## Linguists who count

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An editorial by Miranda Robertson in the Journal of Biology ("Biologists who count", 8(34), 2009), starts this way:

The importance of mathematics in biology is a matter of perennial debate. The squabbles of early 20th century geneticists on the value of mathematics to the study of evolution have recently been revisited in *Journal of Biology* [Crow, "Mayr, mathematics and the study of evolution", *JBiol* 8(13) 2009], and the 21st century has seen an explosion of information from various -omics and imaging techniques that has provided fresh impetus to the arguments urging the need for mathematical competence in the life sciences [Bialek and Botstein, "Introductory science and mathematics education for 21st-century biologists", *Science* 2004, 303:788-790]. While there can be no question about the contribution of mathematics to many fields in biology, there is a curious tendency on the part of numerate biologists (often immigrants from the physical sciences) to insist that it is an essential part of the equipment of a biologist and none should be without it. This seems, on the evidence, extreme.

Robertson concludes that

There seems no need for the snobbery (it is said) of the highly quantitative founding biologists at the Cold Spring Harbor Laboratories, in whose early history ex-physicists played a crucial part, and who are alleged to have referred to their nearby colleagues at Woods Hole as biologists 'who don't count'.

There's an analogous set of arguments in the areas of rational investigation related to language, of which the discipline known as "linguistics" is a regrettably small part. (I mean by this that the balkanization of the field is a Bad Thing, not that the work of psycholinguists, linguistic anthropologists, speech pathologists, speech technology researchers, etc., is inadequate or inferior. But that's a different discussion.)

The role of mathematics in the language sciences is made more complex by the variety of different sorts of mathematics that are relevant. In particular, some areas of language-related mathematics are traditionally approached in ways that may make counting (and other sorts of quantification) seem at least superficially irrelevant — these include especially proof theory, model theory, and formal language theory.

On the other hand, there are topics where models of measurements of physical quantities, or of sample proportions of qualitative alternatives, are essential. This is certainly true in my own area of phonetics, in sociolinguistics and psycholinguistics, and so on.

It's more controversial what sorts of mathematics, if any, ought to be involved in areas like historical linguistics, phonology, and syntax.

I don't have time this morning for a longer discussion, so I'll just baldly state my conclusions, and look forward to hearing the opinions of others.

First, attempts to devise and test formal models of language-related phenomena are often helpful, and sometimes essential, in framing theories clearly enough to be able to see what's right and what's wrong with them. And occasionally, such models lead to genuine insight.

Second, mastery of statistical techniques is useful in all forms of rational inquiry; and networked digital computers are increasing this usefulness in a major way, due to increased availability of data and increased ease of statistical investigation.

Third, mastery of language-relevant mathematics crucially includes knowing when a model is inappropriate, misleading or unnecessary. Yeats said this in a more pointed way, in his 1930 "Letter to Michael's Schoolmaster":

Teach him mathematics as thoroughly as his capacity permits. I know that Bertrand Russell must, seeing that he is such a featherhead, be wrong about everything, but as I have no mathematics I cannot prove it. I do not want my son to be as helpless.

I happen to think that Russell was, on the whole, righter than Yeats was; but it would be wrong for the argument to be won by default due to one side's technical incompetence. If you've ever worked in an interdisciplinary area where mathematical backgrounds are variable, you've probably seen attempts to win arguments by this sort of mathematical default — often promoted by people who don't really understand a technique or program that they've learned to use in a cookbook fashion.

(I certainly don't subscribe to the part of Yeats' letter that says "Don't teach him one word of science, he can get all he wants from the newspapers and in any case it is no job for a gentleman". Alas, many of those who write about science in the newspapers seem to have been educated in accordance with this prescription.)

Anyhow, my conclusion is that anyone interested in the rational investigation of language ought to learn at least a certain minimum amount of mathematics.

