Why are the undefined terms in geometry undefined?

Euclid's introduction of the axiomatic method was formalized over 2 millenia later in work of Hilbert, and it is now the common method of all mathematics. Here is the modern take on how the axiomatic method works.

Roughly speaking, when studying a class of mathematical objects --- Euclidean geometries, vector spaces, abstract groups --- the idea is to try to state the fewest possible assumptions about the behavior of those objects (the axioms) which can then be applied to logically deduce an entire mathematical theory. The format of these assumptions usually goes like this:

• Names for the given objects (also known as "the undefineds")
• Mathematical properties that those objects must satisfy (also known as "the axioms")

So in Euclidean planar geometry we are given the plane, and its points, and its lines, and then we list the properties that these objects must satisfy. The "philosophical" reason that the given objects are undefined is that the mathematical properties of these objects that we wish to study are restricted entirely to the axioms themselves and the theorems that can be proved as a consequence of those axioms. The exact nature of the given objects is unimportant for this process of stating axioms and proving theorems.

Only in the 16th century did Descartes come along and lay down a foundation for defining points and lines using numbers: a point in the plane is an ordered pair of numbers $(x,y)$; a line is the solution set of an equation $Ax+By=C$; and so on. Still, though, this is just kicking the can down the line, because one now begins to wonder how numbers and their arithmetic can be axiomatized, and for that you can take an advanced calculus course.

Of course, some understanding of the nature of the given objects can be helpful to our intuition as we work through the axioms and the proofs. Perhaps Euclid understood this when he wrote his very opening "definition", which translates as: "A point is that of which there is no part". Rather poetic and intuitive, but not really a very good definition from a mathematical standpoint. Your definition of a point is kind of similar, "a location in space", intuitively helpful, but not much of a mathematical definition.

The proposed definitions:

• Point: a location in any space
• Line: a set of all locations that lie in a 1-dimensional space
• Plane: a set of all locations that lie in a 2-dimensional space

Are now using the terms: "location", "space", "set", "lie", "1-dimensional", and "2-dimensional", all of which are themselves undefined terms in the hypothetical presentation. Note that you've now increased the number of undefined terms (versus the original three).

You can't get around the fact that some terms in a work need to start off relying on their natural-language contextual understandings (not formal math definitions). Having it reduced down to just three undefined terms is pretty much minimal.

Euclid does define those terms. For example, he defines a point as "that which has no part", and a line is "breadthless length". These (and your similar definitions) may be useful to give someone an intuitive idea of what Euclid is (or you are) thinking of, but mathematically they are of limited value.

Instead, the approach in mathematics in the last 150 years or so has been to define mathematical objects not by what they "are" but by what they do, by what relationships they have to one another. This approach has been extremely fruitful.

For example, instead of trying to say what a point is, we say what it does: for any point $P$ there is a family of lines through $P$, and for any other point $Q$ exactly one of these lines passes through $Q$. Usually two lines will have one point in common, but never more than one; if they don't we say they are “parallel lines”.

Having made these formal definitions we can prove theorems: if we have any system that has something we can identify as points and lines, and those somethings behave in the ways we required, they must also behave in certain other ways as well. And this holds even if the points and lines don't much resemble the ones we originally had in mind.

If we agree that locations in the plane (which you called "points") and 1-dimensional spaces in the plane (which you called "lines") do behave in the basic ways, then they must behave in certain other ways as well; these are theorems of geometry.

But then the magic happens: The deduction applies to all sorts of other systems of points and lines that may look nothing like the geometric plane. For example, the Fano plane has points and lines, for any point $P$ there is a family of lines through $P$, and for any other point $Q$ exactly one of these lines passes through $Q$. But it has only seven points and seven lines! Nevertheless deductions we made about points and lines, based only on their properties, will tell us something about the Fano plane also, because its points and lines, although very different from geometric points and lines, have the same formal properties.

One example of how this approach bore fruit was the problem of the parallel postulate. Euclid gave five “postulates” and while four of them are very simple, one, the notorious “parallel postulate”, was much more complicated. For centuries mathematicians wondered if it could be disposed of, if perhaps it could be proved from the other properties of geometry. But the answer is no, and the proof is this: we can construct a different system $T$ of points and lines for which all the usual properties of geometry hold, except for the parallel postulate. If we could deduce the parallel postulate from the other properties, then it would have to be true for system $T$, but it isn't, so we can't.

"Point", "line", "plane", etc. in this context are "primitive" notions; thus we don't say what they are but we speak of relationships among them.

Suppose your math teacher tells you that 3 is less than 5.

• Does that mean three apples are fewer than five apples?

• Does it mean three gallons of gasoline is less than five gallons of gasoline?

• Does it mean three dollars is less money than five dollars?

• Does it mean a committee of three persons has fewer members than a committee of five?

• Does it mean three pages of a book are a shorter passage than five pages?

• Does it mean three days is a shorter time than five days?

• Does it mean that three prime numbers are fewer than five prime numbers?

• Does it mean three ideas about how to govern a country are fewer than five ideas?

And so on. As mathematicians, when we say $3<5$ we don't know whether we're talking about apples, or gallons of gasoline, or dollars, or persons, or pages of a book, or days, or prime numbers, or ideas. As Bertrand Russell put it "Mathematics is the subject in which we don't know what we're talking about." Three things are fewer than five things and we as mathematicians don't know what things are; the word thing is undefined. In applications of mathematics, we know what the things are, but the mathematical fact that we apply to "things", the fact that $3<5$, is the same regardless.

