Algebra As Thought Experiment

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Algebra As Thought Experiment

ABSTRACT: This paper addresses the problem of understanding what mathematics contributes to the exceptional success of modern mathematical physics. I urge that we give up the Kantian construal of the division between mathematics (synthetic a priori) and physics (experimental), and that we ask instead how algebra helps synthetic a posteriori mathematics improve our ability to study the world. The theses suggested are: 1) Mathematical theories are about the empirical world, and are true or false just like other theories of empirical science. 2) The air of artificiality in mathematics lies exclusively in the use of algebraic method. 3) This method is constructive much like all fiction is, but this construction is for the purpose of experimental investigation of the physical world to the extent that anything in the world has objects like those in the fictional world of a particular algebra. 4) This is why algebraic techniques are successful even when the assumptions of the system are false: they may still be applicable to some things considered from some perspective. 5) The success of mathematical physics is also due to Descartes' discovery of a remarkable truth: we live in space and time which can be described as a whole. 6) Therefore, what distinguishes modern science from earlier and later philosophy is not a general method of science, but the fact that it happened to find a truth, and a particular way of studying reality which bore fruit.

In the sixteenth century, physics was a part of the general subject known as philosophy. When Galileo wrote the Dialogues Concerning the Two Chief World Systems, he was commenting on some aspects of Aristotle's Philosophy. He was more favorable to the use of mathematics in various problems of physics than was current in his day. He may be described as trying to revive an Archimedean conception of motion in terms of geometry, which may be called "kinematics."

Galileo also introduced the experimental method into physics. The experimental method had been very successfully utilized in Alchemy, which was a low or a clandestine science at the time. Its success in physics brought the issue of experimentation into the spot light. Since there are few claims original to Galileo that we may still wish to defend, we may want to describe his success as the introduction of these techniques which he invented or adapted for subsequent physicists to employ. Experimental techniques have improved considerably since Galileo's day.

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