Unfortunately, the current mathematical curriculum (at least in American colleges and universities) is not very helpful in accomplishing this — and in this respect everyone else is just as badly served as linguists are — because it mostly teaches thing that people don't really need to know, like calculus, while leaving out almost all of the things that they will really be able to use. (In this respect, the role of college calculus seems to me rather like the role of Latin and Greek in 19th-century education: it's almost entirely useless to most of the students who are forced to learn it, and its main function is as a social and intellectual gatekeeper, passing through just those students who are willing and able to learn to perform a prescribed set of complex and meaningless rituals.)

## Twitted by languagelog said,

May 28, 2009 @ 10:03 am

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## D. Sky Onosson said,

May 28, 2009 @ 10:21 am

Fantastic advice. My master's thesis involves research in phonetics, and creating a good statistical analysis of the data is my biggest challenge. I certainly wish I had gotten more math in high school – and I was in an advanced class, to boot!

## Sili said,

May 28, 2009 @ 10:26 am

I happen to enjoy and have enjoyed calculus, but I can only agree that a better (and far far earlier) introduction to hypothesis testing and statistics in a less abstract fashion would have been beneficial.

[(myl) I enjoyed Latin, on the whole, and recommend it to others on cultural and intellectual grounds; but I couldn't in good conscience insist that it ought to be required for medical students, as it was a couple of centuries ago. I feel the same way about calculus.]

This is but a vague recollection but I think David Marjanović has something to say about how the recent(ish) evolution of cladistics in biology should be applied to historical linguistics.

The fact that according to 'these people' Darwin was a biologist that didn't count (to his own regret, admittedly) shows just what a load of bullshit (in the technical sense, if you prefer) that statement is.

## Yuval said,

May 28, 2009 @ 11:15 am

Probability theory, Probability theory, and Probability theory.

The rest is luxury.

[(myl) I'm not convinced. I took an undergraduate course in probability theory, thinking it would help me do science. What I learned was ten ways to prove the Central Limit Theorem, how to do Lebesgue integration, and various ways to axiomatize measure theory -- none of which I ever used, and all of which I've subsequently forgotten.

I'm sure that it did me good, in some abstract moral sense; and it's not that I would have wanted a bunch of canned recipes for t-tests and the like. But I certainly would not recommend this to someone who has (say) two or three semesters of effort available to learn what they need in order to do language-related science better.

(If by "probability theory" you mean how to calculate odds in specific situations or classes of situations, that's different and of course very useful indeed.) ]

## Ryan Denzer-King said,

May 28, 2009 @ 11:19 am

I couldn't agree more with your admonishment to learn more statistics. I am sorely underequipped in that area, and I hope to remedy that soon. I took statistics in high school, but no math in college, and now the best I can manage is to struggle through a chi-square analysis on my old TI-83.

## David Eddyshaw said,

May 28, 2009 @ 11:39 am

Lack of statistical sophistication among doctors has demonstrably led to large scale actual harm.

I think the fundamental problem is that very few of us have any idea as to what kind of statistical approach is valuable in what circumstances, and what needs to be thought about before you even begin your clinical trial (or whatever) if you are to have any prospect of avoiding major errors statistically.

I vividly remember a colleague trying every statistical method on her raw data offered by the software package she was using until she came up with one that came up with a p < 5%. The study was subsequently published in a peer-reviewed journal.

Part of the trouble is that statistics is often very counterintuitive; our brains just don't naturally work in the appropriate ways, and "common sense" all too often leads astray.

## Supergrunch said,

May 28, 2009 @ 11:46 am

I'm not totally familiar with the education system in the US, so I'm not entirely clear of the level at which calculus usually taught, but I get the impression that it's around the first year of college, or in some cases the end of high school. In the UK, assuming you select to do maths of some kind, it's taught a bit earlier, starting with the basics around the ages of 16-17 (the start of A-levels) and getting more complex from there. It's definitely invaluable if you intend to go on to do maths or physics, but I'd argue that the immense number of applications it has across the whole of science (basically anything involving change, however arbitrary or literal) make it worthwhile as a compulsory part of the curriculum – indeed, it could easily be taught earlier.

