I do not want to go OT and start a long defense of Latin and Greek here,

even because what you said is entirely true: those two dead languages were used as a gatekeeper, and that is why Grundtvig called the Classical Lyceum " a School for Death".

I will just sat that Latin and Greek can be taught and learned without this classist stance, and that studying these two languages can be really useful (which is different from "necessary") without resorting trivial nonsenses such as "they shape your head", "they teach you how to think", "they have a inner -and superior- logic"…and so on.

]]>*" . . .whereas there isn't really a lot of creativity in elementary calculus.",*

Au contraire, one needs a very large amount of creativity and imagination to comprehend such basic results as: the existence of functions which are continuous everywhere and differentiable nowhere; or that the infinite limit of a sequence of continuous functions may not itself be continuous; or that there exist continuous one-to-one functions mapping the real line to the real unit square; or that the unit square can be completely filled with a continuous, infinitely-long line such that every point in the square is reachable after traversing only a finite portion of the line.

]]>I suppose it really starts in grade school, too. Continuous functions dominate from elementary school. I wonder if it would feasible to start the foundations of sets from a much earlier age?

]]>What's interesting is that that book makes far greater use of actual analysis, like Landau notation. This seems refute the view, sometimes expressed among engineers, that analysis is only used by pedantic mathematicians to avoid infinitesimals in the proofs in calculus.

I do agree that linear algebra is more useful than calculus, and I am still surprised that it's taught after calculus and considered harder. However, it's not the only assessment of the relative mathematical difficulty of various subjects that I don't agree with. Cohomology isn't as hard as some people make it out to be, for instance.

]]>That said, of course much more time could be spent on actual application versus proofs — pretty parallel to what was said regarding statistics. There is a difference between learning what statistical methods are appropriate in what circumstances and what they can and can't say about a data set, and learning how to derive and prove various statistical axioms. Similarly, there's a difference between how calc is taught now (or at least, how I was taught calc) with its focus on deriving, proving, rearranging and regurgitating various formulas, and learning how calculus can apply to and explain various phenomena (and here one can sneak F=ma back in). In learning *how* to apply calculus to various circumstances, even simple ones, one should learn much of what practically needs to know to understand many everyday phenomena, and certainly (imho) at least this much understanding is required of ecologists in order to intelligently understand certain vital branches in the progress of the field (for example, to be able to tell articles in mathematical ecology that might apply to their system from articles that may be just so much wank). Ecologists should have a working knowledge of stats, linear algebra, *and* calculus. (again, in my young and humble opinion.)

]]>And certainly you need calculus to carry out a Fourier transform of a speech stream. Or a dynamic, evolutionary model of creolisation or language change.

]]>Perhaps I wasn't clear — I'm not arguing that you need to know category theory in order to understand probability or statistics. I am saying that the very notion of "understanding" is domain-dependent and time-dependent, and that what counts as understanding of probability or of statistical theory in one domain or at one time does not necessarily count as understanding of them in another.

It is perfectly possible to understand and to use and even to make major contributions (witness RA Fisher) to the field of statistics without knowing much advanced mathematics. The modern mathematical treatment of probability dates only from the 1920s, although the formal theory of probability itself dates from the 1660s. The first publications on calculus post-date the first publications on probability, and, indeed, calculus in its modern form did not appear until the mid-19th century, some two centuries after probability theory first appeared. So lots of people presumably understood the subject without being able to understand calculus.

It was not until the 1970s that formal non-probability representations of uncertainty began to be studied seriously (by people in AI), thus placing probability theory within a wider context than hitherto, and arguably only then providing the basis for an appropriate understanding of it. Many people working in AI would say that you can't claim to understand probability theory if you don't know anything about these alternative representations for uncertainty, such as Possibility theory and Dempster-Shafer theory. I doubt this is the case for Linguistics, which is precisely the point I am making in para 1 above.

]]>Thanks for your concrete examples above. I do agree that the standard multivariate calculus course taught to non-math majors is often useless, but that is a much more specific claim than the one made in the OP. I in general agree that good math courses are better than bad math courses and there are clearly cases where a good non-calc background is much preferable to a bad background which includes exposure to calculus. What I'm reacting to here was what seemed to be an initial 'calculus is uselsss' outlook. In general, I don't see too much calculus being taught, though there is too much bad calculus teaching. But I don't think a 'no calculus' stance is going to help students, nor is exposure to good calculus teaching a problem for students. We need good math backgrounds and good math teaching, and messing around with 'no X in the curriculum' rules isn't the best way of accomplishing that (though pushing out some of the problematic calculus service courses taught by graduate TAs might not be a bad thing).

]]>I don't see how Feller's book could be comfortably consumed by anyone who doesn't have a working knowledge of basic calculus. Sure, people with really good math intuition could pull it off, but then there is no reason these people would suffer by learning calculus. Indeed, learning calculus is probably the more direct and easiest way of getting the best understanding of statistics rather than trying to avoid it altogether, which just creates more problems, either immediately or later down the line.

Sure, knowing linear algebra is useful, but linear algebra, unlike calculus, is pretty easy to pick up, with courses useful as a back stop to insufficiently abstract ways of thinking about linear spaces, or for an understanding of numerical methods.

Required stat courses were probably the most painful and frustrating parts of my education, in part since the level of mathematical sophistication and abstraction is so low — which leads to a lot of lost motivation and a lack of perspective on what is actually going on and why things are done the way they are.

]]>"I am reminded of the famous criticism of the work of the statistician Ronald Fisher by more mathematically-oriented colleagues, that he wrote papers and books on statistics without knowing any measure theory. Of course, since the work of Efron and Amari, I wonder how anyone can understand statistics without knowing any differential geometry! And, then, how can one understand differential geometry without knowing any sheaf theory or category theory? And what sense does category theory make without a good understanding of n-category theory?

Is it possible to understand something at various different levels, not all of which are appropriate or convenient for any one problem domain."

I don't think there is a good argument that everybody doing applied probability or statistical work should know category theory or even any measure theory. What I do fail to understand is how people can do much original applied work using statistics without a good working knowledge of basic calculus. Simply minimizing likelihoods and understanding where significance levels come from when a model summarizes the data calls for (not very fancy) calculus.

Sure, one can plug data into software programs without understanding what the statistical model actually does — is that what's being advocated here?

[(myl) No. But you can understand the idea of an objective function, and the idea of finding the parameter values that minimize it, entirely in the discrete domain. And these days, optimization is done by numerical methods as often as not anyhow, sometimes because there's no closed-form solution available, and sometimes because the numerical methods are just easier and better.

Similarly for the probability of getting a certain result by chance. In some cases, the calculation involves combinatorics rather than calculus anyhow; or may be done using bootstrap or similar re-sampling techniques. In other cases, you can understand the concept of the cumulative distribution function, and know how to calculate its values, without having to do any integration.

Your attitude reminds me of the traditional approach to digital signal processing, which was to view it as something that could be learned only after several years of calculus-intensive analog signal processing, as a sort of second-best discrete approximation. But most DSP can be viewed, entirely independently, as applied linear algebra — for example, the discrete Fourier transform is just a rigid rotation of coordinates into a space whose orthogonal bases are eigenfunctions of shift-invariant linear operators — and modern DSP texts like McClellan, Schafer and Yoder's DSP First, turn the traditional curriculum upside down, and (as the title indicates) start with the digital version, with only a modest amount of continuous mathematics used along the way. Exactly the same thing can be done with probability and statistics.

The point is not that calculus is irrelevant or never useful, but that if there's a choice between two semesters of math based on calculus, and two semesters of math based on linear algebra, the second is more useful for most students these days.]

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