The definition of a straight-line seems circular?

[Matth.ew]

Hi,I was just wondering if someone could provide clarity on this matter: that if a straight-line is initially defined as "a shape that forms the shortest distance between two points" and conceptualising that shape [that forms the shortest distance between two points] as one that, at an infinitesimal-level, is comprised of smaller shapes; if those smaller shapes are circles, then it seems that a straight-line would not actually be produced in actuality (it would have curves to it) and if those smaller shapes were referred to as triangles and squares and such then, of course, such shapes (that is, triangles and squares and such) are comprised of straight-lines. So, it seems that a definition of a straight-line is circular, so to speak, because the configuration of that shape that forms the shortest distance between two points (that is, a straight-line), seemingly cannot be defined without the reference to shapes that are already comprised of straight-lines.

Any thoughts on this, as to what I might well have overlooked, would be greatly appreciated, because, of course, straight-lines exist in actuality but a definition, specifically a detailed one that also accounts for an infinitesimal-level, seems somewhat elusive (at least, to myself).Kindest wishes,

Matt.

 

[fresh_42]

Hi,I was just wondering if someone could provide clarity on this matter: that if a straight-line is initially defined as "a shape that forms the shortest distance between two points" and conceptualising that shape [that forms the shortest distance between two points] as one that, at an infinitesimal-level, is comprised of smaller shapes; if those smaller shapes are circles, then it seems that a straight-line would not actually be produced in actuality (it would have curves to it) and if those smaller shapes were referred to as triangles and squares and such then, of course, such shapes (that is, triangles and squares and such) are comprised of straight-lines. So, it seems that a definition of a straight-line is circular, so to speak, because the configuration of that shape that forms the shortest distance between two points (that is, a straight-line), seemingly cannot be defined without the reference to shapes that are already comprised of straight-lines.

Any thoughts on this, as to what I might well have overlooked, would be greatly appreciated, because, of course, straight-lines exist in actuality but a definition, specifically a detailed one that also accounts for an infinitesimal-level, seems somewhat elusive (at least, to myself).Kindest wishes,

Matt.

What do you mean by "at an infinitesimal-level, is comprised of smaller shapes"? A line of which shape ever is a one-dimensional object, that has the same structure on every scale.

 

[Mark44]

@Matth.ew, your intuition on what constitutes a straight line is very flawed. A straight line, which as @fresh_42 said, is one-dimensional, and cannot possibly be made up of two-dimensional objects such as squares, triangles, and the like.

 

[Matth.ew]

What do you mean by "at an infinitesimal-level, is comprised of smaller shapes"? A line of which shape ever is a one-dimensional object, that has the same structure on every scale.

I was trying to express (although I obviously could've done more clearly), that if one takes, for instance, a "straight-line" drawn on a piece of paper with a pencil, then a shape of a "straight-line" would be seen by the naked-eye; however, that "straight-line" could presumably be conceptualised to comprise smaller shapes (as, say, pieces of graphite, in this instance), wherein, the shape and orientation of those smaller shapes are presumably going to determine whether a "straight-line" is actually a straight-line.

So, for instance, if the straight-line (under a microscope, or just conceptually) were to be comprised of smaller shapes, to look like:

O00OOO000OOOO000OO(Because of the possible varying shapes of the pieces of graphite, in this instance). Then, although to the naked-eye a "straight-line" is drawn on a piece of paper, it wouldn't necessarily be a straight-line in actuality?

Also, can you please clarify the reference to a "one-dimensional object" because even if one takes (and to continue with the above instance of drawing on a piece of paper with a pencil) the one-dimensional aspect of paper, as an x-axis, then the very thin layer of graphite, as drawn with the pencil, would encompass a second dimension on a y-axis and so a straight-line would presumably be more than one-dimensional?Kindest regards,

Matt.

 

[fresh_42]

Yes, in the real world, there are no points, lines or even planes. What you can draw is always a reduction of the concept. With a microscope at hand, you can't even say what a circle is, or as I like to put it: the real world is discrete.

However, this leads into philosophy rather than mathematics or even physics.

 

[Matth.ew]

@Matth.ew, your intuition on what constitutes a straight line is very flawed. A straight line, which as @fresh_42 said, is one-dimensional, and cannot possibly be made up of two-dimensional objects such as squares, triangles, and the like.