Still there are many other valuable branches of mathematics that are perhaps more neglected, most notably (as others have mentioned) statistics, which is useful in even more fields, and probably would be of help to the general public as well if they were better versed in it. Perhaps then, this kind of maths should be given more importance, but I think calculus is much more likely to be of use than Latin, which is only really helpful to classicists and linguists (though I wish I'd been taught more of it, I'm sure most people rightly couldn't care less). That's not to say, however, that there isn't a lot of useful work that can be done without any maths at all in subjects like biology and linguistics, which don't use maths as a fundamental foundation. I suppose the problem is deciding whether it's worth teaching everybody calculus under the assumption that sufficiently many will need it.

## mgh said,

May 28, 2009 @ 11:54 am

On mathematics in biology:It is worth distinguishing the very different ways of using numbers in biology, and considering which are relevant to linguistics.

First, numbers can assign statistical significance (especially in neuroscience where a 10% change in synapse strength is often considered meaningful); however, as discussed in comments yesterday, when sample size is large, differences may be "significant" yet unimportant. Second, there is the use of numbers for mathematical modeling of complex systems, for example reaction rates and reactant concentrations along a signal transduction pathway. This is what Bialek and Botstein do, in the article quoted here. But does this apply at all to linguistics? Finally, there is simply counting events and looking for large patterns that do not require statistics to interpret. In my opinion, this is the most useful application of numbers, and does not require advanced mathematics, only intellectual rigor. I'll leave it as a discussion point to determine which of those is more easily taught.

On undergraduate teaching of mathematics:I agree with Liberman's points on how and why calculus is taught. Many graduate students (and postdocs) in biology arrive realizing they need a basic statistics course they didn't get in college; this education is often found ad hoc, for example in statistics courses designed for medical students or bioinformatics students.

As an undergraduate, I took a course called "Statistics for Politics and Policy" which was essentially a series of real-world case studies about using large data sets to reach concrete decisions (the instructor's favorite teaching example was the mis-presentation of data that led to the fatal decision to launch Challenger when it was too cold). This course taught how to incorporate r-squared and other common statistics terms into one's intuitive reasoning, without dwelling on the math needed to derive them. Because it was a political science course, not a math course, it appealed to non-math-and-science majors. It might be worth considering this approach as a starting point to better training a wider swath of undergraduates in numerical reasoning.

## wally said,

May 28, 2009 @ 11:54 am

"while leaving out almost all of the things that they will really be able to use"

MYL: What do you recommend for study?

I ask as someone who is the parent of a high school sophomore who is two years ahead of the typical high school math curriculum, and who has the ability to make some real choices especially for his last year. And I have been impressed with your analyses on this blog.

[(myl) At his age especially, I'd start with the premise that the most important thing is for him to have fun. If he's so far ahead of the school curriculum, I imagine that he's learning on his own, perhaps with some individual help from you and others. So he has the freedom to go more deeply into the things that he likes -- and I'd encourage him to do exactly that. ]

My first choice recommendation to him would likely be statistics, backed up by probability theory (e.g. Drake's Fundamentals of Applied Probability Theory, which I would suggest he read on his own before a statistics class ). So I was surprised to see you disparaging probability theory.

[(myl) "Applied" is definitely not what I was talking about -- my complaint was that the course was so focused on fundamentals that I never learned how to apply the ideas to anything. (And anyhow, if he enjoys measure theory and the like, more power to him. My own frustration with the course that I took when I was 18 was that I wanted to understand how to explore the results of experiments, not how to compare alternative axiomatizations of probability.)

I'd strongly suggest (again, consistent with his preferences) finding him a book that does statistics from a computational perspective. Specifically, I'd suggest that he learn the wonderful free-software statistics system R, and try working through one of the many books that use it. One that I'm familiar with is Venables and Ripley,

Modern applied statistics with S(S was the precursor to R); another that may be suitable (though I've never read it) is Peter Dalgaard,Introductory statistics with R. (If these are too advanced to start with, he could try learning basic use of R via one of the many on-line tutorials, and then use it to do the exercises in a more elementary book...) ]## Andrew Carnie said,

May 28, 2009 @ 11:57 am

Here at Arizona, we teach a class called "The Mathematics of Language and Linguistics". It counts

towards the general education mathematics requirement. Strangely (to my mind) it has a reputation of being a hard course. Covers the basics of set theory & a tiny bit of graph theory, 1st order logic and proofs, some formal language theory, and very very basic stats. But we teach it, not the math department.

