# Archimedes finds modern-day calculus text. Would he quickly grasp the concepts? **URL:** \ **Category:** Factual Questions\ **Created:** [December 14, 2006, 4:38am UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079 "2006-12-14T04:38:34Z")\ **Posts on this page:** 8\ **Page:** 2
**Author:** ![Captain\_Carrot](https://avatars.discourse-cdn.com/v4/letter/c/f0a364/32.png) [@Captain\_Carrot](https://boards.straightdope.com/u/Captain_Carrot)\ **Post date:** [December 14, 2006, 11:55pm UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079/21 "2006-12-14T23:55:31Z")
[QUOTE=Mathochist] He all but had integration down, and differentiation isn’t that much harder. [/QUOTE] Huh? I would argue that differentiation is easier. You can differentiate any function, albeit perhaps with great difficulty; not all functions are integrable. (The preceding is taken from last year’s Calc BC class, so it may not be completely accurate.) ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [December 15, 2006, 12:07am UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079/22 "2006-12-15T00:07:25Z")
[QUOTE=Captain Carrot] Huh? I would argue that differentiation is easier. You can differentiate any function, albeit perhaps with great difficulty; not all functions are integrable. (The preceding is taken from last year’s Calc BC class, so it may not be completely accurate.) [/QUOTE] You’ve got it completely backwards. Of the functions you look at in calculus classes given by formulae, that’s correct, but in the real world it’s exactly the opposite. Any continuous function on an interval can be integrated. In fact, it doesn’t even have to be continuous – there can be a finite number of discontinuities and you can still handle it by breaking up the interval. If you define integration the right way (hat tip: Lebesgue) you can even integrate functions with an infinite number of discontinuities like the characteristic function of the rational numbers (1 on every rational numbers, 0 on every irrational numbers). On the other hand if you put all the continuous functions on a dart board and threw a dart, you’d hit a differentiable function with probability zero. In fact, you’d hit a function that has _any_ point where the differential exists with probability zero. Almost all continuous functions cannot be differentiated anywhere. To make matters worse for Archimedes, he knew what an area was, but not what a parametrized curve was. You could talk about integration by slicing up areas into smaller and smaller pieces, but you can’t even give the heuristic definition of a derivative (slope of tangent line) without analytic geometry. ---
**Author:** ![Anne\_Neville](https://avatars.discourse-cdn.com/v4/letter/a/b9e5f3/32.png) [@Anne\_Neville](https://boards.straightdope.com/u/Anne_Neville)\ **Post date:** [December 15, 2006, 1:07am UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079/23 "2006-12-15T01:07:21Z")
[QUOTE=bluethree] Isn’t mathematics supposed to be the universal language? As a brilliant mathematician, he should understand it regardless of what language it’s written in. [/QUOTE] Getting a workable notation for something is extremely important in mathematics, and that’s what’s most likely to vary between cultures. Arithmetic is horribly difficult using Roman numerals and is much easier using a place-value system like the Hindu-Arabic numerals, for example. > [@Carnac the Magnificent!](#): > > lay people (such as myself) are unduly influenced by Hollywood’s portrayal of almost effortless genius, as depicted in “A Beautiful Mind.” An intellect that makes short work of most anything it encounters has almost certainly never existed. Even Newton, Leibniz, Euler, Gauss, Pascal and other titans struggled for years to make sense of work that an above-average math grad student today would consider straightforward. Thank you. I somehow feel better now. I always felt bad hearing about these brilliant people whose intellects could “make short work of most anything they encountered” (Funny thing is, you never got to meet these people, I now suspect kind of like you never got to meet the person who actually microwaved their dog or had a kidney stolen…). > [@Mathochist](#): > > the fundamental concepts of mathematics were extremely different for the Greeks. > > Arithmetic and geometry were two completely separate things. There were “numbers”, like our natural numbers, and there were “quantities” like our (positive) real numbers, but they were completely different things. Even the idea that the integers live on the real line would be new. ISTR reading somewhere that, prior to the Renaissance or so, something we take for granted in math, like that you can have an equation like y = x[sup]3[/sup] + 5x[sup]2[/sup], wouldn’t have made sense, because a squared quantity was an area and a cubed quantity was a volume- how can you make any sense of adding an area to a volume? ---
