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Reading Frege's Grundgesetze
Gottlob Frege's Grundgesetze der Arithmetik , or Basic Laws of Arithmetic , was intended to be his magnum opus, the book in which he would finally establish his logicist philosophy of arithmetic. But because of the disaster of Russell's Paradox, which undermined Frege's proofs, the more mathematical parts of the book have rarely been read. Richard G. Heck, Jr., aims to change that, and establish it as a neglected masterpiece that must be placed at the center of Frege's philosophy.
Part I of Reading Frege's Grundgesetze develops an interpretation of the philosophy of logic that informs Grundgesetze , paying especially close attention to the difficult sections of Frege's book in which he discusses his notorious 'Basic Law V' and attempts to secure its status as a law of logic. Part II examines the mathematical basis of Frege's logicism, explaining and exploring Frege's formal arguments. Heck argues that Frege himself knew that his proofs could be reconstructed so as to avoid Russell's Paradox, and presents Frege's arguments in a way that makes them available to a wide audience. He shows, by example, that careful attention to the structure of Frege's arguments, to what he proved, to how he proved it, and even to what he tried to prove but could not, has much to teach us about Frege's philosophy.
Part I of Reading Frege's Grundgesetze develops an interpretation of the philosophy of logic that informs Grundgesetze , paying especially close attention to the difficult sections of Frege's book in which he discusses his notorious 'Basic Law V' and attempts to secure its status as a law of logic. Part II examines the mathematical basis of Frege's logicism, explaining and exploring Frege's formal arguments. Heck argues that Frege himself knew that his proofs could be reconstructed so as to avoid Russell's Paradox, and presents Frege's arguments in a way that makes them available to a wide audience. He shows, by example, that careful attention to the structure of Frege's arguments, to what he proved, to how he proved it, and even to what he tried to prove but could not, has much to teach us about Frege's philosophy.
316 pages, Hardcover
First published November 29, 2012
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Displaying 1 - 3 of 3 reviews
May 4, 2014
Gottlob Frege is a seminal figure in the history of the foundations of mathematics and mathematical logic. He made the first real advance in logic since Aristotle two millennia before, and he was the first to make a serious attempt at developing a rigorous foundation for mathematics. Frege believed in logicism, according to which mathematics is nothing but a branch of logic. The idea is developed in meticulous detail in Frege's magnum opus, "Die Grundgesetze der Arithmetik" (The Basic Laws of Arithmetic), in which he attempts to derive the axioms of arithmetic from logic.
Frege's work was profoundly influential, and pretty much all later work in set theory and logic built upon his attempt. Yet Frege's magnum opus was left neglected and untranslated for over a century, for a couple reasons. First of all Bertrand Russell found a flaw in Frege's system (Russell's paradox), so for a long time people just saw Frege's book as a waste. Later generations, however, have found ways to repair Frege's original system, and they've found that his book still contained important insights on the philosophy of language, for instance, even if his mathematical system contains a flaw. But the more important reason is that Frege wrote out his logical reasoning not in symbolic form like we do today, but rather in a strange diagrammatic notation that's unwieldy and hard to decipher if you're not accustomed to it. So most philosophers and even mathematicians were not willing to take the effort to learn his notation, even if it meant they'd be missing out on some interesting ideas.
But people have come to realize that Frege is an insightful enough thinker that it's worth putting up with the awkwardness of his mathematical notation, so finally an English translation of Frege's Grundgesetze was released a couple months ago, published by Oxford University Press and written by a serious team of scholars. Although I'd love to, I haven't read it yet, because the notation still makes it intimidating. In the meantime, however. I have the next best thing: Richard Heck's book "Reading Frege's Grundgesetze". Heck has been studying Frege's book for the past two decades (when it was still in German), and this book is a compilation of his papers which try to make sense of Frege's work. These papers are extremely enlightening, because they translate Frege's diagrammatic notation into a form that modern students of logic can understand.
Another good feature is that Heck timed the release of his book to coincide with the release of the English translation of Frege's Grundgesetze, so that he could quote and make reference to the translation. If and when I do decide to tackle the new translation, Heck's book should help me make sense of some of the obscurity of Frege's formal system, so I can grasp the philosophical content of his work.
Frege's work was profoundly influential, and pretty much all later work in set theory and logic built upon his attempt. Yet Frege's magnum opus was left neglected and untranslated for over a century, for a couple reasons. First of all Bertrand Russell found a flaw in Frege's system (Russell's paradox), so for a long time people just saw Frege's book as a waste. Later generations, however, have found ways to repair Frege's original system, and they've found that his book still contained important insights on the philosophy of language, for instance, even if his mathematical system contains a flaw. But the more important reason is that Frege wrote out his logical reasoning not in symbolic form like we do today, but rather in a strange diagrammatic notation that's unwieldy and hard to decipher if you're not accustomed to it. So most philosophers and even mathematicians were not willing to take the effort to learn his notation, even if it meant they'd be missing out on some interesting ideas.