## marie-lucie said,

May 28, 2009 @ 11:57 am

I was speaking recently with a math professor about the fact that high school students (in Canada, at least in the province where I live) are supposed to learn calculus, something that few will use, but are no longer learning any geometry. I never did any calculus and have not missed it but was much happier with geometry than with algebra. I guess this is true of visually-oriented students. I wish I had had some elementary statistics or probability though.

## Mary Kuhner said,

May 28, 2009 @ 12:23 pm

I'd agree with the recommendation for basic statistics. I teach a statistics and computer programming course in a genetics department, meant for incoming graduate students who lack proficiency in one or both, and it fills up every time we offer it. It's difficult to teach because students' backgrounds are so diverse, but discomfort with statistics seems very widespread.

When I teach evolution to undergraduates, I am consistently troubled by students who have no intuition about very basic probabilities–that you should add the probabilities of mutually exclusive alternatives, for example, and multiply the probabilities of independent events. It is hard for them to learn this on the fly as college seniors. They can memorize the rules, but we don't practice enough for them to get a gut understanding, which they really need.

Dungeons and Dragons was a lifesaver for me in this respect: we became obsessed, as a playing group, with dice probabilities and I had to learn to manipulate them. I had players who would roll a few thousand dice to check my work, so I had to be accurate!

## John said,

May 28, 2009 @ 12:55 pm

I think that a lot of us research mathematicians would agree with you that much of what is taught in a first-year college calculus class in the US is useless. But perhaps we aren't saying this too loudly because there are incentives to keep the status quo.

There's a fear that if people started looking too closely at the first-year mathematics curriculum and realizing how much of it is unnecessary, then powers-that-be might be tempted to do away with having these big first-year math classes altogether, and replace them with Physics Math 101 taught by the physics department, Biology Math 101 taught by biologists, and Stat 101 taught by the statistics department. Then math departments would shrink to the size of linguistics departments — which may or may not be a Bad Thing in the grand scheme of things, who knows, but certainly would lead to a lot of unhappy mathematicians in the short run.

I'm also curious what sort(s) of math classes you think should be required of science and engineering majors, and if you think there's a way that mathematicians could argue for a serious curriculum reform without risking our jobs.

## Theo Vosse said,

May 28, 2009 @ 1:39 pm

I always advocate(d) discrete mathematics (a bit of algebra, basic set theory, some combinatorial mathematics, graph theory, and specialized topics such as induction, (boolean) logic, rewriting systems, stuff like that) as a basis for cognitive psychologists and possibly linguists. It helps getting the mind of the student to think more in terms of models, and provides a good starting point for programming and probability theory. I would like to see calculus, discrete mathematics and programming as a prerequisite for all science majors…

## Mark Liberman said,

May 28, 2009 @ 1:49 pm

John:

I'm also curious what sort(s) of math classes you think should be required of science and engineering majors, and if you think there's a way that mathematicians could argue for a serious curriculum reform without risking our jobs.For the biological and social sciences, and for many if not most computer scientists, I'd argue that an undergraduate math curriculum based on linear algebra would make more sense than one based on calculus.

That leaves an enormous amount of latitude — what aspects of linear algebra, and what applications or interpretations, and what additional topics? The range of alternative "semantics" for an inner product or a matrix multiplication is enormous, and includes a large part of practical modern statistics as well as digital signal processing. Throw in a few non-linear operators and you get most of the rest.

A few years ago, I counted 22 different introductory statistics courses at Penn, customized to almost every discipline and then adapted to various levels and backgrounds; and there's no question that the same sort of thing could happen to intro math as a whole. I don't know the history of the proliferation of stats courses, but I suspect that math departments bear at least a modest share of the blame, due to wanting everyone to learn measure theory and the like (though perhaps my perspective is just warped by my own long-ago experience).

