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Title: The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number by Gottlob Frege, J. L. Austin ISBN: 0-631-12694-5 Publisher: Basil Blackwell Pub. Date: 01 January, 1980 Format: Paperback Volumes: 1 List Price(USD): $19.95 |
Average Customer Rating: 4.5 (4 reviews)
Rating: 4
Summary: Frege, You're Not Supposed To Have...
Comment: *The Foundations of Arithmetic*, one of the most durable works of philosophy of mathematics ever produced, is something of a curiosity as presented by J.L. Austin (who translated the work for the use of an Oxford undergraduate course); and perhaps Frege's platonism got the best of Austin, and this work is really just as , well, Kantian as it appears, "a good sight" more Kantian than "standard" Frege is typically allowed to be. Frege's definition of number in terms of equipollence (one-one correspondence of sets) is legendary: that is to say, it is traditionally understood to do a great deal more work than the "thin" version allowed by mathematical logic as reconstructed to avoid Russell's paradox.
But here Frege's work-up of the concept for a general readership is so "genteel" as to suggest that this may not in fact be the case, and that Frege actually partook more heavily of Neo-Kantian bromides than his *theory of arithmetic* suggests; to wit, that this theory was always intended to be situated within a general philosophy of mathematics obeying the strictures of reasoning involving Kantian "intuition" (as is typically said of Frege's last efforts in the field). As such, it would be unfortunate that we cannot effectively read this book (formerly available *en face*, and unfortunately much the worse for the original's omission) in conjunction with its contemporary geometrical counterpart: long out of print, rarely making its way into the philosophical Frege literature, and perhaps in all parts an *anticipatory* if "crochety" rebuke to Hilbertian formalism.
Perhaps Frege was to a certain extent wholly other than the mathematics of his time; perhaps we are not well-served by a Frege "out of time"; we certainly have one of the great prose stylists of English on hand here, and perhaps it would actually do to consider his aptitude for "gold" extraction here as a clue to puzzling out the rest of Frege -- a figure supremely unconcerned with sameness of meaning, and already owing a certain debt to those para-philosophical figures all his work is at cross-purposes with (the German '70s having been quite a time indeed). A great help to understanding number theory, a marvelous thing for a library to have.
Rating: 4
Summary: Excellent work
Comment: His conclusion (p.99e) is that the laws of arithmetic are analytic judgements and consequently a priori.
Note that he is very consistently hard on Mill.
Some interesting quotes: p. 115e #106. "...number is neither a collection of things nor a property of such, yet at the same time is not a subjective product of mental processes either, we concluded that a statement of number asserts something objective of a concept.
... (p. 116e) We next laid down the fundamental principle that we must never try to define the meaning of a word in isolation, but only as it is used in the context of a proposition: only by adhering to this can we, as I believe, avoid a physical view of it.
#107. (p.117e) "A recognition statement must always have a sense."
Rating: 5
Summary: great work
Comment: possibly one of the greatest works in history of philosophy and the founding book of 20th century analytic philosophy... I read it only once and a better appraisal will be coming shortly..I can say right away this is not simply a 'technical' work in philosophy of mathematics but a broad although short philosophical investigation in notions of truth, meaning and identity - although it expressly deals with defining numbers in purely logical terms. continental philosophers who read this work might change some of their negative ideas about where analytic philosophy is coming from.
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Title: The Principles of Mathematics by Bertrand Russell ISBN: 0393314049 Publisher: W. W. Norton & Company Pub. Date: 01 February, 1996 List Price(USD): $17.95 |
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Title: The Frege Reader (Blackwell Readers) by Michael Beaney, Michael Beaney , Gottlob Frege ISBN: 0631194452 Publisher: Blackwell Publishers Pub. Date: 01 May, 1997 List Price(USD): $37.95 |
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Title: Introduction to Mathematical Philosophy by Bertrand Russell ISBN: 0486277240 Publisher: Dover Publications Pub. Date: 14 September, 1993 List Price(USD): $10.95 |
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Title: On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Gödel ISBN: 0486669807 Publisher: Dover Publications Pub. Date: 01 February, 1992 List Price(USD): $6.95 |
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Title: Principia Mathematica to *56 (Cambridge Mathematical Library) by Alfred North Whitehead, Bertrand Russell ISBN: 0521626064 Publisher: Cambridge University Press Pub. Date: 11 September, 1997 |
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