What comes or came first, straight or curved lines?
Hello, & Im sorry if this topic might be totally laughable for some, if not all, & so unworthy of a reply, but Id like to know what yall have to say about whats asked in the O.P.s title, & perhaps (if its needed) a few words in order to further explain your answer, i.e., which one is ultimate, the straight or the curved line, and whats the consequence or significance of that?
Thanks.
Thanks.
Comments (123)
Quoting jgill
When I originally asked which one is first, or ultimate, I was thinking more in terms of independency/dependency, & not information, i.e., in terms of which can be independent of, or doesnt depend on, the other, & not which has either more of less information within its description. Yet, if you will, lets go with that.
So, from your post, I take your meaning to be that, in this case, whichever has less information within its description is ultimate. Please let me know if Ive misinterpreted you here.
Yet if I didnt misinterpret you, Im curious, do things differ from their descriptions, or do they exist only in so far as we describe them?
Has anyone ever drawn a perfectly straight line with no deviations? Has nature ever produced a perfect edge? Non-rotating black holes, however, are supposed to be perfect spheres.
Quoting RogueAI
Yet... that doesnt mean that its existence is impossible. Youd be hard-pressed, my friend, nay, downright defeated in trying to demonstrate that a thing is impossible just because its yet to exist or happen.
In 2001, one of the reasons the monolith is so disturbing is that its dimensions are so exact.
Okay, just to be clear, you were speaking in terms of probabilities, not possibilities. Got it.
Quoting RogueAI
You know that from experience?
Curvature is always measured relative to straight lines, and straightness is the negation of curved. It's two sides to one coin. You can't have one without the other.
Quoting frank
Thats self-contradictory. Indeed, the definition of x doesnt include not-x; thats like saying darkness cant be without light, as if darkness includes light.
Thus, if, according to you, one is the negation of the other, then that itself presupposes that their beings arent mutually inclusive, i.e., two sides of the same coin.
I agreed, at first, then thought that a circle is defined by its radius... one number.
So now puzzling.
Are you asking if I know from experience whether it's impossible to draw/create a long straight line/edge with no deviations greater than the planck length along it's length? I freely admit I have no experience working with planck length measurements. I don't think this subtracts from what I said.
It might not be logically impossible to be that exact with straight lines and edges, but I'm betting it's metaphysically impossible, like exceeding the speed of light.
The OP's question is like asking which came first, left-ness or right-ness? Up-ness or down-ness? The correct answer is, of course, that neither came first. The two can only co-occur as a dyad. This as @frank affirmed.
Like left and right, or up and down, straightness and curvature can only make sense in respect to the other, such that neither is its dyadic counterpart.
But you declare this self-contradictory. Though they occur at the same time as concepts of X and not-X, the two occur in different ways - rather than occurring in the same respect. The LNC is in no way violated, and so there's no logical contradiction involved.
A line is between two points. Between any two points there is an infinite number of possible lines.
As regards the world, no line takes precedence. Therefore, no line is ultimate, and no particular line takes precedence, whether straight or curved.
As regards the mind, the mind judges some lines to take precedence over others. For example, the mind judges that a straight line between two points takes precedence over a line that randomly wanders between the two points through space with no rhyme nor reason. Therefore, as far as the mind is concerned, the straight line is ultimate, and the straight line takes precedence.
Why does the mind judge that the straight line takes precedence over the infinite number of possible lines connecting any two points? Because the mind is of limited mental capacity, and in order to process what it observes in the world it has to reduce a complexity of observations into a few simple and comprehensible patterns. A straight line takes precedence in the mind, not because a straight line takes precedence in the world, but because the straight line has a simplicity that the mind can readily understand.
Quite exactly what a straight line is is open to debate, but people intuitively believe in straight lines even though there may be discussion as to exactly what a straight line is.
The significance of the straight line is that the mind is limited in its intellectual abilities, and always seeks to simplify complex observations into a few simple and readily comprehensible patterns, remembering that these patterns exist in the mind and not in the world of Neutral Monism.
:up:
Now, which is more ultimate, in the sense of (in)dependency?