If straight-lines aren't made up of two-dimensional objects such as squares, triangles and the like, what are they said to be made up of? And also, what is said to actually be a strict definition of a "straight-line"? I'm merely wondering and it is inquisitiveness and not argumentative-ness :)

Kind regards,

Matt.

 

[mfb]

A real sample of graphite cannot form a mathematical straight line. A perfect straight line is a purely mathematical concept. Real-world lines can only be approximations. That is not an issue in mathematics, where you can study ideal straight lines.

what are they said to be made up of?

They are not "made of" anything. They are just mathematical concepts.
What is the number 1 made of?

 

[Matth.ew]

What you can draw is always a reduction of the concept.

I think that clarifies things for myself and was somewhat as I imagined it to be. Thank you for you time!Kindest regards,

Matt.

 

[Matth.ew]

A real sample of graphite cannot form a mathematical straight line. A perfect straight line is a purely mathematical concept. Real-world lines can only be approximations. That is not an issue in mathematics, where you can study ideal straight lines.They are not "made of" anything. They are just mathematical concepts.
What is the number 1 made of?

I think I can begin to understand the premise of your counter-example of "what is the number 1 made of?", however, I think it is pertinent to say that, to most people if they were asked what is a straight-line and referring to a particular example (such as, a "straight-line" drawn on a piece of paper), I think they'd identify, in accordance with "accepted" views, what the "straight-line" was and that it was made up of pieces of graphite, in that instance. As I previously mentioned, I think the comment "What you can draw is always a reduction of the concept" is the sort of clarification that I can begin to make sense of. I can begin to understand that the "number 1", for instance, is essentially a concept and not necessarily of a physical existence (beyond our minds), of course, there seems to be physical manifestations of the number 1 (such as one tree in a field and such) but not of the "number 1" itself, beyond perhaps, of course, [the number] 1 . Anything I've overlooked, please feel free to comment!Kindest regards,

Matt.

 

[Stephen Tashi]

if a straight-line is initially defined as "a shape that forms the shortest distance between two points"

Plane geometry is often presented as a mixture of intuitive ideas connected by a certain amount of logic. (That style goes back to Euclid.) This is in contrast to subjects like calculus and topology where the modern methods of presentation are more rigorous. If you try to reason out the fine points of plane geometry using the intuitive notions of how things are defined (such as your definition of a line), you can't reach any firm mathematical conclusions because you are not dealing with a ideas that are precise enough to meet modern standards of rigorous mathematics. Of course, such attempts to reason about intuitive ideas may help your intuition - perhaps that's your goal.

 

[Matth.ew]

Plane geometry is often presented as a mixture of intuitive ideas connected by a certain amount of logic. (That style goes back to Euclid.) This is in contrast to subjects like calculus and topology where the modern methods of presentation are more rigorous. If you try to reason out the fine points of plane geometry using the intuitive notions of how things are defined (such as your definition of a line), you can't reach any firm mathematical conclusions because you are not dealing with a ideas that are precise enough to meet modern standards of rigorous mathematics. Of course, such attempts to reason about intuitive ideas may help your intuition - perhaps that's your goal.

Thank you for the insights. My goal (I think) was really to try to understand a strict definition of a straight-line because I'm only really aware of a definition of a straight-line as "the shortest distance between two points" and whilst I studied Mathematics at A-Level (in Britain), I am, as I'm sure many would note, far from well-informed on the subject. I certainly gained some clarity on the question I asked, from the replies so far, such as, "real-world lines can only be approximations". To myself, it is somewhat quite easy to convey the concept of a straight-line with, say, a piece of wood and carving it to a "straight-line" but my goal (I think), was to try to begin to understand how professionals within Mathematics strictly define a straight-line because, as I crudely tried to explain my thoughts on it, a straight-line doesn't seem to be necessarily achieved with the definition of "the shortest distance between two points", as it doesn't seem to account for that that forms a "straight-line".

