What is a straight line?

I am not a philosopher; I am an engineer with a reasonable grasp of mathematics. This question has been bothering me for a long time, and I have asked a variation of it to a mathematical community. While some people raised interesting points, others accused me of overthinking. So I thought I would come to the last people on Earth who would ever accuse anyone of overthinking - the philosophers :-)

We all have a very good intuition of what "straight" is. When we stretch a flexible, thin object, such as a piece of string, it becomes straight; and if it is non-elastic, it stops stretching precisely at the point when it becomes straight. As another example, if we want to go from point A to point B, we know that the most efficient way is to move along a straight line. And we certainly don't need anyone to explicitly teach us this principle - in fact, even rather simple animals can apply it.

In his Elements, Euclid provided the following definition and postulate for straight lines:

Definition: A straight line lies equally with respect to the points on itself.

Postulate: One can draw a straight line from any point to any point.

To me - and I suspect that most people would agree - this definition seems vacuous at best. It is essentially an appeal to intuition and does not really define anything.

If we move forward in history from a Euclidean to Cartesian system, we can now use the coordinate plane to define a straight line as a locus of points satisfying an equation of the form y = mx + c, where m and c are constants*. This is more of a definition than Euclid's, but at the end of the day, I still think it provides little more than the equation of the line that happens to conform to our experience of straight. The question remains: what is special about this particular line?

As an attempt at something more meaningful, we can try to define a straight line between two points as the path between them with the shortest length. We can certainly work with this definition mathematically, and from it derive the familiar Cartesian equation of a straight line. All well and good, but we must now look at the premises that we used to arrive at this conclusion. To begin with, what is this path length which we minimized? More likely than not, we have defined the length of a general path as the sum (i.e., integral) of lengths of infinitesimally small, straight segments making up that path. And we have defined the length of one such segment as being the Euclidean norm. But where did the Euclidean norm come from? From Pythagoras' theorem. And how did we prove that? Using triangles constructed with straight lines on a plane - which, incidentally, is another straight object: it is what we get when we stretch a piece of fabric, rather than a string.

Of course in mathematics, as several people have pointed out, we can define a 'straight' line very generally as the shortest path according to a given norm on a given surface. And if the norm happens to be the Euclidean norm, and the surface happens to be a plane, we get our good old 'truly straight' line.

But that still only goes a very short way in satisfying me. It seems like some sort of "introducing extra complexity" fallacy. I guess that the crux of my question concerns these "happens to be".

*Actually, we have already run into a problem here, because a Cartesian plane relies on (at least imaginary) straight lines, along with the concepts of length, parallel and perpendicular.

• I understand your ascent up to the definition in your second-last paragraph. But unfortunately, I do not understand your last paragraph. Could you please explain a bit more the problem concerning "happens to be", thanks. Jan 2 '16 at 17:47
• From a neuroscience perspective, this is a very complex question too. Jan 3 '16 at 2:33
• @Léon Pelletier: What do you mean? Please give some hints, thanks. Jan 3 '16 at 9:37
• "Can we just outrightly say, that the question is purely "overthinking" about this?" @KentaroTomono No. Jan 3 '16 at 21:03
• I think there is actually a mathematical answer to this. I can't, however, post it because the question is protected and I don't maintain an active account on Philosophy. Jan 3 '16 at 21:11

You're overthinking it!

Just kidding, as a fellow engineer, I have an inkling where you are digging at. Welcome to the fascinating underbelly of mathematics, which is constantly churning and mixing with the philosophy of mathematics. It behaves a bit differently than the upper tiers. Instead of trying to make powerful earth shattering statements about our reality, like proving there are countably infinite primes, they're trying to define the most subtle and intuitive assumptions you can make to get there. Your concept that it is an appeal to intuition is valid, because that's what it is.

Once you accept such a definition, such an axiom, it gets brought into a tremendously powerful system. Most of mathematics accepts some form of inductive proof, so the acceptance of an axiom is a very powerful thing. It means you accept any and all consequences that can arise from applying it an infinite number of times. There's a good reason they need to be intuitive. Proof systems that admit induction literally go, "1, 2, skip a few, 99, 100, skip a few more, infinity. See, I proved it!" (Edit: Jan felt this was not a valid use of "literally." For a more precise wording, see the Peano Axioms. A version of "literally" I can use properly is to say that the purpose of induction is to literally skip an infinite number of steps in your proof.) If your definitions are not extraordinarily in line with reality, an infinite series of skipped steps will contort your definition until it looks absurd. Consider Xeno's paradox, which proved you couldn't move anywhere through an infinite number of steps, and yet we move places all the time!

