Wikipedia:Reference desk/Archives/Mathematics/2019 January 18

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January 18

Most recent Mathematics taught in Secondary School (ages 11-17)?

What is the most recent discovery, technique or formula generally taught in Secondary school (let's say the year before algebra through but not including multivariate calculus)? This came from a discussion about an "In the sea of time" story being written that has all of Britain sent back from about 1980 to 1730. While the best Biologists and Chemists from 1730 would seem to have to relearn Secondary school classes from scratch (or even farther back), it would seem that a Mathematician from 1730 would be able to enter college, with their biggest problem being standardiation of notation. Does that sound right?Naraht (talk) 22:22, 18 January 2019 (UTC)

Yeah, sounds fairly reasonable to me. The nature of mathematics means that we know if something is true or not (barring Godelian incompleteness), whereas with the softer sciences what we have is always a best approximation. Newton's laws of gravitation worked pretty well for centuries, but we realised it wasn't quite right and eventually found out about relativity, and potentially whatever replaces relativity. In maths, we know that the quadratic formula will always work, no matter where you are in time and space. Because of that, the foundations of mathematics have been well understood for centuries, and the areas of enquiry are more and more abstruse, to the point that even on a four-year university course you're not going to come across that much from the past hundred years. -mattbuck (Talk) 23:33, 18 January 2019 (UTC)

I generally agree that students of that age rarely learn much post-18th-century math. A possible exception is a (very) small amount of set theory, but they don't really study any of the bit that makes set theory actually interesting (say, Cantor's diagonal argument), so that can probably be subsumed in "notation".

Another possible exception: ε–δ arguments in calculus.

I don't really agree that "the foundations of mathematics have been well understood for centuries". The foundations of mathematics are controversial now. And my prediction is that they probably always will be, in the mortal plane. Maybe we'll find out when we pass beyond the veil.

As for four-year mathematics majors, they may well learn stuff more recent than 1919, particularly if they study logic or theoretical computer science. --Trovatore (talk) 23:55, 18 January 2019 (UTC)

1730 sounds about right to me. Euler first put the calculus of variations on a reasonable basis in 1733 so one can't go past that. Of course most of them won't even have been shown how to solve the Brachistochrone problem or Kepler's laws for instance, or even Pappus's theorem. They probably wouldn't even be able to tell why an ellipse is a section of a cone. Dmcq (talk) 00:29, 19 January 2019 (UTC)

I've actually read a bit of historical math and I don't think it's an easy question to settle. Keep in mind that when you learn say what Newton did in calculus, you're not reading Newton but what some modern author has carefully translated and paraphrased for us. Go back and read the original version of Newton's Principia (or a contemporary translation) before claiming that math in his day would be understandable to us or vice versa. For me the intelligibility cutoff is mid to late 19th century; before that the differences in language and notation make reading a mathematical texts more a question of translation than understanding mathematical ideas. Of course if you do that kind of thing often it would becomes easier with practice, but I'd say reading math in English from 1850 is similar to reading modern math in French or German.

There are some major differences between between what mathematics was then and now, and also some more ineffable differences in what you might call mathematical patterns of thought between then and now. A mathematician from 1730 would almost certainly be intimately familiar with Euclid's elements, to the point of being able to give the book and proposition number of the major theorems. They would also read and write about mathematics almost exclusively in Latin; as I recall the transition from Latin to other language took place in the 18th century with Disquisitiones Arithmeticae, being one of the last significant works in Latin, published in 1801. The contents of that book illustrate some of the issues involved since Gauss had a good practical knowledge of what we might call Algebraic number fields today, but without any of the theoretical terminology like groups, rings, fields, ideals etc. Being so familiar with Euclid, a mathematician from 1730 would be very good at constructing geometrical arguments and would generally prefer constructing a geometric proof even when a more simple (to us) algebraic proof of the same fact existed. And of course a mathematician from 1730 would never have heard of non-Euclidean geometry, Bolyai and Lobachevsky not being born until around 1800. In terms of rigor, a mathematician from 1730 would freely use infinitesimals in ways that would be difficult for us to interpret, and would be unfamiliar with concepts like limits and continuity. Cantor was not born until 1845 so everything about infinite cardinals and related concepts in set theory would be unknown to a mathematician from the 18th century. Many concepts we take as basic now are actually relatively recent, for example a vector as a concept goes back only to around the 1830's. Finally, the most obvious thing an eighteenth century mathematician would lack today is any knowledge of computers or algorithms. You want to use 1980 as a comparison date, so no need to go into the internet, but even in 1980 the computer was becoming an important research tool in mathematics; the computer proof of the four color theorem appeared in 1976.

So if the hypothetical we're trying answer here is whether a mathematician from 1730 could get by in a college math course from 1980, I'd say the answer is it would be difficult. They would have a lot of knowledge about certain areas and probably be able to do some impressive things, for example they could probably handle any word problem from an introductory calculus course. But in addition to having a kind of language barrier about equivalent to an English speaker trying to learn math in Hungarian, they would also have significant gaps in what we would call basic concepts, so e.g. in that calculus course they'd probably have trouble with questions on limits and continuity. Still, they would be much better off then someone in a similar position but with physics, chemistry or biology. --RDBury (talk) 01:21, 19 January 2019 (UTC)

Thank you for all of your comments, there are significantly different ways of looking at things, and definitely would have problems with college courses, but as you said "much better off than someone" in the other sciences. (Of course this gets into the question of whether Mathematics is a science...)Naraht (talk) 15:34, 19 January 2019 (UTC)

I would say statistics, computer science and numerical math. In secondary school children do learn a bit about modern statistical methods, but with less rigor than at university. Children learn about programming and they may also learn about certain modern numerical math techniques e.g. in physics class. Count Iblis (talk) 19:50, 22 January 2019 (UTC)