Kirby,
Lots more interesting stuff and it is appreciated. I'm snipping where I don't see any philosophical issues, though you can imagine me having read along nodding and occasionally saying, "kewl!"
> Also, here's another segue to Wittgenstein and RFM if we
> need it, as
> he uses rule-generated sequences in many examples
> (including in the
> PI) when investigating what it means to "understand" in
> terms of
> continuing to follow some rule. "Now I can go on"
> means I know what
> to write after 162.
Are there also series and sequences corresponding to these constructions and provable in terms of other maths? I would assume there would be...
> Coxeter told Fuller how impressed he was with his formula
> -- on the
> cubic close-packing of balls. And he later took pleasure in
> proving
> it, noting in his diary one day in September 1970: "I saw
> how to prove
> Bucky Fuller's formula," and publishing it in a paper,
> "Polyhedral
> Numbers." Of course more than anything, Coxeter fell in
> love with
> Fuller's geodesic domes.
> """
yes, I turned this up by Googling as I read along:
http://mathworld.wolfram.com/TetrahedralNumber.html I note the Coxeter reference.
> Reassuring has the flavor of showcasing close similarities
> (family
> resemblances) even while making some changes, setting up a
> kind of
> duckrabbit tension between the two approaches (90 degree
> and 60
> degree).
I certainly see how family resemblances between different sorts of structures and different operations, but I am not getting the duck-rabbit connection (though I of course recognize the allusion.)
> In a given lesson plan for teachers, we might invoke
> "n-triangled" and
> "n-tetrahedroned" as swap-in terminology for n-squared and
> n-cubed.
> This is to limber them up regarding our visualizations,
> getting them
> to the point of realizing that (n x n) and (n x n x n) have
> non-
> rectilinear interpretations.
What I am seeing is that these operations have non-rectilinear APPLICATIONS. Is it important for your purposes to also describe these as "interpretations" of those operations?
This may seem like a niggling objection. After all, couldn't we say that every new application of a concept expands the meaning of the concept? We might say such a thing and I won't say that's "wrong".
But I could easily imagine someone being quite at ease with, "Look! Here's a way that we can apply these same operations in a way that is analogous in some respects to this other way of applying them..." but being resistant and feeling that the foundations of mathematics are being challenged when they hear, "See! We don't have to think of 'n x n x n' as meaning this. It can also mean this."
>
> Once they've had the necessary insight, then we maybe
> de-emphasize the
> nomenclature. We go back to saying "n-squared" like
> everyone else,
> but now there's an embedded reminder to think of a
> triangle-based
> model instead.
The point that 'n-squared' may conjure up a clashing picture (where 'n x n' may not) is well taken, yes.
>
> Teacher's who've been through this training or workshop
> have a new
> appreciation for these nuances. We expect them to
> return to their
> rectilinear ways, having sampled our "gypsy math" (back to
> the ethnic
> minority motif)
Just be careful they don't feel "gypped". (I don't know if my having a Gitano partner makes such a joke acceptable, but the joke does offer a reminder of the risks of being accused of "sleight of hand" if differences aren't acknowledged up front. Interesting how metaphors can outrun their intention.)
but with new insights, new gestalts -- an
> important
> aspect of "meaning" per the PI (Part 2 especially).
And especially RPP1&2 and LWPP1&2, where these topics get a much deeper treatment...
I'm still not quite sure about the connection here, not sure that there is a phenomenon here like a change of aspect. An aspect of what? The arithmetical operations? Any particular volume that might be "seen as" made up of tetrahedrons or cubes? Is that a case of "seeing as" or of making other pictures? And perhaps seeing that other pictures may be more effective in dealing with certain problems.
>
> This is not a new pedagogical / andragogical
> technique:
I had to do some searching here
http://teachingadultsonline.pbworks.com/Adult+Learning+Theory
to use some
> interim or "training wheels" vocabulary or notation, and
> then to let
> go of it, having had the requisite insights.
No, of course not. My point about "triangling" was just to warn that as a set of "training wheels", it can create its own difficulties. Just an observation of something you may want to watch for.
With educated adults, I would suspect it not to be an issue.
> Here's another way of looking at it. We do the
> multiplication
> independently, as a calculation per standard algorithm
> (developed from
> the abacus some centuries ago), with no visualization.
>
> Then we take our completed calculation and map it to a
> square and
> triangle respectively, not to figure out an answer, but to
> display an
> already-obtained result.
Yes. The application can go on both directions. But be careful in using "display the result". Countless things can illustrate an arithmetic operation, some more perspicuously than others, which is just to say that arithmetic operations have countless applications. But, "This is what 'n x n' means here," can get us into needless confusion and disagreement if it is then asked, "Does 'n x n' mean the same here or different?"