Euclid's definition of a line is really two pieces. The first doesn't care about how it maps to the real world, so he could have said

A FribbleMoose lies equally with respect to the points on itself.

This is mathematics! We can define anything we bloody well want! A FribbleMoose may be a neat definition to work with. However, it is him electing to call it a "straight line" that suggests that this mathematical definition may have real life implications. This implication is not material to the definition. If it turned out that all of society agreed that Euclid's wording was a poor one, it would at least still have at least as much meaning as a FribbleMoose had, although people may complain about his deceitful naming scheme.

What makes Euclid's straight line interesting is that it is highly effective at making predictions about how our world works. Euclid claims the sum of the angles of a triangle add up to 180 degrees. In 2300 years, nobody has been able to make a triangle with a different sum of angles (without "cheating"), so we're pretty sure that Euclid's definition of straight lines is useful enough to warrant teaching children.

Now I mention cheating. You mentioned alternate topologies on which we may define a "straight line" to have behaviors different from Euclid's. This is just a linguistic issue. If Euclid had called it a FribbleMoose, we'd probably invent a new term for the concept in other topologies. But "straight line" still has an intuitive appeal."

You're an engineer. You're familiar with computing. If you are working as part of a shoestring-budget startup, and you hear someone say "we need more computing power," it has a meaning. It typically means breaking the piggy bank to scrounge together a few GHz worth of processors to do the work. However, if you're working on a BlueGene/Q, and you hear someone say "we need more computing power," it has a different meaning. If you tried to solve their problem by walking in with a few Windows 10 boxen, they'd look at you funny (but the startup might have drooled, and wondered where you got them). In natural languages, the meaning of words often shifts with contexts. For most contexts, Euclid's definition is good enough. For general relativity questions, the term straight-line has another meaning. In fact, it becomes so different in nature that people tend to call the GR version "geodesics!"

So, as a fellow engineer, I think this is where the overthink is happening. The focus of the underbelly of mathematics is coming up with definitions to build all of the rest of mathematics on. It is easier to convey these concepts to non-mathematicians if they have real life implications, so real life terminology leaks in. It also gives mathematicians a great myriad of intuition checks. If something doesn't pass the intuition check, it's worth double checking to see if the meaning of a word has shifted along the way.

Or you can just relabel everything FribbleMoose, and find out whether FribbleMoose or PathosDingbat or any other random word choice you please is applicable to your scenario. However, if it "happens to be" a straight line, that sure as heck makes the rest of our jobs easier!

• Good points, thanks. To be honest, I don't even think Euclid's definition is any sort of definition at all. I think it's either nonsense or a very bad translation: what does "lies equally" mean?
– MGA
Jan 2 '16 at 18:43
• @MGA I'm not entirely sure. However, if I picture a tightrope walker walking from one point to the other, balancing the entire way, "lies equally" starts to have some meaning to me. It might have been a more natural wording given how the Greeks thought. Sometimes translations are horrid. Jan 2 '16 at 19:30
• 'Proof systems that admit induction literally go, "1, 2, skip a few, 99, 100, skip a few more, infinity. See, I proved it!"' - Nope. Not even figuratively. This sounds more like a proof by example, which is invalid in mathematics. Jan 3 '16 at 1:53
• @JanDvorak I believe the official spelling of the axiom is If φ is a unary predicate such that: φ(0) is true, and for every natural number n, if φ(n) is true, then φ(S(n)) is true, then φ(n) is true for every natural number n. Jan 3 '16 at 3:58
• That sounds much better, but it isn't what your answer is saying. Jan 3 '16 at 4:01

This is a very good question, one that brings metaphysics to life. Unfortunately, I do not have a good answer, though I've tried to think about it.

While Cort Ammon and others have given good definitional answers, these are more about the "nature of axiomatic systems," while I suspect what is wanted is more along the lines of, what is a straight line "really?"

A few false starts.

First, a straight line is the "fastest speed." For this is simply another way of saying that it is the "shortest distance" between two points. So the speed of light "C" is physically the "straightest" line. We use it to define the meter. But in physical reality, the linear extension of light, if we may think of it as such, refracts in innumerable mediations. And does so universally through the "lens" of gravitation.

Second, on our apparently spherical planet it came as a slow-moving shock to modern humans to discover that if you travel in a "straight line" away from, say, Lisbon, you will get further and further away. Eventually, when you are as far away as possible, you will find that you have returned to Lisbon! But from the "other side." The Lisbon your ships left returns as the "end of the straight line" backwards, in reverse. The non-Euclidean "straight line" or "great circle."

