Wiggins' question #12
There's an interesting read: Conceptual Understanding in Mathematics by Grant Wiggins. In that text the author proposes "a test for conceptual understanding" which should be given "to 10th, 11th, and 12th graders who have passed all traditional math courses through algebra and geometry."
I would have thought that as a person with higher math education, I should have enough of "conceptual understanding" of high school math$^\dagger$. Yet, the following question, number 12 in the author's list, stumped me.
12) "In geometry, we begin with undefined terms." Here's what's odd, though: every Geometry textbook always draw points, lines, and planes in exactly the same familiar and obvious way – as if we CAN define them, at least visually. So: define "undefined term" and explain why it doesn't mean that points and lines have to be drawn the way we draw them; nor does it mean, on the other hand, that math chaos will ensue if there are no definitions or familiar images for the basic elements.
What is this about? What is undefined and what we "CAN" define visually? In what way the answerer is expected to define undefined terms? How to interpret the last sentence about chaos in mathematics? I don't even...
I would greatly appreciate your help in deciphering author's intentions.
$\dagger$ Note that my skills aren't rusty, I'm an active researcher and as time of writing in top 20 geometry answerers at math.stackexchange; it's not much, but ought to be enough to answer high school geometry question, right?
secondary-education concept-motivation geometry
dtldarekdtldarek
Here's Hilbert, Foundations of Geometry:
Let us consider three distinct systems of things. The things composing the first system, we will call points...; those of the second, we will call straight lines...; and those of the third system, we will call planes....We think of these points, straight lines, and planes as having certain mutual relations....The complete and exact description of these relations follows as a consequence of the axioms of geometry.
The things called points, lines, and planes are just things, which are assumed to exist. No further definition is given. However from the assumptions made about the existence of relations and the properties these have as well as a few other assumptions, Hilbert constructs geometry. Euclid on the other hand gives definitions:
A point is that which has not even one part.
A line is breadthless length.
Euclid defines point in terms of part and line in terms of dimensions. But part, breadthless, and length themselves are not defined. If one were to define those terms, they would be in terms of other undefined terms (unless one were to make a circular definition). The modern way is to start with things -- that is to say, a set (or class) of objects. The elements of the set are considered undefined.
Measurement as a basis for geometry is introduced [G.D.Birkhoff, A set of postulates for plane geometry (based on scale and protractor), Ann. of Math. 33 (1932), 329–345]. Birkhoff further influenced the teaching of geometry by producing a textbook Basic Geometry with R. Beatly. This approach was later adopted for high school in the US (see, for instance, Moise and Downs, Geometry, 1964). Birkhoff uses, without definition, the term undefined:
Undefined elements and relations. The undefined elements are (a) points,...and (b) certain classes of points called (straight) lines. The undefined relations are (c) distance between any two points,... (d) angle formed by three ordered points....
To respond to a question by the OP in a comment to another question, the points and lines are assumed to be given (or supplied) but there is no definition of them. What they are arises out of the axioms that are assumed.
The example given in the comment raises the question whether the $x$ is undefined in $f(x) = x^2$, which seems a different issue if I've understood the point correctly. Ignoring the modern definition of $f$ as a set, one could say that $f(x) = x^2$ is short for "let $f$ be the function defined by the property that for all $x \in {\bf R}$, $f(x) = x^2$. The variable $x$ represents an element from the set ${\bf R}$. As a matter of logic, a variable is indefinite, and I think that is different that being defined or undefined. A closer analogy would be to say that in the definition of $f$, the set ${\bf R}$ is undefined. That is an approach some analysis texts: a set ${\bf R}$ is assumed to exist and further assumed to have certain properties.
P.S. Some quibbles about Wiggins question:
What does Wiggins mean by define, as in "define 'undefined term'"? Whatever is meant by define, undefined means it was not defined. This is not really a serious objection, but the wording, at least in the excerpt, might be confusing.
