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Danielle Macbeth's book tells a thorough and accessible history of the development of Frege's logic which builds up a novel interpretation of that logic. The work is the result of her years as a professor of analytic philosophy at Haverford College.

Traditional interpretations of Frege's work typically dismiss his logic as a failed attempt to do what Russell later did properly (even though neither Frege nor Russell appear to have believed such a thing). Frege's notation has several oddities, most prominently its 2-dimensional notation (rather than the 1-dimension of text). The traditional claim that Frege was trying to achieve the same ends as Russell do not account for these features of his logic. Such accounts also fail to explain many of the claims Frege made in letters he exchanged with Russell (and others) during the logic's development. Many of these claims are quite surprising and make no sense if you try to force Frege into the mould of Russell's logic.

Macbeth provides an alternative interpretation of Frege's logic, which accounts for its notation, Frege's claims about it, and the history of its development. She gives an integrated account of the development of the logic, and how it coincided with other developments in mathematics and logic. Frege's logic is actually quite interesting, and assigns subtly different meanings to logical implication and to the existential quantifier as compared to Russell's logic.

Her work is presented as a historical story, not a logic textbook, and should be read from the beginning rather than in fragments. She does not presume a knowledge of relevant history, philosophy, or logic -- she does an excellent job of explaining precisely those pieces that you need to to know in order to understand her explanation. Concequently, the book may omit your favorite piece of logical history, although I found it quite a relief to read a focused account rather than one which attempted to cover everything. As a result, it is accessible to readers without extensive background in the area, while still accurate and scholarly. It has depth rather than breadth.

Two important books have been published in the last fifteen years, both of which have implications for any discussion of Frege: A. Garciadiego, BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC PARADOXES (1992), and I. Grattan-Guinness, THE SEARCH FOR MATHEMATICAL ROOTS (2000).

Macbeth's book refers to neither of these works. The result is that she marginalizes her own book. Most glaringly, she grants some logical content to Russell's paradox, and so messes up a discussion of the famous interaction of Russell and Frege. This is one historical episode you don't want to get wrong, but she does. She hasn't read Garciadiego, so she doesn't realize that Russell's "paradox" is a meaningless formulation, devoid of logical content.

Because her work is not grounded in math history at all, she also fails to put Frege's ideas in context. Whatever the different ideas of Dedekind, Russell, Frege or Cantor, they all come out of a very ancient mathematical school of "natural" mathematics. This approach--which never manages to form a logical part of any mathematical argument--seeks to "avoid" or "solve" paradoxes. Macbeth has missed the bus, in that she isn't aware that the most important work going on now in philosophy and mathematics is the reexamination of these paradoxes. Garciadiego famously began this work, drawing up a "list" in his book of nonexistent paradoxes, Russell's, Richard's, Burali-Forti's, Cantor's. Among other things, it has destroyed Godel's theorem, which relies--for its distinction between truth and provability--on Richard's paradox having at least some logical content. Richard's paradox has no logical content whatsoever.

This movement has proved so disturbing that it has brought substantive work to a halt, in both philosophy and mathematics, while we see where we are (if we're anywhere!).

And math and philosophy are not the only disciplines which have been brought to a halt. "Natural" mathematics lies at the heart of Sraffa, Kimura, chemistry (in ways which are just beginning to be revealed) and even Einstein, who bought into the "natural" math polemics of a book for which he had high regard, Poincare's SCIENCE AND HYPOTHESIS. Where is it in the relativity of simultaneity? Here:

Up to now our considerations have been referred to a particular body of reference, which we have styled a 'railway embankment.' We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated....People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises: Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A -> B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M' be the mid-point of the distance A -> B on the travelling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M' naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train.

This translation is accurate (the French and Italian are not). Einstein really does say "fallt zwar...zusammen." That is, he says that one point "naturally" coincides with another. The "naturally" reveals the intuitionist expression of the concept, for it reflects the belief that the formulations of geometry do not express facts.

Obviously, the logical problem with it is that, regardless of what Einstein may "feel" about mathematical expressions, nowhere in Einstein's writings--either in the 1905 papers or after--is any meaning assigned to 'naturally.' The failure to do so, destroys the idea, and it is easy to see why. If we retain the concept without meaning there is no logical basis on which to proceed beyond it. If we eliminate it, we wind up with a contradiction: the two assumed coordinate systems collapse into one. What is more, when we place this train experiment next to the various other thought experiments, we see that they are simply translations of the same problem into other terms, just as the false 'paradoxes' turn out to be subject to the same problem Richard indicated (reference to an infinite domain which destroys the meaning). In special relativity, natural coincidence can only be defined by infinitely many words. So the distinction collapses.

Macbeth knows none of this. Typical scholar so deeply into her subject that she loses track of her object.

Of course I didn't see the author's preface: "Although this is a book about Frege's logic, it is not a work of logic in any standard sense....My aim is rather to develop a novel reading of Frege's logical language...." With this in mind the title of her book should have been "Frege's logic: an examination of his notation." I'm a third of the way in and all I get is information on Frege's bizarre logic notation. I'm no scholar of logic. I want to understand Frege's logic as it relates to his contribution to the development of symbolic logic over the years. I wanted to know a bit about his influence on Godel and such. One book I have says that although Russell found a problem in Frege's work, which Frege found totally demoralizing, there are various "fixes" for what they call the Russell Antimony. Okay, what are these "fixes"? I won't find out in this book, I'm sure.