Third, just as the "shortest distance" is the "fastest speed" it is also the "least possible difference." The least "distance" entails two "stances" or "instantiations" or "points." A "dif-ference" is related to the root word for "ferry." To go straight back and forth. The fastest ferry is the least difference. And the shortest route or distance, or "straightest line."

Fourth, a physicist (can't remember which one) said that if we had only thought more carefully about the ambiguous definition of a "point" we would never have been surprised by quantum paradoxes. Our instinct is to think additively. A good corrective to this is information theory, in which knowledge increases by the elimination of "uncertainty." A Hegelian "negation of negation."

Fifth, To expand on the fourth point. We live in "three dimensional" volumes. Scare quotes because we should distrust our axiomatic concepts of "dimension."Two volumes intersect to form a plane. A plane is the intersection or "mutual elimination" of two volumes. A plane is the least thing we can see or imagine. A "line" is the intersection or "mutual limitation" of two planes.A point is the mutual limitation of two lines. And what is the intersection of two points? A circle? An "oscillation?"

Sorry, I am posting as is. Utterly incomplete. Only because I was dissatisfied with the "axiomatic" answers. There is more to it... and the question deserves better. What "are" the Euclidean entities we so readily intuit? I hope this will prompt better answers, and I will edit this mess of an answer as soon as I can.

An excellent book which addresses this subject is the classic Euclid's Elements by Sir Thomas L. Heath. Within it can be found Euclid's definitions:

• "A line is breadthless length.
• "A straight line is a line which lies evenly with the points on itself." (Euclid's Elements, p. 153)

One way to picture his definition is to think of looking down the length of the line, there would be no irregularities or lack of symmetry (see Euclid's Elements, p. 168). To put it in more modern terms, Euclid was essentially saying that a line is a continuous set of points along a single dimensions only. Even though the concept of dimensions or coordinate systems didn't exist as such in his day, I believe that's what he was trying to express with the words "to lie evenly" (ἐξ ἴσου... κεῖται). That idea might be illustrated by means of a stack of bricks, each one lying evenly on top the others such that any irregularity would cause them to topple over.

Whether we illustrate it with bricks or as a line of vision, the idea of direction is assumed. In the case of bricks, it is represented by the vertical direction in which gravity acts on the bricks. The idea of a single dimension is also represented by the word "evenly" because any unevenness would be manifest by an irregularity with respect to the overall direction of the line. In other words, an uneven brick would be out of place in a direction perpendicular that of the line, just as one dimension is perpedicular to another. What is important to note here is that the definition of a line requires the concept of space. The concept of a single dimensions is understood in contrast to other dimensions. We find the same to be true when Euclid gave his definition of a point, which was "that which has no part." (Euclid's Elements, p. 153) A point is a dimensionless entity, which, if it had a "part" or any divisible size at all, it would no longer be a point. Thus, the lack of dimension is understood in contrast to dimension.

Euclid's definitions rely on the fact that people already have an idea of breadth, extension and straightness. In that respect, there is a certain circularity to Euclid's definition, which, as Heath points out, was something that Euclid was trying to avoid, but there was no way around it:

"The truth is that Euclid was attempting the impossible. As Pfleiderer says (Scholia to Euclid), 'It seems as though the notion of a straight line, owing to its simplicity, cannot be explained by any regular definition which does not introduce words already containing in themselves, by implication, the notion to be defined (such e.g. are direction, equality, uniformity or evenness of position, unswerving course), and as though it were impossible, if a person does not already know what the term straight here means, to teach it to him unless by putting before him in some way a picture or a drawing of it." (Euclid's Elements, p. 168)

Any term whose simplicity cannot be reduced by means of a definition consisting of simpler terms is commonly referred to as primitive. Alfred Tarski describes these expressions as follows:

"When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions in this group we call primitive terms or undefined terms, and we employ them without explaining their meanings. At the same time we adopt the principle: not to employ any of the other expressions of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have been explained previously" (Introduction to Logic: and to the Methodology of Deductive Sciences, p. 118)

The necessity for considering some concepts as primitive can be understood in terms of the limitations of communication and other representative forms of knowledge. The words and variables we use in communication, logic and mathematics are only symbols which represent the content to be communicated. They are not vehicles of the content as if they could actually transport meaning; but rather, they are accepted by convention to designate meaning, and thus, they are accidentally related to what they signify. Along with this represented content, communication also consists of the forms of knowledge, which, in turn, consist of relations. Although relations enable us to define content in terms of other content, eventually a limit is reached in which the simplicity of the content is exhausted, and for that reason, we require primitive concepts.

