WI Calculas was invented in 0 AD Roman Empire

All the mathematical prerequisites were there already. Let's say some Newtonesque mathematician invents it in the Roman Empire. Well what would happen? A timeline would be nice.
 
What society with minimal amount of scientist would do with calculus? Anyway, even if there was such knowledge in Roman Empire, it would definitely be lost during eraly Middle Ages.
 
There was no 0 AD. There wasn't even an AD 0.

Did Roman numerals have a way to represent nonintegers?
 
The greatest Roman contribution was the killing of Archimedes.

He, if anyone in the ancient world was to, would have developed calculus. His later works are the same as early questions in a calculus course. The most likely date for an ancient discovery of calculus would be from Archimedes.
 
All the mathematical prerequisites were there already

No they were not. Who ever told you that was lying or ignorant. Zero was not there... It was a scary void. Infinity was not there, but as mind boggling vastness that could not be comprehended. Abstraction is not there. Some of the simpliest things that we tell our kindergardeners would scare the crap out of them mathematically. There were some very deep, very difficult concepts they did not understand and without a big change in their scientific culture and worldview they continue to reject. And their arguments are very persuasive and are what most people without our base of knowledge would probably assume. Then the mathematical culture they based around their own knowledge made it very hard to challenge those assumptions.

With Roman Numerals? You'd need a major change to base 10 (or somesuch) first.

Not necessarily. What they'd really need is the concepts of zero and infinity. Integrals are summations of infinitesimal rectangles over a region. Think of Zeno's puzzle "The Achilles". This is very illustrative of the problems Greco-Roman society had with the fundamental basics that had to be overcome to discover Calculus.

Imagine that Achilles runs at a foot per second, while a tortoise runs half of that speed. Imagine also that the tortoise starts off a foot ahead of Achillies.

Achilles speeds ahead and in one second has caught up to where the tortoise was, but the tortoise who is also constantly moving has moved forward half a foot more. In half a second Achilles makes up that half a foot, but now the tortoise is ahead by a quarter of a foot. In a quarter of a second Achilles makes that up but the tortoise is still ahead an eighth of a foot. And so on and so on. Now the Greeks thought that no matter what Achilles would remain behind the tortoise.

But think about it.. Achilles is faster than the tortoise. He will catch up with him eventually right? And with Calculus we can prove that. Zeno's puzzle is an infinite summation problem. 1/2+1/4+1/8+1/16.... on and on forever. The Greeks thought this race would continue forever, but we know that this summation (the infinite sum of 1/(2^n) converges at 2. In 2 seconds Achilles will pass the tortoise. They broke it up into the infinite steps of the race, saw the infinity and couldn't break the barrier that this infinite amount of something could add up (converge) to a finite sum.

Without a real grasp on infinity and zero there's no way they could break the barrier to Calculus. Not to mention the ingrained fear of the infinite that Pythagoras put into Greek Mathematics by his hatred of irrational numbers. Archimedes was close and had those constraints not been there he could've possibly gotten closer. But even then it would've been very limited and bound to what could be easily graphed out with a stick or drawn on paper and spoken of in geometrical terms. Bigger breakthroughs would have to wait for Diophantes and his creation of Algebra in the late classical period. That brings real abstraction into the picture and that's what's really needed for the higher maths. And helps majorly grasp the problems of zero and infinity.
 
The Greeks thought this race would continue forever

Bright day
I know what you are trying to say- but nobody thought that the race would go on forever. Zeno's paradoxes are mental exercises showing gasps in mathematics of his time.

Sorry, jsut something that icks me when i hear it.
 
Yes.. yes.. and no the everyday people thought it was wrong but they couldn't prove it, because when they also looked at the problem and split it into infinite steps like Zeno did they couldn't parse out that the race would have a limit and thus Zeno's mathematical proof that it would go on forever looked unassailable and stood for centuries. They did think it would go on forever when examined in Zeno's manner, and that's the problem when it comes to discovering calculus.
 
This was not really the question.

Let's say someone invents the concepts of zero and infinity in 0 A.D. Then in 50 A.D. some smart guy invents Calculus. Well how does this affect the Roman Empire?
 
What happens to the guy that invents calculus? Does he publish his results? How? Newton had the advantage of the printing press to get the word out and a lot of interested ears.

Mr. Calculus might just have discovered another bit of mathematical esoterica that gets sort of mystified like the Pythagoreans.
 
This was not really the question.

Let's say someone invents the concepts of zero and infinity in 0 A.D. Then in 50 A.D. some smart guy invents Calculus. Well how does this affect the Roman Empire?

Why is this in Before 1900 and not in ASB then? It's the equivalent of saying what if someone invented the combustion engine in Augustus' reign and asking how that effects the Roman Empire. There's a lot of background knowledge needed and a shift in the culture needed as well.

