexperienced mentality of the adolescent. Flush with elation over his new-found powers of reasoning, one deems oneself invincible, supposing that any problem, be it ever so difficult, will yield to solution with sufficient exertion. Most everyone confronts soon enough the limits of his mettle, but the failure of an individual says nothing about mathematical truth, per se. The ethos of the field dictates trust in the eventuality that someone with enough genius must come along, sooner or later. And so it reliably proved to be, down through the centuries running into the turn of the modern era and the high-water mark of the French Enlightenment.
Thus, Niels Henrik Abel’s (1802-1829) demonstration that there does not exist a solution expressible as a rational function in radicals for the roots of a general quintic polynomial signalizes a watershed in the history of mathematics. For ‘what Abel found is indeed surprising and strangely beautiful’ [p. 3], in the words of the author of the present book, Peter Pesic’s Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability (MIT Press, 2003). At just 153 octavo pages disregarding appendices, a quick read which covers the history of algebra in general, not just Abel’s contribution, starting with the ancient Greeks and culminating in the mathematical world of the 1820’s, then dominated by Paris and Berlin, where, as a promising young upstart from provincial Norway our hero barely managed to secure a foothold in the profession: all the distinguished establishment figures whom Abel called upon dismissed him off-hand, failing to see the value or potential of his work, but the hapless young man did have the good fortune to seal an acquaintance with the amateur and publisher August Crelle, who was impressed enough to be willing to publish several papers of Abel’s in his Journal für die reine und angewandte Mathematik (universally known simply as Crelle’s Journal), then a leading venue for breaking mathematical discoveries.
The down-to-earth exposition is clear and always interesting, if not penned in the eloquent style of a distinguished philologist. A nice feature is the provision of quite informative inset boxes giving worked examples at some length, so the reader can see for himself what is going on in the discussion in the main text. Pesic thinks very much like a trained modern mathematician who is interested in old results from the point of view of what they say about our current understanding of the mathematics itself, not as a historian who cares about the contemporary context. Thus, his book’s genre belongs more to popular science than to the (professional) history of mathematics.
Why, one might well wonder at this juncture, is Abel’s finding of unsolvability more significant than the irrationality of √2? Though the latter is usually portrayed as a disaster in popular accounts, Pesic acknowledges that the extant sources do not permit us to say with confidence that the ancient Greeks experienced it as such, rather than welcome it as a mysterious property of the world. In any case, the irrationality of √2 failed to occasion a change in the concept of number for the Greeks; they always consistently maintained that numbers must be whole integers, which can be considered in proportions. Say, 3:2, which we moderns think of as the rational number 3/2. As it happens, the Greek manner of viewing things lies closer to the technical definition in modern algebra of a field of fractions over a ring in an integral domain. What is more, it can be elaborated into Eudoxus’ elegant theory of continuous magnitudes which hews fairly close (allowing for the anachronism) to Dedekind’s theory of real numbers based on his cuts.
Hence, the balance of evidence indicates that the Greeks did not view the existence of what we would call irrational magnitudes as posing a crisis to their concept of number. Being mainly interested in geometrical demonstrations and possessing at best rudiments of what later became known as algebra, they would not have perceived, say, the irrationality of √2 as meaning the unsolvability in rationals of the quadratic equation x² – 2 = 0. For they would have no doubt that √2 exists as the hypotenuse of a right triangle of unit sides.
Therefore, Abel and his contemporaries found themselves faced with an essentially unanticipated phenomenon:
Viète and Descartes were optimistic about being able to solve polynomial equations of any degree : Descartes recommends reduction to lower degree, for instance, using tricks involving conic sections, but although his method is not general, he supposes something like this should always be possible in principle [p. 58]….The quintic’s intractability was considered a challenge that simply awaited the right assault, and it was widely believed to be solvable until almost 1800. [p. 67]
Carl Friedrich Gauss (1777-1855) was the first to suspect otherwise.
In chapters five through eight, Pesic unfolds his main undertaking, to describe the proximate route by which Abel was able to reach his epochal conclusion that the general quintic is intractible (which does not exclude the possibility that certain special quintics might be solvable; indeed, some are). The major antecedents are due to Joseph-Louis Lagrange (1736-1813) and Paolo Ruffini (1765-1822). For Lagrange had the ingenious idea to contemplate the effect of permuting the roots among themselves, and came up with his so-called resolvent which gets to the heart of the matter. Ruffini realized the relevance of Lagrange’s resolvent to the question of solvability of polynomial equations of higher than fourth degree, but gave a defective proof.