I have to say that the loss of calculus as a shared experience would be a shame. One of the paramount achievements of the human spirit is the application of f=ma, via calculus, to every scale and realm of physical reality. Ideally, that should be at least as important a part of education as Homer and Shakespeare. But the motivation should be the beauty of the ideas, not their applicability in most students' future studies and careers.

And if the choice is whether a student today should know matrix multiplication or the product rule, I'd vote for matrix multiplication. (Of course, a modest amount of calculus, certainly including the product rule, should still be included, just as most undergraduate "calculus" courses do include a little bit of linear algebra.)

At a place like Penn, this is mostly all empty talk, because it would be impossible to change the math curriculum very much as long as medical schools require calculus.

## mollymooly said,

May 28, 2009 @ 2:18 pm

There's the famous cautionary tale of Euler bamboozling Diderot with mathematical gibberish:

"Sir, ( a + b^n )/n = x , hence God exists; reply! "

Sadly, it's a myth.

## Bob Moore said,

May 28, 2009 @ 2:58 pm

Mark,

I am rather surprised this post has accumulated 16 comments without anyone bringing up the central role that statistical modeling has come to occupy in computational linguistics. The discussion in the comments so far about statistics seems to relate only to statistical significance testing, which is definitely important, but quite distinct from the sort of probability and statistics that most computational linguists use on a daily basis.

The question of the importance of calculus is somewhat complicated. If one is not primarily going to do statistical computational linguistics, the depth of knowledge that a linguist should have of probability and statistics may not require calculus. (I believe that every linguist should know something about statistical modeling of language these days, just as every linguist is expected to know something about phonetics and phonology, even if her/his professional interest is in syntax or semantics.)

If one is going to do statistical modeling of language for a living, however, calculus is important, because it is needed to derive so many of the techniques currently used in statistical modeling of language. I find it a constant impediment in my work that I don't remember much about calculus. I got A's in it as a college freshman, but then didn't use it for the next 20 years, so I can't really follow the mathematical proofs justifying many of the statistical methods that I use.

[(myl) This is a good point -- but how important do you think it is for someone to be able to differentiate an objective function, as opposed to just following the reasoning of someone else who has done it?

The discussion here is *not* about hard-core computational linguists, who are likely to have 10-15 semesters of various sorts of post-calculus math courses, just like researchers in population genetics or mathematical ecology. We're talking about a broad range of interdisciplinary scientists -- in various subfields of biology for the JBiol editorial, or in various parts of linguistics, psychology, anthropology, etc., for this post -- who realistically have the space in their undergraduate and graduate education for 2-4 semesters of math and related subjects. (We could argue that they should have more -- but that's a different discussion.)

Given that these students should also learn not only probility and statistics, but also a bit of formal language theory, logic, information theory, and so on, my feeling is that spending two whole semesters on calculus is too much.

It's true that there are plenty of computational linguists who have taken a lot of discrete mathematics, but have forgotten nearly all the continuous mathematics that they ever learned. I occasionally teach digital signal processing to an audience that includes computer science graduate students, and those who have been educated in the U.S. often can't even remember how to multiply two complex numbers. But again, that's a different question. ]

## Nathan Myers said,

May 28, 2009 @ 3:32 pm

John: At my state school, we got basic calculus from the math department, and then each of the subsequent classes (multivariate, etc.) was taught twice, once by the math department using its notation ("i"), textbook, and problem sets, and again by the engineering department ("j", etc.). This was almost certainly the result of turf battles, and may have been wasteful, but the differing perspectives meant that many students got it the second time around without the embarrassment of repeating courses.

## J. W. Brewer said,

May 28, 2009 @ 4:13 pm

What percentage of current freshmen at a highly selective college like Penn these days arrive without having taken some calculus in high school? I assume elementary (i.e. single-variable) calculus teaching at the college level often sucks because the students are by definition those with less intense precollege math preparation or interest who are probably less fun to teach and certainly less likely to major in math, so the "real faculty" beg off and the function gets delegated to the most hapless grad students with impenetrable foreign accents. Or at least that's how it stereotypically worked when I was in college.

[(myl) At Penn, the senior faculty in the math department take undergraduate teaching very seriously, or at least many of them do. The current dean of the (undergraduate) college is Dennis Deturck, who is a famously excellent teacher of undergraduate courses at different levels.

