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

4.18  ·  Rating details ·  779 ratings  ·  34 reviews
The Foundations of Arithmetic is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly ...more
Paperback, 144 pages
Published December 1st 1980 by Northwestern University Press (first published 1884)
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Apr 12, 2007 rated it really liked it
Yeah, yeah, his theory of numbers doesn't really make any sense. But, whatever. You try to define a number.
Jun 02, 2011 added it
I have no recollection of having read this book but I have it on good authority that I did.
Jul 18, 2008 rated it really liked it
Russell's paradox be damned!
Jun 15, 2019 rated it liked it
Shelves: technical-10x
Bloody weird to slap a star rating on this, but there you go, welcome to where nothing is not rateable.
Aug 13, 2012 rated it it was amazing
Shelves: philosophy-read
Originally read many years ago. Michael Dummett cites its #62 as the "most pregnant philosophical paragraph ever written"!
Gerrit G.
A close study of Kant's first critique certainly helps understanding this text (which I lack and consequently do not understand some parts fully). If you come from modern mathematics you'll feel as if you are studying its very inception.
Dr. A
Read this and reviews of other classics in Western Philosophy on the History page of (a thinkPhilosophy Production).

Discovered and popularized posthumously by Bertrand Russell (see Russell and Alfred North Whitehead'’s Principia Mathematica ), Ludwig Wittgenstein, and others, Gottlob Frege is now known as the father of modern logic and of analytic philosophy, and The Foundations of Arithmetic is his most important work.

Frege examines the philosophical
Jun 13, 2013 rated it it was amazing
This was a fun read.

Unfortunately, Frege does some hand-waving throughout the book. He is attempting to show that arithmetic is an extension of logic. Critical to his argument is the definition of number as a concept expresssable by second-order classical logic. In showing his definition of number is in fact correct, he isolates number away from the realm of psychology. There are some minor points throughout the text in which he relies on the reader's intuition and some popular beliefs to 'wave
Alex Jones
Jan 23, 2014 rated it really liked it  ·  review of another edition
Read this for my philosophy of maths course and I really enjoyed it. Not just in an, for an academic book it was pretty readable and not too slow, kinda-way, but in an I'm a massive geek for all things mathematically philosophical or philosophically mathematical and genuinely liked reading it. As the book that led to most modern discussions within the area it is well worth reading, even if the system presented is as wrong as can be, there are still some salient points to be found.
Rick Sam
Jul 22, 2016 rated it really liked it
Shelves: history, philosophy
*Review coming soon*
Dec 26, 2007 rated it really liked it
Recommends it for: Philosophy Nerds
Perhaps some the clearest writing I've ever encountered.
Sep 21, 2008 rated it it was amazing
Written like a knife, this concise exposition of Frege's philosophy of mathematics still stands up well to scrutiny today.
Feb 24, 2019 rated it it was amazing
Shelves: math
Good work, if you've done mathematics you've probably seen the definition of number laid out a bit more clearly (in terms of sets instead of extensions of concepts, but close enough). However the really meaty part of the work is the engagement with philosophers (Leibniz, Mill, etc), and the descriptions of the mathematical difficulties faced along the way which create problems for formalists, etc. A great review of the failures of an older empiricism.
Adrien Mogenet
Mar 19, 2019 rated it liked it
The title is not misleading. This dives *extremely* deep into the foundations of arithmetic. Frege revisits a concept as simple as "Number", but this quickly escalates to something extremely hard to formally define. Throughout his set of axioms, he explores multiple angles and highlight the fallacious reasonings.
Joseph Madison
Sep 19, 2018 rated it it was amazing
Seminal work in the historical development of the logical foundations of mathematics. Frege has a very beautiful logical framework which is described and defended, only to be later dismantled by Russell.
Mar 25, 2017 rated it it was amazing
Agonistic philosophy is rare enough after the Greeks. Frege's bashing of J.S. Mill ranks right up there with Schopenhauer's sarcastic hostility toward Fichte and Hegel, and Nietzsche making fun of everyone (but especially Kant).
Zach Mazlish
Mar 04, 2019 rated it liked it
Didn't understand enough of this to rate it any higher, but still found it interesting/provocative. Don't have much to say as yet just because I don't know enough, but hopefully my thoughts will evolve as I learn more about it.
Walid Amoura
May 05, 2017 rated it it was amazing
Absolutely brilliant.
Jake Yaniak
May 04, 2018 rated it really liked it
Definitely the kind of book that makes you really think. In philosophy it is amazing just how much we take for granted. "What is a number?" such a simple question you would think...
Andrej Virdzek
Jun 27, 2017 rated it it was amazing  ·  review of another edition
If you ever thought that 19th century mathematics has gotten far enough with the constructions of real numbers out of rationals and its continuous zeal for the foundations of analysis, you have not read this one. Herr Frege tries to construct natural numbers out of logic. Whatever one thinks of this book or the importance of the project these days, as a philosophical discussion this has one of the greatest forms that I have witnessed in philosophy.

This one contains discussions of Kant and J.S.
Reinhard Gobrecht
Frege nennt in diesem Buch die folgenden wichtigen Grundsätze:
-Scharfe Trennung von Psychologischem und Logischem.
-Scharfe Trennung von Subjektivem und Objektivem.
-Die Bedeutung der Wörter ist im Satzzusammenhang zu erfragen.
-Der Unterschied zwischen Begriff und Gegenstand ist wichtig.
In diesem Buch unterscheidet Frege die Wahrheiten in analytische und synthetische und ferner in a priori und a posteriori Wahrheiten.
Was ist ein leerer Begriff?
Bejahung der Existenz als Verneinung der Nullzahl!
A very short, very concise, very dense attempt by Frege to address the business of logically deriving the primitive concepts of Peano. Not an inviting writing. I somewhat feel I'm not yet qualified to rate this work. It is nevertheless very entertaining & instructive to read him addressing the theories of others. Perhaps worthy of the hardship of braving through to the last page.
Steven Dunn
Jun 15, 2013 rated it really liked it
Shelves: philosophy
I bought the book at a used bookstore for 15$ (sort of ridiculous). But, after reading Anthony Kenny's exposition of Frege, I can say the book was interesting and not that much of a difficult read.

If you like mathematics, the theory of numbers and topics akin to the like (e.g. logic), then check out the book. If not,

then ignore the mundane review Frege gives to the concept of numbers.
Jun 25, 2015 rated it really liked it
It's hard to read it becouse of all philosophical nonsenses, but it gives you overall knowledge about understanding of natural numbers and Frege gives an amazing explanation about what he thinks that natural number is. I highly reccomend it but it's possible that you will dislikes some parts from the middle of book.
Aug 27, 2015 rated it it was amazing
Highly recommended as the first stop for anyone interested in the Philosophy of Mathematics or interested in understanding the potential relationships between mathematics and concepts.
Jul 21, 2016 rated it really liked it
Not qualified to give a book review.
Oct 16, 2008 rated it really liked it
Just reread this mornings in the bathroom, lovely for clearing my head, not much use otherwise. Still can't understand what those reducing math to logic folks were thinking.
Aria Mia
Apr 01, 2019 rated it it was amazing
World redefining scholarship. Wish this was taught in more philosophy classes At my school. Changed the way I look at philosophy.
Jun 13, 2012 rated it it was amazing
Necessary to understand the core of Ludwig Witggentein's masterpiece "Tractatus logico-philosophicus" who built on this previous, necessary work.
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Friedrich 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 ...more