And even if, the only way without changing basically all of Greek thought is to POD with Archimedes. Let's say maybe Archimedes instead of dying at the Roman soldier's hands is taken back to Rome and is persuaded to establish a school there. And some of his pupils continue his work. (This is a major problem with the possibility of Calculus in the period. Archimedes did do a lot of the the background work and did himself have a good understanding of infinitesimals, but his work was practically ignored until the Byzantine period.) Now out of that you could get a limited Integral Calculus. But that departure is a bit earlier than what was requested.
 
And even if, the only way without changing basically all of Greek thought is to POD with Archimedes. Let's say maybe Archimedes instead of dying at the Roman soldier's hands is taken back to Rome and is persuaded to establish a school there. And some of his pupils continue his work. (This is a major problem with the possibility of Calculus in the period. Archimedes did do a lot of the the background work and did himself have a good understanding of infinitesimals, but his work was practically ignored until the Byzantine period.) Now out of that you could get a limited Integral Calculus. But that departure is a bit earlier than what was requested.

Yeah lets say this happens and calculus is invented sometime after my date.
How is the Roman Empire affected?
 
How about having it invented in India? Indian mathematics were far advanced compared to the rest of the world and they certainly had the concept of zero (not sure about infinity)
 
There was no 0 AD. There wasn't even an AD 0.

Did Roman numerals have a way to represent nonintegers?

No. It was possible to do but only in words (four hundred and thirty eight parts of eight hundred and twenty nine rather than 438/829, for example). Giving them arabic numerals or the like is all but necessary for this.

The greatest Roman contribution was the killing of Archimedes.

He, if anyone in the ancient world was to, would have developed calculus. His later works are the same as early questions in a calculus course. The most likely date for an ancient discovery of calculus would be from Archimedes.

Yes, given arabic numerals I would say that Calculus would have been within his grasp (he actually did some integral calculus the hard way - all from first principles, and in words - the mind boggles).

Yes.. yes.. and no the everyday people thought it was wrong but they couldn't prove it, because when they also looked at the problem and split it into infinite steps like Zeno did they couldn't parse out that the race would have a limit and thus Zeno's mathematical proof that it would go on forever looked unassailable and stood for centuries. They did think it would go on forever when examined in Zeno's manner, and that's the problem when it comes to discovering calculus.

Zeno's paradox is not like that. It's one of a set of four, used to prove that time and space can neither be discrete nor continuous (and hence that something is wrong with their conception of the world, somewhere - it turns out to be their inability to deal with series, as you said). But, beyond the fact that they couldn't deal with series (and series are hard things to deal with: sum 1 - 1 + 1 - 1 + 1 - 1 + 1... for me ;)) it doesn't indicate that their mathematical ability was terribly lacking. Pythagoras had a fear of the irrational, yes, but the later greeks didn't; one guy proved the irrationality of the roots of the nonsquare integers up to (IIRC) 19; he was lauded for this (although he somehow missed the general case).

All in all, I'd say this is hard but not impossible. The obvious POD is "WI Greeks used arab-esque numerals?" with the result that Archimedes can, indeed, invent calculus. The results of this, however, I don't think are too likely to be impressive; its only really practical use at that point is astronomy. Maybe they figure out heliocentricism earlier (although that relies as much on observations as the math, and Kepler and Brahe did it by hand and without calculus - again, the mind boggles.

The other big thing is the greek fetish for geometry - this is a bit of a problem. The immediate pre-calc guys (Descartes especially) were big on using numbers - this was probably a big help for the development of the calculus. Geometry was (and is viewed nowadays) as a thing you do with numbers, not space: a line is ax + by + c = 0, not "a breadthless length". The greeks, by contrast, were big on plane geometry, and the weird, obsessively-but-not-rigourously axiomatic euclidean version thereof. This probably won't help the calculus any. Archimedes is smart enough to invent it anyways, but his successors probably won't find much use for it. it's going to stay the readheaded stepchild of geometry for a long, long time.

In conclusion: it's definitely possible, given a sufficiently abstract POD, but don't expect great things from it.
 
A lot of the problem is that the concept of zero wasn't established in Ancient Greece, so it wasn't going to exist in Ancient Rome. If the Greeks had accepted the concept of zero, this is the start down the path. Archimedes seemed to be on the train for the concept of actual infinite quantities, so all you needed were a few things to happen to get calculus in the Roman world.

1) Algebra: If the Babylonians could get to this point, why not the Greeks? The Greeks were developing algebra.
2) Concepts of zero and infinity: Zero needs to be treated as a number and infinity needs to be treated as a mathematical concept.
3) Bring ideas together: Archimedes was on the path towards doing this. Had Archimedes survived, he may have been able to invent calculus before he died.

Someone may be able to build on the work of Archimedes and it could prompt further development of mathematics and, if someone recognizes it, could be used to start approaching further development of classical mechanics.
 
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