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When Shor’s algorithm has factorized a number, using 10500 or so times the computational resources that can be seen to be present, where was the number factorized?
Aristotle had declared that all valid proofs could be expressed in syllogistic form. But he had not proved this!
Explaining and understanding the world – the physical world and the world of mathematical abstractions – is in both cases the object of the exercise. Proof and observation are merely means by which we check our explanations.
is this: in a reality composed of physics and understood by the methods of science, where does mathematical certainty come from?
We can also see from the above discussion that proof is a physical process. In fact, a proof is a type of computation.
It follows that neither the theorems of mathematics, nor the process of mathematical proof, nor the experience of mathematical intuition, confers any certainty. Nothing does.
the traditional one that a proof can have only a finite number of steps. The intuitive justification of this assumption is that we are finite beings and could never grasp a literally infinite number of assertions.
and none of the known ways in which physical objects can interact would allow for an infinite number of steps to precede a measurable conclusion.
This example shows that because of the possibility of quantum computation, the two notions of proof are not equivalent.
The Turing machine is entirely classical, and does not allow for the possibility that the paper might have different symbols written on it in different universes,
Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing mathematics.
But in principle the reliability of our knowledge of mathematics remains subsidiary to our knowledge of physical reality.
the false theory that yellow light consists of a mixture of red and green light (in the sense that yellow light gives us exactly the same sensation as a mixture of red light and green light does). In reality, all three types of light have different frequencies and cannot be created by mixing light of other frequencies.
We have no choice but to assume that the incomprehensible mathematical entities are real too, because they appear inextricably in our explanations of the comprehensible ones.
For what we are really seeing is only that some abstract structures are fundamental to our understanding of abstractions.
To put it bluntly, the reason why the common-sense theory of time is inherently mysterious is that it is inherently nonsensical. It is not just that it is factually inaccurate. We shall see that, even in its own terms, it does not make sense.
Nothing can move from one moment to another. To exist at all at a particular moment means to exist there for ever.
The various snapshots of the observer do not take it in turns to be in the present. They do not take it in turns to be conscious of their present. They are all conscious, and subjectively they are all in the present.
Objectively, there is no present.
Nothing can move along the line. Time cannot flow.
The origin of this sort of mistake is that we are accustomed to time being a framework exterior to any physical entity we may be considering.
It does not exist within the framework of time – it is the framework of time.
Spacetime is sometimes referred to as the ‘block universe’, because within it the whole of physical reality – past, present and future – is laid out once and for all, frozen in a single four-dimensional block. Relative to spacetime, nothing ever moves.
The determinism of physical laws about events in spacetime is like the predictability of a correctly interlocking jigsaw puzzle.
It is not that people are mistaken when they say that certain physical events are causes and effects of one another, it is just that that intuition is incompatible with the laws of spacetime physics. But that is all right, because spacetime physics is false.
reasoning about causes and effects is inevitably also about variants of the causes and effects. One is always saying what would have happened if, other things being equal, such and such an event had been different.
There is no avoiding the fact that in spacetime exactly one thing happens in reality, and everything else is fantasy.
Logicians call such statements counter-factual conditionals, and their status is a traditional paradox.
So when I travel to the laboratory’s past, I find that it is not the same past as I have just come from.
Typical candidate versions of a quantum theory of gravity not only allow past-directed connections to exist in the multiverse, they predict that such connections are continually forming and breaking spontaneously.
Because no time machine provides pathways to times earlier than the moment at which it came into existence, and because of the way in which quantum theory says that universes are interconnected, there are some limits to what we can expect to learn by using time machines.
emerge from it. What will they tell us? One thing they will certainly not tell us is news of our own future.
call it the knowledge paradox of time travel; here is how the story typically goes.
perhaps replace, the copy he visits. But he can never visit the copy who existed in the universe he started from. And it is that copy who wrote the plays.
is a general rule that after time travel has taken place the total number of copies of me, counted across all universes, is unchanged.
laws of quantum mechanics impose drastic restrictions on which snapshots can be connected to which others. For one thing, two universes first become connected only at a moment when they are identical:
think of all our knowledge-generating processes, our whole culture and civilization, and all the thought processes in the minds of every individual, and indeed the entire evolving biosphere as well, as being a gigantic computation.
is no good trying to pretend that successive explanations are better only in terms of their own paradigm. There are objective differences. We can fly, whereas for most of human history people could only dream of this.
But bringing vigorous and diverse criticism to bear on accepted truths is one of the very purposes of the seminar. Everyone takes it for granted that the truth is not obvious, and that the obvious need not be true; that ideas are to be accepted or rejected according to their content and not their origin; that the greatest minds can easily make mistakes; and that the most trivial-seeming objection may be the key to a great new discovery.
Hugh Everett, then a Princeton graduate student working under the eminent physicist John Archibald Wheeler, first set out the many-universes implications of quantum theory.
But the basis of Everett’s innovation was not a claim that the prevailing theory is false, but that it is true!
The incumbents, far from being able to think only in terms of their own theory, were refusing to think in its terms, and were using it only instrumentally.
Thus they violate a basic tenet of rationality – that good explanations are not to be discarded lightly.
the modern watchwords are just as Popper would have them: experimental testing, exposure to criticism, theoretical explanation and the acknowledgement of fallibility in experimental procedures.
very few philosophers agree with Popper’s claim that there is no longer a ‘problem of induction’ because we do not in fact obtain or justify theories from observations, but proceed by explanatory conjectures and refutations instead.
Popper’s heretical innovation takes the form of a claim that the methodology has been valid all along.
But the point of such objections is not so much to contradict the Darwinian model as to claim that something fundamental remains unexplained in the matter of how the adaptations we observe in the biosphere came into being.
In other words, Dawkins claims that Darwin’s theory of evolution does provide a full explanation of the origin of biological adaptations.
Dawkins elaborated Darwin’s theory in its modern form as the theory of replicators. The replicator that is best at getting itself replicated in a given environment will eventually displace all variants of itself because, by definition,
It is not the fittest species variant that survives (Darwin had not quite realized this) bu...
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