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Monday, April 2, 2007 - 6:47amSanction this postReply
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Wonderful.  Been wanting to add this to my library for years...



Post 1

Monday, April 2, 2007 - 11:33amSanction this postReply
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This is a stunning act of generosity on your part, Stephen, and I thank you for it.



Post 2

Monday, April 2, 2007 - 9:21pmSanction this postReply
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What Andre said.

Ho-ly Cow!

Thanks!

Ed




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Post 3

Thursday, October 16, 2008 - 7:58amSanction this postReply
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I am pleased to announce the completion and installation of the comprehensive SUBJECT INDEX for Objectivity.

Here is a stretch of the “C” section:

 

Chaos. See Dynamics, Chaotic.

 

Charity V2N3 26, 42, V2N5 108, 136

 

Choice. See Decision; Free Will.

 

Cognition. See Perception; Thought; Reason; Knowledge.

 

Concentration. See Attention, Controlled.

 

Concepts V1N1 13–38, V1N3 79–81, V2N6 43–70, 73–99; as Action Instruments V1N1 18, 28, V1N3 9–10, 67–68, 71–72, V2N6 103–4; Adequacy in V1N4 18–19, V1N5 109–12, V2N1 131; in Animals V1N1 14, 22, V1N2 48–49, 54–59, 74–75, V2N4 114, 169, 177; and Beliefs V1N1 23–24, V1N4 4, V2N6 115; Childhood V1N1 13–14, 16–18, 22–24, 28, 30–37, V1N2 38–40; V1N3 9–10, 12–13, 71–72, 81, V1N4 51, V1N5 32–34, V1N6 1–33, V2N2 83–85, V2N4 121–23, 196, V2N6 65, 71, 100–124; and Context V1N1 22, 31, V1N4 36, 52–54, V1N6 15, 17–18, 97, 99, V2N1 97, V2N4 139–40, V2N6 69–70, 80–82; Extensions and Intensions of V1N2 43–44, V1N3 43–44, V2N4 230, V2N6 62, 74–75, 106; Formation of V1N1 24–25, 28–34; V1N2 24–25, 28, 95–97, V1N3 66, 79–81, V1N4 4–5, 9, V1N5 13, V1N6 1–33, V2N1 132, V2N3 22, 104, V2N4 13–14, 25–26, 50, V2N6 44, 48, 51–53, 57–58, 64, 66, 67–68, 98, 103–9, 112–14; and Identification V1N1 13, 20–23, 27–38, V1N4 34, V1N5 112, V1N6 96–97, V2N1 134, V2N2 5, 90, V2N4 139–40, 142, 196, V2N6 44–46, 49–50, 54–58, 64, 67–70, 80–82, 90; v. Images V1N1 25–26, V1N3 70, 81, V2N4 25–26, 162–63, V2N5 40, V2N6 58; Implicit V1N1 13, V1N4 51, V1N6 27, V2N2 134, V2N6 62; and Induction V1N1 29, 35–38, V1N2 36, 42–44, V1N4 33–34, V2N6 81, 115, 117–18, 124; Innateness of V1N1 13, V1N2 34, 51, V1N4 17, V2N2 72–73, 99, V2N4 48, 140; by Intuition v. by Postulation V1N6 41–52; Intuitive v. Abstractive V2N6 50–51, 92–93; and Language V1N1 13, 16–20, 29, 36, V1N2 39–40, V1N3 74–76, 81, V1N4 6–8, 18, 32, 40–43, V1N5 102, V2N3 93, V2N4 25, 114–15, V2N5 14, 39, 42–44, V2N6 6, 48, 50–52, 54–56, 58, 59–63, 65, 69, 80, 104, 106, 114, 123–24; and Mental Economy V1N1 22, V1N2 36, 40, 43–44, 56–57, V1N3 80, V2N1 104, V2N4 108, V2N5 41–42, V2N6 66, 68, 69, 76, 80–82, 97–98; Objectivity in V1N1 36–38, V1N2 24–30, 40, 43, V1N3 17, 105–6, V1N4 6, 10, 14, V1N5 109–12, V2N1 134, V2N2 5–6, V2N6 49–59, 65, 69–70, 76, 79; Primacy of V1N5 109–12, V2N6 70; and Propositions V1N1 22–23, V1N3 44–45, V1N4 11–12, 32, 37, V1N5 110, V2N1 133–34, V2N2 5–6, V2N4 25–26, 50, 142–44; as Prototypes V2N6 79, 83–98, 115, 120, 125; and Recognition V1N1 15–18, V1N6 15, 20, V2N4 139–40, V2N6 58, 62–63, 81–82; and Reference V1N1 13, 19–23, V1N2 22, 25–26, V1N3 14, V1N4 19, V1N5 102–3, 109–12, V2N6 44, 48–55, 62–63, 69–70, 112; Reformation of V1N1 4–9, 34–38; V1N2 13, 15, 17, 27; V1N3 6, 9–10, 14–15, 28, 74–77, 105–6, V2N6 59, 82; and Schemas V1N1 16–18, V1N3 41–42, 69–78, V1N4 52, V2N2 90, V2N4 196, V2N6 103–6; and Sets V1N1 13, 29, 36,  V1N2 43–44, 101–2, V2N4 122, V2N6 90, 105–6; and Similarity V1N1 2, 24, 26, 28–29, V1N3 79, V2N6 41, 43–45, 49–51, 54, 58–62, 64–72, 76, 88–89, 98, 102, 107, 112, 114, 124–26; and Taxonomy V1N1 30–37, V1N3 79–80, V2N6 106, 113; Thematic V1N1 29–31, V2N6 93–94, 102, 112–13; as Unities of Distinct Determinations V1N1 31–34, V1N5 110. See also Measurement, Omission of; Universals.

