Free A Brief Guide to the Great Equations by Robert Crease

from afar. Newton’s stage is spare, but therein lies its beauty and effectiveness. Nature is not to be looked at, as Aristotle had, as a cosmic ecosystem, in which qualitatively different kinds of things act in qualitatively different kinds of ways in qualitatively different domains. It is more like a cosmic billiard table, in which all space is alike, all directions are comparable, all events are motions, and in all changes of motion the same basic kinds of things exert the same basic kinds of forces. In this world, movement involves change in space, not attainment, actualization, or intensification of being. It’s a world where all chandeliers, trapezes, and swings are pendulums, all sport and dance instances of
F
=
ma
, all balls elastic, and all planes go on forever. One can move about this stage anywhere in space and over time – in a car, train, plane, roller coaster, bicycle – and the laws remain the same, ‘invariant under translation’, we would say. If you want to understand what’s happening on this world-stage, according to Newton,here’s what to do: First, quantify positions, speeds, and masses. Then follow the forces. 23
    This is how
F
=
ma
can be both a definition and an empirically discoverable fact. It is a definition insofar as it is part of the warp and woof of the abstract world-stage. It is a fact insofar as, when connected with our world via the right concepts, assumptions, and measurement techniques, it states a quantitative relationship between values found in a laboratory situation. Theories become the vehicles by which to go back and forth between the world-stage and our own, between its ideal values and the real values of our world. To nonscientists, the abstract world might seem strange, something arbitrary and imposed upon nature, a world of fiction – an effective fiction, perhaps, but an invention nonetheless. To scientists, who have been trained to connect this abstract world with our own through concepts, procedures, and measurement practices, the temptation is just the opposite. They can move so confidently back and forth that they can forget how abstract this world-stage is – as if it were not an invented part of the world they live in.
    At this final stop, where
F
=
ma
appears, therefore, we have traveled quite a distance from the beginning of the human understanding of motion. We see the heavens and the earth as the same place. We do not see an inherent distinction between natural and violent motion, or between natural and unnatural places, or between different kinds of forces. All things follow the same laws, and if we think that something behaves strangely, we assume that we do not yet see how the laws we know apply to it. Motion is not an action but a state. There is one kind of motion, and circular motions are to be explained as the result of combinations of components. Aristotle may be right that motions in the heavens seem different from motions down here, but that’s because the resistances in the heavens are different. Nature is a huge space-time determination of forces, accelerations, and motions whose blueprint we can seek through theories. This allows us to view nature as full of quantitative laws to be found through experimentation.
    ‘[Newton] is our Columbus’, Voltaire wrote in 1732, ‘he led us to a new world.’ 24 But it is a strange world. It is not found in our own like a concealed continent. Nor is it revealed by instruments, the way tiny worlds are seen in microscopes, or distant and gigantic ones in telescopes. Newton’s strange new world was found in our world – but it is not our world, either, nor one we could live in. We humans, even the scientists among us, inhabit what philosophers call the ‘lived world’, amid designs, desires, and purposes: we live in an Aristotelian world. The world Newton discovered is an abstract one that appears by changing what we look at and how we look at it. It’s a fishbowl-like world, seen from the outside,