In the Platonic realm, I would reckon that the straight line is more basic, because (in calculus) we think of curves as a collection of infinitely many infinitesimal points. So, a curve is, at the smallest level, a string of straight lines and points, the points perhaps being considered degenerate straight lines? This question shows that this is to a great degree a semantic question, since when we start analyzing things, we find our definitions to be in need of clarification.
In the mental realm, I think we have a discrete equivalent of the above. I think I've read something very relevant in some neuroscientific article, so I will link to it I find it.
A line is a theoretical one diminensional construct, consisting only of length without any width or depth. There are all sorts of ways to express one, but one way might be X=Y and the graph would show an infinitely long diagonal line. X=y^2 would be a parabola, and X=y^3 would be a different shaped curve. In all these instances, none is more "ultimate." They are just different ways to express data.
A line by itself is a non-empirical entity, meaning it does not exist in the observable world. Something without any width or depth can be described, but we don't have physical matter that lacks width or depth where we can actually confirm such a thing exists outside our mental construction of it. This isn't to say there aren't obvious reasons to describe lines when working within the observable world (like when you measure a wall before cutting trim), but that measurement, I'd argue, isn't of a line, but it's of a wall.
This is just to say that X=Y will always (as in every instance) be a more accurate way of representing the line than will be a graphical display of X=Y.
A straight line is also a collection of an infinite number of points. Whether straight, curved, or meandering, the line just describes extension over space, with an infinite number of locations (points) in between.
This is what makes it non-empirical as well. One would expect in the physical universe some limit on the divisibility of a physical object where it cannot be more subdivided, meaning a visible line would not be filled with an infinite number of spaces as a true mathematical line would.
That is to say, in the physical world, if object A is located at coordinate X and object B is located at coordinate Y, and the distance between the two is 0.00000000000000000000000000001 inches, it will be accurate at some point to say they occupy the same location. This is not suggesting we just round off, but it's to say there is some unit no longer sub-dividable and it therefore references the exact physical location (as opposed to mathematical)
My last sentence might not be correct, but it seems to logically follow.
Quoting Hanover
Yes, and a curved line is a collection of infinitely many straight lines. Without straight lines, a curved line could not exist, for it is made of straight lines. Thus, straight lines have a greater degree of basicness. You could have straight lines without curved lines, but you could not have curved lines without straight lines.
I don't think this claim is true according to mathematical calculus. In calculus if the infinitesimal rate of change is constant it can be represented by a straight line, but it does not follow from this that curved lines are made of straight lines, even according to calculus.
In fact straight lines and curved lines appear to be incommensurable in the way you are suggesting. What is curved is not straight and what is straight is not curved, and neither one is made up of the other.
How is X=y^2 (a parabola) a series of staight lines? From any point on the line to any other point on the parabola, there isn't a staight line, but it's a continuous curve, otherwise it would be V shape.
This is to say that if you measured any section of a parabola from point A to point B, its length would be greater than a straight line from point A to point B.
I agree.
You can do the simple math. On a parabola, no two Y coordinates share the same X coordinate, which would be necessary for a straight line.
Your proof for the thesis that curved lines are not composed of straight lines is concise:
Quoting Hanover
Quoting Leontiskos
On the other hand, if one considers the infinite series that constructs pi, it is an iterative relation between two series, an interaffecting of two linear trajectories. Without the prior concept of linear iteration, the series that constructs pi would not be possible.
Don't forget the center.
Quoting Ø implies everything
From the perspective of some computer programs perhaps. However, the length of a contour is defined as such a sum taken to the continuous in the form of an integral.
It's all relative. From one point of view, the straight line is ultimate and from another point of view, the curve is ultimate.
The terms straight and curved are human terms, not something that exist in a world of Neutral Monism.
In human terms, Einstein's theory of General Relativity showed that space-time is not flat but curved by the distribution of mass and energy. Bodies move in curved orbits because they follow a straight path in curved space, ie, they follow geodesics, the shortest path between two points. Bodies always follow geodesics in four dimensional; space-time. In the absence of matter these geodesics correspond to straight lines in three dimensional space. In the presence of matter, four dimensional space is distorted, causing the paths of bodies in three dimensional space to curve.
IE, in the world, the terms straight and curved don't exist, but in human terms, in a three-dimensional space the straight line is ultimate, but in a four-dimensional space-time, the curve is ultimate. So which is ultimate depends on your viewpoint, relatively speaking.