I don't know the statistics, but the math department's page on AP credit says that

"Students taking first semester calculus, MATH 104, are expected to have completed successfully an AB Calculus course or the equivalent. It is strongly recommended that those who have not had a calculus course at the level of AB Calculus or who received a score of 3 or lower on the AB exam take MATH 103 to prepare for MATH 104."]

I do regret not having taken a serious statistics class somewhere along the way, but I'm not sure that calculus is what I would have omitted to "make room." (Maybe music theory my sophomore year of college, although at the time I thought my tradeoff was between that and Sanskrit.) I would hope that "statistics for X majors" classes would attract better teaching, if only because some senior members of the particular department would want to be able to read work product from majors without wincing over the lack of numeracy.

## Robert Daland said,

May 28, 2009 @ 4:20 pm

Speaking as a mathematician who switched fields into linguistics, I can say that statistics and linear algebra are far and away the most *useful* branches of math for my research. I would advocate that we as a field try to increase our undergraduates' numeracy in these areas.

I also use information theory, which is the only place where I have used calculus. Specifically, I once tried an information-theoretic approach to phonetic category learning, and once I derived a measure of word entropy as a function of lexicon size — interestingly, it doesn't really increase once the lexicon gets beyond a certain size)

## Chris Brew said,

May 28, 2009 @ 7:56 pm

In my experience even hard-core computational linguists are typically operating with way less than the 10-15 semesters of post-calculus math that myl presumes. I like computational linguistics students to have three things: fearlessness, good interpersonal skills, and willingness to engage in learning the tools they need to get the job done. It is virtually impossible to read modern computational linguistics papers without a grasp of probability, including joint and conditional probability, independence and conditional independence and a few distributions like the binomial and multinomial. If you also get entropy and cross-entropy at a fundamental level, that helps, as does an ability to think clearly about the search for the optimum of a multivariate function. It is therefore helpful to have been taught calculus (and everything else) by a good teacher, because many of the people who write papers in this area just presume that everyone has that kind of background.

But if you are fearless enough to believe that you can contribute without necessarily being the best mathematician in the room, it isn't really essential to understand every last word. If you have a good goal, and the right social skills to talk effectively with someone who has the preparation that you lack, you can often get to a deeper and more effectively grounded understanding of what the ideas mean for the science of language than the other person. And perhaps pull that person along into a level of understanding that does not come simply from formal facility with the tools.

## Rebecca H said,

May 28, 2009 @ 9:08 pm

I went to a FINE Northeast prep school and never had PHYSICS or CHEMISTRY! I took Calculus in my first yr at Northeastern, lacked the strength in Algebra and Trigonometry necessary to complete the equations. I understood derivatives, and graphs of paraboles if I am remembering correctly. These to me made sense.

As an adult student I was challenged in a class at UNH Finite Mathematics. I took chemistry as well a few years back and enjoyed it. I was a bit bored by some of the labs though, impatience more like. The class not only taught logic and reasoning but how matrix work and the 0000111 system and binary code as the basis for all computing. Today with the computer being a tool like calculators were to me in school, I would think this would be essential. Any thoughts?

Still suck at match and science in NH

## Ran Ari-Gur said,

May 28, 2009 @ 9:43 pm

All of my statistics courses were underpinned by calculus. Without calculus, the courses would have been useless; but perhaps they were useless anyway, by your standard, in that they taught statistics in the same way that (for example) calculus is taught — more about how to do statistics than about how to understand the significance of what you've done. Concepts such as "effect size" and "correlation vs. causation" were not on the curriculum.

In a way, it seems that the debate is between pure math and applied math. Is calculus more important because other pure math is based on it, or is statistics more important because it's more widely applicable?

[(myl) The discussion is certainly about applied math, at least in the sense that the issue is how much biologists should know about applying math to research in biology, or how much linguists should know about applying math to research on language. And any mathematical topic -- whether calculus or linear algebra or statistics or anything else -- can be approached in a whole spectrum of ways, from very abstract foundational stuff to completely mindless turn-the-crank application of specific procedures.

