# How much of modern mathematics is historical coincidence? **URL:** \ **Category:** Great Debates\ **Created:** [November 21, 2005, 3:23am UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027 "2005-11-21T03:23:10Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 21, 2005, 3:23am UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/1 "2005-11-21T03:23:10Z")
In [this thread](http://boards.straightdope.com/sdmb/showthread.php?t=345523), **Topologist** raises an interesting question that I think merits some discussion. > [@Topologist](#): > > But, what if we were to meet a Martian mathematician? If we could recognize it as a mathematician, I suspect that means in part that we would recognize the way it expresses itself, the logical underpinnings of what it does. I wonder, though, about the substance of its mathematics. Would it have algebra, analysis, and topology? Or would it have fields that we haven’t come close to thinking of? How much of what we’ve developed is necessary, and how much is there just because that’s what lay on and near the path we’ve chosen to take, as opposed to a different path the Martians took? Alternate histories are by necessity speculative, but here’s my take on it: Pretty much anything that was developed in antiquity is necessary. I don’t think this will be controversial. Above and beyond that, anything that has immediate applications to commerce–algebra, probability and statistics in particular–is very likely necessary. Once you have algebra, you’d start to wonder if all equations have solutions. This leads you pretty quickly to **Z** , **Q** , **R** and **C**. Whether it’s guaranteed to lead to groups and Galois theory is another question. Calculus is very necessary for any serious physics, and that leads quickly to basic analysis. I don’t see getting very far without linear algebra either, just because of how useful it is. Once you get into the 20th century, and in particular the latter half, things become a lot more murky. Is motivic cohomology natural in any sense? What about category theory? Computation theory? Boolean algebras? I’m not sure. What say you? ---
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [November 21, 2005, 4:35am UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/2 "2005-11-21T04:35:08Z")
First, thanks for starting this thread, **ultrafilter**. It’s a question that occurs to me periodically and I’d be interested to see what others think. It’s late, so I’m not thinking clearly enough to compose my thoughts about possible answers. Let me throw out another aspect of the question, though: We think of much of the applied or applicable mathematics we’ve developed as necessary, as a fundamental part of any understanding of how the world works. And, in Wigner’s famous phrase, we marvel at the “unreasonable effectiveness of mathematics” in the sciences. But, both our explanations of how the world works and our mathematics are developed by us, so perhaps it’s no surprise that they mesh well. Would our hypothetical Martians have developed the same physics as we? Would they need calculus for their explanation of how the universe works, as **ultrafilter** suggests? ---
**Author:** ![E-Sabbath](https://avatars.discourse-cdn.com/v4/letter/e/e19adc/32.png) [@E-Sabbath](https://boards.straightdope.com/u/E-Sabbath)\ **Post date:** [November 21, 2005, 4:36am UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/3 "2005-11-21T04:36:00Z")
> **[Srinivasa Ramanujan](https://en.wikipedia.org/wiki/Ramanujan)** > > Srinivasa Ramanujan FRS (/ˈsrɪnɪvɑːs rɑːˈmɑːnʊdʒən/; born Srinivasa Ramanujan Aiyangar, IPA: \[sriːniʋaːsa ɾaːmaːnud͡ʑan ajːaŋgar\]; 22 December 1887 – 26 April 1920) was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Rama Dur... Srinivasa Ramanujan pretty much independently invented everything from scratch from the end of Trig on up. And he was still able to converse with the outside mathematicians when he crossed paths. Things, all things considered, may look differently, but I think that there will be enough points where convergence happens that the theoretical martian mathematician will still be able to talk meaningfully to us. ---
**Author:** ![Malodorous](https://avatars.discourse-cdn.com/v4/letter/m/e47774/32.png) [@Malodorous](https://boards.straightdope.com/u/Malodorous)\ **Post date:** [November 21, 2005, 5:59am UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/4 "2005-11-21T05:59:53Z")
I would imagine that a (sufficently intelligent) alien given the same axioms would come up with more or less the same systems of mathematics that we have. Presumably logic works the same on mars as it does here in on Earth, after all. It’s conceivable that the axioms we came up with aren’t as obvious as we think they are, and that through some accident of alien history, brain chemestry or something they didn’t choose the same axioms. To take a stab at an example: > [@ultrafilter](#): > > Calculus is very necessary for any serious physics, and that leads quickly to basic