We don't ask whether 'n x n' means something different or means the same in different concrete applications, e.g. a dozen cartons of eggs, a pair of married couples, or a chessboard. But when we are applying one part of a calculus to another part or one calculus to another. After all, we don't see the application as merely contingent!
But then, I am using "calculi" to include systems of calculating with pictures. As I would imagine you would as well. But this is far from typical, even among math educators. (Recall your Midwesterner who seemed to be suggesting that the relationship between the sides and the overall number of triangles was contingent but that the analogous relationship for the squares was necessary!)
No one would suggest that by multiplying to get the total number of M&Ms had changed the meaning of "multiplication" from the previous use of Reese's pieces, nor feel that there was a threat to the consistency of mathematics in doing this.
> We want to motivate a more 60-degree based approach to
> spatial
> geometry more generally. A subculture imbued with
> this alternative
> aesthetic might build houses looking more like this one:
Is the issue of what "n x n" means relevant to this goal? That "n x n" is an arithmetical operation that is applicable here as well would seem to be sufficient.
> > If we say that one is one of many applications of the
> operation n x n. while the other is a picture of what we
> mean by "n x n", no one could object!
> >
> > If we say that we can use the triangle rather than the
> square as a unit of area, this may arouse suspicion. A
> reminder that we can readily convert between the units to
> whatever precision needed (they are only "incommensurable"
> in a philosophically innocent sense) and check the
> calculations made using triangles against calculations using
> squares ought to alleviate that.
> >
>
> And we can use a tetrahedron instead of a cube as a unit of
> volume.
yes
The "turtle" examples were helpful, though the move from multitudes to multitudes misses the point I was hoping to make and even in terms of magnitudes, the areas made by the turtle's "curtain", while demonstrably equal are not congruent, so "the same again" as part of what we mean by multiplication is lost here. Snipping ahead though, you seem to get that...
> > What corresponds to grouping, to correlation, or to
> "the same again" with the triangles?
> >
>
> You're concerned with reading off the right answer from the
> grid,
> thinking of an orchard of X trees in some triangular
> arrangement, and
> wondering how you'd easily compute how many. The rank
> and file
> system, the array, the phalanx, stacks the same number side
> by side,
> repeatedly, is easy to read.
You seem here to be getting the distinction I wish to make but not yet its import.
>
> I'm happy with a "when in Rome" approach, i.e. lets look at
> the
> complementary strengths of both models.
My concern isn't with the relative strengths of the approaches. I had no doubt (though your remarks were interesting and informative) that there are a great many advantages to the triangle/tetrahedro
n approach. The approach is fine.
Actually, it's really frakkin' cool! My appreciation of this stuff may get lost in my examination of some of the philosophical issues.) But I also may not have been sufficiently clear about the reasons I have for drawing such distinctions. Perhaps other remarks here make that clearer, but at the risk of hammering excessively, I'll elaborate tying it to this case.
When it comes to using triangles/tetrahedr
a or squares/cubes as a basis for measuring area/volume, provided there are straightforward conversion rules between these systems (One reason that the entrenched system would be preferred is the ability to bring various data together) and one is consistent with the other, I can see no reason not to choose whichever is most efficient for a given application. (Of course, conventions have a value in themselves, but the transition to adopting an alternate system is a social and educational problem, not a philosophical one.)
And if the claim is only that it is entirely consistent to apply the arithmetic operation "n x n" to triangular units of area or "n x n x n" to tetrahedral units of volume, that's something that can be demonstrated straightforwardly by existing mathematical means.
It is the claim that the _expression_ "means the same" - or even that it "means something different" or "means something similar" - that strikes me as potentially misleading and possibly confused.
If you are saying something about what the _expression_ means in a given application, presumably you don't mean anything like contrasting counting M&Ms with Reese's pieces.
And even if you were contrasting square arrays of M&Ms with squaring the distance in calculating gravitational force, saying that n-squared "means" the same or that it "means" different both seem queer.
The operation is applicable to both cases. And it is the same operation - the same role in the calculus - in either case.
Any application of the operation might be used as a demonstration of that operation. We might them say that with a different method of demonstration, we teach something different and the meaning changes. But if we teach someone to perform the operation using one example and they are not able to apply it to quite a few other cases, then they haven't learned the concept. They do not know how to "go on".
Still suppose we take the use of diagrams of squares and cubes as paradigmatic demonstrations. Would that mean that someone who knew how to perform various calculations and apply them in various cases but had never applied them to squares and cubes didn't really know how to calculate? Why would we say that? Perhaps because an important part of the calculus is in its relationships to certain kinds of measurement.
