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This book illustrates through five examples how certain square roots, which initially seemed impossible or meaningless, have been achieved. In each case the achievement has opened up a whole new vista of mathematics or physics. In each case our understanding and our intellectual power has been dramatically increased as a result. This is a maths book, a book about maths, and also in its later parts a physics book. But, undoubtedly because I am a physicist, even in the earlier parts it is the calculational power which the mathematics provides that interests me most, rather than mathematical rigour per se. These impossible square roots are astonishingly fecund, not just within mathematics itself but within physics too - and elsewhere. Whence cometh, as Eugene Wigner had it, this "unreasonable effectiveness of mathematics in the natural sciences"? It is as mysterious as ever. This is not a book which obsesses about mathematical pedantry. Despite treating some rather "simple" subjects, it is not a book about foundational issues either, though we will be obliged to touch briefly on such matters in chapter 1. In this book we will be heading in the opposite direction: the direction which is primarily about building ever richer mathematical structures. In addressing the square roots which form the principal subject of each of the book's five chapters, I take the opportunity to wax discursive on what springs from them. The subject matter of the book is not merely the roots, but the whole plant which grows therefrom. Well, not the whole plant, of course - that would hardly be possible. But sufficient, I hope, to give some indication of the unexpected luxuriance which unfolds from these roots. It is possible that much of the book will appeal to a general reader without (say) A Level knowledge of maths, though chapter five will be an exception. Certainly, the book is not intended for the expert, who would find the contents elementary and also a rather random selection (being chosen largely by the author's fancy from a virtual infinity of options). The likely appreciative audience (if I may presume that to be a non-null set) will be keen A Level students of maths or physics, or undergraduates of these subjects. I flatter myself that even postgraduate physicists may find things here to entertain or enlighten. Contents1. The Square Root of 22. The Square Root of -13. The Square Root of Probability4. The Square Root of Not5. The Square Root of Geometry12 Appendices

All "Western" societies have become deeply divided, and the divisions have become ever more acrimonious. The reason is that the differences of opinion stem from differences of moral perspective. Where existing works fall short is in explaining why and how this dichotomous morality has arisen, a question which is related to who benefits from it. One of the purposes of this book is to address that question: why has a radically different view of moral rectitude arisen and become so widespread? The answer cannot be morally neutral. Indeed, that moral neutrality is itself a moral error is central to the thesis presented, which is based upon the existence of a true, absolute or natural morality. The central thesis of the book focusses on "moral usurpation", a process of nudging popular opinion over many decades and targeting moral perceptions. The mechanisms deployed are familiar from PR campaigns or outright propaganda, but it is the targeting of the moral sensibilities which produces adherents who are so implacably hostile to contrary views. The cluster of social and political phenomena, the origin of whose peculiar rise to prominence the theory seeks to explain, is known by many names: Progressivism, Cultural Marxism, Collectivism, Identity Politics, Intersectional Feminism, Poststructuralism, Postmodernism, Critical Theory, Victimhood Culture, Social Justice (complete with Warriors), or simply "Woke", or perhaps just "the hard Left". With no pretence of neutrality, all these terms are replaced with one umbrella term: Destructivism, because their common feature is the undermining and destruction of Western culture - hence the book's title. It is not original to observe that annexing the moral high ground can provide a smokescreen for other purposes. But a further aspect of the thesis presented is that capturing ostensible moral peaks (even if fraudulently) functions as a source of social and political power. This has been too little appreciated, but is not new. It is pointed out that ruling elites have always needed to present themselves as upholders of moral principle, and one could always question the extent to which they truly lived up to such pretensions. However, a more virulent problem arises when the moral principles which the public endorse have been nudged away from their valid origins in the true morality. If public morals can be distorted into precepts which happen to favour the elites, then the elites can be relied upon to uphold them. Understanding our predicament requires appreciation that we are not dealing with a single phenomenon or a single group of people, but with an ecosystem of mutually supporting groups who trade between themselves in different forms of power. These forms of power are: money, civic and legislative power, the control of information, and moral standing. Unless the implicit power value of moral standing is appreciated, the overall ecosystem cannot be understood. But the ecosystem forms the Woke Industrial Complex (a term coined by Vivek Ramaswamy). However, the confected morality manufactured by the Destructivists is false, which is why this mutually self-supporting ecosystem of traded power is wreaking cultural destruction as it pumps ever more power to the triumvirate of power groups which are its higher socio-political animals. Identity Politics and allied moral corruptions are providing the perfect moral smokescreens behind which authoritarianism is advancing, which is why these issues are so very pressing.

This is not another quantum mechanics text on conventional lines. Most topics covered in undergraduate quantum physics courses are notable for their absence. Instead this book concentrates upon understanding the intriguing conceptual issues raised by quantum mechanics. It is neither a book for experts nor a "popular physics" book. It is intended for the serious student, but at undergraduate or early postgraduate level. The aim is to demystify ("unweird") quantum mechanics by analysing the many conundrums it presents in terms of the theory's own language, namely Hilbert space algebra. It should be regarded as complementary to standard treatments, not a replacement for them. A rapid reprise of the Hilbert space formalism of quantum mechanics is first presented and then applied to a wide range of topics including, (a) the reason why "which path" information destroys interference, (b) the reason why quantum states cannot be non-destructively copied, (c) what the uniquely quantum "observability of counterfactuals" means, (d) why constantly monitoring a pure quantum state can prohibit its evolution, (e) analyses of interference experiments involving entangled particles and so-called "delayed choice", "quantum erasure" and "delayed erasure" interference, (f) the quantum formulation of entropy, (g) the definition and quantification of entanglement in both pure and mixed quantum states, (h) the two convincing arguments against "hidden variables" in quantum mechanics, and hence the irreducibly indeterministic nature of quantum measurement, (i) what is meant by decoherence and how it can be quantified, (j) why big things tend not to be "quantum mechanical" - and why they sometimes are, and more often than you think, (k) a brief taster for quantum communication and quantum cryptography, and (l) quantum teleportation: what it is and what it isn't. Traditional topics are not ignored entirely. The book includes a rather idiosyncratic treatment of symmetry and conservation laws, the uncertainty principle, and a detailed chapter on algebraic methods in Hilbert space with which all physics students of a theoretical bent should be familiar. In particular the book places great emphasis from the start on appreciating the distinction between pure quantum states and mixed states, which avoids unnecessary confusion. By the end the reader should have a sensible, physicists', perspective on Schrodinger's cat and understand what exactly is meant by being able to extract the result of a quantum computer's computation without turning it on.