analysis. I don’t see getting very far without linear algebra either, just because of how useful it is. Calculus relies on the Completeness axiom, which doesn’t seem 100% obvious to me. I can kinda sorta picture an alien mathematician who didn’t accept the completeness axiom, and developed some alternative to the real number line that contained infintesmals, had no irrational numbers, etc. (I think these things depend on the C. axiom, it’s been a while since I looked at a Real Analysis text though). What’s more, they might even be able to base a workable physics around this. After all, calculus when first developed assumed the existance of infintesmals, so they might be able to get something that gave them more or less the same results as our calculus, using a twisted, non complete version of the Real number line. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 21, 2005, 6:34am UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/5 "2005-11-21T06:34:27Z")
I think the general question of “what would happen if we met a martian mathematician?” is easily answered. As you said, the classical stuff is pretty much universal, so there will be a large body of common mathematics. The concepts of collection and iteration (cardinal and ordinal numbers) provide the first major rosetta stone, for instance. From there, translation of mathematics should proceed relatively smoothly, assuming translation of general language has been taken care of. Now, the first major point of departure is the nature of basic analysis. If the alien mathematics is significantly algebraic, this will look pretty much exactly like ours does, at least in practice if not in conception. This carries through calculus on finite-dimensional manifolds. The other possibility I can see is having started with nonstandard analysis from the get-go. I don’t see this as very possible unless the martians have a _vastly_ different conception of the nature of continua, including of time and space. As for the rest you specifically mention, much of it is category theory. Category theory + algebra theory = Abelian category theory, which is the basis of homology theory. Would they emphasize it? If “it’s really useful” works for saying they’d have linear algebra in the same form, then it should go here too. Boolean algebras might easily be downplayed in favor of Heyting algebras as the basics of logical structures. If they’ve got category theory, they’ll eventually get topoi, so they’ll get Heyting algebras from that. So, would they have category theory? I’m convinced that it’s irresistable once a certain level of algebra has been reached. When you see the proof of the ring isomorphism theorems and realize that they’re almost the same as the ones from groupo theory, you’ll start looking for the common threads. Once you recognize the inherent decategorification we train ourselves to use in “abstracting” numbers from counting, you’re drawn to categorification. In fact, I could see a civilization skipping the decategorification entirely and never establishing abstract numbers as the decategorification of the category of sets. Their mathematics would be based in categories from the absolute beginning, and that would have profound effects on their philosophy and so on. Likely, no. Possible, yes. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [November 21, 2005, 3:24pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/6 "2005-11-21T15:24:35Z")
That Wikipedia article overstates the independence of Ramanujan. Ramanujan didn’t come from a poor family. His family were Brahmins and were really more like middle class. His genius was recognized by his high school teachers and he entered university at 16. He left the university after a year because he didn’t want to study any of his other subjects. He flunked his English course at the university and lost his scholarship and thus couldn’t continue to study there. This was typical of his entire academic career - he studied math and nothing else. After leaving the university, another Indian mathematician found him a job as a clerk which allowed him plenty of time to study and do research in math. Ramanujan frequently talked with mathematicians while in India and even published one paper. A second Indian mathematician told him that he needed to contact G. H. Hardy, since he was working on the same subject as him. The story of Ramanujan is interesting enough in its own right. There’s no reason to pump it up by implying that he had no contact with any other mathematicians before Hardy. ---
