Mathematical preparation for postgraduate studies in Linguistics

Question

I posted this question in https://math.stackexchange.com/ and it was suggested to me that it would be a good idea to submit the question here, too, as there might be more specialists on the matter.

I am an undergraduate student in Mathematics and I would like to continue my postgraduate studies in the harder, more mathematical aspects of Linguistics. Possible areas of interest would include the mathematical aspects of syntax and semantics, as well as computational linguistics, natural language processing, artificial intelligence or machine learning.

As of now, the courses that I have taken have been purely mathematical: real, complex and functional analysis, measure theory, point-set topology, differential equations, group and ring theory, linear algebra. I understand that the analytic courses are not very pertinent in linguistics where data is rarely "continuous" or as nicely behaving as the functions studied in analysis. I would like to hear the community's suggestions about courses that should constitute good preparation for postgraduate studies in the aforementioned areas. The courses need not be mathematical. They can be linguistics, statistics, computer science courses or courses from any other discipline that would best prepare me for the task.

UPDATE: For those that might be interested, I received some answers for this question here, too.

Answer 1

I'm currently completing a Masters of Linguistics, specializing in Computational Linguistics, after a Software Engineering undergrad so I think my experience might be relevant to you.

I think you may be a bit disappointed by the actual math involved in linguistics. Setting aside Computational Linguistics, the only real math I see is statistical analysis used in academic papers (and not something you'll generally need for your course work). Of course, semantics requires some basic set theory and algebraic logic but as a computer programmer by training and [formerly] by trade, it was all fairly basic. My [limited] understanding is that the cutting-edge work in syntax, semantics, and phonology come from examination and theorizing of frameworks involved in language usage.

Computational Linguistics is where you're going to see more advanced math; being a cross-disciplinary field with Computer Science. Machine learning and AI use various mathematical models and statistics. Information Retrieval heavily uses linear algebra and statistics (with some algorithms even using graph theory). Speech recognition and speech production all involve a certain amount of signal processing which you may find interesting. The problem with Computational Linguistics though is that it is typically part of the Computer Science department and this might limit the type of projects you'll get to do and the types of classes you'll get to take.

However, I'm more concerned that you don't seem to have any set goal other than "I did a degree in math and this linguistics stuff looks interesting. I wonder if I can combine the two?" I wasn't much better off when I decided to pursue linguistics but it sounds like you need to do a bit of leg work before you deciding that linguistics is right for you. Linguistics is a very broad and diverse field of study. Your first step should be finding what part you're interested in. Try to audit some introductory linguistics courses and/or skim an introductory textbook. Think about what made you interested in linguistics in the first place.

Once you decide what area you want to pursue then you can start deciding what courses would help prepare you. In general, the math used in linguistics is very applied (as opposed to the more theoretical stuff you're probably used to) so you might be better off taking some engineering or computer science courses. A course on signal processing would be very helpful if you're interested in speech recognition but useless if you're interested in semantics. Similarly, if your interests lie in question-answer systems then some courses in AI would be helpful but would get you nowhere if you're interested in phonetics. Without an area-of-interest it is very difficult to answer your question.

Answer 2

Logic is a big deal in linguistics (especially semantics), and unification is an important idea behind several grammar frameworks like HPSG and LFG. I did both a bit of math and statistics to support my Master's, and unification was only taught at the ling. dept.

Anyway, browse through the table of contents of "Mathematical Methods in Linguistics (Studies in Linguistics and Philosophy)" (1990) by Barbara B.H. Partee et al., that should give you a good overview.

Answer 3

There are a few unrelated subfields of mathematics to choose from. If you're going to focus on machine learning, and related techniques, some of the textbooks mentioned in "Looking for a good beginners reference to learn computational linguistics" should cover what you're looking for.

If you're going to focus on logic and semantics, the two books by L.T.F. Gamut are excellent to begin with. Some researchers who work on formal grammars also use concepts from graph theory and type theory.

Apart from that, you'll probably use concepts from algorithms and complexity theory, the hierarchy of languages, FSM, etc. -- most of which are covered in the book by Jurafsky and Martin.

Based on my own experience, you probably don't need to study any of this before you start your masters. That's what your masters program is meant for anyway. Having had one or two classmates with a background in just mathematics, I'd suggest that you develop competence with some programming language instead; your course might require that you be comfortable with programming already.