As I mentioned, in my mere opinion, a more strict definition of a straight-line couldn't necessarily be achieved by elaborating on the definition of a straight-line of "the shortest distance between two points" by referring to shapes such as, squares, and how it pertains to that that forms a "straight-line", as such shapes, like squares, already have straight-lines. I was thinking that the describing of a straight-line, beyond "the shortest distance between two points", and thus describing that that forms a "straight-line", could use circles, to form a cylinder, as this circumvents the issue of referring to rudimentary shapes which have straight-lines, to describe that that forms a "straight-line"; however, I'm to understand that circles are essentially formed from straight-lines (from rotating a straight-line upon one point)? And so, such a definition of a straight-line, which described in more detail a straight-line, stills seemed somewhat circular; also, I'd imagine to try and elaborate on the basic definition of a straight-line of "the shortest distance between two points" would necessitate an understanding of the fundamental shapes in the cosmos? (As, in my mere understanding, there is nothing to say that all matter, at its most microcosmic level, are circles and cylinders).

Thanks in advance for at least trying to tolerate my naivety!Kind regards,

Matt.

 

[Mark44]

Thank you for the insights. My goal (I think) was really to try to understand a strict definition of a straight-line because I'm only really aware of a definition of a straight-line as "the shortest distance between two points"

This is incorrect. A distance is a number, which is not at all the same as a line. The shortest path between two points is along the line segment that joins the points.

and whilst I studied Mathematics at A-Level (in Britain), I am, as I'm sure many would note, far from well-informed on the subject. I certainly gained some clarity on the question I asked, from the replies so far, such as, "real-world lines can only be approximations". To myself, it is somewhat quite easy to convey the concept of a straight-line with, say, a piece of wood and carving it to a "straight-line"

No, a line (to me, adding "straight" is redundant) is an abstract concept. A piece of wood is three-dimensional, with width, depth, and length. No amount of carving can get it to the geometric concept of a line.

but my goal (I think), was to try to begin to understand how professionals within Mathematics strictly define a straight-line because, as I crudely tried to explain my thoughts on it, a straight-line doesn't seem to be necessarily achieved with the definition of "the shortest distance between two points", as it doesn't seem to account for that that forms a "straight-line".

Any path other than along a line will necessarily be longer.

As I mentioned, in my mere opinion, a more strict definition of a straight-line couldn't necessarily be achieved by elaborating on the definition of a straight-line of "the shortest distance between two points" by referring to shapes such as, squares, and how it pertains to that that forms a "straight-line", as such shapes, like squares, already have straight-lines.

They have edges that are lines, but squares, rectangles, triangles, and other geometric figures in the plane are two dimensional.

I was thinking that the describing of a straight-line, beyond "the shortest distance between two points", and thus describing that that forms a "straight-line", could use circles, to form a cylinder, as this circumvents the issue of referring to rudimentary shapes which have straight-lines, to describe that that forms a "straight-line"; however, I'm to understand that circles are essentially formed from straight-lines (from rotating a straight-line upon one point)?

Yes, but the curve that is generated is in no way straight.

And so, such a definition of a straight-line, which described in more detail a straight-line, stills seemed somewhat circular; also, I'd imagine to try and elaborate on the basic definition of a straight-line of "the shortest distance between two points" would necessitate an understanding of the fundamental shapes in the cosmos? (As, in my mere understanding, there is nothing to say that all matter, at its most microcosmic level, are circles and cylinders).
Lines and circles are already at a microcosmic level, in a sense. A line has no width, being one-dimensional, and the curve that defines a circle is a one-dimensional figure (having only length) embedded in a two-dimensional space -- the plane.

Thanks in advance for at least trying to tolerate my naivety!Kind regards,

Matt.

 

[Matth.ew]

This is incorrect. A distance is a number, which is not at all the same as a line. The shortest path between two points is along the line segment that joins the points.
No, a line (to me, adding "straight" is redundant) is an abstract concept. A piece of wood is three-dimensional, with width, depth, and length. No amount of carving can get it to the geometric concept of a line.
Any path other than along a line will necessarily be longer.
They have edges that are lines, but squares, rectangles, triangles, and other geometric figures in the plane are two dimensional.
Yes, but the curve that is generated is in no way straight.

A definition of a line that then contains the word "line" in it, isn't exactly informative. Anyone whom has worked, or works, in academia understands that basic concept. Yes, I appreciate that distance isn't the same as a line but if you wish to be pedantic, to say "distance is a number", isn't exactly correct either, as distance can be represented by a number but distance isn't a number.

