Friday, 24 August 2012

Is there a word for "mathematics" in English? (part iii)

One of the reasons why I said earlier that English may not be the best language to do maths in has to do with confusing technical concepts for commonly used words. These are words like "set"; "proof"; "rational vs irrational". At a deeper level, sometimes we think that since we are able to read certain words that we immediately comprehend what those words mean.

Take for example human activities (non-human structures, even) that can be described in mathematical terms - (from Barton's book, The Language of Mathematics) like "Pacific navigation" or the kolam patterns drawn by women and girls in many eastern countries (or the hexagonal patterns of honeycombs, spider webs, the fibonacci patterns in nature for that matter) - and actual mathematical discourse. Pacific navigation and kolam drawings are no less beautiful and elegant simply because the practitioners of these "arts" do not describe them in strict mathematical terms - it is their insights and methods that are mathematical and can be described so. In pedagogical terms, especially in construction of maths in an indigenous language, these activities would be a rich source of insights and neologisms.

In fact, I contend that numbers were not the first elements that caught the mathematical imagination of human societies all over the world, geometric shapes are the more primitive in our notions of maths par excellence. The ancient Greeks, Pacific navigators, kolam drawers, etc.

In Inuktitut, the demonstrative morphemes (pointing and locating words called "prepositions" in English) are rather complex and elegant at the same time. In the Inuit languages we can say not only "here/there" but have specific words that denote elevation, proximity, in relation to the speaker, in relation to the spoken to, inside/outside.

Now, going back to that statement that English may not necessarily be the best way to talk maths:

...the Greeks discovered that if you draw a square, say with sides of length unit one, and then you draw the diagonal of that square, then there was no small length that would divide exactly into both the diagonal and the side (Lasserre, 1964)...This was the first irrational number, that is, a number cannot be represented by a fraction. All numbers had previously been thought to be expressible as ratios [emphasis added by me] of two numbers (that is, fractions). They were rational. Now here was a number that could not be written as a ratio [emphasis added by me]. The Greeks thought they were going mad - and there is the origin of the everyday meaning of irrational: the human condition of being without sense or reason. (Barton, The Language of Mathematics, p. 85)

That may be, but there is a sound technical and conceptual difference in meaning of rational and irrational numbers than from the everyday English sense of being rational or irrational. In the mathematical sense the words irrational/rational comes from "ratio" (ie, expressible as a proper fraction); whereas the everyday sense of being irrational/rational comes from "reason" (or lack thereof).

Another word in English maths and everyday English that is the cause of no end to headache is the word "dimension". Many a great thinker struggles with this concept, let alone the laity. Many math-physicists have said without irony that they cannot visualize "four dimensional spacetime" - inadvertantly adding to the mystic of English maths. But what they fail to realize is that the word "dimension" (in maths sense) has two distinct meanings: geometric and arithmetic. The four dimensional spacetime incorporates both senses: the 3-fold space (geometrical) along with the tracking function (arithmetical) that tracks movement/velocity (ie, time as a length). The word "dimension" in the notion of Einstein/Minkowski's four dimensional spacetime is an orthogonal system that has the capacity to track a pointlike particle within that generalized space.

I'm a believer in the "standard model" of "classical" particle physics because I've never seen nor heard a convincing argument for the so-called super-string theory - when someone constructs a convincing argument that one can definitely divide by zero or define a workable boundary of a black hole then I'll believe the string theories.

In The Language of Mathematics, Barton also talks about what he calls "mindlocks" and "metaphors", especially the "Container metaphor" vs "Path metaphor", that sort of implies (perhaps inadvertantly) the futility of trying to use indigenous languages for maths discourse; that English is the best we can do. He sort of skirts around deep and wonderful (in my estimation) philosophical insights that are the purview of European maths/philosophy but to which other traditions may add. But this, like so much of his thesis (whatever that is is still unclear to me), is confusing superficial accepted terms (ie, static word locks) from the conceptual framework (dynamical, evolving discourse) that is mathematics.

My above comments and insights (ie, "dimension" and "rational/irrational" especially), I hope, show that possibilities for cross-pollination between languages are infinitely rich (if somewhat daunting). This is the great strength of conceptual-/meaning-based translation method that has always served me so well. The psychological/spiritual/intellectual worlds of humanity are not language-specific but rather depend on our willingness to learn and discover these great insights that various traditions have to offer freely if we can/would accept them.

Jay