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Do you remember what aspects of Calculus didn't click for you at first, that is when it was presented to you initially as pure mathematics? For example, was it

by j1vms 11y ago

Do you remember what aspects of Calculus didn't click for you at first, that is when it was presented to you initially as pure mathematics? For example, was it the "idea" of a limit, or just not knowing at first why anyone (even mathematicians) would want to know the rate of change of a function?

jeffwass 11y ago
More of the latter. And how to really think of derivatives beyond just the slope of a function. Or seeing an integral as a continuous sum of infinitesimals instead of just the area under a curve (which is only one way of visualising). And what does that 'dx' really mean inside the integrand. And where integration-by-parts comes from. And Taylor expansions too. My high school calc teacher just put the equation on the board for Taylor series one day. No explanation where it came from, or how to think about it. When I asked if he could 'prove' it to give some context and make it seem less like black magic, the other kids in the class said "I'm sure the formula's right if it's in a textbook". But I didn't care about it being right, I cared about seeing where it came from.(and it's not that hard to derive actually). Taylor expansion is like the screwdriver of physics, it's used everywhere. Other than calculus, matrices also didn't make much sense to me until physics. Eg, why is matrix multiplication so complicated, and what do these things actually represent? Especially opaque was the concept of eigenvectors and eigenvalues, until we used them prolifically in quantum mechanics.
- j1vms 11y ago
  Thanks for the response, I asked it out of genuine interest in how we discover new ideas, sometimes ignore or are unable to see their importance, adopt them and then share them. Even forget those ideas and perhaps have to re-discover them later. I personally empathize with the stuff you listed. In particular, I had trouble accepting at first why matrix multiplication is defined the way that it is. As someone who feels "closer" at this point to applied math than pure math - largely, because it is literally more tangible - I admit a fascination with pure math, as if it were the human pursuit of universal beauty, in its ineffable, permanent and probably unattainable form. And that same pursuit of reason and order has brought renown & fame to some, but mania, loss to many. It's both the pride of humanity and terrifying, in equal measure. Like staring at a Van Gogh, and finally perhaps realizing how marvelous, impactful, insightful the statement the artist may have intended to make, yet from such humble circumstances, and with so few tools.

by j1vms 11y ago

jeffwass 11y ago

More of the latter. And how to really think of derivatives beyond just the slope of a function. Or seeing an integral as a continuous sum of infinitesimals instead of just the area under a curve (which is only one way of visualising). And what does that 'dx' really mean inside the integrand. And where integration-by-parts comes from. And Taylor expansions too. My high school calc teacher just put the equation on the board for Taylor series one day. No explanation where it came from, or how to think about it. When I asked if he could 'prove' it to give some context and make it seem less like black magic, the other kids in the class said "I'm sure the formula's right if it's in a textbook". But I didn't care about it being right, I cared about seeing where it came from.(and it's not that hard to derive actually). Taylor expansion is like the screwdriver of physics, it's used everywhere. Other than calculus, matrices also didn't make much sense to me until physics. Eg, why is matrix multiplication so complicated, and what do these things actually represent? Especially opaque was the concept of eigenvectors and eigenvalues, until we used them prolifically in quantum mechanics.

j1vms 11y ago
Thanks for the response, I asked it out of genuine interest in how we discover new ideas, sometimes ignore or are unable to see their importance, adopt them and then share them. Even forget those ideas and perhaps have to re-discover them later. I personally empathize with the stuff you listed. In particular, I had trouble accepting at first why matrix multiplication is defined the way that it is. As someone who feels "closer" at this point to applied math than pure math - largely, because it is literally more tangible - I admit a fascination with pure math, as if it were the human pursuit of universal beauty, in its ineffable, permanent and probably unattainable form. And that same pursuit of reason and order has brought renown & fame to some, but mania, loss to many. It's both the pride of humanity and terrifying, in equal measure. Like staring at a Van Gogh, and finally perhaps realizing how marvelous, impactful, insightful the statement the artist may have intended to make, yet from such humble circumstances, and with so few tools.