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

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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.

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?" got a real answer.

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.

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.

I'm a self-directed learner and had been struggling with generating interest in mathematics for some time. I had bought this book in the hopes that it would bridge my love and interest in logic with a new found interest in mathematics. It did exactly that.

Following his Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens of 1879, Gottlob Frege published Die Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung über den Begriff der Zahl in 1884, of which The Foundations of Arithmetic is a translation. In 1893 and 1903 he published respectively volume one and volume two of his Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet.

A translation of Frege's Begriffsschrifft is available in the source book From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 where it is entitled "Begriffsschrifft, a formula language, modeled upon that of arithmetic, for pure thought". His Grundgesetze der Arithmetik has not been fully translated into English, but a partial translation is available as Basic Laws of Arithmetic.

In The Foundations of Arithmetic [1884], Frege is concerned with the nature of number and the question of what could be the foundation of arithmetic. He begins with an overview and critique of the thinking of various mathematicians and philosophers on these issues and finds them all unsatisfactory. He then presents his own view.

Frege's position links numbers to concepts, which is rather like explaining one mystery through another. He asserts that "the Number which belongs to the concept F is the extension of the concept "equal to the concept F"", adding in a footnote that "I assume that it is known what the extension of a concept is". (pp. 79-80; section 68) [Die Anzahl, welche dem Begriffe F zukommt, ist der Umfang des Begriffes "gleichzahlig dem Begriffe F".]

From this he derives that "the proposition {the extension of the concept "equal to the concept F" is identical with the extension of the concept "equal to the concept G"} is true if and only if the proposition {the same number belongs to the concept F as to the concept G} is also true." (p. 80; I've used the curly brackets to quote propositions which Frege quotes either by offsetting or by quotation marks; they have nothing to do with sets.)

Note that Frege's definition of number (ignoring the translation's distinction between 'number' and 'Number') implies that the concept of number has a number belonging to it, and the number which belongs to the concept number is the extension of the concept "equal to the concept number". It's not evident that Frege has clarified the concept of number, especially since he assumes it is known what a concept is and what the extension of a concept is.

Frege continues by defining equality between concepts. Two concepts F and G are equal if and only if "there exists a relation ~ which correlates one to one the objects falling under the concept F with the objects falling under the concept G." (p. 85; I used ~ where Frege used the symbol for phi.) He needs to define equality because he's defined the number belonging to a concept F as the extension of the concept "equal to the concept F".

If F* represents the concept "equal to the concept F" then for any concept F, the extension of F* is the number belonging to F. So for any concept F, the concept of the relation of being equal to F is the concept F*, and the extension of F* is a number. (This notation is not Frege's but follows his meaning.)

Any relation ~ which correlates, one to one, the objects "falling under" a concept with the objects "falling under" another concept establishes the equality of those concepts. Equality in this sense is numerical and need not mean the objects or the concepts are identical, although it allows for identity since F~F and so every concept is numerically equal to itself. The number belonging to F is the extension of the relation F* of equality with F.

This equality between F and any concept G is defined as being a one to one correlation between "the objects falling under the concept F" and "the objects falling under the concept G". The extension of F* is then the number of objects "falling under the concept F" and also "falling under" any concept G that is equal (under F*) to F. This implies that F~G if and only if F*=G*, where = represents the identity of the extensions of F* and G*.

Any two concepts related to F are related to one another via F. If ~ and ^ are two relations of equality such that F~G and F^H then, abusing notation, G~F^H is a one to one relation between the objects of G and H via F. Since F* is the concept "equal to the concept F", any concept G that is equal to the concept F would presumably "fall under" F* and therefore be in its extension. But the extension of F* is the number belonging to F.

So it seems, unless Frege intends a distinction, that the extension of F* is both a number and all of the concepts equal to the concept F, including F itself. It's tempting to believe that Frege is is talking indirectly about what is now called an equivalence relation, with the number belonging to the concept F being the equivalence class of all concepts equal to F. But is he?

Proceeding to define specific numbers, Frege defines zero as "0 is the Number which belongs to the concept "not identical to itself"." (p. 87; section 74) [0 ist die Anzahl, welche dem Begriffe "sich selbst ungleich" zukommt.] (The German 'ungleich' could mean "not equal" and so Frege could intend that 0 is the number which belongs to the concept "not equal to itself". I suspect this is what he meant, since he's defined equality but not identity. It makes a difference, but to shorten the review, I'll follow the translation.)

If we refer to this concept "not identical to itself" as the concept Q, then 0 belongs to Q. So the concept "not identical to itself" is the concept Q, and the concept Q* is the concept "equal to the concept Q" which translates as Q* is the concept "equal to the concept "not identical to itself"". The extension of Q* is 0.

If we understand the extension of the concept Q* as the number of "objects falling under" the concept Q, then the only way this is going to come out right is if there are no "objects falling under" Q. This appears to mean that Q has no extension, or has a so-called empty extension. So does there exist a one to one relation # such that Q#Q? Even though Q is the concept "not identical to itself", Q as a concept is identical to itself (you can see Russell with his pipe and his paradox approaching briskly from just over the hill), and since Q is identical to itself it must be equal to itself, if identity implies equality. So # must exist.

However, Frege has defined equality between concepts as dependent upon a one to one relation, and if Q has no "objects falling under it" how can there be such a relation? Frege gets around this by asserting that any relation whatsoever suffices to demonstrate Q's equality with itself since "no object falls under it". (p. 89) Although Q has no extension, the extension of Q* is 0, and since Q#Q, it seems that Q "falls under" Q*.

Frege defines a successor n of a number m such that "there exists a concept F, and an object falling under it x, such that the Number which belongs to the concept F is n and the number which belongs to the concept "falling under F but not identical with x" is m". (p. 89; section 76)

Replace x above with 0. Replace F above with the concept "identical with 0". (p. 90) However, following Frege, replace 'falling under F but not identical with x' not with 'falling under the concept "identical with 0" but not identical with 0', but rather with 'identical with 0 but not identical with 0', which comes to the same thing, since whatever "falls under" the concept "identical with 0" is presumably identical with 0.

We then have: there exists a concept "identical with 0" and an object 0 falling under it, such that the number which belongs to the concept "identical with 0" is n and the number which belongs to the concept "identical with 0 but not identical with 0" is m. ["gleich 0 aber nicht gleich 0"] Frege tells us that 0 is the number which belongs to the concept "identical with 0 but not identical with 0", and that 1 is the number which belongs to the concept "identical with 0". (p. 90)

Q is the concept "not identical to itself". Q* is the concept "equal to the concept Q". The extension of Q* is 0, the number belonging to Q. The concept Q is empty.

If we represent the concept "identical with 0" by E, then E* is the concept "equal to the concept E". The extension of E* is the number belonging to E. Frege tells us that 1 is the number belonging to E. So the extension of E* is 1. The concept E is not empty.

The extension of Q* is 0. The extension of E* is 1. If the concept "identical with 0" is equal to 0 then E~Q* where ~ is a one to one relation. In modern set theoretical terms, it appears that Frege is almost saying that "0={}" and "1={{}}". If this is correct, then the object "falling under" the concept E is the number belonging to the concept Q. This makes sense of the assertion that E is the concept "identical with 0", as well as the relation E~Q*.

See the book for more details.

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."

*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.