And the goal, I think, is to assign priorities to the concepts and skills that a researcher in a given area would benefit from knowing; and then to try to figure out the best way for them to get as far down the list as possible, given the amount of time they can devote to it.

My problem with calculus is not that it's irrelevant, but that most of what's taught in the first few semesters of undergraduate calculus is very far down the list of priorities, as far as I can tell; and that there are many critical things that are left out. The sad thing is that most of the stuff that's left out is much easier than calculus, and in fact could easily be taught in high school.]

## Useful Math? « Cheap Talk said,

May 29, 2009 @ 12:00 am

[...] 28, 2009 in Uncategorized | Tags: education, maths | by jeff A post at Language Log explores the use of mathematics in linguistics. It closes with Anyhow, my [...]

## stefan said,

May 29, 2009 @ 12:18 am

I'm puzzled how one is supposed to understand and do probability and statistics without calculus.

[(myl) How many pages of Feller's

Introduction to Probability Theory and its Applicationsrequire calculus? I'm sure that there are some, but flipping through my copy, I don't easily find any. And that's before the development of statistics courses based on computer exercises, e.g. Venables and Ripley. Nearly all practical statistics these days, as far as I can see, belongs to discrete rather than continuous mathematics, and so is much more strongly based on linear algebra than on calculus. ]## linguist.in.hiding said,

May 29, 2009 @ 7:15 am

I agree that most math and most statistics including stochastics are by and far not needed in linguistics. Just to illustrate this I have two cases.

The first one includes the use of Kolmogorov complexity in measuring linguistic complexity. Recently I had an exchange of comments with a mathematician (this got started on that the mathematician claimed that some languages are more logical than others). I said that you couldn't measure linguistic complexity (I at least implied "in a reasonable manner"). Even though the possible measuring examples were all from morphology and my pointing out that this does not measure linguistic complexity, only morphological complexity, this vital fact was not taken into consideration. Nevertheless, this had me going a bit deeper into Kolmogorov complexity. I soon saw that precisely this was offered as something wonderfull to linguists by net loons. I also saw that this was already done by linguists on studying morphological complexity. Sorry for bashing the writer again but the link is:

http://maxbane.com/wp-content/uploads/2007/09/wccfl-slides.pdf

I wrote the following:

"So we have a complexity figure of 35.51 % for Latin, 19.51 % for Dutch, 16.88 % for English, 0.05 % for Vietnamese and so on. Now, this indeed describes even the intuitive sense of complex vrs simple morphology.

Sorry for not being impressed but how has the assigning of percent values with the use of the Kolmogorov thingie advanced our knowledge of linguistics… You could as well have measured them the old-fashioned way like counting the mean of affixes to a "word" in a text (of course there are difficulties in that too). I doubt the order of the languages would have changed considerably. And even if it did… So what? I fail to see that that method is intrinsically superior."

The other is just a general remark that some linguists (here, phoneticians included) who use math or statistics do not understand what they are doing. A year ago I was in a phonetics conference. There I listened to a speech by one participant. She mentioned she used Cronbach's alpha in her measurements. I wondered where I had heard that. Then I remembered that in my statistics lectures the lecturer was highly critical of the use of Cronbach's alpha. I looked at the evidence and agreed with the lecturer. At the very least you should say that the use of Cronbach's alpha is controversial. I was a coward and didn't say anything about that to the speecher. I know I should have.

## Sascha Griffiths said,

May 29, 2009 @ 7:39 am

I'm unsure what is meant by maths here. I have a first degree in mathematics and it had absolutely nothing to do with practical statistics or formal logics (as such). I think it would have done better at it if it had.

However, I sometimes do get worried when I hear that people are advocating more "maths" in some fields if that is reduced to statistics. I have sometimes wondered when attending research seminars of experimental-quantitative "social sciences" how many people seem to start switching off brains as soon as we've got quantitative data and statistical tests. They start worrying more about chi square, fischer test,… than to wonder if the question makes sense.

## linguist.in.hiding said,

May 29, 2009 @ 8:01 am

> They start worrying more about chi square, fischer test,… than to wonder if the question makes sense.

