The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number

by Gottlob Frege

Book Overview

Part of the "Longman Library of Primary Sources in Philosophy," this edition of Frege's Foundations of Arithmetic is framed by a pedagogical structure designed to make this important work of... This description may be from another edition of this product.

• Format: Paperback
• Language: English
• ISBN: 0810106051
• ISBN13: 9780810106055
• Release Date: December 1980
• Publisher: Northwestern University Press
• Length: 144 Pages
• Weight: 0.45 lbs.
• Dimensions: 0.4" x 5.5" x 8.5"

Customer Reviews

Foundations of Arithmetic

This subtitle, "A Logico-mathematical Enquiry into the Concept of Number," indicates very well the nature of the work. The first three quarters of the book are devoted to a critical analysis of the idea of previous writers (Kant, Leibnitz, Grassmann, Mill, Lipschitz, Hankel, Jevons, Cantor, Schröder, Hobbers, Hume, and others) on the subject of number, and Frege does not find the ideas of any of these philosophers and mathematicians entirely satisfactory. His conclusions is "that a statement of a number contains an assertion about a concept," and his definition of number is: The number which belongs to the concept F is the extension of the concept "equal to the concept F." Frege regards the number zero as belonging to to the "natural" or "counting" numbers, whereas we subscribe to the view that zero is not a counting number at all (the first of the counting numbers being 1) and is only properly used when we regard a number as a "relative-magnitude," zero being the relative-magnitude of two equal counting numbers. This work of Frege's has considerable historical interest as a forerunner of the work of Whitehead and Russell. The translation is excellent and the printing leaves nothing to be desired.

The first escape from the Elencus...

You know how _frustrating_ it is, reading a platonic dialog? Some question like "What is virtue?" or "What is justice" is asked, and Socretes goes on for pages showing that the so-called "experts" don't have a clue about what it really is? But what's _really_ frustrating is that you're all expecting, at the end of the dialog, after following a hard line of argument, that you'll be rewarded with THE definitivie definition of 'virtue' or 'justice' or whatever--only to be disapointed. All you get in the end is a new appreciation of your own hopeless ignorance... ...well, imagine a platonic dialog which started the same as any other platonic dialog, but with the question "What is a number?" Only this time, at the end of the dialog, you actually get an answer to the question? In retrospect, its pretty amazing that Plato didn't write a Socratic dialog concerned with the question "What is number?' After all, Plato considered numbers more real than physical objects, and people like the Pythagorians were going around claiming that everything _was_ made out of numbers. But what the heck _is_ a number, anyways? Perhaps the reason was that everybody thought they already understood what numbers were. But Frege, like Socretes before him, realized that this so-called knowledge was really just a collective ignorance. So Frege starts out this book with a thorough, merciless review of what his coleages and predicessors were saying about what numbers were, showing that they ranged from cocksure to confused, from pompously-wrongheaded to just plain silly. But then Frege does something really amazing--for the first time in history, he goes on give a real answer to the question "what are numbers?" Building on the work of Hume, he gives a sustained argument now known as "Frege's theorem" which shows how numbers can be grounded on an understanding of one-to-one correspondence. Unfortunately, this work had to wait almost a century for the rest of us to really catch up to its significance. Russell found a contradiction in the arguments presented here, and for the next 80 years attention shifted elsewhere. But first Charles Parsons, in 1964, and then Crispen Wright and others in the 80's and 90's begain to realize that Frege's theorem could be reconstructed without the paradox. This sparked a whole flurry of neo-Fregean studies which is one of the most active branches of analytic philosophy today. This revival means that Frege's importance, and the importance of reading and comming to grips with the arguments presented by Frege in this book, are going to continue to grow. Although tragically Frege didn't live to see the day, we now realize that the line of reasoning he followed in this book was one of those signature moments in human history, every bit as profound as the invention of the wheel or the discovery of the pythagorian theorem--it was the moment where, for the first time ever, the question "what the heck _are_ numbers, anyways?" g

Excellent work

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."

great work

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.

A Must for Any Philosopher of Mathematics

This book written by Gottlob Frege is one of the most influential books of the 20th century philosophy of mathematics. In here Frege establishes the nature of arithmetics as founded in logic, which is his logicist proposal. For that, he refutes the assertion that logic as such is founded on psychology.Sometimes he distorts a little bit what others say about logic, so he argues against those thinkers more effectively. In here he establishes the anti-psycology difference between concept and object; though he has not made a difference yet between sense and reference. He also refers to a principle called the contextual principle, in which the word makes reference to something depending on the context. Afterwards after he wrote the book, he would reject this principle, because of his doctrine of sense and reference: the sense of the words determine the sense of the sentence; and the reference of the words determine the reference of the sentence.This is a great philosophical work, and I would suggest it to anyone who is starting to study Analytic philosophy (philosophy of mathematics, logic and language), and also those who want to consider the platonist proposal.