What does Wiggins mean by draw? This seems more serious (see also below). I know very well what it means in art class. Mathematically, I am uncertain what can be meant by "the way we draw them." There are several problems with the way I draw them. I've not drawn a mathematical line ever. Two points do not determine a straight line, because there is always some error; if I draw a line twice through the same two points, I get two different lines. Likewise, extending a straight errs, when I do it. But I imagine two points determine a line and I can imagine my figures are perfect. I dismiss my error and do not think my lack of perfection disproves the axioms. Maybe Wiggins is after that sort of explanation, and that mathematically lines need not be drawn that way (in which, say, lines have a thickness or are not "straight," whatever that means). I was taught in H.S. that we cannot draw a geometric point or line, which is only a slightly different explanation. It also sounds like Wiggins might be opening the door to models of non-Euclidean geometry. The might be that we do not have to imagine that straight lines follow our intuition of straightness, but that we should deduce properties only from the axioms.
P.P.S. For Proclus (Comm. Eucl., 400s CE), "leading geometry out of Calypso's arms, so to speak, to more perfect intellectual insight and emancipating it from the pictures projected in imagination" would be the "perfect culmination" of studying Euclid. This shows that the concern that images and what we imagine about them might lead us astray in our reasoning goes back to ancient times. Proclus' take is somewhat opposed to the perspective in Wiggins' question. Wiggins seems to think his readers will find comfort in familiar images and definitions for every thing. His readers might fear not having them. The term "chaos" might be a pun as it tends to imply formlessness, which would be an ungeometrical state. I think the question is asking for an explanation of why mathematics is not invalid just because of undefined terms and "straight" do not have to be what we imagine as straight.
P.P.P.S. As a side note, in both Euclid and Hilbert, but not Birkhoff, a line is not a set of points, which unnerves most college students. Some find it liberating, though.
$\begingroup$ Thank you for your answer, please see my comment to Benjamin Dickman's post. $\endgroup$ – dtldarek Jun 1 '14 at 10:53
$\begingroup$ Alright, I think I got it, the problem was I've never treated primitive notions in terms of defined/undefined, as defining a primitive notion didn't make sense to me. Your example with function and $\mathbb{R}$ helped a lot. I would say that premises and theorems are "undefined", as I see all the theorems as (example for set theory) "if you give me set theory, then I will give you a theorem that works in it", or (for geometry) "if you tell me what points, lines, etc. are, then I will tell you a theorem these satisfy". These are not undefined, the definitions are there, only deferred. $\endgroup$ – dtldarek Jun 1 '14 at 12:55
$\begingroup$ @dtldarek No, in this abstract geometry, points, lines and planes are not defined. They can be anything: cars, people, abstract concepts; as long as they satisfy the axioms. Only "concrete" math and -- using that: physics -- give a proper definition: they are affine subspaces of $\mathbf{R}^3$. $\endgroup$ – Toscho Jun 1 '14 at 16:54
$\begingroup$ I think the last four sections of this answer add more confusion than clarity. Also, @dtldarek your remarks after "I think I got it" are not very precise. I don't know of a theory in which theorem is undefined or a primitive notion; this is not the case in Real Analysis (regardless of which of the four or so main constructions one uses to build up $\mathbb{R}$). $\endgroup$ – Benjamin Dickman Jun 2 '14 at 6:29
$\begingroup$ @BenjaminDickman I do agree with you on all accounts (although it was this answer that helped me more). The issue I had was that I would not view axioms as true in general, only as properties of some structure we are talking about. For example, I have no idea if the axiom of choice is true, but if you give me set theory in which it (and several other things) is true, then I will prove Banach-Tarski paradox. In such case the empty set is not undefined, because (when giving the structure) you need to tell me what it is. Cont. $\endgroup$ – dtldarek Jun 2 '14 at 7:00
I think you might find some value in Raymond Wilder's classic textbook Introduction to the Foundation of Mathematics. A free copy is archived here. Here is a short excerpt from p. 10 of Wilder's text:
Now, you could broach this using the traditional point and line (or blob and streak) model. But here is an alternative way that uses something like SET cards (limited to a single shape and filling):
Here the model is not the standard one we think of for plane geometry, at least in its presentation above, yet still satisfies the mathematical axioms without anything chaotic ensuing. Note further that I have "defined the undefined terms," namely, I have given definitions for both "point" (which consists of some number - one, two, or three - of colored dots) and "line" (this is inspired by the game of SET).
