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Basic Laws of Arithmetic: Exposition of the System
lxiii, 144 pp., paperback, previous owner's name to title page, remainder marks to page edges else text clean & binding tight. *Buyer is responsible for any additional duties, taxes, or fees required by recipient's country* - If you are reading this, this item is actually (physically) in our stock and ready for shipment once ordered. We are not bookjackers.
- GenresPhilosophy
210 pages, Hardcover
Published January 1, 1964
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About the author
Gottlob Frege
126 books172 followersFriedrich Ludwig Gottlob Frege (German: [ˈɡɔtloːp ˈfreːɡə]) was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. While he was mainly ignored by the intellectual world when he published his writings, Giuseppe Peano (1858–1932) and Bertrand Russell (1872–1970) introduced his work to later generations of logicians and philosophers.
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October 8, 2024
A "FOUNDATIONAL" WORK IN MODERN LOGIC
Friedrich Ludwig Gottlob Frege (1848-1925) was a German mathematician, logician and philosopher, who was one of the founders of modern logical theory. Giuseppe Peano and Bertrand Russell introduced his work to later generations of logicians and philosophers.
Frege explains in his Introduction, "The ideal of a strictly scientific method in mathematics, which I have attempted to realize, and which might indeed be named after Euclid, I should like to describe as follows. It cannot be demanded that everything be proved, because that is impossible; but we can require that all propositions used without proof be expressly declared as such, so that we can see distinctly what the whole structure rests upon. After that we must try to diminish the number of these primitive laws as far as possible, by proving everything that can be proved. Furthermore I demand---and in this I go beyond Euclid---that all methods of inference employed be specified in advance; otherwise we cannot be certain of satisfying the first requirement." (Pg. 2)
He states, "It will be granted ... that the laws of logic ought to be guiding principles for thought in the attainment of truth, yet this is only too easily forgotten, and here what is fatal is the double meaning of the word `law.' In one sense a law asserts that is; in the other it prescribes what ought to be. Only in the latter sense can the laws of logic be called `laws of thought': so far as they stipulate the way in which one ought to think... laws of logic... have a special title to the name `laws of thought' only if we mean to assert that they are the most general laws, which prescribe universally the way in which one ought to think if one is to think at all...
"[B]eing true is different from being taken to be true... There is no contradiction in something's being true which everybody takes to be false. I understand by `laws of logic' not psychological laws of takings-to-be-true, but laws of truth. If it is true that I am writing this in my chamber on the 13th of July, 1893, while the wind howls out-of-doors, then it remains true even if all men should subsequently take it to be false. If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this that they have authority for our thought if it would attain to truth." (Pg. 12-13)
He adds, "The question why and with what right we acknowledge a law of logic to be true, logic can answer only by reducing it to another law of logic. Where that is not possible, logic can give no answer. If we step away from logic, we may say: we are compelled to make judgments by our own nature and by external circumstances; and if we do so, we cannot reject this law---of identity, for example; we must acknowledge it if we wish to reduce our thought to confusion and finally renounce all judgment whatsoever. I shall neither dispute nor support this view; I shall merely remark that what we have here is not a logical consequence. What is given is not a reason for something's being true, but for our taking it to be true." (Pg. 15)
He announces the purpose of his book as follows: "I sought to make it plausible that arithmetic is a branch of logic and need not borrow any ground of proof whatever from either experience or intuition. In the present book this shall now be confirmed, by the derivation of the simplest laws of Numbers by logical means alone. But for this to be convincing, considerably higher demands must be placed on the conduct of proof than is customary in arithmetic." (§0, pg. 29)
In Appendix II, he admits, "Hardly anything more unwelcome can befall a scientific writer that that one of the foundations of his edifice be shaken after the work is finished. I have been placed in this position by a letter of Mr. Bertrand Russell just as the printing of this [second] volume was nearing completion. It is a matter of my Basic Law (V). I have never concealed from myself its lack of self-evidence which the others possess, and which must properly be demanded of a law of logic.... I should gladly have relinquished this foundation if I had known of any substitute for it. And even now I do not see how arithmetic can be scientifically founded... unless we are allowed... the transition from a concept to its extension...
"But let us come to the point. Mr. [Bertrand] Russell has discovered a contradiction, which may now be set out. No one will want to assert of the class of men that it is a man. Here we have a class that does not belong to itself... Now let us fix our attention upon the concept `class that does not belong to itself.' The extension of this concept ... is accordingly the class of classes that do not belong to themselves.... Now let us ask whether this class C belongs to itself... a contradiction... Are we to suppose that the law of excluded middle does not hold for classes?... If these difficulties frighten us off from regarding classes (and hence numbers) as improper objects, and if we are nonetheless unwilling to recognize them as proper objects... then there is indeed no alternative but to regard class-names as pseudo proper names, which would thus in fact have no denotation." (Pg. 127-129)
He concludes, "Thus there is no alternative at all but to recognize the extensions of concepts, or classes, as objects in the full and proper sense of the word, while conceding that our interpretation hitherto of the words `extension of a concept' is in need of correction." (Pg. 130)
Persons wanting to work through Frege's actual book will need to get the German original, of course; but those of us simply wanting to understand something of the "idea" behind his thought, can benefit from reading this book.