From this, it follows that communication is only made possible provided that the recipient already possesses, to some extent, the rudiments of the content to be communicated. Words and the variables can represent spatial concepts such as straightness, length and area, but they cannot communicate them without the recipient knowing a priori their significance. Immanuel Kant spoke of content in terms of its form (which should be distinguished from the forms of knowledge or judgements) and its matter. Whereas the matter is received by means of sensation, the spatial characteristics are represented within the constraints of possible geometric form which are possessed a priori:

"Space and time are the pure forms [of intuition]; sensation the matter. The former alone can we cognize a priori, that is, antecedent to all actual perception; and for this reason such cognition is called pure intuition." (Critique of Pure Reason, A41/B59)

As a point of clarity, the recipient must also possess the means of representing the "matter" of content. Although we receive the matter from sensation, qualities such as colors, tastes and sounds must be represented phenomenally since they are too primitive to be communicated by any symbolic means.

I fully agree with Cort Ammon's answer but I'd like to add a few points on the topic of definitions.

There are different things we can call definitions.

A strict definition associate a term with an equivalent statement, in such a way that the term can be substituted with the other statement in any sentence without changing the meaning of the sentence.

This can work for terms that are derived, but at some point you'll need primitive terms that cannot be defined this way but will be used to define other terms (you'll reach the bottom of your conceptual system).

For primitive terms, there are implicit definitions, which are actually a set of axioms where the terms are used. It's a kind of definition through usage. Take for example Newtonian physics: "mass" and "force" are never defined, they are primitive terms. However the axioms of Newtonian physics relate masses and forces in such a way that we can grasp what they are, even if they were not strictly defined.

The same goes for euclidean geometry: lines, points segments and angles are the primitive terms and they are not strictly defined, but implicitly through axioms which say how points and lines behave relatively to each other.

An axiomatic system is a good description of an intended domain if all the consequences we can derive from it seem to be true of the objects of that domain that we are intuiting and of course intuition plays a role here. In the case of lines, the axioms are good because all consequences (theorems...) are true of our representations of lines in euclidean space.

This leaves open the question of the origin of the intuitions (what are those lines we are trying to define?). In the case of geometry our intuitions certainly come from physical space. In order to know that the theorems of geometry are true of physical space we need what Poincare called coordination postulates: for example the assumption that light travels in straight lines in vacuum. They map primitive terms of an axiomatic system to our observations (Poincare saw them as kinds of conventions). Coordination postulates can be thought of as part of the implicit definition of physical lines but it's important to remember that an axiomatic mathematical system is independent of the way it is mapped to our observations and in a sense, being a line in a mathematical sense is nothing more than obeying the corresponding axioms of geometry.

EDIT I think it is problematic to define lines as functions (y=ax+b) because a coordinate system is not a geometric space. It has an origin, a length unit and preferred directions, which geometric spaces lack. One should rather see a coordinate system as a way to assign sets of numbers to geometrical points and a function is a way to define a sets of numbers that can correspond to lines but all this already requires other concepts, so the right way to define a line is by using purely geometric axioms.

"Straight line" is a sequence of words, and as a society we have implicitly agreed upon certain conventions as to which sentences we can correctly use the words "straight line" in. That's all.

This is not a very useful answer, so let me add to it. In most modern formulations of Euclid (Euclid's original axioms had some subtle holes, so they were patched up in various ways in the 19th century), "straight line" is taken as an undefined term. There have to be undefined terms in any theory, since, if you try to define all words in terms of other words, you get an infinite regress. An undefined term isn't given any meaning, but there are axioms about the various undefined terms saying what ways the undefined terms can be correctly used with each other. For example, one of the axioms is that, given any two distinct points ("point" is also an undefined term), there is exactly one straight line both points are on (the relationship of a "point" being "on" a "straight line" is also an undefined term). From the axioms, one can logically deduce various theorems. Some of these theorems are complicated to state using only the original undefined terms, so one also makes definitions, which state that some word or phrase is shorthand for a longer combination of undefined terms (or other definitions, which could then be further expanded).

Here, an undefined term has no specific meaning, but it has a context which gives how it can be used correctly. In some sense, this context is the meaning of the term. In any case, you can get no more from the theory itself.

Let me back up for a moment; I used to word "theory" in a technical way which I should elucidate. A theory is a collection of undefined terms and axioms.