**Author:** ![Zsofia](https://avatars.discourse-cdn.com/v4/letter/z/7bcc69/32.png) [@Zsofia](https://boards.straightdope.com/u/Zsofia)\ **Post date:** [November 21, 2005, 3:27pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/7 "2005-11-21T15:27:07Z")
I’m not a mathematician by any stretch of the imagination, but I often wonder about this - if the Martians had, say, twelve fingers, and bilateral symmetry was unknown to them - say they were, I dunno, three sided with a round giant eyeball on top or something, so they don’t think in terms of even and odd numbers, etc. Would that change the way they thought of math? I assume a species would work in whatever base corresponds to their number of fingers, or is that a silly assumption? Anyway, my question is - would these things just be arithmatical differences, only an issue in the most basic of math and ceasing to be an issue once you get to more theoretical concepts, or would they change everything about math for the Martians? ---
**Author:** ![smiling\_bandit](https://avatars.discourse-cdn.com/v4/letter/s/e9a140/32.png) [@smiling\_bandit](https://boards.straightdope.com/u/smiling_bandit)\ **Post date:** [November 21, 2005, 3:41pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/8 "2005-11-21T15:41:39Z")
> [@](#): > > Would that change the way they thought of math? All mathematical counting system are equal numerically; they are merely organized differently. But the same basic principles can be applied to all of them, more or less. It might be harder to use certain operations in some counting systems, but they are never impossible. As for asking whether or not this would change their psychological makeup? Probably not. Humans do not all use base 10. (Well, they mostly do today, but didn’t always and did fine with other bases). The Babylonians, who had different bases (12?) practically invented the earliest systematic forms of math. ---
**Author:** ![SentientMeat](https://avatars.discourse-cdn.com/v4/letter/s/43a26b/32.png) [@SentientMeat](https://boards.straightdope.com/u/SentientMeat)\ **Post date:** [November 21, 2005, 4:10pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/9 "2005-11-21T16:10:58Z")
I consider [this book](http://www.amazon.com/gp/product/0465037704/102-5697468-9731304?v=glance&n=283155&colid=&coliid=) to be essential reading on the subject. Its central thesis is that our conception of mathematics literally comes from the world around us. Our “logic” that a statement cannot be simultaneously true and false comes from the fact that _eg._ an object cannot simultaneously be inside and outside a vessel or region. Our “number line” is a metaphor for a _literal_ line comprising a number of footsteps. Operators like “+” or “x” represent putting more objects in a collection (and doing that a number of times for “x”, and doing _that_ a number of times for powers, and _never stopping_ that for infinite series), while division represents _sharing objects out_ amongst parties. So, since Mars (and, indeed, every other planet) is three dimensional and has increasing entropy, there is no reason to believe that Martians won’t have similar mathematics to ours. Perhaps a Planck-scale organism or a black hole resident might have significantly different maths, I guess. ---
**Author:** ![E-Sabbath](https://avatars.discourse-cdn.com/v4/letter/e/e19adc/32.png) [@E-Sabbath](https://boards.straightdope.com/u/E-Sabbath)\ **Post date:** [November 21, 2005, 4:36pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/10 "2005-11-21T16:36:40Z")
The article struck me as a bit hinky, but it covered the basics, that a number of the advanced concepts he independently invented were the same as what people in a fairly different culture had also invented independently. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 21, 2005, 7:10pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/11 "2005-11-21T19:10:51Z")
I’m not so sure that Ramanujan is a good case study. He was exposed to the same ideas as anyone getting a math education at the time, and at the very most he was doing early 20th century stuff. > [@Mathochist](#): > > So, would they have category theory? I’m convinced that it’s irresistable once a certain level of algebra has been reached. When you see the proof of the ring isomorphism theorems and realize that they’re almost the same as the ones from groupo theory, you’ll start looking for the common threads. I can buy that, but is that level of algebra really necessarily going to be discovered? > [@Malodorous](#): > > Calculus relies on the Completeness axiom, which doesn’t seem 100% obvious to me. I can kinda sorta picture an alien mathematician who didn’t accept the completeness axiom, and developed some alternative to the real number line that contained infintesmals, had no irrational numbers, etc. (I think these things depend on the C. axiom, it’s been a while since I looked at a Real Analysis text though). Modern calculus depends on completeness, but Newton’s calculus was actually developed using infinitesimals. The move away was due to the fact that infinitesimals don’t work well unless you treat them rigorously. ---