I don't think adding the adjective "straight" is redundant. You say that "any path other than along a line will necessarily be longer"; that isn't true, as a path along a "straight" line, that joined the two end points (or rather, start and end point of that path), isn't longer than a "squiggly" line (of said two end points).

I agree, "The curve that is generated is no way a straight line", as the curvature itself doesn't represent a straight line, however, a distinct tangent to the circles could form a straight line.

I don't know if some people have taken my humility as an excuse to try and insult my intelligence; clearly such attempts to tell me I'm wrong, without providing sound reasoning as to why I might have overlooked things is rather sad. I'm yet to see anyone actually provide a strict definition as to what a straight-line is; I'm not necessarily expecting such a definition but it's rather disappointing that some people try to just say it is "an abstract concept". From previous replies, and my initial thoughts, I appreciate that a straight-line is an abstract concept; however, simply because a straight-line is an abstract concept should not be a cop-out for not having a definition of it because evidently some definitions of the abstract concept of a straight-line already exist. My questioning was on whether there is a more strict definition of a straight-line that more accurately describes the concept of a straight line. So, to me, such a definition of a straight-line as "the thing that constitutes the shortest distance between two points", or "the thing that forms the shortest distance between two points" and such, seems rather inadequate at describing the concept of a straight-line.Kind regards,

Matt.

 

[Mark44]

A definition of a line that then contains the word "line" in it, isn't exactly informative. Anyone whom has worked, or works, in academia understands that basic concept. Yes, I appreciate that distance isn't the same as a line but if you wish to be pedantic, to say "distance is a number", isn't exactly correct either, as distance can be represented by a number but distance isn't a number.

I wasn't trying to define the word "line" -- I was trying to correct your misconception.

From the wikipedia article on lines:

All definitions are ultimately circular in nature since they depend on concepts which must themselves have definitions, a dependence which can not be continued indefinitely without returning to the starting point. To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive. In those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy.

If we're talking about the distance between two points in the plane, that distance is a pure number -- no units. If we're talking about the distance between two cities, then the distance would have units associated with it, such as miles or kilometers or the like. In mathematics, including geometry, distances are most often given without units; i.e., they are numbers.

I don't think adding the adjective "straight" is redundant. You say that "any path other than along a line will necessarily be longer"; that isn't true, as a path along a "straight" line, that joined the two end points (or rather, start and end point of that path), isn't longer than a "squiggly" line (of said two end points).

I don't think you understand what I wrote. If you connect two points in the plane with any curve, the length along that curve will be longer than the path along a line segment between those points.

I agree, "The curve that is generated is no way a straight line", as the curvature itself doesn't represent a straight line, however, a distinct tangent to the circles could form a straight line.

Yes, but so what? The circle has curvature. The tangents have zero curvature.

I don't know if some people have taken my humility as an excuse to try and insult my intelligence; clearly such attempts to tell me I'm wrong, without providing sound reasoning as to why I might have overlooked things is rather sad.

No one here has tried to insult your intelligence. Instead, people responding here are trying to correct your misconceptions; e.g., "a line is the shortest distance between two points." and the disconnect between something drawn with a pencil versus the geometric concept of a "line."

I'm yet to see anyone actually provide a strict definition as to what a straight-line is; I'm not necessarily expecting such a definition but it's rather disappointing that some people try to just say it is "an abstract concept". From previous replies, and my initial thoughts, I appreciate that a straight-line is an abstract concept; however, simply because a straight-line is an abstract concept should not be a cop-out for not having a definition of it because evidently some definitions of the abstract concept of a straight-line already exist. My questioning was on whether there is a more strict definition of a straight-line that more accurately describes the concept of a straight line. So, to me, such a definition of a straight-line as "the thing that constitutes the shortest distance between two points", or "the thing that forms the shortest distance between two points" and such, seems rather inadequate at describing the concept of a straight-line.

There are multiple definitions for the term "line." One definition, from analytic geometry, says a line in the plane "is the set of points that satisfy a given linear equation." Further, a linear equation is an equation either of the form x = a or y = mx + b.

