I want to write a brief post on the Search for Mathematical Roots with some concluding thoughts on it. There is a lot in the book I didn’t talk about in my other posts on the book. I haven’t touched on the development of set theory in the book. I also haven’t talked at all about the disputes between the so-called part-whole theorists and set theorists. Before I get to the concluding thoughts, I want to talk about the role of the syllogism and the senses of logic in the book.

One trend that Guinness-Grattan goes over decently is the rejection of older syllogistic logic as exhausting logic. This comes to the fore in the sections on De Morgan. He was an early modern critic of logic as syllogism. He gave some inference forms that could not be captured in Aristotle’s syllogistic that were good inferences, such as: ears are parts of donkeys, so ears are parts of animals. Similar sorts of criticisms seemed to come out in the sections on algebraic logicians. One thing that I did not get from this part of the book is a sense of who held on to the syllogism as logic. It looks like mathematicians had abandoned it as constraining their arguments long before De Morgan. There were some mathematicians cited in the book as criticizing logic on the grounds that it does nothing for mathematics. I want to say that Cantor was among them, but I’m not sure. By the 19th century, syllogistic logic had developed beyond what was found in Aristotle, but Grattan-Guinness doesn’t indicate what the conception of it at the time was. This is important for seeing what exactly was being rejected.

Something that is not emphasized, though does receive mention in places, is the diversity of senses of logic in play and how they were narrowed down. There was logic in the sense of syllogism, in the sense of laws of thought, in the sense of good reasoning, in the sense of developing concepts, and in the sense of organizing science. Various logicians from Boole onward mixed several of these senses. Frege receives brief mention for criticizing the psychologistic takes on logic, although Grattan-Guinness does not make much of this. It is the Polish logicians, Lesniewski in particular I think, that get credit for a shift to logic being about a relation of implication rather than a psychological process of inferring, which were mixed in Russell. Of course, one need not take inference to be psychological, but such was the state of play apparently. There is a small note about reactions to this shift, as some philosophers were (are?) unclear about what logic is for if it is not for teaching us good reasoning practices, as this was traditionally taken to be a goal of logical study. Grattan-Guinness doesn’t make this explicit, but it gives one a sense of what was gained,and what given up, in the mathematizaton of logic, and a sharper view of what this means.

To close out the note on the Search for Mathematical Roots, I will say that while I have found it valuable, it is not for everyone. There is an abundance of uninteresting detail, with interesting detail mixed in. There is some philosophical commentary, though not a lot. The philosophical commentary on the figures I am most familiar with was for the most part poor, but a goal of the book was to emphasize the role of some logicians that do not receive much attention in the standard philosophical canon. Where the book really shines is the history and evolution of different strands of thought in the early development of logic, particularly how they were related to developments in the math of the time. Cantor, Russell, and Peano all receive fairly detailed discussions, with Russell receiving the most. The section on Hilbert, like the section on Polish logic, was short and would’ve benefited from more commentary. I had thought Hilbert was a bigger figure in the early development of logic than was indicated in the book. I hope to track down some books on the Polish logicians and Hilbert at some point.

I can recommend this book to anyone interested in the early development of modern logic, as long as one is okay with it being heavy on historical details and light on philosophical commentary. The writing can be a bit dry and the book is quite long, but it gives one a good sense of the developments of the times, which was, for me, quite valuable.

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