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Students do seem to have a sense of how individuals “do science,” and they discover it correspondingly less intimidating. With this in mind I actually have been working to convince my students that math usually works pretty much like science. Joseph Silverman lays this out in his e-book A Friendly Introduction to Number Theory (which I am instructing from this semester, and he says this in chapter 1,accessible right here) actually nicely–in number concept, we gather data by computing lots of examples.

Aristotle defined arithmetic as “the science of quantity” and this definition prevailed until the 18th century. Three leading forms of definition of arithmetic at present are known as logicist, intuitionist, and formalist, each reflecting a different philosophical college of thought. All have extreme flaws, none has widespread acceptance, and no reconciliation seems possible. The first girl mathematician recorded by history was Hypatia of Alexandria (AD 350–415).

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