Nevertheless, Ruffini’s work, transmitted through the medium of Augustin-Louis Cauchy (1789-1857), supplied Abel with his starting point. By illustrating Abel’s method under the simpler case of the quadratic equation, Pesic motivates the reasoning behind Abel’s 1824 paper, announcing unsolvability of the quintic. From this formulation it becomes evident why solvability is related to symmetry under permutation of the roots, a subject followed up upon very nicely in chapters seven and eight, by means of ample diagrammatical illustrations. Here we meet another Romantic hero, Évariste Galois (1811-1832), who in a remarkable fit of inspiration completes the program Abel began. Here is how Pesic explains the inner connection between the two gifted mathematicians:
Galois’ work agrees entirely with Abel’s results. What is new in Galois is a turn toward abstraction in an essentially modern way, leading to a complete understanding of solvability, which Abel lacked. [pp. 108-109]
If the reader is alert, perhaps he will hit upon the same question that occurred to this reviewer in the course of Pesic’s exposition: if radicals do not suffice to express the general solution to polynomials of degree five or higher, maybe one could attain the desired formulae upon enlarging the collection of functions one is willing to entertain beyond nested radicals? The answer turns out to be, yes:
Now we can complete the odyssey of the quintic. It turns out that their solutions can be identified with the so-called elliptic modular functions (defined as sums of powers times trigonometric functions). These functions are based on the ‘theta functions’ that Abel and Jacobi began to study in 1827, though it was not until 1858 that Charles Hermite, Leopold Kronecker and Fernando Brioschi independently showed their relation to the quintic. By 1870, Jordan had proved that any polynomial equation, of whatever degree, could be solved by using these generalized functions. [p. 146]
But in one of those paradoxes of which the history of mathematics is full, these startling results have been pretty well bypassed and overlooked, owing to a shift in emphasis by practicing mathematicians. Chapter nine, The Order of Things, traces the rise of a self-conscious discipline of algebra in the aftermath of Abel and Galois. Progress towards the modern view, self-evident to everyone today, was halting. Pesic dwells on some of the stations along the way, such as, preeminently, the Irish mathematical physicist William Hamilton’s discovery in 1833 of the noncommutative algebra of quaternions, which, due to its connection to the rotations of 3d Euclidean space, lies at the inception of the vector analysis so familiar to generations of physicists ever since.
In the latter half of the nineteenth century, algebra came into its own. Perhaps most memorable, in retrospect, is the impetus Felix Klein imparted to the field by stressing the inner connection between algebra, in the guise of symmetry groups, and novel perspectives on the problems of geometrical space. Thus, a rich vein was opened up that would continue to be mined with great success all through the twentieth century, not least in theoretical physics with Einstein’s general theory of relativity, on the one hand, and Yang-Mills’ nonabelian gauge theory, on the other.
Pesic summarizes the prevailing situation, thus:
The historical unfolding of the notion of noncommutativity implies larger questions: what is its significance? Why should we care about it? In 1872, not long before Cayley set out group theory in full abstraction, the great mathematician Felix Klein gave a lecture at Erlangen, in which he proposed that group theory be considered the heart of mathematics, more fundamental in its import than geometry or algebra alone. He argued that each geometrical figure should be considered more in terms of its fundamental symmetry—specified by the group governing it—than by any figure or equation in itself. In Klein’s view, considering ever more general groups would be the royal road to understanding ever more complex spaces, reaching past three dimensions to manifolds of higher dimensions. The ‘Erlangen Program’ was the clarion call of a new abstract vision, going past the traditional starting points of mathematics to reveal the deepest, most general concepts underneath. It was an influential rallying cry that still resounds in contemporary mathematics. [pp. 138-139]
(The interested reader may refer to Felix Klein’s 1893 lectures given at Northwestern University in Evanston, Illinois, our review here; and to his Grundlegung der Geometrie zum Erlanger Programm, published as part of the first volume of his collected works, our review here.)
To conclude: Pesic is to be applauded for a fine piece of popular literature in the history of mathematics. Enjoy!