This is basic geometry. A curved line is fundamentally incompatible with a straight line because the one has two dimensions while the other only has one dimension. In other words, one has an attribute or property which the other cannot have. To make matters worse, there is a basic incommensurability between the two distinct dimensions as demonstrated by the irrational nature of the ratio between two equal length perpendicular lines, and also the irrational nature of the ratio between the circumference and diameter of a circle.
Attempts by mathematicians to produce compatibility between the two, such as saying that a circle is composed of an infinite number of straight lines, are just smoke and mirrors sophistry. Such mathematicians are better known as mathemagicians.
Quoting Metaphysician Undercover
What if we said that a circle is composed of an infinite series of ratios between straight lines?
I wouldn't dare! Although the proposed label has a certain appeal.
I don't know, can you say what those ratios would be?
Quoting jgill
You have to admit that what people can do with mathematics is extremely fascinating, even magical. I think it is magic, plain and simple.
Interesting, can you say more? My math is rusty. According to Wikipedia the "infinite series" and the "iterative algorithm" are two different approaches among many. (link)
Quoting RogueAI
So, what, then, is the basis of what you said, let me guess... , probabilities?
Quoting RogueAI
Id like to know how you distinguish between metaphorical & logical possibility, i.e., how do they differ?
Quoting javra
Its really not the same, indiscernible.
Quoting javra
Can we observe darkness without light, & vice versa? If so, hows it then that they must co-exist, like right-&-left, up-&-down?
Quoting RussellAQuoting RussellAQuoting RussellA
By what mean or means have you come to know the world, as opposed to the mind, & the like viewpoints?
Quoting Hanover Express what data?
Quoting Hanover
... representing the line ... . What here is the relationship between the representation & whats represented?
Quoting Ø implies everything
Assuming the platonic realm is a petitio principii. Hows there any other realm than the platonic?
You might want to spell that out.
You never answered my question in post #3.
Quoting ItIsWhatItIs
Why didnt you answer it, if I may so ask in addition to asking you again to answer it?
Descriptions can vary in detail. We can describe the path of a cannonball mathematically but that is a far cry from being on the receiving end of that trajectory.Physical things exist independent of our descriptions but ideas exist within their descriptions.
It may sound like I know what I am talking about, but I don't. :cool:
Yet, I must ask, so why talk or make declarations without knowing what youre talking about?
Quoting jgillFrom whence are ideas, which exist within this or that description, had?
I need to make the meaning of "world" more explicit.
From my personal observations, backed up by science and all founded on Innatism, my belief is that some things are animate having minds such as life and some things are inanimate not having minds such as rocks. Rocks exist in that part of the world that is mind-independent, and humans exist in that part of the world that includes minds.
I accept that my beliefs may be wrong, but in a sense, I don't need to know whether they are true or not as long as my beliefs allow me to successfully integrate into the world that I observe. This is a pragmatic philosophy, that whether my belief works or not is more important than whether my belief is true or not.
IE, I have come to believe in a world that is part mind-independent and part mind through personal observation, backed up by science and all founded on Innatism.
If a curve is made of infinitely many points, then it is is made of infinitely many pairs of consecutive points.
A straight line is defined as two points, with their gap (if there is any) being filled with a row of rectilinear points. So, a curve is at minimum composed of infinitely many two-point straight lines.
Not sure what you mean. I never claimed that the Platonic realm exists, although by talking about it, it seems I'm implying it exists. However, I only talked about it as a "if it exists, I think it would be like this". In reality, I don't think the Platonic realm exists as an actual reality, but I refer to it a lot as really a collection of constructs as how we think of them, as opposed to constructs as how they are. Constructs as objects impacted only by their definition, and nothing else; no imperfections, that is.
As for your second question, ask Plato. I can't find the source right now, but I think I read he was something akin to a substance trialist; believing in the Platonic realm, mental realm and physical realm.
A straight line is not defined as two points. It's defined as the distance between two points. A point has no dimensions. No length, no width, no depth. Any line would have an infinite number of points.
You're envisioning physical space, like a computer screen with a finite resolution, where a line has a finite number of pixels and a pixel has a very small length.