**Author:** ![Malodorous](https://avatars.discourse-cdn.com/v4/letter/m/e47774/32.png) [@Malodorous](https://boards.straightdope.com/u/Malodorous)\ **Post date:** [November 21, 2005, 7:26pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/12 "2005-11-21T19:26:32Z")
> [@ultrafilter](#): > > Modern calculus depends on completeness, but Newton’s calculus was actually developed using infinitesimals. The move away was due to the fact that infinitesimals don’t work well unless you treat them rigorously. That’s sorta my point. Aliens coulda thrown out completeness as nonsense and developed a calculus based on a rigirous treatment of infintesmals. I know non-archemidian mathematics have analogs to the Intermediate Value Thereom and such, so they could make some sort of calculus, and then try and relate that to physics problems the same way we relate our calculus to real world phenomenon. Of course I have no idea if you could actually use nonarchemdian calculus as a useful tool for analysing the real world, but it seems at least somewhat belivable to me. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 21, 2005, 7:31pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/13 "2005-11-21T19:31:17Z")
I’ve seen a calculus book based on non-standard reals, and it had pretty much the same results as a standard calculus book. The reason I find it difficult to believe that you’d get something like that historically is because non-standard analysis relies very heavily on model theory, which is a mid-20th century branch of mathematics, and therefore far enough down our path that slight divergences early on might have pushed us away from it. ---
**Author:** ![E-Sabbath](https://avatars.discourse-cdn.com/v4/letter/e/e19adc/32.png) [@E-Sabbath](https://boards.straightdope.com/u/E-Sabbath)\ **Post date:** [November 21, 2005, 7:46pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/14 "2005-11-21T19:46:32Z")
Ramanujan may not be a good case study, but can you think of a better? He was isolated, he did have a slightly alien mindset, and he came up with understandable concepts in more advanced mathematics. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 21, 2005, 9:52pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/15 "2005-11-21T21:52:41Z")
> [@SentientMeat](#): > > I consider [this book](http://www.amazon.com/gp/product/0465037704/102-5697468-9731304?v=glance&n=283155&colid=&coliid=) to be essential reading on the subject. Its central thesis is that our conception of mathematics literally comes from the world around us. In its primitive basics, that book may well be accurate about mathematics. However, like so many books on philosophy of mathematics it stops at the beginning of the 20th century, at least to judge by the table of contents and the index. It leaves out homology, category theory, topoi (whose logic is often radically different from that you describe), a lot of the meat of modern model theory, and a host of other topics which simply didn’t exist or weren’t emphasized at the time the percieved major debate occurred. I’ve said this before: platonism vs. formalism is dead and buried. The terms barely have relevance to how _actual_ mathematicians do _actual_ mathematics anymore. Anyhow, all the pre-20th century stuff falls into what **ultrafilter** and I said about having a certain universal core that would line up and provide a rosetta stone for the rest. The differences, if there are any in content, would be beyond the scope of that book. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [November 22, 2005, 4:11am UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/16 "2005-11-22T04:11:00Z")
E-Sabbath writes: \> Ramanujan may not be a good case study, but can you think of a better? That’s the point. There isn’t any better case of a mathematician working on his own. There are _no_ cases of someone re-inventing some substantial field of mathematics up to the state it was in at his time. (Let’s restrict this to 1800 or later. I don’t know whether anyone did this before then either. I can almost conceive of someone doing it in the eigthteenth century though.) That’s why the movie _Good Will Hunting_ is nonsense. I don’t think that there is any contemporary case of someone with, say, a high-school education reading mathematics by themselves up to the point where they could make a significant contribution to mathematics. Heck, I know that I’m untypical in being a mathematician who grew up in a working-class family. I’ve met just a few other mathematicians from working-class backgrounds who (like me) went straight into college wanting to study math. More commonly, I’ve met mathematicians who came from working-class families who started planning to do something else. One joined the Army out of high-school because he didn’t want to go to college and nobody he knew thought he should go to college. He studied electronics technology on his own because the Army didn’t want to train him in it. He discovered that he liked the math he was learning in the course better than the electronics, so he went to college and eventually got a masters in math. Another was a guy who got some technician’s degree and worked in a factory for a while when he realized he wanted to study math. He went to college and also got a masters in math. A third was a woman who became a nurse because she was told that was what smart women did. She also decided she liked math better, went back to school, and got a masters in math. ---
