IN THE BEGINNING: The Art of Slowing Down and Thinking Things Through (Chapter 1, Part 5)
This is the fifth of a six-part series. Below is a table of contents I'll update as each part is posted to Minds.
2. The Evolution (or Transformation) of Language
5. Mathematics and Symbolic Logic Are Languages Too (you are here!)
6. Why Thinking Through Language is the First Topic of This Work
E. Mathematics and Symbolic Logic are Languages Too
Aside from the idea that humans think verbally and visually (and, as I’ve mentioned earlier, the value or prevalence of visual thought absent language is difficult to assess because often language is an interpreting filter applied to any image we might be perceiving or mentally constructing), there are those who suggest that mathematical ability is characteristically different from verbal ability. This worldview is reflected in our approach to standardized testing, such as with the SAT exam having historically distinguished between verbal and mathematical proficiency. And it makes sense to test for mathematical ability separately from English proficiency (much as you might separate French proficiency from English proficiency), does it follow that mathematics is not, in actuality, just another kind of language?
At first glance, scholarly opinion seems to support the idea that mathematical thinking – whatever it may be – is distinctly different from language ability. Returning again to Martin Nowak’s paper, Evolutionary Biology of Language, he writes: “Our language performance relies on precisely coordinated interactions of various parts of our neural and other anatomy, and we are amazingly good at it. We can all speak without thinking. In contrast, we cannot perform basic mathematic operations without concentration.” Leone Burton, in Mathematical Thinking: The Struggle for Meaning, begins by apparently making the argument that mathematical processes evolve out of a distinct “kind of thinking:”
Most schools assume that by teaching mathematics compulsorily and over a number of years they are providing the conditions through which pupils will develop their mathematical thinking. This assumption, usually unchallenged, rests on a view of mathematics as a logically developed discipline, together with the expectation that the logic will spill over and be absorbed by the pupils into all aspects of their lives as they pursue a study of the content of mathematics, for example, in learning number, geometry, trigonometry, or algebra. Experience, however, tells a very different story…Certainly an inordinate amount of time in schools is spent teaching mathematical content and techniques while the process, the means through which mathematics is derived, receives little attention…Exploring process is not very profitable when teachers do not understand the kinds of thinking from which process springs.
If we dig in a bit deeper, however, the apparent distinctions Burton makes between “mathematical thinking” and thinking in general (which I’ve tried to establish as relying significantly upon and being intertwined with our language faculties) begin to evaporate (emphasis in the following quote being the author’s own):
The process is initiated by encountering an element with enough surprise or curiosity to impel exploration of it by manipulating. The element may be a physical object, a diagram, an idea, or a symbol, but it must be encountered at a level that is concrete, confidence inspiring, and amenable to interpretation. A perceived gap between what is expected from the manipulation and what actually happens provokes tension that provides a force to keep the process going until some sense of pattern or connectedness releases the tension into achievement, wonder, pleasure, or further surprise or curiosity that drives the process on. Although the sense of what is happening is vague, further manipulating is required until the sense can be expressed in an articulation.
Expressing an articulation, you say? That process sounds eerily similar to the entire point of language. Yet Burton goes on to state:
Pupils need tools to help them structure their responses so that they can build their reflective powers. Further, they need engagement to capture their feelings at the moment of expression. Consequently, students of all ages have been encouraged to develop the use of particular words that reflect their responses as they tackle questions. These words can then act as triggers to further thought as well as providing mental markers…Without in any way insisting on a particular choice of words – indeed, the more personal the choice the more likely they are to prove useful – [research has] found that the action of writing such annotations both facilitates results and stimulates an awareness of mathematical thinking…. The key to recognizing and using mathematical thinking lies in creating an atmosphere that builds confidence to question, challenge, and reflect. Behind such behavior is an acknowledgement of the need to
· query assumptions
· negotiate meanings
· pose questions
· make conjectures
· search for justifying and falsifying arguments that convince
· check, modify, alter
· be self-critical
· be aware of different approaches
· be willing to shift, renegotiate, change direction
If core concepts of language ability such as negotiating meanings, posing questions, facilitating further progress via a framework of “particular words” tagged to responses generated by tackling questions, writing these “particular words,” and ultimately arriving at articulations are all critical parts of the supposedly unique process of mathematical thinking, how is this in any way actually distinct from the process of language generally? Moreover, when Burton claims ideas like being “self-critical,” being “aware of different approaches” and being “willing to shift, renegotiate, [and] change direction” are also integral, how is that any different Obama’s similar admonishments earlier in the broader context of understanding how language shapes reality generally? While we might not always perceive of mathematics as a language in the same way that the perceive of English or Arabic as languages (perhaps in part because the concepts of mathematics, though somewhat universal, can be expressed in different languages both orally and orthographically), the mechanisms at play are strikingly similar. Maybe part of the reason we hold mathematics in such high regard due in part to its unusually high “fitness” in terms of Nowak’s theories about language. The number “one” can only refer to the number “one,” whether you are saying or writing “one” in English, “uno” in Spanish, or “un” in French; there is decidedly much less ambiguity in mathematics than there are in more, shall we say, “conventional” languages.
At its core, language is a system of symbols. Sometimes these symbols refer to phenomenon experienced through our senses – we might refer to a particular visual phenomenon outside of our window as a “tree,” for example, “tree” being the symbol associated with the phenomenon. (Even “window,” in this case, is such a symbol.) Sometimes these symbols are more abstract, like the idea of numbers. While we can apply the idea of a number to other symbols, such as “I see one tree” or “I see two trees,” it’s harder to think of directly experiencing a number as a phenomenon in and of itself. In any case, since we can understand languages as systems of symbols, it may behoove us then to also consider the discipline of symbolic logic.
When symbolic logic was becoming codified as a science in its own right, all the way back in 1942, Edmund C. Berkeley, in the work “Conditions Affecting the Application of Symbolic Logic,” described it in the following manner (emphasis my own):
…Symbolic logic, is, in its broadest sense, a new science which studies through use of efficient symbols the nature and properties of all nonnumerical relations, seeking precise meanings and necessary conclusions. As an applied science, it holds immense promise. For example, it may give us an unambiguous language for political, economic, and social fields, which will conveniently reflect the structure of these fields and make discussion and analysis easy.
Insofar as symbolic logic is defined as seeking “precise meanings” and supplying us with an “unambiguous language,” it should be easy to predict that I am going to argue that symbolic logic is not particularly different, then, from language in general. Thus, it too owes no higher claim to The Truth than any other language. Berkeley continues to elaborate on the applications of symbolic logic, further demonstrating its similarities to language in general: “We observe first that symbolic logic can define certain ideas which neither mathematics nor the dictionary can possibly define; for example, symbolic logic can define number.” If the power of symbolic language is in its ability to define, well, that appears to be a characteristic and defining nature of language generally.