Let us return, in a postscript, to Hilbert: as a supreme master of contemporary algebra and number theory, he is well aware of Abel’s result and of its rich repercussions in Galois theory. Hence, for him, mathematical practice aims, among other things, to show existence resp. non-existence of a solution to a problem, along the lines originally proposed. Indeed, his renowned Nullstellensatz of 1893 boils down to proof of the existence of a finite resolution of the radical of an ideal in a polynomial ring, by non-constructive means! Therefore, he is perfectly well prepared to accept either existence or non-existence. For Hilbert, the salient point is that one can decide between the two alternatives, either way. This conviction leads him to formulate his formalist agenda, surmising that every conceivable theorem should be susceptible of proof or disproof (in elementary arithmetic, for starters). A further and natural stipulation would be consistency, viz., that no false proposition be derivable from the axioms.
It is commonly opined that this formalist program was wrecked by the young Austrian logician Kurt Gödel in 1931, with his celebrated incompleteness theorems. In a sense, yes, but not quite a refutation as it is normally taken to be. For what Gödel establishes scarcely plunges us into an abyss of unknowing; it merely indicates that the path of knowing is far subtler than anyone in modern times guessed (leaving aside Parmenides the pre-Socratic, who already saw with amazing penetration the substance of the matter, if not perhaps all the technical details, as of the early fifth century BC). The mathematician of today will be forced to recognize, contrary to naïve expectation, that consistency of arithmetic cannot be proved from within itself—a setback, indeed, but far from a terminal diagnosis! Gödel tells us, instead, that we must seek out other and stronger self-evident principles, if we wish to continue the program of complete comprehension of the logical structure of mathematical reasoning. Nowadays, questions such as these can be investigated in the burgeoning discipline known as the theory of higher cardinals, also sometimes called proof theory. To date, a surprising number of deep discoveries have been puzzled out, and there is no end to progress in sight. (See our review of John Stillwell’s The Story of Proof: Logic and the History of Mathematics, here.) Thus, the clear-eyed mathematical logician currently has no cause for despair on account of Gödel’s initially discouraging breakthrough, but should rather steel himself for further hard work and accede to what Hilbert himself declares, with unimpaired validity, in the sentence immediately preceding the passage quoted above from his 1930 radio address:
Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus gefallen.
Read
December 10, 2025Fikk denne boka av en som studerte matte på UiO på åpen dag der. Jeg må si at jeg likte den mer enn jeg trodde jeg skulle. Jeg har jo ikke doktorgrad i matte akkurat så var flere ting jeg ikke skjønte eller en gang hadde sett før, men var veldig kult når jeg følte at jeg skjønte ting jeg ikke har lært før.
Boka fortalte også om historien som ledet opp til Abel (og etter Abel) som jeg syns var spennende.
Føler ikke jeg kan rate denne boka med stjerner, men den var helt fin. Spennende noen steder, mens andre ganger falt jeg nesten helt av.
Boka fortalte også om historien som ledet opp til Abel (og etter Abel) som jeg syns var spennende.
Føler ikke jeg kan rate denne boka med stjerner, men den var helt fin. Spennende noen steder, mens andre ganger falt jeg nesten helt av.