Given a theory, one can have models of the theory. A model is a realization of a theory in the context of some other mathematical theory. For Euclidean geometry (the theory of straight lines and points 3 outlined 3 paragraphs above), there is a standard model of Euclidean geometry in Cartesian geometry. When a theory T has a model M inside some other theory U, then the undefined terms of T can be given definitions in terms of the undefined terms of U in such a way that the axioms of T become theorems of U. This is what it means to realize one theory in terms of another. So, in terms of the model of Euclidean geometry inside Cartesian geometry, a "straight line" is the "set" of consisting all "ordered pairs" of "real numbers" satisfying some equation ax+by=c (for fixed a, b, and c). You can trace this definition down, but, as you observe, eventually you run into the undefined terms of Cartesian geometry. You can further make a model of Cartesian geometry inside some other theory, and ultimately in Zermelo-Frankel set theory, but that too has undefined terms.

Keep in mind that Euclidean geometry has other models that are not Cartesian geometry, and even models in Cartesian geometry that are different from the standard model. For example, I believe it's possible to construct a model where, to our eyes, all straight lines bend when they cross the y-axis (like light rays bending when they cross into water). (For a humorous model of projective geometry, see http://arxiv.org/abs/1406.5157)

At heart I'm basically a nominalist, and I would say that's all you get. I would say that you can continue to work and get more refined and detailed and useful ways to describe what "straight line" means, but you never get a complete clarification(*). My favorite analogy is to building a foundation for a building. We build a foundation in the ground, and we can dig deeper to anchor the building more securely, and for taller buildings we require for good reason more secure foundations, but we don't insist on connecting our buildings with the center of the Earth (and indeed it would be folly to try).

There are also realists of one sort or another, who hold some of these undefined terms refer to actual objects. Generally considered the most extreme of realists are Platonists, who hold that some theories (to them the actually true theories) have undefined terms that refer to objects in some Platonic realm where they actually have the relationships given by the axioms. Realists differ in the scope of what undefined terms they consider to have reality, how they have reality, and so on.

On the other hand, there are also idealists of one sort or another, who hold that some undefined terms (the ones in actual theories) live as specific patterns of thought encoded into our minds.

In my opinion, it's hard to be an empiricist about mathematics with the Banach-Tarski paradox, but there are some.

Some ideas specific to mathematics and the philosophy of mathematics can explain more to you, but at the end of it, you get to choose between the various answers for "What is anything, really?" Just keep in mind it's possible to have a different answer for mathematical objects than for physical, material, or social objects.

(*) This sentence IS intended to refer to a remark in Philosophical Investigations, somewhere around number 100, but I don't have my copy with me.

• As Wittgenstein said to a friend who wished him well, "please, think about what you are saying." A nominalist may well claim that a rose by any other name... etc. But I would love to see an adequate nominalist critique of the "line" dividing "two sides" of a property. The question involves the famous "peculiar efficacy" of mathematics. And similarly the "nominalist" has to explain the peculiar efficacy of proper names. Otherwise it is just laziness. Jan 3 '16 at 2:49
• As for "peculiar efficacy", I think it's just a property of mathematical definitions being created by humans for human purposes, combined with selective memory. (We don't notice the cases where mathematical ideas don't have peculiar efficacy.) I am aware others disagree. Jan 3 '16 at 2:59
• I think the "line" dividing the plane into "two sides" is actually a theorem in most proper axiomitizations of Euclidean geometry. (You need betweenness and separation axioms to make proper sense of this.) It should be in Hilbert's book (which I have not read). If I'm wrong, I'd also expect that to be in Hilbert's book. Jan 3 '16 at 3:11

We all have to admit there are no straight lines in the world. So what would it mean to say what one was, really, in terms of anything in the real world other than ourselves?

That implies that from within mathematics, or from objective reality, you are never going to get any ideas about what is important about straight lines. You have to seek an answer from what makes us up: psychology and evolution.

Mathematics is not about real things. It is about idealizations upon which human minds can generally be made to agree.

Let me establish this perspective on mathematics a little.

For a simpler example, let's back off to a number, say 5. You can show me sets of five things and say you have identified what five means, in the real world. But that does not explain how you can also have a length of five centimeters, because you will never be able to draw a line exactly five centimeters long, or cut a length into exactly five separate one-centimeter pieces. Nor could I tell if you did so. Even worse, try linking the five objects to an area of five square centimeters.

It is only the fact that we can hook up those intuitions to one another via simple explanations that makes them all aspects of the same thing, the number 5. But as Socrates and the Pythagoreans had already demonstrated long ago, at base, those simple explanations make no sense unless you are already predisposed to understand them.

That number is a bunch of very distinct things that fit together in your mind, and have a certain ideal quality. That area of five square centimeters is never exactly five square centimeters, after all. You just know what it would mean if it were.

We know what a straight line is the ideal version of: a path that heads from one point to another as quickly as possible and extrapolates the result infinitely in both directions. And we know that it is not real, but a linguistic convention.

But it is a very special sort of linguistic convention -- one that comes to humans exceptionally naturally. More naturally than real, observable things come to us when we really observe them.

