Three Roads To Quantum Gravity

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IN THE FUTURE WE SHALL KNOW MORE 31 called it other names. In its ®rst forms it was called `intuitionistic logic'. More sophisticated versions which have been studied more recently are known collectively as `topos theory'. As a mathematical formalism, topos theory is not easy. It is perhaps the hardest mathematical subject I've yet encountered. All of what I know of it comes from Fotini Markopoulou-Kalamara, who discovered that cosmology requires non-standard logic and found that topos theory was right for it. But the basic themes of it are obvious, for they describe our real situation in the world, and not only as cosmologists. Here in the real world, we almost always reason with incomplete information. Each day we encounter statements whose truth or falsity cannot be decided on the basis of what we know. And in the forms of our social and political life we recognize, often explicitly, that different observers have access to different information. We also deal every day with the fact that the truth or falsity of statements about the future may be affected by what we choose to do. This has very profound implications for a whole host of issues. It means that to judge the rationality of our decisions, we do not have to pretend that there is some supernatural observer who knows everything: it is enough to demand that the different observers report what they see honestly. When this rule is followed we discover that when we and another person each have enough information to decide whether something is true or false, we always make the same decision. Thus, the philosophers who attempted to ground ethics and science in the ultimate judgements of an all-knowing being were mistaken. We can live rationally without having to believe in a being who sees everything. We need only believe in the ethical principle that observers should communicate honestly what they see. If we stick to this, then the fact that there will always be questions that we cannot answer need not prevent us from coming to an agreement about how to understand those aspects of our world which we share in common. So topos, or cosmological, logic is also the right logic for understanding the human world. It, and not Aristotle, must be the right basis for economics, sociology and political
32 THREE ROADS TO QUANTUM GRAVITY science. I am not aware that anyone in these areas has taken up topos theory and tried to make it the foundation of their subject, although George Soros's approach to economics, which he calls the theory of re¯exivity, is certainly a start in the right direction. But we should not be surprised if both cosmology and social theory point us in the same direction. They are the two sciences that cannot be formulated sensibly unless we build into their foundations the simple fact that all possible observers are inside the systems they study.
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CHAPTER 11 ........................
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CHAPTER 12 ........................
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THE HOLOGRAPHIC PRINCIPLE 171 gener
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THE HOLOGRAPHIC PRINCIPLE 173 FIGUR
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THE HOLOGRAPHIC PRINCIPLE 175 the g
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THE HOLOGRAPHIC PRINCIPLE 177 In th
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HOW TO WEAVE A STRING 181 thing!" I
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HOW TO WEAVE A STRING 183 other. Th
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HOW TO WEAVE A STRING 185 What is r
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HOW TO WEAVE A STRING 187 for a pic
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HOW TO WEAVE A STRING 189 in Einste
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HOW TO WEAVE A STRING 191 p in the
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HOW TO WEAVE A STRING 193 may be th
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WHAT CHOOSES THE LAWS OF NATURE? 19
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EPILOGUE: .........................
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EPILOGUE: 209 And quantum gravity i
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EPILOGUE: 211 of systems containing
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POSTSCRIPT 213 These cosmic rays ar
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POSTSCRIPT 215 Three explanations h
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POSTSCRIPT 217 light, which is the
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POSTSCRIPT 219 This is good news in
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POSTSCRIPT 221 But there is a secon
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POSTSCRIPT 223 Chopin Soo and Marti
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POSTSCRIPT 225 speed of light with
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GLOSSARY 227 cannot be greater than
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GLOSSARY 229 event In relativity th
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GLOSSARY 231 Newton's gravitational
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GLOSSARY 233 spin The angular momen
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SUGGESTIONS FOR FURTHER READING ...
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SUGGESTIONS FOR FURTHER READING 237
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SUGGESTIONS FOR FURTHER READING 239
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INDEX .............................
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INDEX 243 black hole, 70, 73±4, 17
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INDEX 245 black hole formation, 189
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Three Roads To Quantum Gravity

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