**Author:** ![Carnac\_the\_Magnificent](https://avatars.discourse-cdn.com/v4/letter/c/ccd318/32.png) [@Carnac\_the\_Magnificent](https://boards.straightdope.com/u/Carnac_the_Magnificent)\ **Post date:** [December 15, 2006, 1:08am UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079/24 "2006-12-15T01:08:17Z")
[QUOTE=Nanoda] I don’t think he’d grasp everything right away (I for damn sure didn’t w/o lots of class time), but the reason the mathematicians you mentioned had such a tough go of it was they were inventing the fields in question. I was actually more annoyed by an episode of ST Voyager (hate the whole series, too BTW) where it was pretty much said that because they had a huge base of science, they were smarter than Leonardo Davinci (there due to some holodeck malfunction or something) despite none of the cast being able to hold a candle to him. It would no doubt still be difficult for him, but most texts (mine at least) have useful diagrams and pictures that I think would be sufficient (given a strong mathematical and logical intellect as Archimedes had) to work out the symbols for new operations and such. I’d wonder more if he would have taken the time. I seem to recall someone saying that there wasn’t much effort taken towards the application of ‘scientific thinking’ to any physical effect. If so, it would mean that if he didn’t find it interesting, he wouldn’t pursue it. [/QUOTE] That’s what I was getting at. 😉 I think we’re getting back to the “standing on the shoulders of titans” proposition. Today’s math grad student finds calculus a breeze only because of the breakthrough understandings made before him. Another factor alluded to: Archimedes couldn’t then speak our modern mathematical lingua franca. He lacked both the notational tools and the understanding of algebra. He was trying to crack the calculus nut using geometry. ---
**Author:** ![Carnac\_the\_Magnificent](https://avatars.discourse-cdn.com/v4/letter/c/ccd318/32.png) [@Carnac\_the\_Magnificent](https://boards.straightdope.com/u/Carnac_the_Magnificent)\ **Post date:** [December 15, 2006, 1:14am UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079/25 "2006-12-15T01:14:46Z")
[QUOTE=Mathochist] The language and concepts of what we call “The Calculus” are so radically different from the way mathematics was done then that it’d be [strike]Greek[/strike] gibberish to him. [/QUOTE] I didn’t see your (far better) response before I responded. In a sense, Archimedes lacked the Rosetta stone to decipher the mystery text before him. He wouldn’t have the keys to unlock the code. He might intuitively understand some of the concepts, after many hours of study, but couldn’t rigorously prove them or use them in a meaningful way. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [December 15, 2006, 1:21am UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079/26 "2006-12-15T01:21:34Z")
[QUOTE=Anne Neville] ISTR reading somewhere that, prior to the Renaissance or so, something we take for granted in math, like that you can have an equation like y = x[sup]3[/sup] + 5x[sup]2[/sup], wouldn’t have made sense, because a squared quantity was an area and a cubed quantity was a volume- how can you make any sense of adding an area to a volume? [/QUOTE] That’s the biggest stumbling block in identifying the geometric constructions as abstract algebraic operations. In fact, since the Greeks only thought up to volumes even Heron’s formula was an enormous leap forward. The classical proof (in Heath’s edition of Elements, in the commentary after IV.4) requires making two areas, then considering the proportion of their _squares_, though a “square” really was the area of a square on a line segment. To consider the “square” on an area was a huge leap of faith in formalisms. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [December 15, 2006, 1:23am UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079/27 "2006-12-15T01:23:16Z")
[QUOTE=Carnac the Magnificent!] I didn’t see your (far better) response before I responded. In a sense, Archimedes lacked the Rosetta stone to decipher the mystery text before him. He wouldn’t have the keys to unlock the code. He might intuitively understand some of the concepts, after many hours of study, but couldn’t rigorously prove them or use them in a meaningful way. [/QUOTE] Actually, even Newton and Leibniz couldn’t rigorously prove them. They just used the intuitive notion of “differentials”, which is where we get the abusive notation “dy/dx” from. Limits were a far later addition to the theory in order to put analysis onto a rigorous foundation. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [December 15, 2006, 3:30am UTC](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079/28 "2006-12-15T03:30:10Z")
[QUOTE=Captain Carrot] Huh? I would argue that differentiation is easier. You can differentiate any function, albeit perhaps with great difficulty; not all functions are integrable. (The preceding is taken from last year’s Calc BC class, so it may not be completely accurate.) [/QUOTE] (to add to what **Mathochist** already said) You’re right if by “differentiate” and “integrate” you mean “find a formula for the derivative of” and “find a formula for the antiderivative of.” But (1) that isn’t what mathematicians mean by the word “integrable,” which, if I may put it very sloppily, has to do with whether the “area under the curve” exists\*, and (2) the ideas of functions and algebraic formulas and such would have been foreign to Archiimedes. \*(If you want to get technical, whether or not a function is integrable depends on what kind of integration you’re talking about (i.e. whose definition you’re using)—Riemann, Lebesgue, etc.) [Previous page](https://boards.straightdope.com/t/archimedes-finds-modern-day-calculus-text-would-he-quickly-grasp-the-concepts/384079.md?page=1)