On 3/8/2017 12:10 AM, Jeffrey Brian Downard wrote:
I'm trying to interpret Peirce's remarks about the importance of stating the mathematical hypotheses of a system precisely for the purpose of drawing conclusions with exactitude.
I certainly agree. And the point I was trying to make is that the *creative insight* comes first. That is the discovery of the diagram. The diagram gives you *exactitude*. Formalization is a convenient notation -- and as Peirce noted, even algebra is a linear diagram. Note that tic-tac-toe, chess, go, bridge, and similar games have very well defined diagrams. The rules are usually stated in natural languages, but they are (or can be) expressed with as much precision as any system that is called mathematics.
That, I take it, is the kind of advance that was made by Euclid and his predecessors in stating the postulates, definitions and common notions with considerable (although still far from perfect) precision.
I agree that Euclid's systematic treatment was an important advance, and it stimulated a very active school in Alexandria. But the Sumerians, Babylonians, and Egyptians had sophisticated math for centuries before Pythagoras, who was a couple of centuries before Euclid. In fact, Plato was considered a better mathematician than Aristotle, who was primarily a biologist. But Aristotle's systematic way of organizing and presenting his writings inspired Euclid to organize his _Elements_.
The philosopher, on the other hand, must accept the vague conceptions that are part and parcel of his inquiries-- warts and all.
All perception begins with vagueness. Through experience, certain aspects (icons) are distinguished as more important than others. Those are the things that are named. Languages -- or symbols in general -- "grow from icons". Language is the great advantage of our species. And mathematics is just a systematic refinement of certain kinds of language games. It didn't spring into existence with Euclid. Scratches on bone and monuments such as Stonehenge show that mathematics evolved for millennia before Euclid. The reason why philosophy is not as precise as physics is that physicists have been studying the easy stuff -- things that can be clearly distinguished, measured, and organized. Quantum mechanics was a painful shock for many physicists. John
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