Science

Mathematics in V1N1 5, 15, 27–28, V1N2 1, 30, V1N3 14, 39–40, 48–49, 104–5, 107–8, V1N5 13–17, V2N1 32–44, V2N2 27–28, 116–19, V2N3 52–54, V2N4 96, 105, 109, 123–27, 146, 149–57, 168, V2N5 9–10, 18–19, V2N6 131–86.

V1N1 p.28  “Capturing Concepts”

Solids are distinguished from fluids in virtue of the fact that they have moduli of rigidity that are not (too close to) zero; any solid is capable of withstanding shearing forces up to some particular measure. The particular modulus of rigidity of a particular solid object is part of what constitutes its individual identity, and the fact that its particular modulus is not zero is what qualifies it for the class solid.

V2N2 pp.27–28  “Space, Rotation, Relativity”

It was Huygens who named the tendency of a body in circular motion to recede from the center the centrifugal (“flees from center”) tendency. He succeeded in quantifying the strength of this tendency; he gave us the formula of what we, after Newton, take to be centrifugal force. Imagine sitting on the edge of a merry-go-round holding a plumb bob. The bob hangs vertically initially while the merry is still, but moves outward from the hand as the speed of the merry increases. Let the speed of the merry be made constant. Huygens computed from pure kinematics that were the bob released, rather than held by a wire, it would travel outward a distance proportional to the rotation speed squared divided by the merry’s diameter and directly proportional to the time interval since release squared. This is so provided that time interval is small, and small is all we require for a formula of the instantaneous parameters of circular motion.

 

Huygens knew that distance in one of Galileo’s expressions for free fall is also proportional to the square of the time interval. Huygens still thought of the tendency of bodies to fall and their centrifugal tendencies as an inherent force, or power, they exhibit in those situations. He did not yet have clearly our Newtonian concept of force as external cause acting on the body. A whirling body has a centrifugal tendency, and like the falling tendency of bodies, it yields not a uniform motion but one proportional to the duration squared, at least for short durations. So for Huygens, as for everyone after him, a body rotating at uniform speed is [classified as] undergoing an acceleration. Huygens initiated what we should now call the dynamics of circular motion, as he quantified the centrifugal tendency, by determining what rotational speed the merry would need to for the bob to have a centrifugal tendency equal to its gravity.

V2N3 pp.53  “Space, Rotation, Relativity”

Newton imagined a square inside of which is circumscribed a circle. Let the sides of the square be banks of a square billiards table (without pockets). Launch a ball so as to strike and rebound at a point, on each of the four banks in consecution, midway between corners; the very point at which the imaginary circle touches the square. The angles of incidence and reflection will be 45°. In making one complete circuit, Newton figured, the force that the ball exerts on the banks in reflections is to the force of the ball’s linear motion (the ball’s linear momentum) as the path length of the ball’s circuit is to the length of the circle’s radius. Newton then showed that whatever regular polygon is fitted about the circle (replacing the square about the circle), the ratio of the ball’s reflecting forces exerted in a complete circuit to the force of the ball’s linear motion is always as the circuit’s length to the circle’s radius. Then if we allow the sides of the polygon to become infinitely short and numerous, the polygon becomes the circle, and we have that the ball’s force exerted on its circular container in one revolution about that circle is 2ðr/v, where r is the circle’s radius. Then the instantaneous force the ball exerts outward when in circular motion, the endeavor of the ball to flee the center, is m(v·v)/r, as Huygens had earlier found, unbeknownst to Newton at the time of his own discovery.

[Newton on planetary orbits: this.]

V1N3 p.14  “Induction on Identity”

Inference to the existence of atoms is a case of induction in the genre of what William Whewell (1794–1866) termed the consilience-induction. By 1900 atoms and molecules were evidenced by Dalton’s law of multiple proportions, Gay-Lussac’s law pertaining to the volume of gases, Avogadro’s law (which made possible the determination of molecular weights), and the kinetic theory of gases (which could approximately predict molar heat capacities). After 1908, when Jean Baptiste Perrin published his results on the sedimentation distribution of (visible) particles suspended in a still liquid and his measurement of Avogadro’s constant, the existence of atoms could not be reasonably doubted. [Note also.]

V2N2 pp.116–19  “Three Chances”

V2N1 pp. 32–44  “Chaos”

V2N6 131–86  “Invariance, Electrodynamics, and the Special Theory”

(Available also here.)

V1N3 107–8  “The Complexion of Number” —David Ross

Every analytic function is a solution of Laplace's differential equation, the most important equation in mathematical physics. The class of analytic functions provides an abundance of solutions to problems in steady state fluid dynamics, electromagnetic theory, elasticity, and minimal surface theory. In fluid dynamics, free boundary problems (problems in which no wall contains the fluid) and airfoil problems can be treated particularly effectively and elegantly. In all these cases, the highly structured nature of analytic functions allows us to infer a lot of qualitative and quantitative information about the behavior of these solutions. Analytic functions can also be regarded as mappings of the complex plane into itself. They are referred to as conformal mappings in this context. This geometrical interpretation aids in the construction of solutions of Laplace’s equation in geometrically complicated regions, e.g. regions with corners or holes.

 

The Laplace-equation / conformal mapping techniques constitute the most striking and direct contribution of analytic function theory to mathematical physics. However, this theory has probably contributed much more in another way. Often, a problem involving functions of a real variable can be viewed as a special case of a more general problem involving complex analytic functions. This permits us to bring all that is known about the qualitative structure of analytic functions to bear on the problem. Virtually every branch of mathematical physics has benefitted from this technique.

 

An example of this is the technique developed in the 1970s by Garabedian and co-workers at NYU for designing shockless transonic airfoils. When an airplane moves at a high subsonic speed (as many commercial planes do), the air accelerates past the sound speed as it crosses the wing. It is then decelerated and compressed suddenly by a standing shock wave as it leaves the wing. This deceleration causes considerable drag, which could be eliminated if the air could be decelerated smoothly. Until the early ’70s, no airfoil shapes that would decelerate air smoothly at transonic speeds were known. By extending the equation of momentum conservation to complex values and applying complex variable techniques to solve the extended equation, the NYU group was able to generate the first shockless airfoils.

V1N2 p.30  “Philosophy of Mathematics” —Merlin Jetton

There have been parts of mathematics developed with no knowledge of the relevance to the real world . . . . An example is Riemann's geometry. But it turned out to be essential for Einstein's geometrical theory of gravity. Another example is group theory (a kind of hyperabstract algebra). It turned out to be very useful in quantum mechanics about a hundred years later. Without group theory, the unification of the electromagnetic force and the weak nuclear force would not have been achieved.

V1N3 pp.39–40  “Induction on Identity”

Under the supposition that time is homogeneous, we can derive from Hamilton's principle

. . . conservation of energy. The conservation of mass-energy is a very robust principle, experimentally and theoretically. . . . The conservation of mass-energy can be derived also as a consequence of Einstein's field equation and one of the geometrical identities known as Bianchi identities. . . . On the left of [Einstein’s field equation], we have geometric structure of spacetime; on the right, we have matter (stress-energy tensor), the source of the geometric structure. Bianchi identities on the left correspond to conservation of mass-energy on the right.

V1N5 pp.13–17  “Can Art Exist without Death?” —Kathleen Touchstone

(Available also here.)

V2N4 pp.123–27 and 149–57  “Mathematics and Intuition” —Kathleen Touchstone

(§VII “Computational Synapses” and §XI “Neural Networks”)

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See also: “Functions of Mathematical Description in Astronomy and Optics, Illustrations from Antiquity”