November 2, 2019
The formula for the solution of the general equation of the third degree by the Italians Del Ferro, Tartaglia, and Cardano, in the 16th Century, was one of the triumphs of Renaissance mathematics, and one that was a clear improvement upon the achievements of the Ancients. Soon thereafter Cardano's student Ludovico Ferrari obtained the solution of the quartic equation. And at this point matters rested, with repeatedly failed attempts to get a formula for the solutions of the general equations of the fifth and higher degrees, until the 1824 paper by the Norwegian mathematician Niels Henrik Abel, then twenty one years old, settled the matter for good, by proving such a formula cannot exist. This great little book tells the story of this intellectual quest from the very beginning, with the Pythagoreans shocking discovery of the irrationals, and proceeding with the work of the Arabic and Italian algebraists on the solution of equations and other algebraic problems, not least among them the introduction of appropriate notation. The slow but steady development of ideas, with contributions by Viéte, Descartes, Newton, Gauss, Lagrange, and Ruffini, among others, resulted in the brilliant result by Abel (a translation of which is printed in Appendix A.) Also covered in the text is the aftermath of Abel's work, in particular his 1828 paper on the relation between solvability and noncommutativity, and the immense extension of this idea by Galois, with the development of the concept of group and the explanation of the solvability of algebraic equations in terms of the commutativity properties of certain quocient groups derived from the equation. The centrality of certain abstract algebraic notions subsequently introduced by Hamilton, Grassman, Gibbs, Sylvester, Cayley or Boole is also touched upon in this book. In short: this is a remarkable work that, although written for the educated lay person, is not shy to present and comment upon "real" mathematics (mainly in the boxes scattered throughout the text and in the appendices) and could very well serve as the backbone of an advanced course in the History of Algebra, guiding the study from the earlier examples of Babylonian mathematics to the development of several concepts of number (integers, rationals, reals, complex, quaternions,...), the notion of unsolvability, the development of mathematical notation, the gradual creation of the objects and concepts of present day linear and abstract Algebras. All of these themes could be introduced by starting with an appropriate part of this little gem of a book, and then take off from there, exploiting exciting events in this part of the intellectual history of mankind, and then get back again to this great little book to gain context and take off again a little later, and a little wiser...
December 30, 2020
I ought to begin with the positives: Pesic clearly presents specific mathematical ideas, embeds some bits of cute narrative throughout, and is able to cite a genuine abundance of sources in his endnotes.
However, Pesic's approach to historical narrative serves to obfuscate the meaningful mathematical points. A few themes emerge in and out of focus, such as the interplay between arithmetic and geometric ways of thinking---and indeed, some of his exposition of Abel's and Galois' proofs is quite illuminating in terms of how symmetries among polynomial roots relate to the Platonic solids. But Pesic is too committed to his narrow strand of "unsolvable problems" through an undefined canon of great works to dwell on such matters. Instead, the book tends to feel like a parade of great men in math through a certain narrow scope.
The second half of the title, on the "meaning of mathematical unsolvability," would have hinted at connections to broader trends in the history of mathematics with regards to intellectual thought. Indeed, Pesic buries tantalizing clues to such a reading, discussing various trends in the reception of algebraic notation, but promptly abandons those in favor of the same over-wrought biographical details one finds in every other source on Galois (or Abel). So, the book is laden with hints of what it could have been, had Pesic's history matched some of his own later writings.
However, Pesic's approach to historical narrative serves to obfuscate the meaningful mathematical points. A few themes emerge in and out of focus, such as the interplay between arithmetic and geometric ways of thinking---and indeed, some of his exposition of Abel's and Galois' proofs is quite illuminating in terms of how symmetries among polynomial roots relate to the Platonic solids. But Pesic is too committed to his narrow strand of "unsolvable problems" through an undefined canon of great works to dwell on such matters. Instead, the book tends to feel like a parade of great men in math through a certain narrow scope.
The second half of the title, on the "meaning of mathematical unsolvability," would have hinted at connections to broader trends in the history of mathematics with regards to intellectual thought. Indeed, Pesic buries tantalizing clues to such a reading, discussing various trends in the reception of algebraic notation, but promptly abandons those in favor of the same over-wrought biographical details one finds in every other source on Galois (or Abel). So, the book is laden with hints of what it could have been, had Pesic's history matched some of his own later writings.
December 28, 2024
(4.2 stars) A quick, interesting history tracing the evolution of math through the history of trying to solve polynomial equations. The ancients could solve for quadratic equations, it took until the renaissance (1500s) for cardano, tartaglia and del ferro to solve cubic and quartic equations, and then in the early 1800s abel proved quintic equations cant be solved using just radicals, and then galois proved the same for all higher order equations.
The book itself is nicely written. I particularly liked the little boxes that gave more detailed math info. I also liked that the story was told through advances made on solving polynomial equations. Its pretty incredible how much of modern math owes to just trying to solve these sort of pointless and pure math problems. I also really liked that the author included abels paper in an appendix along with commentary. Clearly this book was a real labor of love, and the effort and interest of the author show themselves very clearly.
Ultimately being a math history book it is inevitably a bit dry, and much of the actual math done towards solving polynomial equations becomes very tedious and complicated very quickly. The author does a good job of showing just enough of this complication but not requiring the reader to plough through monstrous computations to understand the story.