The very special thing about a straight line is that if you draw an approximation to it, another human being is strongly predisposed to actually perceive a better approximation than you drew. They might be tempted to correct your drawing, or if they copy it, they will draw a version that produces the intended figure in their head more properly than you managed in reality.

So we have to admit that a straight line is a shared human habit, something to which our visual and explanatory mechanisms are drawn by our nature. (I would claim that these shared human habits are the real subject matter of mathematics.)

People will do the same thing with a harmonized interval, or a catchy rhythmic figure. We can more easily get behind these and see the sensory simplifications that the "more proper" forms allow. They are more easily reproduced, because of the simpler waveforms we can sense through resonance in attempts to reproduce the harmonies, or the uses of inertia and muscle memory in the body that we can use to reproduce the rhythmic figures.

I would propose that music and geometry have this in common, that we are drawn to seek ideal forms that are the easiest forms to mechanically repeat without tools.

From that point of view, the straight line does have a clear definition: It is the ideal form of the drawn or otherwise impressed figure that is easiest for a human being to approximate without tools when copying it.

The ideal form is straight because straightness involves solving an optimization problem, if we miss, we can improve our approximation by removing variation. If we fix it enough, we can always make it good enough to convey the right idea. No specific curve has that quality, for us.

It is also not of a given length, because we can always be wrong about the length of a line, and we can fix that, too. But that is hard. So the perfect line would be one with no specific length.

Conveniently, we have an optical mechanism with perspective, so we have the idea of "vanishing into infinity", to pack arbitrarily large visual spaces into finite representations. And we can imagine a line that would just vanish into infinity in both directions.

I agree with Cort Ammon in principle, but I want to undercut his anthropocentrism. Yes, straight lines are useful and beautiful. But this compactness of representation is what makes the idea of a straight line useful to us, and that compactness is 'compact' because we are us.

It is easy to imagine that some species might have chosen circles as the basic geometric figure, if they had the extreme symmetry of an octopus who might whip out a circle by drawing with all arms in concert -- or hyperbolas if they had a wormlike ability to stretch themselves out into arbitrarily long strands at a fixed rate. And these would lead to equally useful, very different perspectives that could just as firmly base science and engineering.

Euclids definition, if I'm interpreting it correctly, is a local definition; this is why he's talking of the points on the line, lying equally.

In a modern context, we would say that the line is locally flat; it's also used as the definition of a manifold in that it is a space that is locally flat.

From here, we can as he suggests in his postulate draw a line from one point to another; whilst tracking our motion locally as moving in a straight line; this is parallel transport.

Both this, and the other definition you suggested - the variational definition are both used in modern geometry, and in (mathematical) physics.

I'm not sure if this will satisfy you, but it's another (rather obvious) way to think about it -- a line is some function the derivative of which is a constant, or the second derivative of which is zero. When you look at the collection of infinitessimal line segments you described earlier, does each one have the same slope -- the same tangent -- as the ones before and after it?

I guess the concepts of slope and tangent are also based in the concept of the line, but this still appeals to my intuitions.

It's not altogether absurd to call a circle a "line" that is biased towards a center. In this case, every infinitessimal segment follows a pattern set by the one before it, where its "path" changes according to that pattern, that bias towards the center. A "straight" line is one with no such bias -- where every line segment is just a transposition of the last, with no transformation.

My ideas are starting to bleed into one another, but I guess that's fine, because it doesn't seem like you're looking for an answer, so much as scratching of an itch. Hope I helped.

• A circle is obviously a line, and even a "straight line" on a spherical topology. I would suggest that instead of attempting to rethink the mathematics, you think about what a line "really" is. Whatever that means. For example, we use "lines" all the time. But they technically have no "width." One could never actually "see" the line dividing Germany and Poland, for example. If you "draw" a line it is no longer technically a line. Just to make you itchy again. Jan 3 '16 at 2:55
• If you "draw" a line what you've drawn is a depiction of a line segment. Same as if you draw a tiger, only the depiction of the line segment is much closer to the real thing, and two dimensions too many (assuming your lead or ink is made of three-dimensional particles). Jan 4 '16 at 3:53
• As for your comment about topology -- I would say that the topology is the same as the "bias" as I mentioned it. A "straight line" is a straight line on a "neutral" topology -- that is to say, a flat one, although using the word "flat" or saying it should be "two dimension" might be begging the question. This is mostly enough to satisfy my intuitions on the topic, but I certainly can't blame you for your continued skepticism of such a fundamental aspect of our understanding of the world. Jan 4 '16 at 3:55

He didn't define what a straight line is!

