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Originally Posted by DChristopher  Berkeley's criticism of Calculus was very fair and very well-done.
Mathematicians of the time knew two things very well:
(a) that Calculus worked; and
(b) that Calculus was on shaky theoretical footing.
They knew what they meant, but they didn't know how to explain it or prove it formally. Oh, the verbal hoops they jumped through.
In the 1800's they fixed it, vindicating (to some extent) Berkeley's complaints.
[Aside: I have no idea what Berkeley wrote on philosophy, I'm only familiar with his famous rant against "ghosts of departed quantities"  ] |
Berkeley believed that things don't exist unless they are perceived. For example, Berkeley would say: a tree will vanish if no one's looking at it. Which is why I think overall, I don't see how Newton was required to give a better defense of the ideas than to merely say that we all stopped looking at the quantities, and they simply vanished. I don't see what basis Berkeley would have in arguing the point any further.
It's true that in the 1800's, Calculus was placed on a more rigourous foundation that used limits, and epsilon-delta definitions. But this didn't really answer the questions Berkely raised as much as it avoided talking about the problems that lead to the questions. IMO it was just an evasive technique to silence the critics.
Even later (1960's?), Abraham Robinson's work on infinitesimals and non-standard analysis showed that Newton's approach was generally workable, in that we can define infinitely small quanities and we use them consistently. In other words, there was nothing terribly wrong with Newton's basic approach aside from some vagueness ... IMO the primary problem was just that most people (esp Berkeley) did not understand Newton's overall thought process. Newton didn't help matters, as he seemed to hate publishing his ideas or dealing with questions about them.
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Originally Posted by bns He's western. The ancient Greeks, as well as the Romans are considered to be part of western culture.
Easy there, Godwin.  |
But Heidegger really was a nazi ... this isn't an analogy.