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René Descartes is often called the "father of modern philosophy." Born into the French aristocracy in 1596, Descartes was fascinated by mathematics and founded the field of analytical geometry. He wanted to build a system of philosophy that was as clear and certain as mathematics. To do this, he decided that he would need to question all of his presuppositions about the world. One of the presuppositions that he doubted was whether or not he could believe that what he observed through his senses was true. Consider this passage from Descartes' Meditations (Melchart, 2007):
Of course, whatever I have so far accepted as supremely true I have learned either from the senses or through the senses. But I have occasionally caught the senses deceiving me, and it's prudent never completely to trust those who have cheated us even once. But, while my senses may deceive me about what is small or far away, there may still be other things that I take in by the senses but that I cannot possibly doubt-like that I am here, sitting before the fire, wearing a dressing gown, touching this paper. And on what grounds might I deny that my hands and the other parts of my body exists?-unless perhaps I liken myself to madmen whose brains are so rattled by the persistent vapors of melancholy that they are sure that they're kings when in fact they are paupers, or that they wear purple robes when in fact they're naked, or that their heads are clay, or that they are gourds, or made of glass.
Take a moment to reflect on this passage, and then discuss the following:
A sphere of radius R is uniformly charged to a total charge of Q. It is made to spin about an axis that passes through its center with an angular speed ω. Find the magnitude of the resulting magnetic field at the center of the sphere.
A resistor is in the shape of a cube, with each side of resistance R . Find the equivalent resistance between any two of its adjacent corners.
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