But people have come to realize that Frege is an insightful enough thinker that it's worth putting up with the awkwardness of his mathematical notation, so finally an English translation of Frege's Grundgesetze was released a couple months ago, published by Oxford University Press and written by a serious team of scholars. Although I'd love to, I haven't read it yet, because the notation still makes it intimidating. In the meantime, however. I have the next best thing: Richard Heck's book "Reading Frege's Grundgesetze". Heck has been studying Frege's book for the past two decades (when it was still in German), and this book is a compilation of his papers which try to make sense of Frege's work. These papers are extremely enlightening, because they translate Frege's diagrammatic notation into a form that modern students of logic can understand.
Another good feature is that Heck timed the release of his book to coincide with the release of the English translation of Frege's Grundgesetze, so that he could quote and make reference to the translation. If and when I do decide to tackle the new translation, Heck's book should help me make sense of some of the obscurity of Frege's formal system, so I can grasp the philosophical content of his work.
June 16, 2013
Heck's primary argument through the book is that a substantive reading of Grundgesetze requires a careful examination of the proofs, and while he does offer some general conceptual sketching in accessible terms in the first few sections, once you get beyond chapter 5 or 6 it becomes unreadable without serious technical proficiency.
One of the notes in the early sections in that the loss of emphasis on the proofs in Frege is no doubt in part due to the obscurity of Frege's notation. Heck, as a result, translates most of Frege's arguments into the contemporary notation. For that alone, it seems that this is a substantial accomplishment in studying Frege.
I'm working hard, personally, to understand Frege because I think it's necessary to understanding good philosophy of mathematics. Heck argues in some sections for the relevance of Frege to later theories (particularly Tarski's theory of truth) and these arguments are good reinforcement for that view. That said, Heck is not attempting to provide either a companion to Grundgesetze or an introduction. The book is by a Frege scholar for logicians and philosophers of logic and mathematics.
Reading Frege's theory of arithmetic in isolation is a little odd, because its relation to a theory of language is important. I've always understood the two as intimately related, but Heck barely treats Frege on language at all, so that's an area that may require some filling out later. Overall, though, a good and fulfilling read.
One of the notes in the early sections in that the loss of emphasis on the proofs in Frege is no doubt in part due to the obscurity of Frege's notation. Heck, as a result, translates most of Frege's arguments into the contemporary notation. For that alone, it seems that this is a substantial accomplishment in studying Frege.
I'm working hard, personally, to understand Frege because I think it's necessary to understanding good philosophy of mathematics. Heck argues in some sections for the relevance of Frege to later theories (particularly Tarski's theory of truth) and these arguments are good reinforcement for that view. That said, Heck is not attempting to provide either a companion to Grundgesetze or an introduction. The book is by a Frege scholar for logicians and philosophers of logic and mathematics.
Reading Frege's theory of arithmetic in isolation is a little odd, because its relation to a theory of language is important. I've always understood the two as intimately related, but Heck barely treats Frege on language at all, so that's an area that may require some filling out later. Overall, though, a good and fulfilling read.
May 4, 2014
I’ve been eagerly (though sporadically) reading this work, so far up to chapter 9. The publication of the translation of the Grundgesetze itself has sidetracked me temporarily, but Amazon’s robots have been nagging me to review this on the grounds that no-one else has done so. So I’ll note the obvious, that this is a magisterial work of enormous benefit to anyone who is, or will be, reading one of the greatest achievements of the human intellect, and I’m outraged at the academic(s) who delayed it by telling Professor Heck that he wouldn’t get tenure if he worked on Frege. [footnote three of preface]
Much as I’m enjoying it, I do have a few presentational quibbles. The general comment is that I would have appreciated more foreshadowing; sometimes when the author makes a comment about the point to a proof I’ve just struggled through, I would have liked to know the point before the struggle, e.g., at the end of §6.7, and the top of p. 222. Slightly more substantively (but only slightly), the elucidation of (56) on p. 147 is in a slightly different order from the formula elucidated, transposing two conjuncts in the middle line. I’d also suggest the following index entries: <: 187; FA: 7, 11; Grace, Amazing: 200.
[Disclaimer: I was at Oxford with Professor Heck, but haven’t spoken with him since.]
Much as I’m enjoying it, I do have a few presentational quibbles. The general comment is that I would have appreciated more foreshadowing; sometimes when the author makes a comment about the point to a proof I’ve just struggled through, I would have liked to know the point before the struggle, e.g., at the end of §6.7, and the top of p. 222. Slightly more substantively (but only slightly), the elucidation of (56) on p. 147 is in a slightly different order from the formula elucidated, transposing two conjuncts in the middle line. I’d also suggest the following index entries: <: 187; FA: 7, 11; Grace, Amazing: 200.
[Disclaimer: I was at Oxford with Professor Heck, but haven’t spoken with him since.]
Displaying 1 - 3 of 3 reviews