If members of a tribe knew how to count heads, hands, fingers, toes, rocks, beans, trees, and so forth, could work out that a pile of 7 beans and a pile of 5 beans together made the same number as 3 piles of 4 beans, and so forth, but did not apply these methods to distance, time, area, volume, and the like, we could certainly say that theirs was a more primitive game, but we could still have every reason to translate one of their words as "plus", another as "times", and so on.
We might nevertheless treat a certain picture as a paradigm for what we mean.
(Compare, swatches of color preserved as standards for our use of color words: someone who had never traveled to see those swatches might still learn to use our color language with perfect eloquence, albeit having learned with different examples.)
The use of pictures of squares and cubes as paradigms of multiplication, squaring, and cubing might be a way to give sense to the idea that we "mean the same" (or "different") in applying "n x n" "n-squared" and so on to one case or another. That seems to me to be the use of the picture your Midwesterner had in mind.
And in some ways, it has sounded as if you saw the triangles and tetrahedrons as a competing picture that might serve the same paradigmatic role. Talk of different tribes and such reinforced that impression for me.
If you wish only to defend the legitimacy of applying arithmetic in a way that seems odd because it is analogous in some respects but different in others and to defend the merits of an alternative system of measurement, then I don't see any value in a war over the whether a particular picture can serve just as well as a paradigm. But the talk of "meaning" and "interpretation" rather than "technique" and "application" seems to suggest that.
But if you want to treat this as a matter of pictures seeking equal legitimacy as paradigms of calculation, then the problems I raised apply. Someone who sees the pictures of squares and cubes as paradigms of multiplying, squaring, and cubing will not find the triangles and tetrahedra to be equally legitimate for THAT purpose (though they may be quite useful for other purposes).
It isn't about the ease, per se, of using one picture or another; it is about the perspicuity with which the picture represents different aspects of the calculation for which it is to serve as a paradigm. My talk about "grouping", "correlation"
, and so forth was easily mistaken for a remark on the ease with which a picture can be used (and the answer, "but this picture is easier to use for other things," is then a reasonable one).
The orthogonal arrangement is easier to work with for some purposes (not for all) but the point of treating it as a paradigm is that it is not just a picture of the concept of multiplication, but clearly shows the related concepts of grouping, of correlation, and repetition, the relationship between multiplication and addition, and so on. The value of clarity here is not simply a matter of ease of use in a particular application, but ease of understanding as an overview of the concepts.
your
> Midwestern interlocutor
> >
> > He is right to object to "it means the same here",
> though he offers no clear grounds that I can see.
> >
>
> Yes, I agree with "not the same" but then fall back on
> "family
> resemblance" and even "close family resemblance"
.
>
And as I've probably hammered enough by now, I'm now wondering whether making it a question of "n x n" MEANING the same, similar, or different isn't just counter-productive.
But if it is important to speak of an "interpretation" of "n x n" rather than simply of applications, then yes, family resemblances and specifically emphasizing differences as well as similarities (so as to avoid the suggestion of fallacious arguments) would be well-advised.
>
> Here's an opportunity to show where philosophy is making a
> positive
> contribution in some practical sense, helping our math
> teachers co-
> develop a more up-to-date and relevant spatial geometry
> curriculum.
>
> Philosophy of mathematics to the rescue!
(You may want to catch the postscript to my reply to Sean regarding "cognitive science" if your interested in some of my views related to this.)
>
> Thanks again for your assistance,
I'm pleased to be able to help.
hope you will continue
> our dialog.
> Maybe others will join in.
I'm somewhat surprised they haven't. But inevitably, tastes and interests vary and with a philosopher whose work was as varied as Wittgenstein'
s, a common interest in his work may not indicate a common interest in much else.
(I've noticed professed Wittgensteinians also to be more varied in their religious beliefs, political positions, ethical attitudes, and aesthetic sensibilities than those who are influenced by other philosophers. I wonder if that reflects a wider appeal or just a greater flexibility in Wittgensteinian thought. Certainly, Ayn Rand is more popular in many circles, but Randians seem to be in agreement about quite a lot. Likewise, Popperians, Perhaps also the widespread disagreements in how Wittgenstein is to be understood plays a role here. But I digress...)
I've been working on a couple of posts on two still different topics. One concerns the grammar of pictorial images and the other an examination of some research on sexual arousal (I know: cue inane Beavis and Butt-head laughter. Right?) I'm not sure how much attention they'll receive, but thoughtful feedback is certainly useful.
JPDeMouy
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