In this post, I would like to expand on the
point that I addressed in my first blog post (http://blogforling1.blogspot.com/2014/10/abstraction-how-do-linguists-think.html)
on how the thought processes and approaches of mathematicians and linguists
coincide. In the first half of the course, phonology and morphology served as
good examples of how abstraction is integrated in the study of languages.
Throughout the second half of the course, our discussion of syntax and
semantics have introduced even more ways of mathematical thinking present in
linguistics.
In mathematization theory, a branch of
meta-mathematics that deals with the general philosophy and methodologies of
mathematics, mathematical thinking is defined as a comprehensive concept that
incorporates observation, induction, finding patterns, generalization,
abstraction, formalization, structuralization, algorithmization, proofs and
deductions, modeling, etc. Out of these overwhelming number of components, I
would like to focus on the rather essential abstraction (considered in the
first post) and formalization.
Formalization is a process that follows
abstraction. The term describes the action of representing the extracted
essence into formal expressions and logical systems. One may remember from high
school geometry how we formulate our proofs for theorems using a few of
assumptions such as “we can draw lines and circles”. The basic assumptions upon
which we draw proofs of theorems are called “axioms”, and hence the term “axiomatism”
for the approach. Beginning from Euclid’s discussion of plane geometry, axiomatism
has been the hegemonic method that mathematicians adopted; modern mathematics
is built upon axioms of Zermelo-Frankel set theory and formal logic.
What is the upside of formalizing our
thoughts and logical deductions into sequences of strings that obey certain
rules and follow from pre-determined axioms? Formalization does not necessarily
facilitate our reasoning and help drawing new theorems; in fact, it usually
prevents our intuitions from expanding freely, and therefore often hinders the
process of actual mathematics. However, by fitting mathematics into a formal
system we obtain a way to systematically consider which statements can be deduced
(syntax) and evaluate the meaning of created statements (semantics). We can
also assure ourselves that the exhaustive discussion of math is consistent,
provided that the formal system has no flaws in itself. Hence a formal language
equipped with a logical background provides mathematics a solid and consistent
foundations on which mathematicians can restart their free intellectual
adventures.
Formalization is observed widely in
linguistics, especially when we begin analyzing and interpreting sentences. As
an example from syntax, consider the construction of grammar in syntax and how
we used phrase structure rules to analyze the syntactic structure of a given
sentence. Defining phrase structure rules consists of assigning ways to break a
given expression in a certain lexical category into a combination of different
lexical categories. Such rules are symmetric in the sense that when we apply a
rule once to a given sequence of lexical categories and get a new sequence on
which we can apply the rules again. With a pre-determined set of lexical
entries, these rules give a recursive algorithm to decide which sentences are
validly constructed in a given sentence. This resonates with the importance of
formal logic and its ability to ensure that the proven sentences are
consistent. The only difference would be that in linguistics such system is
used to initiate a top-down analysis on a given sentence, whereas in
mathematics the axioms and the rules of inference are mostly used to create new
statements, or sentences.
Compositional semantics also provides a
good example of formalization in linguistics. Transforming sentences like “Sally likes dog” into a sentence in
formal language, e.g. ∀x∈Dog[Love(Sally,
x)], allows one to fit infinitely diverse
sentences into a system with few assumptions and rules. By establishing an embedding
of natural language into formal language, one is in a much better shape to
begin analyzing relationships between sentences (entailments, contradictions,
etc.) as they can be computable in formal languages. Such computability makes
the process of deriving new propositions from a given set much more systematic
and hence the facilitation in the semantic analysis on the implications of
certain sentences.
By incorporating mathematical thinking,
linguistics obtains a special degree of assessable validity and computability
among the possibly ambiguous social sciences. It is quite interesting and
moreover exciting to discover how seemingly distant subjects converge in their
essential philosophy.
I like how you link mathematics and linguistics through fomalization Jae. I wonder, however, if what you are describing is an approach used not only by mathematicians but by the scientific community at large. Linguists are very similar to physicists, biologists and chemists. Each performs formalization by representing the extracted essence of an observed phenomena and placing it into formal expressions and logical systems for analysis and theoretical development. The beauty of the sciences, be they the social or non social variety, is that the processes for experimentation, advancement and discovery tend to follow the same roadmap. Each must incorporate observation, induction, find patterns, create generalization, abstraction, formalization, structuralization, algorithmization, proofs and deductions. In many ways Linguists speak the same language as mathematicians and practitioners of the non-social sciences. In order produce cutting edge research, they must all adhere to the same precepts of scientific discovery that you've outlined in your post.
ReplyDeleteGreat read Jae! I think it's very interesting that you link mathematization with linguistics! To be quite honest, I feel that your points are very accurate, there is a very strict relationship between the formalization we see in linguistic theory and the mathematical process. What's more interesting to me is how this affects computational linguistics, when we start talking about things such as feature vectors and parse trees and other NLP items. I find it fascinating that it seems that mathematics becomes the de-facto way to analyze the massive linguistic data we have gathered over the years. Moreover, the recent developments in Neural Networks end up making everything mathematical recurrences in the simulated neurons. My question is, if Neural Networks are so effective (and are becoming even more effective) at decrypting spoken language using mathematics, isn't mathematics an inherent part of our cognitive process too? Is there a way to mathematically replicate a brain? If we look at your questions in that light, then maybe we arrive at the conclusion that indeed, mathematization is our default strategy simply because it's how our brains are wired! I find this topic to be personally engaging, and it's very related to philosophy of mathematics.
ReplyDeleteSolid, perceptive thoughts and ideas just like the last post on abstraction. I definitely felt that the connection between mathematics and linguistics was more apparent the second half of the quarter. For example, when it comes to defining grammars, the structure of the rules are comparable to recursively defined functions in mathematics. Additionally, problems such as finding constituents are formalized in an analogous manner to mathematical tests to determine things such as convergence or divergence. All in all, I really resonate with a mathematical approach to linguistics as it is my approach of choice for problem solving in any discipline. However, unlike subjects such as painting, linguistics is definitely more compatible with a mathematical mindset. Consequently, processing the material mathematically has been effective in understanding the concepts and solving problems.
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