Math is a rational enterprise, isn’t it? Pure logic without the encumbrance of words. Reason’s pinnacle. So is science, right? Maybe all of us should devise our life maps on the rational projections afforded by science and math. Live, for example, a statistically happy life. Meet the mean, mode, and median, and then succeed in skewing the curve to an above-average, even superior, lifestyle as defined by what is rational and scientific, and countable in sets. What if, however, the math-science projection has a flaw? In fact, mathematical proofs cheat a little. No one writes all the details of a proof because such writing could go on ad infinitum. Instead, mathematicians use mathematical induction, with lots of implication: One statement implies the next. And now mathematics has evolved to incorporate numbers that are much too big for most of us to understand, like the Monster Group, that is, the 10^55 elements that vertex operator algebra has given us.
What did the ancients like Euclid of Megaris and Euclid of Alexandria do to us?
They didn’t just give us a way to describe geometric figures. They set in motion a way of thinking about the world that has influenced Western thought for more than 2,000 years. They gave us reason as a guide and made the principles of proof the value system we hold dear, not just for understanding shapes like squares and triangles, but for understanding the dots of life. We think in Euclidian terms. We are geometers of life. We prove by constant justification, and we justify by references to axioms, assumptions, and sets. We love to hear summaries of statistics about what people think, what they do, how they live, how they get sick, and how they die. It’s our intellectual inheritance, and, in our brain’s grid cells, a built-in pattern. Except...
Except, as Kurt Gӧdel explained, a “logical” or a “mathematical” system will always be incomplete because it cannot ultimately prove itself. So, when we make an informed and rational decision, we should remember that the system that provided the information and the rationality itself might—rather, probably will—have a basis in ignorance and assumption. We think axiomatically, don’t we? We accept certain axioms, and we go about proving our theorems of life. Take polls and statistics as examples. If you get Lou Gehrig’s disease, you are, statistically speaking, “not long for this world.” Now, think of a person who has that disease, like Stephen Hawking. Famous for his theoretical physics and popularizations of those physics, Hawking got the disease in his twenties. Decades later, he is, as I pen this, still theorizing, writing, and making public appearances. Wasn’t he supposed to die young and unaccomplished?
You are not a number. You could defy any numerical description or statistical projection. Euclid might say “No” from a “reasoned" argument. All the Euclidean thinkers might say, “No, you can’t; no, you shouldn’t; no, you couldn’t.”
You can respond, “Yes, you could, Euclid.”
What did the ancients like Euclid of Megaris and Euclid of Alexandria do to us?
They didn’t just give us a way to describe geometric figures. They set in motion a way of thinking about the world that has influenced Western thought for more than 2,000 years. They gave us reason as a guide and made the principles of proof the value system we hold dear, not just for understanding shapes like squares and triangles, but for understanding the dots of life. We think in Euclidian terms. We are geometers of life. We prove by constant justification, and we justify by references to axioms, assumptions, and sets. We love to hear summaries of statistics about what people think, what they do, how they live, how they get sick, and how they die. It’s our intellectual inheritance, and, in our brain’s grid cells, a built-in pattern. Except...
Except, as Kurt Gӧdel explained, a “logical” or a “mathematical” system will always be incomplete because it cannot ultimately prove itself. So, when we make an informed and rational decision, we should remember that the system that provided the information and the rationality itself might—rather, probably will—have a basis in ignorance and assumption. We think axiomatically, don’t we? We accept certain axioms, and we go about proving our theorems of life. Take polls and statistics as examples. If you get Lou Gehrig’s disease, you are, statistically speaking, “not long for this world.” Now, think of a person who has that disease, like Stephen Hawking. Famous for his theoretical physics and popularizations of those physics, Hawking got the disease in his twenties. Decades later, he is, as I pen this, still theorizing, writing, and making public appearances. Wasn’t he supposed to die young and unaccomplished?
You are not a number. You could defy any numerical description or statistical projection. Euclid might say “No” from a “reasoned" argument. All the Euclidean thinkers might say, “No, you can’t; no, you shouldn’t; no, you couldn’t.”
You can respond, “Yes, you could, Euclid.”
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