"In the absolute universe all events can be regarded
as absolutely deterministic, and if we can’t perceive the greater structures,
it’s because our vision is faulty. If we had a real grasp of causality down to
the molecular level, we wouldn’t need to rely on mathematical approximations, on
statistics and probabilities, in making predictions. If our perceptions of
cause and effect were only good enough, we’d be able to attain absolute
knowledge of what is to come. We would make ourselves all-seeing." (Robert
Silverberg, "The Stochastic Man", 1975)
"Whenever the Eastern mystics express their
knowledge in words - be it with the help of myths, symbols, poetic images or
paradoxical statements-they are well aware of the limitations imposed by
language and 'linear' thinking. Modern physics has come to take exactly the
same attitude with regard to its verbal models and theories. They, too, are
only approximate and necessarily inaccurate. They are the counterparts of the
Eastern myths, symbols and poetic images, and it is at this level that I shall
draw the parallels. The same idea about matter is conveyed, for example, to the
Hindu by the cosmic dance of the god Shiva as to the physicist by certain
aspects of quantum field theory. Both the dancing god and the physical theory
are creations of the mind: models to describe their authors' intuition of
reality." (Fritjof Capra, "The Tao of Physics: An Exploration of the
Parallels Between Modern Physics and Eastern Mysticism", 1975)
"It seems that truth
Is progressive approximation
In which the relative fraction
Of our spontaneously tolerated residual error
Constantly diminishes." (R Buckminster Fuller,
"And It Came to Pass - Not to Stay", 1976)
"Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate." (Gregory Bateson, "Number is Different from Quantity", CoEvolution Quarterly, 1978)
"Until now, physical theories have been regarded as merely models with approximately describe the reality of nature. As the models improve, so the fit between theory and reality gets closer. Some physicists are now claiming that supergravity is the reality, that the model and the real world are in mathematically perfect accord." (Paul C W Davies, "Superforce", 1984)
"The fact that [the model] is an approximation does not necessarily detract from its usefulness because models are approximations. All models are wrong, but some are useful." (George Box, 1987)
"When it comes to very highly organized systems, such as a living cell, the task of modeling by approximation to simple, continuous and smoothly varying quantities is hopeless. It is for this reason that attempts by sociologists and economists to imitate physicists and describe their subject matter by simple mathematical equations is rarely convincing." (Paul C W Davies, "The Cosmic Blueprint: New Discoveries in Nature’s Creative Ability to Order the Universe", 1987)
"A distinctive feature of mathematics, that feature
in virtue of which it stands as a paradigmatically rational discipline, is that
assertions are not accepted without proof. […] By proof is meant a deductively
valid, rationally compelling argument which shows why this must be so, given what
it is to be a triangle. But arguments always have premises so that if there are
to be any proofs there must also be starting points, premises which are agreed
to be necessarily true, self-evident, neither capable of, nor standing in need
of, further justification. The conception of mathematics as a discipline in
which proofs are required must therefore also be a conception of a discipline
in which a systematic and hierarchical order is imposed on its various
branches. Some propositions appear as first principles, accepted without proof,
and others are ordered on the basis of how directly they can be proved from
these first principle. Basic theorems, once proved, are then used to prove
further results, and so on. Thus there is a sense in which, so long as mathematicians
demand and provide proofs, they must necessarily organize their discipline
along lines approximating to the pattern to be found in Euclid's
Elements." (Mary Tiles, "Mathematics and the Image of Reason" ,
1991)
"A model is something one tries to construct when one has to describe a complicated situation. A model is therefore an approximate description of reality and invariably involves many simplifying assumptions. […] models are convenient idealisations." (Ganeschan Venkataraman, "Chandrasekhar and His Limit", 1992)
"Symmetry is basically a geometrical concept. Mathematically it can be defined as the invariance of geometrical patterns under certain operations. But when abstracted, the concept applies to all sorts of situations. It is one of the ways by which the human mind recognizes order in nature. In this sense symmetry need not be perfect to be meaningful. Even an approximate symmetry attracts one's attention, and makes one wonder if there is some deep reason behind it." (Eguchi Tohru & K Nishijima , "Broken Symmetry: Selected Papers Of Y Nambu", 1995)
"To select an appropriate fuzzy implication for approximate reasoning under each particular situation is a difficult problem. Although some theoretically supported guidelines are now available for some situations, we are still far from a general solution to this problem." (George Klir, "Fuzzy sets and fuzzy logic", 1995)
"Science is more than a mere attempt to describe
nature as accurately as possible. Frequently the real message is well hidden,
and a law that gives a poor approximation to nature has more significance than
one which works fairly well but is poisoned at the root." (Robert H March,
"Physics for Poets", 1996)
"The role of science, like that of art, is to blend proximate imagery with more distant meaning, the parts we already understand with those given as new into larger patterns that are coherent enough to be acceptable as truth. Biologists know this relation by intuition during the course of fieldwork, as they struggle to make order out of the infinitely varying patterns of nature." (Edward O Wilson, "In Search of Nature", 1996)
"Perhaps the most common complaint about the weakness of the random-walk theory is based on a distrust of mathematics and a misconception of what the theory means. 'The market isn't random', the complaint goes, 'and no mathematician is going to convince me it is'. [...] But, even if markets were dominated during certain periods by irrational crowd behavior, the stock market might still well be approximated by a random walk. The original illustrative analogy of a random walk concerned a drunken man staggering around an empty field. He is not rational, but he's not predictable either." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)
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