**Author:** ![SentientMeat](https://avatars.discourse-cdn.com/v4/letter/s/43a26b/32.png) [@SentientMeat](https://boards.straightdope.com/u/SentientMeat)\ **Post date:** [November 22, 2005, 12:11pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/17 "2005-11-22T12:11:12Z")
> [@Mathochist](#): > > I’ve said this before: platonism vs. formalism is dead and buried. The terms barely have relevance to how actual mathematicians do actual mathematics anymore. Just as early hominid’s evolutionary linguistics as explored by Chomsky, Pinker and the like has little to do with Shakespeare, surely (or indeed, abiogenesis and early life struggles to explain _homo sapiens_)? Just because modern mathematics is _far removed_ from its original cognitive basis does not make the statement “mathematics is ultimately explained by cognitive science” false. If Martians had not yet put together certain metaphors in the way Kurt Godel did, that would still qualify as an “accident of history”, IMO. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 22, 2005, 4:38pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/18 "2005-11-22T16:38:09Z")
> [@SentientMeat](#): > > Just as early hominid’s evolutionary linguistics as explored by Chomsky, Pinker and the like has little to do with Shakespeare, surely (or indeed, abiogenesis and early life struggles to explain _homo sapiens_)? > > Just because modern mathematics is _far removed_ from its original cognitive basis does not make the statement “mathematics is ultimately explained by cognitive science” false. If Martians had not yet put together certain metaphors in the way Kurt Godel did, that would still qualify as an “accident of history”, IMO. Platonism and Formalism aren’t the cognitive basis. They’re more like Beowulf if modern mathematics is Shakespeare. My point is that I don’t trust a book about mathematics written by a linguist who thinks the last interesting things happened a hundred years ago. They may well tell some plausible stories, and some of them may even be somewhat accurate. They aren’t explaining most of what mathematics is or how it works today. And as I said before, to whatever extent they’re correct it jibes with what has been said here before: there is a certain core which is abstracted from experience and couldn’t really be otherwise. How it’s gone from there may have diverged at some point, but the basics provide enough common ground for each side to learn the other. ---
**Author:** ![SentientMeat](https://avatars.discourse-cdn.com/v4/letter/s/43a26b/32.png) [@SentientMeat](https://boards.straightdope.com/u/SentientMeat)\ **Post date:** [November 23, 2005, 11:08am UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/19 "2005-11-23T11:08:26Z")
> [@Mathochist](#): > > My point is that I don’t trust a book about mathematics written by a linguist who thinks the last interesting things happened a hundred years ago. I’m not sure that’s an accurate summary of the book, actually. Perhaps the _index_ might be more useful than the table of contents in showing that the authors do consider some 20th century maths, just as a dinosaur evolution textbook might make a few references to crocodiles. > [@](#): > > They aren’t explaining most of what mathematics is or how it works today. How is this different to the Intelligent Desginer’s argument that evolution might explain _some_ things, but not the vast complexity and diversity of the present day? ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [November 24, 2005, 7:08pm UTC](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027/20 "2005-11-24T19:08:11Z")
> [@SentientMeat](#): > > I’m not sure that’s an accurate summary of the book, actually. Perhaps the _index_ might be more useful than the table of contents in showing that the authors do consider some 20th century maths, just as a dinosaur evolution textbook might make a few references to crocodiles. If you’ll read carefully, I already mentioned looking through both the table of contents and the index. > [@](#): > > How is this different to the Intelligent Desginer’s argument that evolution might explain _some_ things, but not the vast complexity and diversity of the present day? I’m not making vague references to “diversity” or “complexity”. I’m referring to specific fields and viewpoints. Homology theory and category theory are completely missing, for one thing. Off of that comes topos theory and the structural/categorical view which actually describes how mathematicians do mathematics. The concept of a group did exist a hundred years ago, but almost never in the abstract. Now the abstraction has passed completely into structure and away from instantiation. None of this is even touched upon. [Next page](https://boards.straightdope.com/t/how-much-of-modern-mathematics-is-historical-coincidence/332027.md?page=2)