That doesn't apply to what we're talking about. Imagine the resolution being infinite, meaning between any two points, there are an infinite number of points.
Sorry, my mistake.
Quoting RussellA
A thing can not have a mind & yet still not be mind-independent, such as objects within our dreams. So, to say that x doesnt have a mind, therefore x is mind-independent is fallacious. The formers demonstration doesnt entail the latters.
Quoting HanoverWhats the distance between two co-existent dimensionless points?.....
You'd have to measure it.
If point A is at 0,0 and point B is at 4,0, the distance would be 4.
Regards, stay safe 'n well.
Trying to fit in to the intellectual atmosphere here. :smile:
And practicing what one of the few professional philosophers on the site said was an approved notion: You can be wrong as long as what you say is interesting. :cool:
Can we measure a dimensionless thing? If so, by what means?
Quoting jgill
L.o.l., thats funny. I get what you mean, ha.
Quoting jgill
You see, thats where I think a line should be drawn. If you (Im referring to any reader in general) know that youre (again, not you particularly) wrong, dont say it; but if youre unsure, argue for it as fervently, yet respectfully, as possible.
... & the hips dont lie, am I right, Torus? L.o.l..
Quoting Torus34
... thanks, & you too, Torus.
A thing that is dreamt about, such as a rock, which in our dream doesn't have a mind, in our dream is mind-independent. Yet this thing, this rock, because it is in our dream, has been created by the mind, and because created by the mind, cannot be independent of the mind, cannot be mind-independent.
A more complete sentence would have been: "We may dream about a thing that can not have a mind & yet still because this thing has been created by the mind will not be mind-independent."
We're not measuring the point. We're measuring the distance between points, which is a line, which does have a dimension, and that dimension is length.
A point has location, but has no dimension. If it did, it would occupy more than one location.
How is this complicated?
No, the distance is the length of the line, which is explicitly not the line itself. A line is an object of geometry; its length is merely a number corresponding to that object.
However, that's all merely irrelevant pedantry. You mention the infinite resolution of lines, which for lines in [math] \Bbb R^{n>0} [/math] follows from the density of the real numbers (and the continuity of such lines follows from the completeness of the reals). Okay, so what? I mentioned that too. You are clearly missing my point.
As I said, since lines a composed of infinitely many points, they are then composed of infinitely many pairs of points, because of the lines continuity, in conjunction with the existence of the bijection [math] \Bbb R^2 \mapsto \Bbb R^4 [/math]. Now, those pairs of points; are they lines (or to be more specific, line segments)? Well of course. If you disagree, please explain how these pairs of points are not line segments.
Quoting Ø implies everything
In an infinite series of points, no two are immediately adjacent to the other. There will always be a point in between.
If I plot a point at 0,0, what point do you claim is immediately adjacent, and what is the length you claim between the two points?
Quoting Ø implies everything
No two points are next to each other without an infinite number of points being between those points, so when you identify point A and claim there is a line to another point B, what is the distance from A to B? How long is that line segment you reference? Is it 1/X as X approaches infinity?
https://en.wikipedia.org/wiki/Surreal_number
Not complicated, just unclear. So, if you oblige me a little longer, Id really appreciate it.
Quoting Hanover
Uh... measuring from one thing to another includes both of them within the measurement, i.e., measuring where they start & stop. If the points, from which your line supposedly starts & stops, arent included within your observation, how can you determine your lines length?
You see, for instance, this is what I mean by unclear.
Quoting RussellA
How dont these two parts of your latest posts first paragraph contradict each other?
No. The point has no length. The point defines only a location. It's not like measuring a board where you have to cut on the other side of the pencil dot to be sure it's the right size. The point itself literally takes up no space because it is dimensionless. It's like the world's tiniest dot, so small, it has a length of zero.
And to be clear, there is no suggestion here that the physical world comports to this mathematical construct. It is not possible to infinitely divide a board, or any piece of matter.
I'm late to the party, but curve is the more general notion and you could define straight lines as a special case, curves of constant curvature 0.
@jgill?
Sure. I work with contours in the complex plane all the time, and rarely are they straight lines.