He defined what in his set of axioms (the Euclidaen topology) a concept named 'straight line' is. Coincidentally this definition is intuitive and usually implicitly used without specifying what topology we are using -- even in the 'real world'. But at that point we are leaving the mathematics and asking ourselves what a straight line mighy be and, stumbling over the neat and intuitive definition that Euclid provided, just accept is in the majority of the cased.

I could at any time invent a new topology that has a new concept of something called 'straight line' that's entirely different but will still be correct as long as it's not self-contradictory.

You are not overthinking it, you are just not leaving the real-world behind you enough. Maths doesn't really care whether a definition is sane or intuitive (that's only what mathematicians do).

Becuse humans do maths we usually try to -- for convenience -- use names we actually have used and seem intuitive.

It all boils down to the naming, so better ask yourself:

• What is a 'straight line' in 'our world' (and not in a convenient set of axioms)?
• Why did mathematicians name X 'x' for every mathematical concept ;-)
• >It all boils down to the naming, I can not agree more.
– user13955
Jan 3 '16 at 15:20

A straight line is simply an object of perception which we "know when we see it". That's enough. The important thing is that we have enough of an intuition to be able to fix axioms about straight lines which seem obviously true to everyone. From those axioms we can then deduce properties of straight lines, including that they can be represented by equations of the form ax + by + c = 0 (which can be deduced from theorems about similar triangles and the triangle area formula, if I remember correctly). Being willing to accept certain axioms about physical objects is the price of entry to applying mathematics to the real world.

Note that it wouldn't be satisfying at all to just say that the Cartesian equation is the definition of a straight line, since then how do you know that say, the edge of my desk is a straight line? You could measure it, but aside from being a very ugly ad hoc solution, it's copping out of admitting the fact that really, you were pretty sure it was a straight line before you ever measured.

The same problem arises with circles. A statement like "a circle is the set of points a fixed distance from the center" isn't really a definition. The definition of "circle", outside of technical mathematical speech, is "that object which we can all intuitively visualize which we call a circle". You just know a circle when you see it, and that's enough to get the mathematical ball rolling. You fix a property like "every point is the same distance from some center" which seems to obviously apply to this shape that you see all around you and call a "circle", and then deduce further theorems from that. The "shortest distance" definition for a line is similar. The definition of a straight line is still just "things that are obviously straight lines when you look at them", but "the shortest distance between two points" is a reasonable seeming axiom about those objects.

It's very interesting that the visual patterns - shapes, in other words - which we pick up on most intuitively (lines, circles) also happen to be the ones with simple mathematical definitions and which turn out to be of fundamental importance. For instance, "lines" are simply a kind of pattern in our visual inputs that we're able to easily pick out - we know them when we see them. Yet by some strange fate they turn out to have a simple analytic definition by a linear equation, and in fact we now understand that they're just one case of the more general notion of a hyperplane in a vector space, which is fundamental in all kinds of contexts beyond just the geometry of physical space. This seems like a hell of a stroke of luck - the shapes which ancient humans decided were the "nicest" and "simplest" turn out to be important in advanced mathematical theories developed in the 19th and 20th centuries!

What I think is going on here is basically that nature is mathematical. It therefore follows that concepts that are important in mathematics will manifest themselves in the physical laws governing reality. The examples most relevant to us and straight lines are the ways that one-dimensional subspaces of euclidean space interact with the laws of optics and mechanics (objects move in a straight line when undisturbed, light travels in a straight line). Therefore natural selection prepares us with an intuitive grasp of some of these important mathematical concepts, so that we can navigate the physical world. These intuitions give us the ability to guess at true mathematical properties of these objects (axioms), from which we can reconstruct the general mathematical laws and phenomena that started the whole ball rolling in the first place - completing a kind of cosmic cycle, if you're willing to indulge a bit of poetry.

I don't know how workable such a definition would be in practice (an important consideration), but you might define a straight line in terms of a subset of the Euclidean plane P and the distances between points in P as follows:

A subset L of P is a straight line if and only if there exists distinct points x and y on P such that every element of L is equidistant from x and y.

Similarly, you might define a plane in terms of a subset of the Euclidean 3-space S3 and the distances between points in S3 as follows:

A subset P of S3 is a plane if and only if there exists distinct points x and y in S3 such that every element of P is equidistant from x and y.

EDIT: In general, you could define straight lines and planes in a similar way for any metric space.

Before we get onto what a line is, lets think about what we interpret other things.

First of all, there's a simple mathematical definition error in your question. From my elementary school math, as I recall, a line-segment has two end points, a ray has one starting points, and a line goes on indefinitely. I think what you wanted to ask in your question is about line-segments instead of lines.

