Five different calculuses
1. I understand from recent background reading that ancients had procedures for approximating at successive points.
2. I also understand that Newton and his contemporaries developed an abstract method to simulate continuity in these matters.
3. At age 15 I encountered dy over dx (which the ratty teacher irrationally insisted, doesn't cancel to y over x) and a snake shape. It wasn't admitted that these had any purpose, nor was the "procedure" explained.
(Prior to that age I loved my trig, and my quadratics, and cross multiplying, etc)
4. Some second hand books that I have leafed through, seemingly of the period when I was at school, averred that calculus was all about areas and speeds, though that had never had anything to do with the lessons I had "had". Is this a class thing? Reality for secondary modern and technical college, and gobbledy gook for grammar school (which I only scraped into).
5. Recently published material I encounter has the advantage of reverting to the ancient approximation goal, but the continued disadvantage of keeping their abstract method impenetrable.
It strikes me the ancients had a good instinct because at the atomic level, and in human sciences, all is mathematically granular. Grateful for your comments, reminiscences, inspirations etc.
2. I also understand that Newton and his contemporaries developed an abstract method to simulate continuity in these matters.
3. At age 15 I encountered dy over dx (which the ratty teacher irrationally insisted, doesn't cancel to y over x) and a snake shape. It wasn't admitted that these had any purpose, nor was the "procedure" explained.
(Prior to that age I loved my trig, and my quadratics, and cross multiplying, etc)
4. Some second hand books that I have leafed through, seemingly of the period when I was at school, averred that calculus was all about areas and speeds, though that had never had anything to do with the lessons I had "had". Is this a class thing? Reality for secondary modern and technical college, and gobbledy gook for grammar school (which I only scraped into).
5. Recently published material I encounter has the advantage of reverting to the ancient approximation goal, but the continued disadvantage of keeping their abstract method impenetrable.
It strikes me the ancients had a good instinct because at the atomic level, and in human sciences, all is mathematically granular. Grateful for your comments, reminiscences, inspirations etc.
Comments (14)
3. Were you aware that in the 18thC your incredulity was famously supported by Bishop Berkeley in The Analyst? And that despite his religious motivation for doubting science, he seems to have been exonerated by...
4. reforms of the calculus in the 19th C? In terms of limits, as explained copiously hereabouts by @fishfry, e.g. https://thephilosophyforum.com/discussion/comment/184240. That's probably what 20thC textbooks are trying to explain, although many people find that no less challenging, and accept the older and apparently questionable notational shortcuts.
5. The twist in the tale, 20thC, is that maybe Newton and Leibniz were (entirely) right all along: Berkeley's dreaded infinitesimals were rehabilitated. There is a thread about that somewhere. Whereas I would expect that recent material 'reverting to the ancient approximation goal' is more likely in the spirit of 4. Which is still the consensus.
Quoting Fine Doubter
Yes, # me too!
25
Would you expand on this or provide links?
Velocities and areas are applications of calculus, as in Newton's fluxions and fluents, corresponding to today's derivatives and definite integrals. In pure math, one only uses velocity and area as illustrations to help students understand, but they're irrelevant to the mathematical content.
Find Doubter, nice pun on "Find Outer."
Quoting bongo fury
Thanks for remembering :-)
If I had my druthers I would have high school math teachers end with a decent introductory course in analytic geometry, rather than do what they might do to calculus.
What I suspect is the people who developed calculus started with shapes and moved to notation and as they learned the subject would use mostly notation. As a student, thinking in terms of shapes can help. But there is the extra layer of knowing your calculators capabilities and limits. And sometimes pencil and paper work too. I sometimes work simple problems just so I don't forget the basics. I'm not someone who uses it at work so I just find my own comfort level.
All his works were lost.