In an algebra course you may learn that a "group" is a set of "things" on which a certain kind of binary operation exists, so that there is a "multiplication" table, thus:

$$ \begin{array}{c|cccccccccccccccc} \times & e & p & q & r & y & z \\ \hline e & e & p & q & r & y & z \\ p & p & e & y & z & q & r \\ q & q & z & e & y & r & p \\ r & r & y & z & e & q & q \\ y & y & r & p & q & z & e \\ z & z & q & r & p & e & y \end{array} $$ What are these "things" called $e,p,q,r,y,z$? In doing this kind of algebra, we don't know. They are "primitive"; they are undefined. In applications to other areas of mathematics, we may know what they are: in some cases, they are permutations of a set of three objects and the "multiplication" is composition of permutations. In some other cases, they are geometric transformations of one kind or another.

How many partitions of a set of four "things" exist? A mathematician can tell you the answer is $15$ without knowing what the undefined "things" are:

1. $abcd$

2. $abc/d$

3. $abd/c$

4. $acd/b$

5. $bcd/a$

6. $ab/cd$

7. $ac/bd$

8. $ad/bc$

9. $ab/c/d$

10. $ac/b/d$

11. $ad/b/c$

12. $bc/a/d$

13. $bd/a/c$

14. $cd/a/b$

15. $a/b/c/d$

Mathematics generally works in that way with undefined "things".

As for geometry, not all applications of geometry are applications to physical space. The "points" in such applications may not be points in physical space. For example, in statistics, suppose each number in a list of numbers is decomposed into the average of the numbers in the list and the deviation from the average, thus:

$$ (99, 97, 103, 101) = (100,100,100,100) + (-1,-3,+3,+1) $$ Looking at this geometrically, we say that the vector in which every component is $100,$ the average, is the orthogonal projection of the point we started with onto a $1$-dimensional space, and the vector of deviations from the average is the projection onto the complementary $3$-dimensional subspace. Such geometry enables us to prove in some contexts that the sample mean and the sample standard deviation are independent random variables, and knowing that is crucial to finding the probability distribution of a statistic used in finding a confidence interval for the population mean.

Thus in some appliations the "points" in geometry may be locations in physical space; and in some they may be statistical data sets, but the geometry is the same either way.

In modern mathematics geometry is defined as an axiomatic system. It consists of a set of primitive or undefined terms (including point, line and plane); a set of relations between or statements about those terms, which are assumed to be true (the axioms of geometry); and whatever propositions can be derived from the axioms using the formal rules of logic. There are various ways of representing geometry as an axiomatic system - one of the best known is Hilbert's axioms of Euclidean geometry, which German mathematician David Hilbert published in 1899.

In an axiomatic system each primitive term is defined by the axioms in which it appears. We do not need to provide a separate definition for each primitive term because either it would duplicate the definition implicit in the axioms or it would add to the implicit definition - in which case the axioms are incomplete - or it would contradict the implicit definition - in which case the system is inconsistent.

Expanding on my comment (as requested by @DanielCollins), if we know enough linear algebra, we can give proper definitions. Most of what I write below is from Geometry by Audin.$\newcommand{\E}{\mathcal E}$

Let $k$ be a field. We say a set $\E$ is an affine space over $k$ if there is a vector space $E$ over $k$ and a function $\Theta\colon\E\times\E\to E$ satisfying the following two properties:

1. For a fixed $A\in\E$, the function $\Theta_A\colon\E\to E$ sending $\Theta_A(B)=\Theta(A,B)$ is a bijection.
2. For any $A,B,C\in\E$, the following holds:$$\Theta(A,B)+\Theta(B,C)=\Theta(A,C)$$

Intuitively, $\E$ is a vector space which has "forgotten" its origin. Once we fix a point $A$, then it essentially becomes a vector space. In fact, writing $\vec{AB}$ for $\Theta(A,B)$ will make the notation even more familiar (even for the physics people), as the second condition can be seen as the more famous "triangle law of vector addition", and then usual stuff like $\vec{AB}=-\vec{BA}$ holds.

Some examples include a vector space itself with $\Theta(u,v)=v-u$ (check!), or even the empty set $\emptyset$ (why?).

So long as we have a nonempty affine space $\E$, the associated vector space $E$ is unique upto isomorphism (why?). Hence,

We call a nonempty affine space $\E$ as $n$-dimensional if the associated vector space $\E$ has dimension $n$.

The final thing before we go into defining points, lines and planes is the notion of a subspace.

Let $\E$ and $E$ be as above. A subset $\mathcal F\subseteq\E$ is called an affine subspace if $\mathcal F=\emptyset$ or there is $A\in\mathcal F$ such that $\Theta_A(\mathcal F)\subseteq E$ is a linear subspace.

Essentially, $\emptyset$ is an affine subspace, and a nonempty affine subspace is one for which fixing a point in it, we get a linear subspace.

For example, for the vector space $\Bbb R^2$ viewed as an affine space, any usual line (like $y=2x+1$), even if it doesn't pass through the origin, is an affine subspace.

Note that a nonempty affine subspace has a dimension, and thus, we can define:

Let $\E$ be an affine space. A point is a $0$-dimensional affine subspace, a line is a $1$-dimensional affine subspace, and a plane is a $2$-dimensional affine subspace.

In general, the book by Audin does a very good job at providing rigorous definitions for affine spaces including things like angles via isometries (which is very interesting!) and proves some classical theorems along the way too. One could check that book out for more details.

Hope this helps. :)