We also live in a more than 3d universe. Some argue that it has 10 or more dimensions. What can this mean? We still can't exactly know. If we take a 2d universe, everything is on a flat plane, infinity stretching or not. To turn 2d into 3d, we must add another dimension by expanding it in a different way. As we happen to know (guess), 1d is where only a straight line exists. If that 1d line has been transported into 3d space, it still needs to undergo many expanding in different directions, unimaginable by the human mind. 3d -> 4d = line -> square; 4d - 5d = square -> cube; 5d -> 6d = cube -> whole timeline of the cube (as I like to call it) or rapid prototyping (as others like to call it). We have gone far enough. If we take a 2d triangle, with edges that add up to 180 deg (duh!), into 3d space, the angles no longer add up to 180 deg. The same will happen with a line. If it shall remain a line, it will become curved, so it is no longer a line? The definition is up to you to edit in the next dictionary :)

My answer is going to be more of a semi-pseudo-scientific approach... if that's how you actually spell that out.

Personally, I would say there is no such thing as an accurate, physical manifestation of what we'd classify as a straight line; at least, in the way we'd normally envision it. Given the Heisenberg uncertainty principle, we cannot measure exact momentum and location at the same time; so, in a sense, your line may never be complete, or even straight.

But, if I were to look at it from a normally observable size, I'd still say there is no such thing as a truly straight line. The only way I would actually consider anything truly straight is if the creator completely calculated out the earth's curvature when creating the "straight" object. In the practical and literal sense, that's about as close to a "straight" line as I can envision. While we could conceptualise a number of things in our head, we'd need to put theory into practice before considering it a good enough representation, like the everything else the world.

On a more philosophical note, I'd have to repeat the original statement: there is no such thing in the physical or observable universe. A "straight" line is nothing more than a mental image that we've conceptualised and then created in the physical world. Actually defining a straight line in a way that one can understand in a physical way almost reminds me of trying to define emotions, logic, or rationality. You cannot make the "shortest distance" without pausing time, freezing every single particle in space, and moving them all by hand.

Broken down into a much more simple, yet still complex way: a straight line is nothing more than an idea that is explained with noises by humans, attempted to be recreated accurately in the physical realm, but ultimately defies the laws of perceivable physics as the concept itself is flawed. The idea that you create the "shortest" path means that you actually know the shortest path. How many dimensions do we need to navigate? Can we use a wormhole? When defining "straight" as the shortest path between two points, if we negate what most people think of being "straight" (as in no curves or deviations from a direct and never changing course), you could actually bend the mind a little. If we add another dimension to our normally perceived 3 dimensions (x, y, and z axis), and we insert say... the w axis... now we would begin to question the meaning of "straight" in an entirely new way.

Is there really such a thing as a straight line? Conceptually, I'd say yes. Existing in the physical world? I think you're going to have to make the silicon-28 sphere look like a boring fact in science class to complete nerds.

• Well, if you read your post over, I suspect you will admit, it looks a bit mixed up, or at least not entirely coherent and concise. Work in progress. I take your view of the matter to be "nominalist," as is Alexander Woos's. At the same time, you interject "physics," where, of course, Einstein already basically disproved the absolute "linearity" of light. We can go back to Euclid, and say, yes, it's all axiomatic or "nominalist" yet instinctively ask, why does it work? Jan 3 '16 at 3:33
• I did preface it by stating this is a more pseudo-scientific approach. That may be the wrong term to use, but I thought it would help put it all into perspective. Regardless, it does make complete sense to me. If you would like some clarification on something, please ask. If you brought up light's linearity due to my mentioning of Heisenberg's uncertainty principle, I think you might have taken it in the wrong direction. As far as being nominalist, I'd say that is oversimplifying my answer, as I tried to stay somewhat concise; but I digress. Jan 3 '16 at 3:56

"Straight" is an abstract concept, and a very broad one at that.

There is no single definition of what it means for a line to be straight, because there are many things such a concept would depend on, such as what topology are these two points going to be taken on. What about the dimension of the space? What kind of space is it, euclidean, Cartesian, polar? To define what it means for a line to be straight, you'd have to narrow it down much more. For a two-dimensional Cartesian plane, it'd be the shortest possible path between two points. However, there's still a number of mathematical axioms to satisfy as well...

Contemplating an abstract concept directly, without any grounds to implement the concept on, would lead you to paradoxical thinking, misleading thoughts, and outright confusion. To reiterate, you would have to narrow down what and where this straight line would be; you'd have to implement the concept in some form or fashion. Find a system of rules to satisfy to define the straight line.

After all, a world without rules is chaos.