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The resulting "non. Euclidean geometry" turned out to be of great use in science. In fact, Einstein employed it in his general theory of relativity a theory of gravitation , and, as mentioned, we now know that Euclid's geometry is actually wrong in the real world: crudely speaking, space is curved by gravity. Euclidean geometry is still taught in schools because it remains a very good approximation under most circumstances.

The lesson of this story, however, is that it is unwise to consider any axioms as so self-evidently correct that they could not possibly be otherwise. It is generally agreed that logico-deductive arguments constitute the most secure form of reasoning, though I should mention that even the use of standard logic has been questioned by some. In so-called quantum logic, the rule that something cannot both be and not be such-and-such is dropped. The motivation for this is that in quantum physics the notion of "to be" is more subtle than in everyday experience: physical systems can exist in superpositions of alternative states.

Another form of reasoning that we all employ is called "inductive. The prediction that the sun will rise tomorrow is an example of inductive reasoning based on the fact that the sun has faithfully risen every day so far in our experience. And when I let go of a heavy object, I expect it to fall, on the basis of my previous experiences with the pull of gravity. Scientists employ inductive reasoning when they frame hypotheses based on a limited number of observations or experiments. The laws of physics, for instance, are of this sort. The inverse-square law of electric force has been tested in a number of ways, and always confirmed.

We call it a law because, on the basis of induction, we reason that the inverse-square property will always hold. However, the fact that nobody has observed a violation of the inverse-square law does not prove it must be true, in the way that, given the axioms of Euclidean geometry, Pythagoras' theorem must be true. No matter on how many individual occasions the law is confirmed, we can never be absolutely certain that it applies unfailingly.

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On the basis of induction, we may conclude only that it is very probable that the law will hold the next time it is tested. The philosopher David Hume cautioned against inductive reasoning. That the sun has always been observed to rise on schedule, or that the inverse-square law has always been confirmed, is no guarantee that these things will continue in the future. The belief that they will is based on the assumption that "the course of nature continues always uniformly the same.

Mind of God | Book by Paul Davies | Official Publisher Page | Simon & Schuster

True, it may be the case that a state of affairs B e. For in what sense might B have to follow A?


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We can certainly conceive of a world where A occurs but B doesn't: there is no logically necessary connection between A and B. Might there be some other sense of necessity, a sort of natural necessity? Hume and his followers deny that there is any such thing. It seems we are forced to concede that conclusions arrived at inductively are never absolutely secure in the logical manner of deductive conclusions, even though "common sense" is based on induction.

That inductive reasoning is so often successful is a remarkable property of the world that one might characterize as the "dependability of nature. As we shall see, there is no logical reason why the world may not have been otherwise. It could have been chaotic in a way that made inductive generalization impossible. Modern philosophy has been strongly influenced by the work of Karl Popper, who argues that in practice scientists rarely use inductive reasoning in the way described.

When a new discovery is made, scientists tend to work backward to construct hypotheses consistent with that discovery, and then go on to deduce other consequences of those hypotheses that can in turn be experimentally tested. If any one of these predictions turns out to be false, the theory has to be modified or abandoned. The emphasis is thus on falsification, not verification. A powerful theory is one that is highly vulnerable to falsification, and so can be tested in many detailed and specific ways. If the theory passes those tests, our confidence in the theory is reinforced.

A theory that is too vague or general, or makes predictions concerning only circumstances beyond our ability to test, is of little value. In practice, then, human intellectual endeavor does not always proceed through deductive and inductive reasoning. The key to major scientific advances often rests with free-ranging imaginative leaps or inspiration. In such cases an important fact or conjecture springs ready-made into the mind of the inquirer, and only subsequently is justification found in reasoned argument. How inspiration comes about is a mystery that raises many questions.

Do ideas have a type of independent existence, so that they are "discovered" from time to time by a receptive mind? Or is inspiration a consequence of normal reasoning which takes place hidden in the subconscious, with the result being delivered to the conscious only when complete?

Dr. Jay Lombard – The Mind of God

If so, how did such an ability evolve? What biological advantages can such things as mathematical and artistic inspiration confer on humans? A Rational World The claim that the world is rational is connected with the fact that it is ordered. Events generally do not happen willy-nilly: they are related in some way. The sun rises on cue because the Earth spins in a regular manner. The fall of a heavy object is connected with its earlier release from a height. And so on. It is this interrelatedness of events that gives us our notion of causation.

The window breaks because it is struck by a stone.

