Why integrals took 2000 years to come up in a rigorous manner?

[Adesh]

TL;DR Summary

It’s more a discussion about history of mathematics than about the actual mathematical problem. Moderators if you think it doesn’t fit here you may move this thread to “General Discussion”.

Archimedes Riemann integral is one of the most elegant achievements in mathematics, I have a great admiration for it. Mr. Patrick Fitzpatrick commented on it as

Archimedes first devised and implemented the strategy to compute the area of nonpolygonal geometric objects by constructing outer and inner polygonal approximations of the object. It is attributed to the German mathematician Bernhard Riemann beacuse he in 1845 placed the approximation strategy of Archimedes in a general, rigorous mathematical context applicable to problems much more general than the computation of area. Riemann’s contribution was made more than 2000 years after Archimedes computed the area of parabolic and circular regions by the construction of ingenious elementary geometric devices. Archimedes calculated the area of the circle of radius 1 and provided accurate error bounds for his approximation; he calculated $\pi$ with an error bound of $1/500$


Anybody want to share his/her feeling about rigorous definition of integrals or want to comment on it?
Please write your precious opinion about why it took 2000 years for integrals to come up in a rigorous manner?

Please during this discussion somebody please teach me what “error bound of $1/500$” mean :-) does it mean that error of 1 digit in 500 digits ?

 

[fresh_42]

$\varepsilon= 1/500=\frac{1}{5}\cdot \frac{1}{100}=0.2 \%$ So an error bound $\varepsilon$ on $\pi$ means he calculated $\pi \pm 0.2\%$ and also provided this information about his accuracy.

 

[Adesh]

$\varepsilon= 1/500=\frac{1}{5}\cdot \frac{1}{100}=0.2 \%$ So an error bound $\varepsilon$ on $\pi$ means he calculated $\pi \pm 0.2\%$ and also provided this information about his accuracy.

Can you please explain what does $\pi \pm 0.2 \%$ actually mean? I see it very often, that $\pm x \%$ thing but don’t understand it. Explain it with an example please.

 

[fresh_42]

If $p$ is the number Archimedes actually calculated, then
$$
p\in [\pi-0.2\%\cdot \pi\, , \,\pi+0.2\%\cdot\pi]\approx [\pi-0.0062832\, , \,\pi+0.0062832]\approx [3.13531 \, , \,3.147876]
$$

Edit:
Maybe it is better to phrase it the other way around:
$$
\pi \in [p-0.2\%p, p+0.2\%p]
$$
since he didn't know $\pi$. He only knew that it was in this interval, depending on his calculation $p$ plus minus the error margin.

 

[fresh_42]

Please write your precious opinion about why it took 2000 years for integrals to come up in a rigorous manner?

This was - as always - a luxury problem. There was no need to calculate precise volumes and surfaces. Measurement has been sufficiently accurate for trade. And that was what people had been busy with: taxes, agriculture and trade. Nobody has had the time to think about e.g. integration.

Archimedes could do so, as his society relied on slavery. Newton could do so, as his society had improved yields on agriculture, and because he wrote the King's horoscopes. And of course there were always the hope to synthesize gold at the time. The 2,000 years in between had been filled with survival. The Romans who also bought their achievements from slavery weren't interested in science except it helped solving their mechanical problems, i.e. constructions. Obviously integration isn't needed to build a road or a cathedral. Measurement was the crucial issue.

 

[Adesh]

@fresh_42 frisch herr, that’s an excellent answer, really! Integral was not developed because it was not needed. That’s really what’s happening these days too (but I won’t go into that).

But this opens up a new question: Was it needed for Mr. Riemann (your paisan) to develop the rigorous definition of integration?

 

[fresh_42]

As soon as calculus (Newton and Leibniz) was formulated, people started to look at differential equations, which described their physical processes. And in order to solve them, you need integration tools. So instead of integration you should ask about differentiation.

We are talking about 1,700 AD. This is when the age of enlightenment began in Europe. Hence we must ask what triggered it. I guess this question alone fills volumes.

We may also ask why it didn't happen in other cultures. India brought as the number system, China was way ahead of Europe for thousands of years, the Arabic world had centuries (800-1,200) of thriving science, and in South America they had true astronomic experts. It seems as other ventures had a higher priority: exploration, navigation, measurement, all related to trade.

 

[Adesh]

described

Is that “d” striked-off or is it only to me that it’s looking like that ?

 

[fresh_42]

Is that “d” striked-off or is it only to me that it’s looking like that ?

It was shorter than "described / describe" as differential equations are still how we model physics. So with or without a "d", the sentence allows both.

 

[Stephen Tashi]

Changes in integration developed according to what people were trying to integrate. Archimedes was dealing with his contemporaries idea of "curves" which didn't include things like fractal curves.

To evaluate Riemann would require knowing what he and his contemporaries were trying to integrate. Going by what modern texts attempt, we can Riemann integrate functions whose graphs are more complicated that the curves of Archimedes. For example, depending on the exact version of Riemann integration that is defined, we can integrate functions that are discontinuous at a finite number of points.

Integration had to progress beyond Riemann. For example, consider the following (theoretical) game of chance. A player tosses a fair coin. If it lands heads, he receives a score of 1/3 and the game ends. If the coin lands tails a random number selected from a uniform distribution on [0,1] and that number is the players score.

Graph the probability distribution $p(x)$ of the players score. Find the expected value of the players score by using integration.