 

Conditioning; Classical V1N5 38, V2N4 198–99, 201; with Delayed Reinforcement V1N5 12; Operant V1N5 38, V2N1 115, V2N4 202; with Probabilistic Reinforcement V1N2 55–56, V2N4 128–30

 

Consciousness V1N5 29–64, V2N1 93–106, 116–19, V2N2 136–38, V2N6 1–38; and Animal Action V1N2 47–63, V1N5 30, 36–42, 46, 52–54, 102, V2N1 103, 105–6, 112–13, 117, 119, V2N2 77–78, 88–89, 92–95, 136, V2N3 20–21, V2N4 191, 197–98, 200–202, V2N6 16–17, 20–21, 70–71; at Birth V1N1 14, V2N2 101; Dissociation of V1N5 59–60, V2N6 12, 20; Efficacy of V1N2 52, V1N5 45–46, V2N1 99–106, V2N4 12–13, V2N6 16–17; Evolution of V1N2 54, 69, V1N5 22, 36–42, 46, V2N2 88–96, V2N6 4–5, 36; First-Person v. Third-Person V2N6 5–8; Focal V1N1 20–22, V1N2 14, 24–25, V1N3 79, V1N5 43–44, 47, V2N1 119–21, V2N2 136–38, V2N3 21–22, V2N4 152–54, V2N6 11–12, 32; and Free Will V1N2 47–63, V1N5 79, V2N1 104–5, 111–21, V2N3 21–22, 101–2, V2N4 193, 195–96, 202, V2N6 12, 32, 34; Hard Problem of V2N6 1–2, 8, 34; as Identification V1N2 33, V1N3 44, V1N4 34, V2N1 134, V2N4 110, 195–98, V2N6 5; Intentionality of V1N4 34, 36–37, V1N5 29, 35–36, 102, V2N1 96, 119, V2N5 37–38, V2N6 6; and Mind V1N2 52, V1N5 51–52, 100–105, V2N1 95–96, 111–19, V2N4 115–16, V2N6 11–27, 34–37; Multiple-Drafts Model of V1N5 55–56, V2N6 14–17; Multiple-Realizability of V1N5 101–6, V2N6 4, 21, 34–37; Neuronal Bases of V1N3 81, V1N5 37–39, 41, V2N1 94, 98–99, 104–6, 111–17, 124–26, V2N2 85–88, V2N4 152–57, V2N6 16, 20–27, 29–32, 34–35; Perceptual v. Conceptual V1N1 3, V1N4 61, V1N5 47, V2N1 113–18, 132–34, V2N2 87–90, V2N4 194–96, V2N6 31–32, 66–68; Primacy of V1N1 11, V1N2 23, 60, V1N3 27, 64–65, V1N4 36, V1N5 30, 109–13, V2N4 99–100, V2N6 7–8; Primary v. Higher-Order V2N1 116–19, V2N2 88–89, 138, V2N4 195; Reality of V1N2 52, V1N4 37, V2N1 93–99, V2N2 9, V2N6 1–3, 20; and Self-Consciousness V1N2 48–51, 59–61, V1N5 31–32, 52, V1N6 29–31, V2N1 116–18, 133–34, V2N2 88, V2N3 98–99, V2N4 191, V2N6 12, 33; and Self-Mirror V1N2 67–75, 80, V1N3 96–97; and Time V1N1 11, V1N2 10–11, V1N3 19–20, 64, V1N4 24, V2N1 111, 118, V2N2 88–89, V2N4 155, V2N6 12–18, 184–85; Vigilant V1N5 20–22, 37–38, 48, V2N4 153–54; Volitional Degrees of V1N2 47–63, V1N5 29, 42–47, 49–51, V2N1 118, 120–22, V2N6 70–71; and Zoological Taxa V1N2 69, V2N2 137. See also Perception; Attention; Emotions; Concepts; Subconscious.




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Post 4

Friday, November 14, 2008 - 8:13amSanction this postReply
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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”)

 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 

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




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