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I have been prompted to write this because of the recent article in The Daily Sceptic which takes a comical look at a paper written in 2017 that goes by the title “Living Mathematx : Towards a Vision for the Future”. Even amongst the wokest of the woke indoctrination centres we call schools I suspect it’s probably seen to be a bit, erm, shall we say, “alternative”?
What the paper does do, however, is give a fascinating insight into the extreme woke “mind” and how it perceives the world.
More generally, it prompts us to ask the question what, exactly, is mathematics?
But before I delve into my non-mathematician’s “answer” to that question, let’s have a bit of a chuckle and see what’s in that ground-breaking paper on Mathematx. The author is one Rochelle Gutiérrez who is a Professor of Education at the University of Illinois. She describes herself in the paper
As a Chicanx scholar, a cis gender female with Rarámuriiii roots, I seek to decenter the field’s overreliance on Whitestream views.
That bastard whitey again - just completely fucking everyone over again. How dare they have actually gifted the world most of the liberal precepts the woke value so much. I just hope these woke arseholes use “indigenous ways of knowing” to get their cataracts fixed.
After this obligatory statement of positionality so that her views can be correctly slotted in the appropriate place in the intersectional matrix she goes on to describe her ideas more fully.
I introduce the concepts of In Lak’ech, reciprocity, and Nepantla to suggest we learn from other-than human persons . . .
Does she mean aliens here? No, she’s talking about animals, plants and, erm, rocks. Are Mr and Mrs Shale home today? Yes, but you’ll have to come back tomorrow because they’re feeling a bit sedimentary today.
You can get an idea of where she’s coming from (and, perhaps, what she’s been smoking) from her impassioned argument
That is, because a Western worldview does not consider plants, animals, and rocks as living beings of equal value with the same rights to this universe as humans, the result is that plants, animals and rocks suffer the same treatment as Indigenous peoples have endured throughout time. For example, like American Indians who were stripped of their lands and communities and forced to live in boarding schools, plants are yanked from their families and forced to assimilate into Western ways of doing things (e.g., to become suburban gardens).
Pity those poor, oppressed, potatoes - just stripped from their families and forced, yes forced, to assimilate into Western ways of doing things.
I’ve just been out to my back yard to apologize. I had a chat with my juniperus communis. Xe was a bit pissed off, to be honest, and told me “If I could get out of this fucking pot, I’d give you a bloody good hiding”.
I’ve promised xim a good layering of well-rotted manure as partial reparation for having imposed my Western way of doing things on xim.
At this point your Acme Loon Early Warning Detector is laying in a smouldering heap in the corner; you’ve overloaded its circuits.
Now realize that this, this, is someone who wants to tell us how mathematics ought to be practiced and to change the meaning of what it is.
Dr Dolittle was said to be able to talk to the animals, but one assumes that Dr Gutiérrez, by virtue of her indigenous Chicanery, can talk to the rocks1.
Having realized in all of the paper’s preamble that this is exactly the kind of person we want to be telling us how to do mathematics, we learn that she’s drawing on some “indigenous” concepts. She describes herself as a Nepantlerx, and you’ll have to attempt to figure out what she means by that by reading her paper. The best I could do was “viewing all sorts of contradictory shite as valid”. A more charitable view would be to suggest it’s all a bit “yin-yangy”.
Apparently this Nepantlerx view, which she privileges, is important in forming this shiny new mathematics she envisages. Mathematics is a performance and the actual output comes way down the list
Second, whereas mathematics tends to be thought of as a noun (e.g., a body of knowledge, a science of patterns, a universal language), mathematx is performance and, therefore, a verb. Just as identity is not something that you are, but rather something you do (Butler 1999), mathematx emphasizes the guiding principles and the process as opposed to the product. Drawing upon the concept of reciprocity, mathematx is an intervention-in-reality (action) as opposed to a representation-in-reality (explanation)
I’m so glad she clarified that for me.
What is “living” mathematx then?
Living mathematx means both that we live a version of mathematx as well as we are a living version of mathematx. This framing is consistent with Nahua metaphysics that suggests one is both in Nepantla and one is Nepantla. Living mathematx means moving through the world with other living beings, acknowledging, appreciating, and reciprocating the patterns produced. If we look to animals and plants for some insight, we see that Brassica oleracea (Romanesco cauliflower) performs itself in both utilitarian (compact) and non-utilitarian (pleasing) ways that may get us to pay attention to its form and to continue to cultivate it.
I had some cauliflower cheese the other day. It performed rather well.
I’d probably better stop with the quotes because you’re going to get brassicad off. So I’m going to kale it a day. It’s bean fun and there might even be a kernel of truth coniferred along the way. It’s an amaizing paper, built on a salad foundation, but we need to romaine calm. It’s thyme to stalk about proper mathematics.
Much has been made of the whole “2 + 2 = 5” thing. It’s a simple way of describing some of the woke absurdity, but it really doesn’t capture very much (if anything) of what mathematics is actually about.
So, let’s consider the process of summation.
Start off at 1, halve it. Then halve that result. Then halve this new result, and so on. You end up with the sequence of numbers
What happens if we add these numbers up?
