Here's a poser for you: if you had to define mathematics from scratch, how would you do it? This is not an idle question. Mathematics is undeniably successful, but why is it successful? Could we ever build something else that would work just as well, but would be different? What would it look like? What elements would it share with mathematics?
I'll tell you why I think this is important. Consider this: "numbers have lives of their own. We can't control them.... once we decide what we mean by them we have no say in how they behave. They obey certain laws and have certain properties, personalities, and ways of combining with one another, and there's nothing we can do about it." Mathematics begins with a few basic, autonomous entities (numbers) and a few basic, simple principles (additivity, commutativity, substitutivity), and then allows these to (conceptually) interact. We discover the relation between mathematics and the world; we don't define it. Compare what we would have if we began with, say, "a system for determining the placement of the planets relative to ourselves".
Evolution doesn't begin with an elephant. Mathematics doesn't begin with calculus. Chemistry doesn't begin with plastics. Innovation doesn't begin with applications. Networks don't begin with bandwidth optimization plans. True science begins with something interesting, creates ways for it to interact and combine, and then steps back discovers where it take us. I would suggest that this is the best way to develop people, too.
I'll tell you why I think this is important. Consider this: "numbers have lives of their own. We can't control them.... once we decide what we mean by them we have no say in how they behave. They obey certain laws and have certain properties, personalities, and ways of combining with one another, and there's nothing we can do about it." Mathematics begins with a few basic, autonomous entities (numbers) and a few basic, simple principles (additivity, commutativity, substitutivity), and then allows these to (conceptually) interact. We discover the relation between mathematics and the world; we don't define it. Compare what we would have if we began with, say, "a system for determining the placement of the planets relative to ourselves".
Evolution doesn't begin with an elephant. Mathematics doesn't begin with calculus. Chemistry doesn't begin with plastics. Innovation doesn't begin with applications. Networks don't begin with bandwidth optimization plans. True science begins with something interesting, creates ways for it to interact and combine, and then steps back discovers where it take us. I would suggest that this is the best way to develop people, too.
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