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A review by taicantfly
The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number by Gottlob Frege
challenging
informative
medium-paced
3.25
From an objective standpoint this book is one of the most influential in all of modern philosophy. It's the catalyst for the linguistic turn, it's (arguably) one of the most novel and impressive philosophical inquiries into the epistemology and ontology of mathematics, it's the standard reference point for many students of the philosophy of mathematics (myself included, which is why I read it).
I recognise the risk of turning this review into simply an essay in which I disagree with the book. Fair enough. But genuinely it astounds me despite how innovative Frege's ideas were that a lot of the time he was just plain wrong, and when he wasn't plain wrong he was just unconvincing. Frege says that the axioms of geometry are synthetic and the axioms of arithmetic are analytic, but doesn't apply the methodology used to find the syntheticity of the axioms of geometry when it comes to analysing arithmetic. Frege says that Mill is wrong for his agglomerative account of number, but then radically misunderstands the Millian view in a way that is almost embarassing. Frege says that Hume's Principle is not a sound starting point for derivation but then picks something not only similar in nature but (as later shown) just paradoxical! Frege says that number is not a property of objects (like colour is), but doesn't recognise the intricacies of the application of colour terms function with the same complexities as the application of number terms!
The writing is at times clunky; I understand that I read a translation and that some of the flow is likely to be lost, but his Grundgesetze (at the time of writing this review I have only read the Grundgesetze partially) is much more elegant with symbolism and able to express his points in less ambiguous ways. It seems like this book was written more as a polemic, aimed at convincing those with some philosophical background that the Fregean view of mathematics was correct. But if this is the case and he wished to avoid overly technical content, Frege could have easily avoided clunky sections such as the section on parallel lines and the one on the infinitude of the naturals. If it was aimed as, to some extent, a genuine logical manual, then it just gets outshined by the Grundgesetze he writes later. And no work exists in a vacuum, so the fact he was able to express his ideas so much more effectively just makes this seem a little disappointing in comparison. But there would be no Grundgesetze without this - so it holds up as a titan even with its flaws.
This was a very frustrating read as someone who aligns themself more with empiricism. However the fact it found ways to annoy me in about ten different related topics is probably a sign of its wide-ranging merit as a text. Overall I have very mixed feelings; I would recommend it, but I would recommend it largely because a) it's very necessary context for the modern discussion and b) it gives you free reign to be a hater.
I recognise the risk of turning this review into simply an essay in which I disagree with the book. Fair enough. But genuinely it astounds me despite how innovative Frege's ideas were that a lot of the time he was just plain wrong, and when he wasn't plain wrong he was just unconvincing. Frege says that the axioms of geometry are synthetic and the axioms of arithmetic are analytic, but doesn't apply the methodology used to find the syntheticity of the axioms of geometry when it comes to analysing arithmetic. Frege says that Mill is wrong for his agglomerative account of number, but then radically misunderstands the Millian view in a way that is almost embarassing. Frege says that Hume's Principle is not a sound starting point for derivation but then picks something not only similar in nature but (as later shown) just paradoxical! Frege says that number is not a property of objects (like colour is), but doesn't recognise the intricacies of the application of colour terms function with the same complexities as the application of number terms!
The writing is at times clunky; I understand that I read a translation and that some of the flow is likely to be lost, but his Grundgesetze (at the time of writing this review I have only read the Grundgesetze partially) is much more elegant with symbolism and able to express his points in less ambiguous ways. It seems like this book was written more as a polemic, aimed at convincing those with some philosophical background that the Fregean view of mathematics was correct. But if this is the case and he wished to avoid overly technical content, Frege could have easily avoided clunky sections such as the section on parallel lines and the one on the infinitude of the naturals. If it was aimed as, to some extent, a genuine logical manual, then it just gets outshined by the Grundgesetze he writes later. And no work exists in a vacuum, so the fact he was able to express his ideas so much more effectively just makes this seem a little disappointing in comparison. But there would be no Grundgesetze without this - so it holds up as a titan even with its flaws.
This was a very frustrating read as someone who aligns themself more with empiricism. However the fact it found ways to annoy me in about ten different related topics is probably a sign of its wide-ranging merit as a text. Overall I have very mixed feelings; I would recommend it, but I would recommend it largely because a) it's very necessary context for the modern discussion and b) it gives you free reign to be a hater.