Is it possible to define "Straight-line" logically? If it is possible, How you will define it?

Question

Recently I am studying the "ELEMENTS" of Euclid. It is a translation of SIR THOMAS L. HEATH. In the definition part of the first book, the second definition is, "A line is Breathless length". My question is what we understand by Length And Breadth? Is it not straight line? how we can define line with the help of the concept of straight line? Can we define a "Dimension" logically?

Answer 1

A straight line is nowadays defined in a abstract way, using axioms:

• By two distinct points of the plane passes one and only one straight line.

• Any straight line contains at least two points.

• There are at least three non-aligned points.

• If B lies between A and C, then B lies between C and A.

• Let B and D be two distinct points. There are three points A, C, E of (BD) such that B ∈]AD[, C ∈]BD[ and D ∈]BE[.

• Given three points of a straight line, one and only one of them lies between the two others.

• ...

The straight line is in fact not given a definition in the sense you have in mind. Only essential properties of lines and points are listed.

Answer 2

In axiomatic geometry, generally one does not give a definition - or in particular, the definition cannot be mathematical. Definitions like those in Euclid have no mathematical flavor, but are rather in human language, and meant to convey intuitive concepts and imagery so as to connect what the axioms are talking about to understandable experience and make their intent clear. The situation is not any different in the modern mathematical treatments - the difference is that our axioms are logically airtight for the manner in which modern mathematics is carried out and moreover that in a modern treatment one is up-front and honest that the terms like "straight line", etc. cannot be defined mathematically. They are what are called "primitive" concepts. Intuition, and intuitive understandings and concepts are, in my mind, VERY important, but they have to be guided with rigor. Rigor is the scaffolding, intuition is that in which it is dressed. The rigor cannot be truly understood without the intuition (and much problem solving is done on an intuitive level first), and intuition can lead one wrong if one does not also make sure it can be followed in rigor.

That said, if one takes the opposite approach of using analytic geometry, in which geometry is constructed out of the real number line using coordinate planes or spaces, then one can define the concept of a direction in space using calculus, and say a straight line is a curve that preserves a constant direction. The analytic approach, of course, is equivalent, but it admits of more generality in that it can be generalized to highly non-Euclidean, fluid spaces; it is the only approach in which things like Einstein's theory of general relativity can best be formulated.

(Historical note: This is, by the way, not saying Euclid was bad. The mathematics in his cultural area and time simply did not have the same methodology as now. His axioms are more like recipes for stating what sort of figures one is allowed to construct, but not so much recipes for how they are to behave and to interact. In particular, some scholars analyzing Euclid have suggested that his axioms, where they did more than prescribe constructions to make, were designed to convey what we would call the "metrical" - measuring or distance - aspect of Euclidean geometry, while the "topological" part - i.e. intersections - was to be handled by the use of diagrams and the diagram itself was an important reasoning tool. One can actually axiomatize diagrams separately to use them in a modern framework, but this approach is a bit wordier than it needs to be and most approaches just combine the topologic and metric axioms together. Though even with that, I still think a case can be made Euclid was truly incomplete, as some of his contemporaries or slightly later successors criticized his work on incompleteness grounds as well, for example, one criticism amounted to that he did not really know that a straight line could not meet with another in more than one point.)

Answer 3

Yes, you can define dimension in a rather consistent way across all categories see there. The only thing all these definition have in common is that the cartesian vector space $R^n$ has dimension $n$.

Be warned however of some un-intuitive facts, like negative Lebesgue dimension of topological spaces or non-integer Hausdorff dimension of metrical spaces.

Note also that the cohomological dimension is more than a unique number, as you need not only to measure the dimension of the space, but the dimension of the holes in it.

Answer 4

Whether you can define "line" depends on what other undefined terms there are (every theory has to have some primitives, after all).

I once played with developing axioms for Euclidean geometry where the only primitives were "point" and the relation "Points A and B are closer together than points C and D" (symbolized as $AB \prec CD$). Then defining

• Equidistance: $AB \sim CD$ if $AB \not\prec CD$ and $CD \not\prec AB$. Equivalence classes of $\sim$ defined lengths.
• $B$ is between $A$ and $C$ if among all points $D$ with $BD \sim BC$, it holds that $AD \preceq AC$.
• $A, B, C$ are colinear if one of the points is between the other two.
• A linear space is a collection of points such that for every two points in it, all other points colinear with them are also contained in it.
• The span of a collection of points is the smallest linear space containing the entire collection.
• A line is a linear space spanned by two points. A plane is a linear space spanned by three points, etc.

There were no logical problems with the project, but I abandoned it because the axioms necessary to make sure everything behaved properly were getting out-of-hand, which made this approach conceptually much harder than the more common approaches.