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Modern science could not advance until the ideas of force and mass were understood. Newton provided this understanding by stressing the role of momentum and by creating his three laws of motion. But notions were not rigorously based and left much undefined until Mach gave a detailed operational method for defining mass and force. Newton's genius and insight was illustrated by the fact that he got everything right in spite of this. When force and mass were examined with rigorous logic, they were found to be in accord with Newton's intuition.

An account is given of the Forrest‐Armstrong theory of number (Peter Forrest). Natural numbers are argued to be relations holding between a certain property and a certain mereological whole (black swan on the lake now, and the whole that these swans make). With the rational numbers and the real numbers the relation becomes one of proportion, they are the units that measure the proportion. It is pointed out, however, that this view is largely to be found in Isaac Newton, and is even anticipated in Aristotle. What it is for a mathematical entity such as a number to be ‘instantiated’ is considered.

This chapter deals with experimental philosophy, as represented in Gilbert on magnetism, Hobbes on the air pump, and Newton on the production of the spectrum. It is shown that experimental philosophy differs from mechanism in quite radical ways. In particular, it has explanatory success but in apparently very localized domains, and it construes causation not in terms of underlying causes but in terms of causes acting at the same level. Its difference from mechanism is manifest in the contrast between Descartes' and Newton's accounts of the production of the spectrum: Descartes provides a fully geometrical account of the separation of coloured rays, but then shifts into a different register, a qualitative and speculative one in attempting to provide a micro-corpuscularian account of the physical basis of colour production; Newton manages to account for the spectrum without leaving the phenomenal geometricized level, eschewing any recourse to ‘underlying’ causes.

This chapter looks at attempts to quantify natural phenomena and, in particular, forces. Early efforts along these lines — notably by Galileo and Descartes — tried to extrapolate from statics to dynamics, whereas later in the century kinematics, as pioneered by Galileo, was taken as the model by Huygens and Newton. Newton, building on Hooke's suggestion that planetary orbits were not a given and unquestionable feature of the cosmos, was able to show how such orbits were generated and clarify the dynamics needed to account for the processes involved. In this way, mechanics, traditionally excluded from natural philosophy in the Aristotelian sense, is transformed not only into a natural-philosophical discipline, but into what was in many respects the natural-philosophical discipline par excellence.

This chapter argues that Albert Einstein was not the first to use the relativity principle (RP) as a postulate in the treatment of a problem in physics. Christian Huygens had done so over two hundred years earlier in his treatment of collisions. It is shown that even Newton in his pre-Principia writings viewed the RP as having the same axiomatic status as his laws. But the prominent role played by the principle in Einstein's 1905 paper on moving bodies in electrodynamics marked the beginning of a new attitude concerning the foundational status of symmetries in physics.

This chapter focuses on how Einstein arrived at his special theory of relativity. It discusses Einstein's postulates, his derivation of the Lorentz transformations, and experimental evidence for the Lorentz transformations. The chapter then addresses the question of whether Einstein's inertial frames are the same as Newton's.

What, then, is space? It seems strange since it is part of the physical world but unlike other physical things; in some sense it is the ‘place’ where they all are. This chapter first explores how the founders of modern science attempted to solve the problem: Descartes thought that space was just the objects that occupied it; Newton thought it some kind of separate container, ‘absolute space’; Leibniz simply denied that there was such a thing, instead claiming that all motion is relative, ‘relationism’. Newton showed that these philosophical debates have deep significance for physics: his ‘bucket’ thought‐experiment emphasizes that, according to his mechanics, things have motion distinct from their relative motion. The chapter ends by defending relationism against Newton's critique.

Newton says shadows are mere absences of light. Since the visual response to a lack of stimulation is black, all shadows would then be black. In his attack on Newton's optics, Goethe drew attention to subtle conditions under which there appear to be blue shadows. According to Goethe, all hues are colored shadows. Later, color scientists, most famously Edwin Land (founder of the Polaroid Corporation), appear to produce shadows of virtually every hue. Most contemporary color scientists inconsistently accept both Newton's account of light and standard, textbook demonstrations of colored shadows. They should regain consistency by distinguishing between shadows and "filtows."

The main aim of this chapter is to clear the ground of objections to the notion that time, ‘of itself, and from its own nature, flows equably without relation to anything external’. The three objections to the passage of time: logical, scientific, and epistemological, are discussed. It is argued that time passes, and that in virtue of that passage things change. There are no good logical, scientific, or philosophical arguments that cast doubt on the passing of time, and there are no impediments to representing in present physical theories that time passes.

The major iterative algorithms based on functional iteration techniques for solving nonlinear matrix equations are described, analysed, and compared in this chapter. First, the basic concepts on fixed point iterations are recalled, then linearly convergent iterations are treated, and finally Newton’s iteration is considered.

Some specialized structures are investigated in this chapter and some of the algorithms in previous chapters are adapted to the specific cases. Markov chains with limited displacement (non-skip-free processes) are solved by means of functional iterations and cyclic reduction. Markov chains of M/G/1-type are reduced to a special QBD process with infinite blocks and treated with cyclic reduction. Finally, three different algorithms for tree-like stochastic processes, relying on fixed point iterations, Newton’s iteration, and cyclic reduction, are introduced and analysed.