The oak tree grows because the acorn is planted. The invariable conjunction of causally related events becomes so familiar that we are tempted to ascribe causative potency to material objects themselves: the stone actually brings about the breakage of the window. But this is to attribute to material objects active powers that they do not deserve. All one can really say is that there is a correlation between, say, stones rushing at windows and broken glass. Events that form such sequences are therefore not independent. If we could make a record of all events in some region of space over a period of time, we would notice that the record would be crisscrossed by patterns, these being the "causal linkages.

Without them there would be only chaos. Closely related to causality is the notion of determinism. In its modem form this is the assumption that events are entirely determined by other, earlier events.


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  • Determinism carries the implication that the state of the world at one moment suffices to fix its state at a later moment. And because that later state fixes subsequent states, and so on, the conclusion is drawn that everything which ever happens in the future of the universe is completely determined by its present state. When Isaac Newton proposed his laws of mechanics in the seventeenth century, determinism was automatically built into them.

    For example, treating the solar system as an isolated system, the positions and velocities of the planets at one moment suffice to determine uniquely through Newton's laws their positions and velocities at all subsequent moments. Moreover, Newton's laws contain no directionality in time, so the trick works in reverse: the present state suffices to fix uniquely all past states. In this way we can, for example, predict eclipses in the future, and also retrodict their occurrences in the past.

    If the world is strictly deterministic, then all events are locked in a matrix of cause and effect. The past and future are contained in the present, in the sense that the information needed to construct the past and future states of the world are folded into its present state just as rigidly as the information about Pythagoras' theorem is folded into the axioms of Euclidean geometry. The entire cosmos becomes a gigantic machine or clockwork, slavishly following a pathway of change already laid down from the beginning of time.

    Ilya Prigogine has expressed it more poetically: God is reduced to a mere archivist turning the pages of a cosmic history book already written. Standing in opposition to determinism is indeterminism, or chance. We might say that an event happened by "pure chance" or "by accident" if it was not obviously determined by anything else.

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    Throwing a die and flipping a coin are familiar examples. But are these cases of genuine indeterminism, or is it merely that the factors and forces that determine their outcome are hidden from us, so that their behavior simply appears random to us? Before this century most scientists would have answered yes to the latter question.

    They supposed that, at rock bottom, the world was strictly deterministic, and that the appearance of random or chance events was entirely the result of ignorance about the details of the system concerned. If the motion of every atom could be known, they reasoned, then even coin tossing would become predictable. The fact that it is unpredictable in practice is because of our limited information about the world. Random behavior is traced to systems that are highly unstable, and therefore at the mercy of minute fluctuations in the forces that assail them from their environment.

    This point of view was largely abandoned in the late s with the discovery of quantum mechanics, which deals with atomic-scale phenomena and has indeterminism built into it at a fundamental level. One expression of this indeterminism is known as Heisenberg's uncertainty principle, after the German quantum physicist Werner Heisenberg. Roughly speaking, this states that all measurable quantities are subject to unpredictable fluctuations, and hence to uncertainty in their values.

    The Mind of God: The Scientific Basis for a Rational World

    To quantify this uncertainty, observables are grouped into pairs: position and momentum form a pair, as do energy and time. The principle requires that attempts to reduce the level of uncertainty of one member of the pair serves to increase the uncertainty of the other. Thus an accurate measurement of the position of a particle such as an electron, say, has the effect of making its momentum highly uncertain, and vice versa.

    Because you need to know both the positions and the momenta of the particles in a system precisely if you want to predict its future states, Heisenberg's uncertainty principle puts paid to the notion that the present determines the future exactly. Of course, this supposes that quantum uncertainty is genuinely intrinsic to nature, and not merely the result of some hidden level of deterministic activity.

    In recent years a number of key experiments have been performed to test this point, and they have confirmed that uncertainty is indeed inherent in quantum systems. The universe really is indeterministic at its most basic level. So does this mean that the universe is irrational after all? No, it doesn't. There is a difference between the role of chance in quantum mechanics and the unrestricted chaos of a lawless universe. Although there is generally no certainty about the future states of a quantum system, the relative probabilities of the different possible states are still determined.

    Thus the betting odds can be given that, say, an atom will be in an excited or a nonexcited state, even if the outcome in a particular instance is unknown. This statistical lawfulness implies that, on a macroscopic scale where quantum effects are usually not noticeable, nature seems to conform to deterministic laws. The job of the physicist is to uncover the patterns in nature and try to fit them to simple mathematical schemes.

    The question of why there are patterns, and why such mathematical schemes are possible, lies outside the scope of physics, belonging to a subject known as metaphysics.