Five different calculuses

Fine Doubter

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.

bongo fury

↪Fine Doubter

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.

reminiscences, — Fine Doubter

Yes, # me too!

25

jgill

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 — Fine Doubter

Would you expand on this or provide links?

fishfry

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? — Fine Doubter

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."

reforms of the calculus in the 19th C? In terms of limits, as explained copiously hereabouts by fishfry — bongo fury

Thanks for remembering :-)

Mark Nyquist

↪jgill

Something that comes to mind is Zeno's paradox. Zeno of Elea born c. 495 BCE. A detail I recall, but can't reference, was that it was intentionally meant to be counter intuitive. Zeno understood the problem and used it to provoke a reaction. This would have been pre-Socratic Greece. So that would be the earliest example I know of someone dealing with the basics of calculus. I'm sure you knew that but it might be a starting point for anyone interested.

jgill

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. — Fine Doubter

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.

Fine Doubter

↪jgill

That was a smattering of web pages, plus a few newish second hand books I saw. I suppose the people that write these things don't understand "elementary".

Fine Doubter

↪jgill

I'd have loved more maths of all kinds, but authorities tell you what not to do when you are stressed out at age 15 or 16 and have no guidance in assertiveness. I left school with only one (poor) A level but subsequently managed to get back in contact with languages (fortunately) but you know how it is one sits and thinks if only I had been a chemist / architect . . .

Fine Doubter

↪bongo fury

Thank you for the link to the conundrums thread, I'm going to have huge enjoyment in that! :yum: :cool: :starstruck:

Fine Doubter

↪Mark Nyquist

Yes, most of the "paradoxes" from that period are just riddles. Another example I like is "all Cretans are liars" (a play on a prominent personality of the time) which is meant to illustrate the difference between an opposite and a contrary to "some Cretans are liars" and the real meaning is "not all Cretans are liars". I wish most of the people around today would understand this vital piece of logic!

Mark Nyquist

↪Fine Doubter

Here is my experience with calculus. A few years after taking undergrad Calculus I and II at a state university (about 40 years ago) cheap graphing calculators were available. I got one and relearned by graphing problems.
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.

jgill

↪Mark Nyquist

Eudoxus (370 BC) laid the foundations for integral calculus by approximating areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. Occasionally a calculus course is taught from this historical perspective, with integration first, then differentiation.

Mark Nyquist

↪jgill

Eudoxus of Cnidus. This is a good one...astronomer, mathematician...connections with Socrates, Plato and taught Aristotle...Built an astronomical observatory plus what you said. Worked with proportions...that's interesting. Wow.
All his works were lost.

Mark Nyquist

↪Fine Doubter

I could add there is a lot of free math content online now. Tutorials, calculators, simulators, on and on.

TheMadFool

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Five different calculuses

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