You can get an idea of what might happen by using a picture.
You can intuitively see that this process of halving, and then adding it in, is always going to give you something contained in the (big) square.
You can see that no matter how many times we do this halving then adding process we’re going to stay within the big boundary - and furthermore we can see that as we continue doing this our sum of all of these pieces is going to get closer and closer to the area of the big square.
Our intuition, then, says that if we start with 1 and do this process of halving and adding, and never stop, we’re going to end up with 1 + 1 = 2.
Notice just how ridiculous it would be to assert that 1 + 1 = 3 here. Or that we could assert this. The numbers, the symbols, represent something very specific. There is no “indigenous way of knowing”, no amount of Nepantlerxing all over the place, that is going to result in being able to generate a larger area than the big square when adding up all the halved squares and rectangles.
In math squiggle language we’d write this summation process like this
What’s that weird “lim” thing in there?
This is the mathematical concept of a limit. Basically, if you run at a wall you get closer and closer to it. The limit happens when you actually knock yourself out. It’s quite a tricky concept because you can’t actually sit down and write out all of the numbers in this sum - there’s an infinite number of them. And there we have yet another mathematical concept - one that’s even trickier to get your head round2.
Although we are definitely interested in this specific result, a mathematician will instantly get this irresistible urge to make things far more complicated. This particular infinite sum converges on a particular value. Do all infinite sums do that?
Nope - just add 1 to itself an infinite number of times.
OK, then, what kinds of sums converge on a finite value?
How can we test for whether one of these summations (they’re called infinite series) actually converges on a definite value?
These are the sorts of questions that get a mathematician reaching for their pencils in paroxysms of delight.
One weird result from these kinds of investigations is that there exist certain kinds of infinite summations that can give you any result - so you really can (sort of) have the result 2 + 2 = 5 with these things (except the infinite part here is critical - this won’t happen with any finite sum - and it only happens with certain kinds of infinite series).
Although we’re talking about summing things - look at how much more involved we’re getting now. These delightful “involvements” are what gives a mathematician a reason for living.
We’re a long way from 2 + 2 = 5 territory now Dorothy.
Most mathematicians don’t give a flying homeomorphism about the utility of all of this. Let those dull clods, the physicists and the engineers, worry about whether this stuff is useful or not.
That’s a nice bridge, I suppose, but my theorem is magnificent.
How do we prove the result we had from our general intuition?
We note that the series above, before we let things go infinite, can be written more generally (in our example above we have a = 1 and r = 1/2) as
Multiplying this by r we get
If we look at S(n) - rS(n) you can see that all the “middle” terms are going to cancel out, so that we end up with
A bit of algebra gives us the result
Now that’s a nice result in itself (doesn’t work when r = 1, though) but we want to take things to infinity (but not beyond). The r in our example is just 1/2 and as n gets bigger and bigger this r to the power of (n+1) term gets smaller and smaller. By the time we’ve got 1,000 terms in our series (n = 999) this bit is tiny and so when we “take the limit” and hit our heads on the brick wall of mathematics we obtain the result for positive values of r less than 1 that
Even though this is a specific result for a specific kind of series (the geometric series) it’s kind of neat. We’ve been able to prove that we can sum up an infinite number of terms to get a definite, fixed, value (provided r is less than 1). In the example above we had a = 1 and r = 1/2 and we can see that the value of this infinite sum is 2 - which is in line with the intuition we had from drawing squares and rectangles.
Even this example doesn’t really properly capture what maths is “about” - although it gets us a hell of a lot closer than “2 + 2 = 5”.
There’s no ethics here, none of the “responsibility to others” (including non-human persons like rocks) that Gutiérrez gibbers on about. There are no “alternative” answers to the question of what this infinite series sums to. There’s no “indigenous” result that would differ (and be correct).
Going from mathematics to mathematx is a case of going from the sublime to the ridiculous (via brassicas, it seems).
I’m not a mathematician and so I apologize to any resident mathematician if I’ve misrepresented you here. I have the profoundest respect for what you do, even if I do think you’re a bit weird.
Cue a chorus of Anything You Can Do, I Can Do Better
Thinking about infinity is said to have driven Georg Cantor off his trolley, or since we’re in maths and vegetable territory here, out of his gourd.
"lim" is for lime, and it performs itself in both utilitarian (fertilizer) and non-utilitarian (cocktail) ways that may get us to pay attention to its form and to continue to cultivate it. Am I doing it right?
As a mathematician, I can very much relate to the joy of doing mathematics just for the sake of itself. Doing mathematics is like performing music. Doing mathematx seems to be more like never getting to the performance because black notes bound to white sheets are white supremacy exploiting black bodies for fun, or something.
No! Please tell me Rochelle Gutiérrez is just another (even wittier and more clever) Peter Boghossian. Then again, as a white American, i don’t even deserve to opine on such things. I am like the invasive bamboo that creeps across the landscape; neither of us chose our place of birth, but we both take our insidious hold and strangle out indigenous ways of knowing and doing in the fertile soils of the imagination slash reality. -Diana (she/hurling things at the computer monitor in disbelief)