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As a family life educator for over 25 years, I also heard numerous parents ask how to keep their love alive with toddlers running around and little money for babysitters. In my mind "Rekindle the Passion" passes both the professional and personal test of a book well worth reading. If for no other reason than just chuckling over the short quotes at the beginning of each chapter, it's worth the money.
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The Iwasawa and Bruhat decompositions and the Weyl group construction are shown to hold for non-compact groups in chapter 5. The Borel-Weil theorem is proven for compact connected Lie groups using the results of the chapter. The Harish-Chandra decomposition fo linear connected reductive groups is proven in chapter 6. The author shows clearly the role of holomorphic representations in obtaining this result and the construction of holomorphic discrete series. The principal series representations of SL(2, R) and SL(2, C) are use to motivate the notion of an 'induced representation" in chapter 7. The theory of induced representations involves the Bruhat theory and its use of distribution theory, and relates via the 'intertwining operators', irreducible representations of two subgroups.
The author discusses the notion of an admissible representation in chapter 8, which are representations on a Hilbert space by unitary operators and each element in K has finite multiplicity when the representation is restricted to K. Equivalence of admissible representations are discussed via the concept of an "infinitesimal equivalance", which is the usual notion if the representation is unitary and irreducible. The Langlands classification of irreducible admissible representations is discussed in detail. The Langlands program shows to what extent irreducible admissible representations of a group are determined by the parabolic subgroups. The construction of discrete series, used throughout the proof of the Langlands classification, is then done in detail in the next chapter. Ths concept of an admissible infinitesimally unitary representation plays particular importance here. Here the representation operators act like skew-Hermitian operators with respect to an inner product on the space of K-finite vectors. If one reads this chapter from a physics perspective, the representations constructed using discrete series are somewhat 'exotic' and will probably not enter into applications, in spite of the fact that physical considerations do dictate sometimes the use of noncompact groups.
Chapter 10 addresses the question as to the completeness of irreducible admissible representations using discrete series. If there not enough discrete series representations this will show up in the Fourier analysis of square integrable functions on the group. In the compact case, Fourier analysis proceeded via the characters of irreducible representations. The author shows how to do this in the noncompact case via 'global characters' of representations, which are well-behaved generalizations of the compact case. The well-behavedness of global characters comes from their being of trace class, with the result of the trace being a distribution. The author gives explicit formulas for the case of SL(2, R), and shows hows differential equations can be used to limit the possibilities for how characters behave. In fact, the author shows to what extent characters are functions, proving that the restriction of any irreducible global character of G to the 'regular set' is a real analytic function.
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Well balanced, insightful review.
This book includes arguements against extreme and moderate scepticism. On the first of these scepticisms his arguements are cogent. He does not, however, overcome the force of Hume's arguements for moderate scepticism.
Next, Meynell explains his version of the correspondence theory of truth, based on how we come to know. Sense-data are taken to be metaphysical simples, from which we articulate intelligable theories about the world. These theories approximate reality, Meynell claims, because they can predict how things would have been if they were true. Since we may combine this feature of expirimentation with intelligent thinking and our experience of a designed world we can have knowledge about the world.
Although Meynell does not state Theism at the beginning of the book his theory does not seem plausable without it, due to the fact that his realism is permeated with idealism. If there were no Necessary being what reason do we have to hold that the world, reached through our experience of knowing, is a world for knowing? In short, Meynell's world looks like a world made to be known. His order of presentation leads me to think that he wanted to show the reverse of this point: that because the world is knowable it is made.
Meynell then claims that this view of correspondence implies a radical Cartesian internalism and a Thomistic metaphysics of immaterial essences along with mind\body interactionist deulism. By the end of the book his realism surely does require these positions, however I cannot see how Meynell arrives at his Thomistic conclusion from his initial chapterrs on truth, reality and data. It seems to me that Meynell's love for God makes him exagerate the implications of his premises. Naturally, my claim here instanciates both the fallacy of "to the man" and the so-called "genetic fallacy." Here I hope not to argue against Meynell, but simply to give my impression of his book.
I also think he fails to explain the metaphysics of causation, in most of the particular details of his Lonerganian philosophy.
This book is fun and worth reading because of its broad outlook and its innocent, though not uninformed, and speculative view of philosophical problems. Indeed, correspondence theories of truth are not dead, as long as we have articulate and acute Englishmen like Meynell to endorse them. Meynell also seems like a good person.