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Instructional concepts for a saner approach to the global precarious.
Has it been fully fifty years and more of a constructivists' paradigm, in the driving seat of mathematics, since a revolution around 1969? Without answering this question exactly, herein we present an imagined conversation between Stanley, a character from real life, a New Zealand primary school teacher about to undergo her duty-bound retraining in 1969, and Livostein from the department that was overseeing the switch at the time.
Inspired by CIV-2019-485-707, a Wellington High Court proceedings, R Z Christensen v Attorney-General. (It may postulated in explaining the trend of this proceeding that a generation or two of lawyers have not been schooled properly in the mathematical arts. They are too willing to consign mankind to a devotee status when it comes down to the counting of joules.)
Stanley : Numbers and quantities are ontologically prior. I didn't get sets until I was fourteen. Now you want us to start the new entrants on sets. Five years old and they're going to be one-to-one corresponding! No! What a headache!
Livostein : Numbers and quantities come in sets or classes or whatever. We've got to have these containers of things, surely, on the canvas, in the mind.
Stanley : Oh, yes. Well we've only been classifying just about everything under the sun already. Even with the new entrants, for heaven's sake. Only it doesn't work like your sets. When we except every member out, we get zero, we don't get the null set!
Livostein : Well, I'm telling you. From now on, it's the null set.
Stanley : O.K. Well I understand you are going to be paying the wages.
Livostein : Look! We're peopling the tide, OK?
Stanley : I would say: Alas you are peopling the tide!
Livostein : Aw, so what's so precious? Surely the government should inject a little zing factor, when so very many literati are enthused!
Stanley : Alas, zing factor could be a horse wrongly whipped. The advance of Man is a horse, you know, it can buck. It can throw Man off. Souls can be lost en masse from its mule-like qualities. For primary school teachers like me, the students see mathematical relations but you want them to see sets instead ... sets, correspondences and functions, etcetera. Well, what you want them to see is missing a few boughs; it is a grim trim. How about the fundamental modal dichotomy? Where has it gone?
Livostein : What dichotomy is that? Did we miss something?
Stanley : Miss! I would say there has been some stealing going on! Eventually in seeing better into relations, one must be able to say something like "K is single-branched in x" with x as variable and K as relation. There are just so many things one might want to say about a relation. Therefore one contends that a bare equation does not state a fact; only modally elevated propositions like the descriptions of relations are facts. But then for primary school teachers, like me, the statement of implication with the two-lined arrow symbol - that makes a modally elevated statement as well and there is much to say about that. The dichotomy for us is one between the equation in the sense of an antecedent or consequent and the statement of implication as an expression of fact. Before that, it is a dichotomy between nomenclature and algebra and a dichotomy between cases and facts. With your new curriculum, I see that algebra has stolen the subject matter of nomenclature. It could remind me of two children playing. And now the government is failing to see that one child has stolen the other's toy.
Livostein : So! We demand that you follow us into the turpitude! Ha! But think of the Peano axioms! Surely they are expressions of fact, even if they are only a variety of presumption.
Stanley : Well, yes and no. Relations are just so ubiquitous. And so intimate with the study of systems! And when it comes to the advance of Man and systems, well the two seem to be inescapably entwined. So with the Peano axioms, we acknowledge a hidden antecedent. It is acknowledged in one way with the first four axioms and in another way with the fifth. We can see that it does not need to be fully expressed. It is a repository for whatever assumptions one will find when asking the like of what the first four axioms may depend upon.
Livostein : I think we must be optimistic. This matter of an old modal dichotomy cannot be leading us to worry, not now. Leaders have spoken. We are with the tide and we will find our nourishment in the set theory.
Stanley : You have heard it called 'naive set theory', have you not?
Livostein : The term is not unknown to me. I have thought it to be a subjective expression. You must know: the vision is very strong.
Stanley : Whose strong vision are you saying? Their vision that we shall be led? Our enterprise to be subjugated? And further that they shall enjoy leading us by a crooked allocation?
Livostein : The subject matter you mean is not necessarily the property of your old nomenclature studies. We've put in the zing factor and given the new set theory a go. Now your duty is to get in the mood and adopt an optimistic outlook.
Stanley : Ha! We must find our sweet spots by retraining our tastes and predilection!
Livostein : Aye, well there's a new generation on the way. You can look forward to your retirement and then, you know, in short order it will be for the younger ones. They will man up, don't worry!
Stanley : You will people the tide with new teachers but the tide can take souls, waste souls and you have not looked closely at this tide. You have merely crowned it from a noticing of decoration. They are decorated masters who will vouchsafe for this new enterprise.
Livostein : Look, it makes sense if you want to put in some zing, to give the job of zinging to your masters. Anyway, what is the worst that can happen? I think you may be over-rating the danger in allowing the passion to press its hands on the wax.
Stanley : The relation between antecedent and consequent cannot be decided without knowledge of nomenclature. It has long been a priority for us primary school teachers to draw a distinction between exception and subtraction, so students can understand two fundamentally different ways of taking away. In the proposition "A implies B", there is an assertion about excepting out of the places where A is the case. What happens if we draw exceptions for all the places where B is the case? This is in the domain of nomenclature and mathematics cannot cut itself off without a self-harm. You maybe do not see this but algebra can weave only half a canvas.
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