My observation exactly.

## peter said,

May 29, 2009 @ 8:21 am

stefan said (May 29, 2009 @ 12:18 am):

I'm puzzled how one is supposed to understand and do probability and statistics without calculus.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.

## Sascha Griffiths said,

May 29, 2009 @ 8:35 am

I confused Fisher test and Fischer test (btw); the Hans-Fischer Test is only used in gymnastics. I wonder if that would be a useful requirement for linguists. ;)

## Rebecca H said,

May 29, 2009 @ 8:45 am

I wish I could have edited my post above sorry …

Mathematics in biology ? Fractals are fascinating to me…

## John Cowan said,

May 29, 2009 @ 9:33 am

A few years ago, an economist (of all people) said to me, "If you can't explain it

withoutmathematics, you don't understand it."## stefan said,

May 29, 2009 @ 12:08 pm

peter writes:

"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.]

## stefan said,

May 29, 2009 @ 12:41 pm

"(myl) How many pages of Feller's Introduction to Probability Theory and its Applications require calculus? I'm sure that there are some, but flipping through my copy, I don't easily find any. And that's before the development of statistics courses based on computer exercises, e.g. Venables and Ripley. Nearly all practical statistics these days, as far as I can see, belongs to discrete rather than continuous mathematics, and so is much more strongly based on linear algebra than on calculus."

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.

## stefan said,

May 29, 2009 @ 1:05 pm

myl, we are posting here at the same time.

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).

## peter said,

May 29, 2009 @ 3:40 pm

Stefan —

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.

## john riemann soong said,

May 31, 2009 @ 9:58 pm

Ehh, I don't see how calc is useless for most college students, though I think other fascinating fields of mathematical theory are overlooked too much. For one, you need it to do physics, and you need physics to do biochemistry, and you need biochemistry to do medicine.

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.

## J said,

June 1, 2009 @ 10:23 pm

As an ecologist, I have to say I think calculus is something every *ecologist* should know. We still use a lot of continuous time models, or that is to say, some of the basic premises and dynamics are often expressed as such. Discrete time models are also very very common, as are numerical methods, but as we try to understand the changes in n-dimensional systems interacting with each other, it is often useful to understand the calculus behind our models of it.

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.)

## Jason Merchant said,

June 2, 2009 @ 3:14 pm

To return to one of the phrases from Mark's post–linguists "who count": just wanted to point out that when I took Math Methods from Geoff Pullum in grad school (a great course, of course, and required for the PhD), using Partee, Wall, and ter Meulen, we did lots of math, but no counting. (I'm not sure PWtM even has any quantitative math in it, but I'm on the side of Mt Olympus doing fieldwork so I can't check).

## Robert said,

June 2, 2009 @ 3:52 pm

Feller's volume 1 uses generating functions for various calculations (although rational generating functions can be derived by long division, treating x^2 as smaller than x, i.e. in the x-adic metric). He also touches on the Fokker-Planck equation at one point.

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.

## john riemann soong said,

June 3, 2009 @ 5:26 am

I suppose linear algebra was developed "later" than calculus; in a way linear algebra is a bit more abstract. Whereas some calculus theorems are sort of intuitive (Rolle's theorem, mean value theorem, etc.) and some identities are tedious but straightforward to prove directly, when I studied it it seemed to me that linear algebra theorems had fantastically magical results in a e^(pi*i) sort of way. To understand what is really going on with eigenvectors and with n-dimensional spaces, you really needs lots of imagination and creativity, whereas there isn't really a lot of creativity in elementary calculus.

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?

## Wednesday Round Up #66 « Neuroanthropology said,

June 3, 2009 @ 12:11 pm

[...] Liberman, Linguists Who Count Nice reflection on the quantitative impulse and the rigors of research not just in linguistics but [...]

## peter said,

June 4, 2009 @ 7:00 pm

john riemann soong said (June 3, 2009 @ 5:26 am)

" . . .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.

## uqbal said,

November 12, 2011 @ 9:18 am

Oh, thanks for taking Latin and Greek into the discussion (I'm a Classics teacher in Italy).

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.