Physics is the study of dynamics. Dynamics is the description of the actual forces of nature that, we believe, underlie the causal structure of the Universe and are responsible for its evolution in time. We are about to embark upon the intensive study of a simple description of nature that introduces the concept of a force, due to Isaac Newton. A force is considered to be the causal agent that produces the effect of acceleration in any massive object, altering its dynamic state of motion.

Newton was not the first person to attempt to describe the underlying nature of causal ity. Many, many others, including my favorite ‘dumb philosopher’, Aristotle, had attempted this. The major difference between Newton’s attempt and previous ones is that Newton did not frame his as a philosophical postulate per se. Instead he formulated it as a mathemat ical theory and proposed a set of laws that (he hoped) precisely described the regularities of motion in nature.

In physics a law is the equivalent of a postulated axiom in mathematics. That is, a physical law is, like an axiom, an assumption about how nature operates that not formally provable by any means, including experience, within the theory. A physical law is thus not considered “correct” – rather we ascribe to it a “degree of belief” based on how well and consistently it describes nature in experiments designed to verify and falsify its correspon dence.

It is important to do both. Again, interested students are are encouraged to look up Karl Popper’s “Falsifiability 29 and the older Postivism30. A hypothesis must successfully withstand the test of repeated, reproducible experiments that both seek to disprove it and to verify that it has predictive value in order to survive and become plausible. And even then, it could be wrong!

If a set of laws survive all the experimental tests we can think up and subject it to,

we consider it likely that it is a good approximation to the true laws of nature; if it passes many tests but then fails others (often failing consistently at some length or time scale) then we may continue to call the postulates laws (applicable within the appropriate milieu) but recognize that they are only approximately true and that they are superceded by some more fundamental laws that are closer (at least) to being the “true laws of nature”.

Newton’s Laws, as it happens, are in this latter category – early postulates of physics that worked remarkably well up to a point (in a certain “classical” regime) and then failed. They are “exact” (for all practical purposes) for massive, large objects moving slowly com pared to the speed of light3³1 for long times such as those we encounter in the everyday world of human experience (as described by Sl scale units). They fail badly (as a basis for prediction) for microscopic phenomena involving short distances, small times and masses, for very strong forces, and for the laboratory description of phenomena occurring at rela tivistic velocities. Nevertheless, even here they survive in a distorted but still recognizable form, and the constructs they introduce to help us study dynamics still survive.

Interestingly, Newton’s laws lead us to second order differential equations, and even quantum mechanics appears to be based on differential equations of second order or less. Third order and higher systems of differential equations seem to have potential problems with temporal causality (where effects always follow, or are at worst simultaneous with, their causes in time); it is part of the genius of Newton’s description that it precisely and sufficiently allows for a full description of causal phenomena, even where the details of that causality turn out to be incorrect.

Incidentally, one of the other interesting features of Newton’s Laws is that Newton in vented calculus to enable him to solve the problems they described. Now you know why calculus is so essential to physics: physics was the original motivation behind the invention of calculus itself. Calculus was also (more or less simultaneously) invented in the more useful and recognizable form that we still use today by other mathematical-philosophers such as Leibnitz, and further developed by many, many people such as Gauss, Poincare, Poisson, Laplace and others.

In the overwhelming majority of cases, especially in the early days, solving one or more problems in the physics that was still being invented was the motivation behind the most significant developments in calculus and differential equa tion theory. This trend continues today, with physics providing an underlying structure and motivation for the development of much of the most advanced mathematics.


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