When one's written extensively enough one sometimes realizes that towards the end of a piece that one has run out of gas - have exhausted one's resources and knowledge of the subject to such a degree that one is running on fumes. By the time Chapter 7 in Barton's The Language of Mathematics has come around he has definitely reached that point.
He claims, with little supporting evidence or arguments (whether historical or operational), that mathematics is like JRR Tolkien's Middle Earth - that profound and wonderful mathematical insights are nothing more than "fictions":
Mathematics is a created world, a world of the human imagination, and, like Middle Earth, we can write about it, film it, become part of it in our minds and emotions. Also like Middle Earth mathematics has been expanded upon by others apart from Tolkien (despite his family's best attempts to preserve copyright)...
...Once we have the number 1 and the number 2, then no mathematical Tolkien could have written anything other than 1 + 1 = 2. Once we construct a circle and its diameter, and then draw a triangle on the diameter to a point on the circumference, it is not just geometric poetic licence that says that the angle at the circumference will be a right angle. And it is not just a muse's whisper that requires that a right-angled triangle to have sides that obey Pythagorean relationship. These things must be so.
The mistake is to think that this situation does not exist for Middle Earth. If you are a hobbit of Middle Earth, and you get yourself into deep trouble with the Forces of Evil, then, in your moment of dire need, lo, the Elves will come to your aid. It cannot be otherwise. For if it were otherwise it would not be Middle Earth!!...
...In the same way, if 1 + 1 does not equal 2, then we are not talking about the world of mathematics, we are in some other world. The number objects 1 and 2 were [emphasis added by me] created into just the relationship embodied by 1 + 1 = 2. That is what mathematics is. Circles and triangles and angles were also [emphasis added by me] created into their relationships.
But when Tolkien wrote Lord of the Rings, he had all the relationships and consequences worked out in advance. As the mathematicians write mathematics, the consequencs of some of their supposed [emphasis added by me] imaginative constructions are still being discovered, many are suspected but not yet proven, and still more are not yet known - or so the hundreds of budding mathematicians hope. (pp. 121-122)
The sword Barton welds is too big for him. In his supposed anti-Eurocentric discourse on maths, he has squarely put back the Eurocentricism onto maths (rigour analysis vs geometric structures): ask any person unfamiliar with Tolkien's or Jackson's work what a hobbit is, and the more likely response would be a blank stare; ask anyone in the world who can count what 1 + 1 equals to, and the answer would be immediate.
Teach a child "pebble notation" and ask them to arrange and rearrange them into certain geometric shapes and they'd begin to realize that a given "set" cannot be arranged into just any, old shapes but just so. At a slightly more sophisticated level, they'd be able to demonstrate Pythagorean's theorem using only lengths of strings that by "squaring" the lengths of the two shorter sides will always equal the length of the longest one "squared" (or that a2
+ b2 = c2 in analytic terms).
The increasing separation of European-style maths (analytic) from their origins (geometric) gives the impression that mathematics is just a word game (as per the abuse Barton wreaks upon the great Wittenstein) with all their wherefores and whatnots, but mathematical proofs are always based upon the foundations of what defines "number" or basic relational/structural aspects of geometric constructs of interest.
When Newton first wrote up his physics and his discourse on optics he hid much of the calculus (analytic) and chose to demonstrate his insights in geometric terms to the Royal Society; the geometric treatment of Einstein's theories of relativity are also simple enough for highschool students to understand and take something from them, though the differential and integral calculus of them are quite difficult.
Mathematics, if taught in a school at all, is never really taught in this way. Rote memorization of multiplication tables, log tables (if it reaches that far) are more the norm. But there is that old chestnut of the youth Gauss burning through the summing of numbers from one to a hundred.
Gauss realized immediately that he could add up the first and last numbers to 101, same thing with 2 and 99, and so on... if this story is based on reality he may have been too young to express this insight in a pat equation but his geometric (mathematical) intuition had never had the chance of being choked off by lesser mortals up to that point, so he visualized a triangular structure (called a partition) and came up with the answer in that brilliant flash of genius (the columns all add up to the same number if you reproduce the original triangle and match it up with the reproduction). That poetic image of the innocent being the vessel of the divine...
Barton is a fool. A dangerous fool, but a fool nonetheless.