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Thursday 27 August 2020

For more than a decade I've posted annual micro reviews of books I've read in the year in question. Lately I've read fewer books from cover to cover, dipping into quite a few books, and reading a lot of articles and pieces of escapist fiction on the Internet. Unsure how to record these increasingly haphazard activities, I stopped keeping a record in 2018. As I write this, in July 2020 I'm still unsure how to proceed, but am inclined to make comments on each book when I stop reading and replace it on the bookshelf, whether I've finished it or not. I may also refer to some material on the Internet.

Ayer provided a lucid review of various analyses of probability and its possible relevance to the confirmation of theories, pointing out difficulties in some widely held accounts.

He made several good points that I don't recall seeing anywhere else in the lierature. Reading the book has helped me sort out my own ideas.

This is another book I managed to finish, though, excluding preface and index, it is only 139 pages long

Published in 1982, seven years before Ayer's death, the book was intended as a sequel to Russell's *History of Western Philosophy*

Although I bought the book not long after it was published, until recently I hadn't done more than dip into it. Now that I'm tackling it seriously I find it's filling gaps in my knowldge of Philosophical developments in the first half of the last century. Although familiar with their names, I knew little about C. I. Lewis, Schlick, and Neurath apart from Lewis's work on modal logic, and the fact that Schlick, and Neurath were members of the Vienna Circle. I'd also been unaware that Godel and Quine had both attended meetings of the circle.

Ayer's discussion of Russell and Moore persuaded me that both have been overestimated. In my late teens and early twenties I was captivated by Russell, but I now realise that I let his lively and elegant prose style distract me from some very innaccurate reasoning. I'd gradually been coming to the conclusion that G. E. Moore was rather dull, and Ayer's discussion of him confirms that. Moore devoted most of his effort to the logical analysis of concepts, yet the only analysis offered was Russell's theory of descriptions. On the other hand Russell and Moore deserve credit for addressing genuine questions and avoiding the muddled wool gathering of F. H. Bradley, Prichard, and Cook Wilson.

I quite surpsised myself by finishing this book

Conway proposed generalising the idea of a Dedekind cut, to produce a much broader entity which he called a 'game'. Games he defined recursively, starting with the simplest game, called 'endgame' becuase there are no moves. One player has just won without either making any move.

The players are referred to as Left and Right, or sometimes Black and White. Left, or Black, always moves first. In endgame in which there are no moves Black is the winner. A Games may be described by two sets: {L | R} L is the set of moves available to player L, or Black and R is the set lf moves available to Right or white. Each time a move is made that converts the game into another.

Certain rather simple games are called 'numbers' for example Endgame is { | }, and is called 0, { 0 | } is called 1. Conway was able to produce analogues for all the real numbers, but was also able to generate the transfinite ordinals, and some infinitesimal numbers. An infinitessimal corresponds to a cut that defines a 'number' greater than zero, but less than every positive real number. It is impossible to produce such a definition with a Dedekind cut, but it is possible using Conway's more relaxed definition of a cut.

Conway called his number analogues 'surreal numbers'. They constitute a proper Class, not a set, and satisfy the field axioms. He called them the Class No, using the capitals to emphasize that they constite a Class, not a set.

I've found an summary of Conway's theory of surreal numbers. Although less comprehensive than the treatment in his *Numbers and Games* it is easier to follow and so a better starting point than the book.

First published in 1968 this volume was one of several that replaced G. H. Hardy's *Pure Mathematics* which had dominated the teaching of Analysis for several decades. My copy of Burkhill was given to me by a former student after he'd completed a course in 1978. I expect it has in turn been superceded by now, but it still gives me something to think about.

The Second course is a sequel to *A First Course in Mathematical Analysis* by J. H. C. Burkhill. The First Course was directed at first year undergraduates, and I found it easy reading. The Second Course generalised the treatment in the first course to deal with any metric space rather than just real numbers, and I find it quite challenging so I've started to make notes and to attempt some of the exercises.

I actually managed to read this book from cover to cover, something I rarely do these days

Quinton examined the question: what constitutes a 'thing' ? To what sorts of subjects can we refer?

Quinton makes a plausible case for a version of materialism.

In the course of the book he has something to say about most of the problems usually associated with Philosophy. I found that reading the book helped me to clarify my own ideas, though the book would be tough for someone with no prior knowledge of Philosophy. Some of Quinton's sentences are very complicated. More commas would help, but some of his long sentences would be easier to understand if replaced by several shorter ones.

I particularly liked:

"Malcolm's theory of dreams is the philosophical equivalent of the Charge of the Light Brigade"

and of Wittgenstein's *Philosophical Investigations*:

"The inviolate character of its theoretical virginity is fiercely protested by Wittgenstein's partisans, who regard his work as a mystery only to be understood by those who have undergone a complex ritual of initiation."