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→Kursk invasion
:Merriam-Webster's definition of infinity is unusually inadequate: "an indefinitely great number or amount." The essence of infinity is not that it is indefinite.--[[User:Aschlafly|Andy Schlafly]] ([[User talk:Aschlafly|talk]]) 00:36, 18 May 2018 (EDT)
:Merriam-Webster's definition of infinity is unusually inadequate: "an indefinitely great number or amount." The essence of infinity is not that it is indefinite.--[[User:Aschlafly|Andy Schlafly]] ([[User talk:Aschlafly|talk]]) 00:36, 18 May 2018 (EDT)
== Bernhard Riemann ==
''Inherent in the concept of infinity is the concept of infinitesimal: infinity small. Bernhard Riemann identified the sum of infinitesimal strips of area under a function to establish a rigorous definition of the integral, now known as the Riemann Integral. ''
I think that this misrepresents Bernhard Riemann's ideas:
The infinitesimals were tools of the creative geniuses of the 18th century, as Euler, Leibniz, and Newton. Riemann took a step away: he does not talk about things which ''are'' infinitely small, but which ''become'' infinitely small - using the idea of limits. Therefore, the integral is not the sum of infinitesimal strips, but the limit of finite sums of small stripes, becoming smaller and smaller.
That may sound like hair-splitting, but his approach could be rigorously formalized quite easily, while this was much more difficult for infinitesimals.--[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 10:36, 30 July 2018 (EDT)
:Not sure if there is a meaningful distinction here. Infinitely small is inherently a limit. But you make a very interesting point and I'll think about it further. Also, did Euler, Leibniz or Newton use the concept of infinitesimal?--[[User:Aschlafly|Andy Schlafly]] ([[User talk:Aschlafly|talk]]) 12:24, 30 July 2018 (EDT)
#Fermat had already used infinitesimals to calculate the tangents of polynomials. For the parabola y=<math>x^2</math>, he would look at the secant through the points (x,<math>x^2</math>), (x+D,<math>(x+D)^2</math>) which are separated at the x-axis by the infinitesimal distance <math>D</math> and then calculate <math>slope = \frac{(x+D)^2 - x^2}{D} = </math> <math>\frac{2Dx+D^2}{D} = </math> <math>2x+D = </math> <math>2x</math> Calculations like this drove philosophers like Hobbes crazy: how can you say something is zero at one time and works not as zero another time. For us, it is easy to reconcile this by introducing limits, but those were not thought of for quite a while!
#Euler used infinitesimals creatively, Leibniz called them <math>dx</math> etc. If I recall correctly, Newton did not generally use them for integration: he transformed the function which he wanted to integrate into a power series, integrated the parts of the series (as it was well known how to integrate polynomials), and at last, tried to transform the resulting power-series back into a well-known function. The first and the third step often required quite an ingenuity
#In my opinion one of the great things about Riemann-Cauchy-integration is that it transformed an art into a craft: only few had the talent to use infinitesimals in a way that provided correct results!
--[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 15:06, 30 July 2018 (EDT)
::That's fascinating history, August! Sounds like Fermat et al. used tiny numbers, but Cauchy and Riemann made it infinitesimal rigorous. Note that Merriam-Webster says the term infinitesimal did not exist in English until the early 1700s, which would have been late in the game for Newton.--[[User:Aschlafly|Andy Schlafly]] ([[User talk:Aschlafly|talk]]) 00:15, 31 July 2018 (EDT)
:::#No, Fermat, Newton, Leibniz did not use tiny numbers: they used quantities which differed from numbers - and they used them with various degrees of discomfort.
:::#"infinitesimal" as a noun exists since the 1650s in the English language - but that is of not much relevance, as Newton published some of his work in Latin, and furthermore, used the term "moments" instead of "infinitesimals". In his later works, he tried to avoid infinitesimals.
:::#No, Cauchy and Riemann got rid of infinitesimals, thereby putting the theory on sound footing. Abraham Robinson with his non-standard analysis made infinitesimals rigorous - but this was in the 1960s
:::#This is all quite basic stuff. I should be surprised that someone who claims to be able to identify the greatest modern mathematician just by looking on a chart of published papers knows so little about the history of the mathematics (and the mathematicians) he is talking about. But this being Conservapedia, and the someone being you, the surprise is somewhat infinitesimal...
:::--[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 02:34, 31 July 2018 (EDT)
== Multiplication of the Loaves ==
As we are not still drowned in loaves, I suppose only I ''finite'' number of loaves (and fishes) were created... --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 16:15, August 25, 2021 (EDT)
==Kursk invasion==
I heard a good analysis the other day about the [[NATO invasion of Russia]]: the strategic thinking and military logic behind the Kursk invasion is like trying to divide infinity by two. [[User:RobSmith|RobS]]<sup>[[User talk:RobSmith|''The Truth. Just Putin It Out There'']]</sup> 08:13, September 5, 2024 (EDT)