The book itself is nicely written. I particularly liked the little boxes that gave more detailed math info. I also liked that the story was told through advances made on solving polynomial equations. Its pretty incredible how much of modern math owes to just trying to solve these sort of pointless and pure math problems. I also really liked that the author included abels paper in an appendix along with commentary. Clearly this book was a real labor of love, and the effort and interest of the author show themselves very clearly.
Ultimately being a math history book it is inevitably a bit dry, and much of the actual math done towards solving polynomial equations becomes very tedious and complicated very quickly. The author does a good job of showing just enough of this complication but not requiring the reader to plough through monstrous computations to understand the story.
July 20, 2023
An interesting history of the insolvability of the quintic. I was expecting a somewhat dry explanation of the proofs and instead read a pleasant history of how mathematical thought changed over time, the introduction of algebra into mathematics and more. The explanations of the proof were just accessible enough for me to understand, and the original proofs are included in the back of the book as well.
February 24, 2019
Best book I've found on Abel-Ruffini.
April 5, 2015
Fascinating.
An interesting history of the proof that there is no general solution of a fifth degree polynomial equation. The story extends from the discovery of irrational numbers by the Greeks, through the work of Lagrange, Ruffini, Abel and Galois.
Galois discovered the same proof but in complete generality. Consequently most algebra textbooks present Galois theory but not Abel’s proof. After studying Galois theory elsewhere, I was curious about Abel’s proof and found a good resource here.
Little is said about the impact this proof has had upon mathematics since Abel. For example, nothing is said about Lie groups, which provide a natural framework for analyzing the continuous symmetries of differential equations in much the same way as permutation groups are used in Galois theory for analyzing the discrete symmetries of algebraic equations.
The book has a few typographical errors and awkward wording in the mathematical explanations which make it difficult to follow some of the derivations, especially in the appendices.
An interesting history of the proof that there is no general solution of a fifth degree polynomial equation. The story extends from the discovery of irrational numbers by the Greeks, through the work of Lagrange, Ruffini, Abel and Galois.
Galois discovered the same proof but in complete generality. Consequently most algebra textbooks present Galois theory but not Abel’s proof. After studying Galois theory elsewhere, I was curious about Abel’s proof and found a good resource here.
Little is said about the impact this proof has had upon mathematics since Abel. For example, nothing is said about Lie groups, which provide a natural framework for analyzing the continuous symmetries of differential equations in much the same way as permutation groups are used in Galois theory for analyzing the discrete symmetries of algebraic equations.
The book has a few typographical errors and awkward wording in the mathematical explanations which make it difficult to follow some of the derivations, especially in the appendices.
March 27, 2012
It was always going to be a difficult project to write a short book about the non-existence of a quintic formula for a general audience. Peter Pesic has probably done as well as anyone could. The appendices offer the primary sources, and the notes include a lot of helpful references.
As far as I can see, the text is accurate, though translation from MathSpeak to English necessarily introduces some imprecision. The author introduces not only Abel's ideas but a fair amount of the history of algebra. I'd recommend the book to a non-mathematician particularly for that purpose -- to pass on the idea that mathematical symbolism is really technology, just as much as a calculator is.
As far as I can see, the text is accurate, though translation from MathSpeak to English necessarily introduces some imprecision. The author introduces not only Abel's ideas but a fair amount of the history of algebra. I'd recommend the book to a non-mathematician particularly for that purpose -- to pass on the idea that mathematical symbolism is really technology, just as much as a calculator is.
October 13, 2016
Very interesting read. One might read all the math involved if interested or just skip those bits and yet not lose the thread of the essay. I am now interested in reading some of the suggested references too!
June 2, 2016
Great read, easy to follow and with good details inside.
May 1, 2009
Really good!
April 21, 2017
Nonostante il libro inizi con la "solita storia" della matematica dall'antica Grecia fino agli italiani del tuo 1500, rischiando quindi di suonare un po' come "more of the same", evolve in una interessante e ben svolta discussione dei lavori di Abel e Galois e delle ripercussioni sul mondo della matematica e della fisica, facilmente comprensibili... almeno per un "matematico non specializzato" come sono io. Immancabile, anche se tostissimo, il testo della dimostrazione completa tradotto e commentato, che è stato molto piacevole da esplorare e "afferrare".
Displaying 1 - 13 of 13 reviews