I think the author just wants you to think about how the axioms of plane geometry are generally interpreted by, e.g., secondary school textbooks in the same way, despite the undefined terms having nothing that forces a point to be thought of as a single dot, and so forth. If you are to construct a different model that satisfies these axioms, even if it looks different, it still ought not to lead to any chaos in the sense of contradictions. This is because the axioms are consistent insofar as they are satisfiable.
As a side-note: By satisfying the third Axiom above, the model presented here gives an indication as to how one answers the "fun question" I posed on ncr's MESE answer of SET as a game for learning mathematical concepts. In particular, each pair of SET cards uniquely determines the third card that forms a set. The full game has four characteristics (shape, color, filling, number) and three choices for each, hence the full deck has $3^4 = 81$ total cards. The total number of sets, then, can be found as follows: You have $81$ choices for the first card, $80$ choices for the second, and $1$ for the third; but this over-counts by $3!$. So the total is $81\cdot80/6 = 1,080$ possible sets.
Another note: See this paper (Davis and Maclagan) for other SET-math connections.
Benjamin DickmanBenjamin Dickman
$\begingroup$ Ok, it seems I'm beginning to understand what the question is about, i.e. I was able to narrow down what was confusing me. When dealing with group theory, a group is given as a structure $\langle A, \bullet, \bullet^{-1}, e\rangle$ (or sometimes just $\langle A,\bullet\rangle$) that satisfies certain properties. The carrier set $A$ may seem undefined, but it is not, it is a part of an input (e.g. premises). Isn't it the same with geometry, that is, what lines are is supplied with the geometry? It would be like saying that $x$ in the definition of real-valued function $f(x) = x^2$ is undefined. $\endgroup$ – dtldarek Jun 1 '14 at 10:51
$\begingroup$ @dtldarek In group theory, the set $A$ is probably conceived of within, say, naive set theory, where you have certain primitive notions; in particular, the empty set, $\emptyset$, is an example of a primitive notion. (Cf. en.wikipedia.org/wiki/Primitive_notion) $\endgroup$ – Benjamin Dickman Jun 1 '14 at 11:18
$\begingroup$ In the "geometric axioms" supplied in the excerpt from Wilder, the undefined terms (i.e., primitive notions) are point and line. In colloquial language, we have an idea as to what constitutes a point and a line, and geometry textbooks generally draw them in similar ways. For the sake of the axiomatization, though, it is only important that point and line satisfy certain axioms. $\endgroup$ – Benjamin Dickman Jun 1 '14 at 11:18
$\begingroup$ It started to make sense. Thank you for your answer and comments. $\endgroup$ – dtldarek Jun 1 '14 at 12:57
This question is simply testing if the students
sees a difference between mathematical objects and their relations (drawing undefined objects)
can work with the relations regardless of the mathematical objects (question of undefined)
realizes, that Euclidean Geometry is not the only theory satisfying geometric axioms (points and lines don't have to be drawn that way)
knows, that working only with relations will only yield relational results. Concrete results will only be achieved if working with mathematical objects or feeding these to the achieved relational results. (math chaos will ensue without concrete objects)
ToschoToscho
Coxeter, in his book titled Projective Geometry, describes a dictionary game called Visch (short for "viscious circle"):
Point=that which has position but not magnitude
Position =place occupied by a thing
Place= part of space...
Space=continuous extension..
Extension=extent
Extent=space over which a thing extends
Space=continuous extension
The word "space" is repeated so we have Vish in Seven.
Coxeter then remarks that the only way to avoid vicious circles is to regard certain primitive concepts being fundamental and left undefined.
When one understands geometry as an axiomatic system with undefined terms, one is ready to think about models of the axiomatic system. A model is an concrete interpretation of the undefined terms where the relationships postulated by the axioms hold true in the concrete setting.
(Tables, chairs, and beer mugs: https://math.stackexchange.com/questions/56603/provenance-of-hilbert-quote-on-table-chair-beer-mug).
Then one can think about properties of the geometry such as consistency, completeness, and categoricalness. In the van Hiele model of mathematics education, this would be operating at the highest level, where "students understand that definitions are arbitrary and need not actually refer to any concrete realization."
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