By Marc Lange
No longer all medical factors paintings by way of describing causal connections among occasions or the world's total causal constitution. a few mathematical proofs clarify why the theorems being proved carry. during this publication, Marc Lange proposes philosophical money owed of many forms of non-causal factors in technology and arithmetic. those issues were unjustly overlooked within the philosophy of technological know-how and mathematics.
One vital type of non-causal medical clarification is called rationalization by way of constraint. those factors paintings by means of offering information regarding what makes yes proof particularly inevitable - extra helpful than the standard legislation of nature connecting explanations to their results. evidence defined during this approach go beyond the hurly-burly of reason and impression. Many physicists have appeared the legislation of kinematics, the good conservation legislation, the coordinate alterations, and the parallelogram of forces as having factors by way of constraint. This booklet offers an unique account of reasons via constraint, focusing on numerous examples from classical physics and targeted relativity.
This booklet additionally deals unique debts of a number of different sorts of non-causal medical clarification. Dimensional reasons paintings through displaying how a few legislations of nature arises in basic terms from the dimensional relatives one of the amounts concerned. quite statistical causes comprise causes that entice regression towards the suggest and different canonical manifestations of probability. Lange presents an unique account of what makes definite mathematical proofs yet now not others clarify what they turn out. Mathematical rationalization connects to a number of different very important mathematical principles, together with coincidences in arithmetic, the importance of giving a number of proofs of an identical consequence, and typical houses in arithmetic. Introducing many examples drawn from real technology and arithmetic, with prolonged discussions of examples from Lagrange, Desargues, Thomson, Sylvester, Maxwell, Rayleigh, Einstein, and Feynman, Because with no Cause's proposals and examples should still set the schedule for destiny paintings on non-causal explanation.
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Additional resources for Because without cause: non-causal explanations in science and mathematics
An explanation begins with the fact that temperature is a continuous function. That is, roughly speaking, as you move along the equator, temperature changes smoothly rather than jumping discontinuously. Now imagine placing your two index fingers on a globe at two antipodal points on the equator. Take the temperature on Earth at the location x that your left index finger is touching minus the temperature at the antipodal location that your right index finger is touching. This difference function D(x) must change continuously as you move your two fingers eastward, keeping them at antipodal points on the equator (since a function is continuous if it is the difference between two continuous functions).
However, some distinctively mathematical explanations appeal to contingent natural laws. Here is an example. Suppose we make a “simple double pendulum” by suspending a simple pendulum from the bob of another simple pendulum and allowing both bobs to move under the influence of gravity (which varies negligibly with height) while confined to a single vertical plane (see fig. 3). 3), where a “configuration” is fixed by the angles α and β. 3) and then to determine the configurations in which both bobs feel zero net force.
This fact has a distinctively mathematical explanation: it is just an instance of the Honeycomb Conjecture. By the same token, “word problems” in mathematics textbooks are full of allusions to facts that have distinctively mathematical explanations—for example, the fact that if Farmer Brown, with 50 feet of negligibly thin and infinitely bendable fencing, uses his fencing to enclose the maximum area in a flat field, then he makes his fencing into a circle. 3 The simple double pendulum (left) and its four equilibrium configurations (right); only the first equilibrium configuration is stable.