mathematical concept which does not have meaning and so which is not assigned an interpretation
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What are the 4 undefined terms in geometry?
What is an example of an undefined term?
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.
Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property โundefinedโ, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.
To put matters straight: Division is a function $$q:\quad{\mathbb R}\times{\mathbb R}^*, \qquad(a,b)\mapsto q(a,b)=:{a\over b}\ ,$$ whereby $q(a,b)$ is the unique number $x\in{\mathbb R}$ such that $b \>x=a$.
When we say that $\displaystyle{a\over0}$ is undefined then this means no more and no less than that the pair $(a,0)$ is not in the domain of the function $q(\cdot,\cdot)$.
Now to your three ways of understanding "undefined" in the realm of division by $0$:
If $\displaystyle{a\over0}$ could be any number, say $=13$, then this would enforce $13\cdot0=a$, which is wrong when $a\ne0$.
This is even worse. Why should $\displaystyle{7\over0}$ be the Eiffel tower?
There are circumstances where division by zero makes sense, e.g. in connection with maps of the Riemann sphere, or with meromorphic functions. There one has $\infty$ as an additional point in the universe of discourse. But these circumstances require special exception handling measures, and the "usual rules of algebra" are not valid when dealing with $\infty$.
In geometry we say that points, lines, and planes are not defined only described. But isn't a description a definition? A square has 4 parallel sides in equal length with 4 right angles. That's the definition but isn't that also a description? How are points, lines, and planes different?