Friedrich Ludwig Gottlob Frege (1848-1925) was a German mathematician, logician and philosopher, who was one of the founders of modern logical theory. Giuseppe Peano and Bertrand Russell introduced his work to later generations of logicians and philosophers.
Frege explains in his Introduction, "The ideal of a strictly scientific method in mathematics, which I have attempted to realize, and which might indeed be named after Euclid, I should like to describe as follows. It cannot be demanded that everything be proved, because that is impossible; but we can require that all propositions used without proof be expressly declared as such, so that we can see distinctly what the whole structure rests upon. After that we must try to diminish the number of these primitive laws as far as possible, by proving everything that can be proved. Furthermore I demand---and in this I go beyond Euclid---that all methods of inference employed be specified in advance; otherwise we cannot be certain of satisfying the first requirement." (Pg. 2)
He states, "It will be granted ... that the laws of logic ought to be guiding principles for thought in the attainment of truth, yet this is only too easily forgotten, and here what is fatal is the double meaning of the word `law.' In one sense a law asserts that is; in the other it prescribes what ought to be. Only in the latter sense can the laws of logic be called `laws of thought': so far as they stipulate the way in which one ought to think... laws of logic... have a special title to the name `laws of thought' only if we mean to assert that they are the most general laws, which prescribe universally the way in which one ought to think if one is to think at all...
"[B]eing true is different from being taken to be true... There is no contradiction in something's being true which everybody takes to be false. I understand by `laws of logic' not psychological laws of takings-to-be-true, but laws of truth. If it is true that I am writing this in my chamber on the 13th of July, 1893, while the wind howls out-of-doors, then it remains true even if all men should subsequently take it to be false. If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this that they have authority for our thought if it would attain to truth." (Pg. 12-13)
He adds, "The question why and with what right we acknowledge a law of logic to be true, logic can answer only by reducing it to another law of logic. Where that is not possible, logic can give no answer. If we step away from logic, we may say: we are compelled to make judgments by our own nature and by external circumstances; and if we do so, we cannot reject this law---of identity, for example; we must acknowledge it if we wish to reduce our thought to confusion and finally renounce all judgment whatsoever. I shall neither dispute nor support this view; I shall merely remark that what we have here is not a logical consequence. What is given is not a reason for something's being true, but for our taking it to be true." (Pg. 15)
He announces the purpose of his book as follows: "I sought to make it plausible that arithmetic is a branch of logic and need not borrow any ground of proof whatever from either experience or intuition. In the present book this shall now be confirmed, by the derivation of the simplest laws of Numbers by logical means alone. But for this to be convincing, considerably higher demands must be placed on the conduct of proof than is customary in arithmetic." (§0, pg. 29)
In Appendix II, he admits, "Hardly anything more unwelcome can befall a scientific writer that that one of the foundations of his edifice be shaken after the work is finished. I have been placed in this position by a letter of Mr. Bertrand Russell just as the printing of this [second] volume was nearing completion. It is a matter of my Basic Law (V). I have never concealed from myself its lack of self-evidence which the others possess, and which must properly be demanded of a law of logic.... I should gladly have relinquished this foundation if I had known of any substitute for it. And even now I do not see how arithmetic can be scientifically founded... unless we are allowed... the transition from a concept to its extension...
"But let us come to the point. Mr. [Bertrand] Russell has discovered a contradiction, which may now be set out. No one will want to assert of the class of men that it is a man. Here we have a class that does not belong to itself... Now let us fix our attention upon the concept `class that does not belong to itself.' The extension of this concept ... is accordingly the class of classes that do not belong to themselves.... Now let us ask whether this class C belongs to itself... a contradiction... Are we to suppose that the law of excluded middle does not hold for classes?... If these difficulties frighten us off from regarding classes (and hence numbers) as improper objects, and if we are nonetheless unwilling to recognize them as proper objects... then there is indeed no alternative but to regard class-names as pseudo proper names, which would thus in fact have no denotation." (Pg. 127-129)
He concludes, "Thus there is no alternative at all but to recognize the extensions of concepts, or classes, as objects in the full and proper sense of the word, while conceding that our interpretation hitherto of the words `extension of a concept' is in need of correction." (Pg. 130)
Persons wanting to work through Frege's actual book will need to get the German original, of course; but those of us simply wanting to understand something of the "idea" behind his thought, can benefit from reading this book.
March 14, 2025
Important to logic and the philosophy of mathematics but I have no idea why I read it, since Russell (and later Gödel) brought up inconsistencies and ultimately destroyed the logicism project.
Displaying 1 - 2 of 2 reviews