On Sat, Mar 6, 2010 at 9:45 AM, Joe Niederberger <email@example.com> wrote: > > Kirby, how do you see types in relation > to this Wittgensteinian concept? >
What Wittgenstein does in philosophy is wade in to ordinary language looking for prime / telling examples of actual usage. He calls this "investigation" (which it is) and one purpose is to disentangle confusions and/or escape from unproductive debates premised on an unclear understanding of the meanings involved.
You and I could undertake such an investigation with respect to "types" (we wouldn't need Wittgenstein's help necessarily). We're both aware of a lot of computer science swirling around, but we should, out of respect for our subject area (early math learning, lets say pre-college), focus on how "types" are embedded in ordinary language or "layman's language" if you like.
Along those lines, we're all familiar with "types of object" where we categorize by type. Filtering criteria come into play. Then we have "edge cases" where we might be uncertain about which type something is. The whole idea of a taxonomy or tree enters the picture, taking us into biology especially.
We could talk about body types, personality types, types of teacher, and immediately get plenty of student attention, dontcha think? Even just allusions along these lines would go a long way, because of all the resonance with ordinary teen talk (which we consider an asset, not a liability).
As soon as we spell this out, it becomes clear where computer science is getting its metaphors in the first place. "Types of animal" and "types of math object" are not far-apart concepts.
We could talk about a "math object zoo" (similiar to "particle zoo" in high energy physics). Types of math object include: rational number, vector, polyhedron, polynomial, quaternion, integer etc.
> I have always disliked strong typing in programming > languages. Also the notion of static class hierarchies in > OO. >
What's attractive about the object oriented approach in my view, is this bold attempt to map to ordinary language, to take grammatical patterns already deeply ingrained in our everyday thinking patterns, and bring those into a somewhat arcane and difficult domain, where we attempt to solve problems using machine-executable logic.
When explaining about objects, how they inherit characteristics from parent types, we can afford to be very biological in our presentation. This gives us the valuable distinction between a type definition or blueprint (a template for an object) and the objects themselves.
For example, we might write a Vector class with a few lines of code, and explain how this is the generic blueprint for all Vectors. However, we then have *instances* of the Vector type, which are the actual vectors used in whatever computations.
I think this "class template versus instances" distinction is relevant to students of mathematics as it's all about generalizing to the abstractions, looking beyond the instances to the abstractions. What is a generic tiger? What makes some animal a tiger (not just stripes, zebras have those)? What makes some animal a mammal? You have thousands of instances to categorize, but you're looking for commonalities.
Given our thread about multiplying, let's walk the talk and be honest about what "be fruitful and multiply" means. Clearly, when animals multiply, it's not quite like the algorithms for addition or multiplication. You often start with two and so its a binary operation, although when individual cells split, that's a unary operation.
We would and could do a lot more with these analogies in math class if we weren't ruled by puritans who eschew all allusions to animals mating, no matter the cost (we lose them to daydreaming because the approved dreams are all stale and bereft of much substance -- even history has been purged, rendering math largely antiseptic and joyless, by design).
Wittgenstein's concept of "family resemblance" reminds us to not fall into the trap or pitfall of always thinking there's some essential distillation to just one feature or core definition. What do all "games" have in common, such that we agree to call them games? The answer might be: no one specific feature, no one commonality, just a lot of partially overlapping features and characteristics. That may sound somewhat "loose" (by design) but it is not anti-mathematical thereby.
> On the other and, I am a fan of teaching ADTs as a > topic in their own right, apart from OO. >
The above article in Wikipedia is useful in that it topic-wise points back to a debate I've been having on math-thinking-l, a list frequented by functional programmers.
This camp dislikes "imperative programming" philosophy, and when someone like me holds out the prospect of funding high school classes based on books like Math for the Digital Age and Programming in Python (Skylit publishing), they tend to go ballistic (check the February archive).
My response to this altercation is akin to that of the intelligent design community (I see those smirks in the bleachers) where I say "teach the controversy". In other words, it'd represent a huge step forward if debate-minded high schoolers felt included in some of these more abstruse "math war" topics and, truth be told, this is how many an adult gets clued in as well, by snooping on what we're telling to teenagers (at least, a reading level they're able to comprehend as well).
I think one reason our economy is stagnating, with future visioning circling apocalyptic scenarios, is our collective ability to debate technical topics is so poor. Journalists, bloggers, columnists, stick to the usual fluff about whether kids should memorize the times tables, or use a calculator, should use direct instruction, or constructivism, because they've learned the code words and want to fight culture wars through them.
Questions about what to do with computer technology in math class, how to bring "smart houses" on-line (dwelling machines that promote awareness of energy usage) don't get much bandwidth. The level of technological sophistication we'd need, in order to accommodate our billions at a decent living standard, is not attainable so long as we indulge in so much willful dumbing down.
In the olden days, we looked to philosophers, other polymaths, certified generalists, to point the way. But academic philosophy has become mired in arcana to the point of irrelevance. Technology overtakes us, events engulf us, yet our level of discourse remains dangerously side-tracked. What about the religious authorities? Do we still have any of those?
My 15 year old daughter, quite intelligent, avid debater, is eagerly learning all about jury nullification so she might debate its merits and demerits. But mathy debates don't get any brains cycles, take up no bandwidth. I think part of the reason for that is math is historically presented as a fait accompli at the high school level, is where the adults exercise their authoritarian (dare I say patriarchal) demeanor and brook no "but but but..." attitudes. Math is a fortress, math is secure.
This is why tetrahedral mensuration (as I sometimes call it) gets little airplay either. It's undermining i.e. logically consistent and mathematically sophisticated (in terms of opening doors to sciences and engineering), yet looking back over the decades, even centuries, it's not evident, hasn't been promulgated, including by orthodox religious authorities.
The idea that basic mathematics could contain anything "new" that the "controller adults" (aka high priests) don't already have complete mastery over, upsets the whole psychology of a traditional high school math curriculum. Traditionalists have a big investment in not having math be "debatable".
Of course wherever one encounters authoritarian tendencies, one tends to encounter the opposite. Constructivism, letting students construct their own understanding, tends to be spearheaded by a lot of refugees from traditionalist curricula who harbor hatred and resentment against those who wanted to railroad them, force them, along some "one right way". They feel that's no way to treat defenseless children.
> Doesn't the Wittgenstinian concept support these views? > Obviously, I've only just become introduced to it. >
Wittgenstein encourages laymen, not-philosophers, to feel competent and authorized to undertake their own investigations, including into the foundations of mathematics.
You don't need to be a "doctor of philosophy" with mastery over some particular language game or logic (likely abstruse and arcane, cryptic and intimidating by design), to explore these foundations, because ultimately that's not what "foundations" consist of.
Ordinary language, the every day world, is what's to bounce off.
On the other hand, the ordinary world consists of intricate patterns (like the particle zoo) and designs (like lacework on a mosque) so in keeping it experiential, we don't lose our natural mathematical capabilities at the end of the day, and some of us grow into becoming the next contributors to the mathematical and/or philosophical literature.
> It also reminded me immediately of some AI problems > in knowledge representation adn categorization, and I was > not at all surprised to immediately find some research going on > on exactly that. What was surprising was it was so current. > See: http://www.springerlink.com/content/e78g586716j65806/ > > joe N. >
Yes, current and topical.
> - ----- Original Message ----- > From: "kirby urner" <firstname.lastname@example.org> > > > Yes, "family resemblance" is a relevant meme here. Wikipedia > > has an entire entry on the concept, as defined (through use) > > by the philosopher Ludwig Wittgenstein: > > > > http://en.wikipedia.org/wiki/Family_resemblance