 

[PeroK]

I'm yet to see anyone actually provide a strict definition as to what a straight-line is;

The definition of a straight line is covered in GCSE Maths (Geometry). For example:

http://www.examsolutions.net/tutori...aight-line/?level=GCSE&module=gcse&topic=1239

 

[Stephen Tashi]

but my goal (I think), was to try to begin to understand how professionals within Mathematics strictly define a straight-line

The situation (perhaps embarrassing from the point of view of a logician) is that most mathematicians do not think of the concepts of plane geometry in a completely rigorous fashion. So when they discuss concepts like "straight line" they don't have a formal definition for it on the tip of their tongue. By contrast, if you ask about the definition of something like "derivative" or "limit of a sequence" they know familiar standard definitions.

There are developments of plane geometry in a formal manner. (E.g. "Elementary Geometry from an Advanced Standpoint" by Edwin E. Moise). However most mathematicians don't think of geometry that way.

however, I'm to understand that circles are essentially formed from straight-lines (from rotating a straight-line upon one point)?

I've never see a circle defined that way.

And so, such a definition of a straight-line, which described in more detail a straight-line, stills seemed somewhat circular; also, I'd imagine to try and elaborate on the basic definition of a straight-line of "the shortest distance between two points" would necessitate an understanding of the fundamental shapes in the cosmos? (As, in my mere understanding, there is nothing to say that all matter, at its most microcosmic level, are circles and cylinders).

If you want to define "straight line" (or any other concept) with mathematical formality, you must begin by stating what you will used as undefined terms and then make your other definitions using those terms. For example, in the book by Moise, the term "line" is an undefined term! The fact that terms are undefined things does not preclude a system of mathematics stating assumptions that declare relations among such things. For example, Moise takes "point" and "line" as undefined but his system states the assumption that a line is set of points and it uses the assumption "Given two different points there is exactly one line containing them". ( Of course, in order to describe a system of mathematics that gives Euclidean plane geometry, further assumptions are required).

The history of trying to make Euclidean geometry rigorous gave rise to non-Euclidean geometry (https://en.wikipedia.org/wiki/Non-Euclidean_geometry) and the realization that the concept of "distance" has more than one mathematican model. "Lines" in different geometries have different properties.

 

[Matth.ew]

The situation (perhaps embarrassing from the point of view of a logician) is that most mathematicians do not think of the concepts of plane geometry in a completely rigorous fashion. So when they discuss concepts like "straight line" they don't have a formal definition for it on the tip of their tongue. By contrast, if you ask about the definition of something like "derivative" or "limit of a sequence" they know familiar standard definitions.

There are developments of plane geometry in a formal manner. (E.g. "Elementary Geometry from an Advanced Standpoint" by Edwin E. Moise). However most mathematicians don't think of geometry that way.

I've never see a circle defined that way.
If you want to define "straight line" (or any other concept) with mathematical formality, you must begin by stating what you will used as undefined terms and then make your other definitions using those terms. For example, in the book by Moise, the term "line" is an undefined term! The fact that terms are undefined things does not preclude a system of mathematics stating assumptions that declare relations among such things. For example, Moise takes "point" and "line" as undefined but his system states the assumption that a line is set of points and it uses the assumption "Given two different points there is exactly one line containing them". ( Of course, in order to describe a system of mathematics that gives Euclidean plane geometry, further assumptions are required).

The history of trying to make Euclidean geometry rigorous gave rise to non-Euclidean geometry (https://en.wikipedia.org/wiki/Non-Euclidean_geometry) and the realization that the concept of "distance" has more than one mathematican model. "Lines" in different geometries have different properties.

Thank you! I appreciate your time and patience in replying. In my mere opinion, it seems that you are quite well read on the topic; I shall certainly look through the suggested readings. That's certainly insightful that a "line" is an undefined term and that most mathematicians don't think of geometry in a formal manner; I appreciate that some terms within professions are undefined, such as, with legislative matters, the word "fair" is often left undefined or the ambiguity of a basic description is at least noted. I just find it interesting how such basic concepts, such as a straight-line, isn't formally defined, much the same way that when angle measurement is taught at schools/college, it doesn't seem to often (if at all) be described how to first of all, draw 360 degrees in a circle from scratch; still to this day, like many things in mathematics, I have no idea how to produce 360 degrees in a circle from scratch. I understand that starting with 3 equilateral triangles and halving them and then halving each triangle within the circle, would result in, say, 384 or 768 same-size triangles emanating from the centre of a circle (which is essentially what a degree is, but, of course, what I described isn't a degree). Anyway, I digress. Thank you though!

Kindest regards,

Matt.