The Cambridge Companion to Hobbes (Cambridge Companions to Philosophy)

Hobbes saw geometry as the "'onely Science it hath pleased God hitherto to bestow on mankind'" (87). Consequently he took "geometry as the model science and as the foundation of his entire philosophy" (110-111). Indeed, "some of his apparently oddest ideas were in fact faithful echoes of the mathematical mainstream of centuries past" (126).In social philosophy, Hobbes started from the assumption that "life in the state of nature is 'solitary, poore, nasty, brutish, and short'" (216), and found that the only reasonable strategy would be to stipulate a few basic principles that enable mankind to avoid this state by a "contractual escape route" (225). These are the axioms of the theory, as it were, and the laws of the land "'are but Conclusions, or Theoremes concerning what conduceth to the conservation and defence of themselves'" (223).But the mathematical parallel goes deeper than this. For a crucial aspect of classical geometry is its constructive character, seen in the fact that it constructs everything using motions, i.e., using instruments such as ruler and compass:"Demonstrations are flawed unless their conclusions are demonstrated by construction, that is, by description of figures, that is, by the drawing of lines. For every drawing of a line is motion, and so every demonstration is flawed, whose first principles are not contained in the definitions of motions by which figures are described." (88)By analogy, "'civil philosophy is demonstrable, because we make the commonwealth ourselves'" (104), just as in geometry we make the figures of which we speak using ruler and compass.Note the anti-Platonic nature of this stance: things have meaning only through our constructions of them; there is no Platonic ideal world of truths independent of human actions.In social philosophy, this means that morals are socially constructed. "'Whence it is to be understood that they, who consider men by themselves and as though they existed outside of civil society, can have no moral science because they lack any certain standard against which virtue and vice can be judged and defined.'" (180)In geometry, the same principle means that there is no "ideal" breadthless line and the like, since geometry is only what we can make physically. "[Hobbes's] most insistent challenge to the ancient legacy was at its very foundations, the definitions that underlie the Elements. Again and again he criticized Euclid's definition of a point as 'that which has no part'. A geometrical point (he urged) is a visible mark, and so has quantity, and so is potentially divisible into parts, although such parts are 'not considered' in demonstrations. Similarly he balked at Euclid's definition of a line as 'breadthless length.' For 'lines are not drawn but by motion, and motion is of body only', so that a line must have a width, although this too is always negligible in practice. These pronouncements encapsulate much of Hobbes's philosophy of mathematics. They place him in sharp opposition to the mainstream view that the objects of geometry are abstractions from, idealizations of, sensory experience. He saw geometry as a quasi-physical science of extended body, and his insistence that its objects are produced by physical motions was profoundly characteristic." (112)This perspective served Hobbes well in optics, where the idea that a light ray have some breadth enabled him to explain refraction as resulting from the fact that one part of the ray enters the new medium sooner than the other (134).In mathematics itself, the emphasis on geometric construction means that "Hobbes earned much notoriety, in his own time and later, for his hostility to the mathematicians' rapidly increasing use of algebra. ... He had no patience with algebra's 'scab of symbols' the shorthand that made a mathematical page look 'as if a hen had been scraping there'. He conceded that these symbols might be useful, even necessary, aids to demonstration, but 'they ought no more to appear in public, than the most deformed necessary business which you do in your chambers'." (114)In physics the analog of the construction paradigm is that "'There is no Effect of Nature, ye Cause whereof does not consist in some motion.'" (90) In other words, all physics should be explained in terms of bodies bumping into other bodies.The same principle also applies in psychology. For instance, sensory impressions are mechanical actions that literally strike up a motion in the mind. But since "'when a thing is in motion, it will eternally be in motion, unless something else stay it'" (158) it follows that impressions remain in our mind (as inertial motions) until a new impression strikes. Consequently, "'any object being removed from our eyes, though the impression it made in us remain; yet other objects more present succeeding, and working on us, the imagination of the past is obscured, and make weak. ... From whence it followeth, that the longer the time is, after the sight or Sense of any object, the weaker is the imagination. For the continual change of man's body destroys in time the parts which in sense were moved.'" (308)In physics we also test theories by checking its predictions against empirical data. Hobbes's physical, anti-Platonic conception of mathematics means that mathematics is no different in this regard. Social philosophy is once again much the same, with history taking the role of empirical data: "History is the laboratory of the hobbist philosopher; ... the verification par excellence of a political and philosophical theory is the kind of history it can produce" (325).