4 ideas
18085 | Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy] |
Full Idea: When the successive absolute values of a variable decrease indefinitely in such a way as to become less than any given quantity, that variable becomes what is called an 'infinitesimal'. Such a variable has zero as its limit. | |
From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4 | |
A reaction: The creator of the important idea of the limit still talked in terms of infinitesimals. In the next generation the limit took over completely. |
18084 | When successive variable values approach a fixed value, that is its 'limit' [Cauchy] |
Full Idea: When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the 'limit' of all the others. | |
From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4 | |
A reaction: This seems to be a highly significan proposal, because you can now treat that limit as a number, and adds things to it. It opens the door to Cantor's infinities. Is the 'limit' just a fiction? |
1558 | Clearly the gods ignore human affairs, or they would have given us justice [Thrasymachus] |
Full Idea: The gods pay no attention to human affairs; if they did, they would not have ignored justice, which is the greatest good for men; for we see that men do not act with justice. | |
From: Thrasymachus (fragments/reports [c.426 BCE], B8), quoted by Hermias - Notes on Plato's 'Phaedrus' 239.22 |
1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
From: Herodotus (The Histories [c.435 BCE], 2.123.2) |