Whatever is "decided" here will have little impact on mathematical practices. :cool:
One thing I was thinking -- assuming we're playing this game for whatever reason -- is that it's a question of whether you think of getting a curve by bending a straight line, or getting a straight line by straightening out a curve. If you allow that there already is a sense in which the straight line is curved, just with the degenerate curvature of 0, then that's a way of explaining in what sense a straight line is bendable: it already has a curvature and what we call bending or straightening is modifying that curvature. (I think I'm sort of doing differential topology on 1-spaces, but it's been so many decades.)
And thus the curve is the more fundamental notion, and comes first. If you start with straight lines, you've no justification for the concept of bending.
The other obvious approach would be to talk about geodesics, but that's enough silliness.
Now, Im no mathematician myself, but all this talk of defining straight and curved lines by points brings to mind the alternative notions of point-free topology and point-free geometry.
Which seems to beg the question, what is ontologically primary: point-based lines or point-free lines?
Yes, points were historically conceived of first, but no one has ever seen one, being that theyre volumeless and such. Which to me indicates that theres something to be said for the ontological primacy of point-free lines - both of the curved or straight varieties. These we've all seen.
:roll: :nerd: :razz:
I dont think that youve rightly understood what was meant by my previous posts only question, perhaps I wasnt clear enough. Would you like me to repeat &-or explain it? If not, thats alright too. Im not being condescending really. Honestly, just trying not to be pressing/pushy.
Quoting Hanover
So, infinite division is impossible because things, as matter, arent made up infinitely of parts?
Quoting Srap Tasmaner
As its said, better late than never, right? You dont have to answer that, l.o.l..
Quoting Srap Tasmaner
So, youre saying that straightness is to curvature as a special case is to a general notion?
I assume we accept that rocks in the world don't have minds.
I assume we accept that when we dream about a rock, the rock we dream about has been created by the mind.
Where is the problem ?
The relationship between the mind and a world outside the mind is critical to answering your original question: "which one is ultimate, the straight or the curved line".
As I wrote: "As regards the world, no line takes precedence........... As regards the mind, the mind judges some lines to take precedence over others."
Your original question cannot be answered without taking into account the difference between the mind's attitude to what is straight and curved and the existence of straight and curved in a world outside of any mind.
I think I did. You asked if point A and point B would be included in a line that went from A to B, asking then if A and B would be included in the length from A to B, correct?
My answer is no, A and B add nothing to the length of AB because they have no length themselves, but are simple addresses (so to speak) of the beginning and ending of the line.
It's not as if points are the atomic subparts of a line. A line is not composed of points. A line simply has infinite locations along its expanse. A line is composed of nothing in the physical sense. It's a concept describing the length from A to B.
The question of the op is not what is the current practise, but which came first.
Quoting Srap Tasmaner
Yeah sure, straight lines are already curved, just like squares are already circular.
Quoting Srap Tasmaner
Right, start with a self-contradicting premise, "the straight line Is curved", to get the logical conclusion you desire. That's rigorous logic at its finest.
I don't think it's possible to have any lines in nature, straight or curved if one has a reductive view of reality. It's like Mandlebrot with fractal geometry in nature; something can look perfectly straight at one scale, but if you zoom in to the scale of molecules you have tons of bumps and valleys. If you zoom in even closer you have vast areas of empty space between the "points" of atomic nuclei and seething electron clouds. Likewise, a ball of string can look like a tiny point from far enough away, a 3D ball from closer, a line when considering any given length of string from arm's length, or a point again when zoomed in down to the scale of atoms. The dimensionality is variable, and fractional.
But you also have self similarity at different scales (e.g., things appearing as points when zoomed in or out enough, be it atoms or star systems). It's like how the coastline of the UK looks the same at different scales and any true measurement of its length becomes impossible because the more you zoom in the more little inlets and peninsulas show up. So, dimensionality itself gives way to patterns of recursion, self-similarity at intervals.
Plus, we should really be questioning if '"elementary particles," exist from which to construct our "lines" (which can't be true lines because they can't be dense). These "particles" seem to be more a sort of robust stability in process, or "energy well stabilities." In QFT, such particles don't exist, so any straight line reduces to a cloud of probabilities across some field at a fine enough grain.
Or for your example, we can also describe a black hole as two dimensional (indeed we need to in order to get its entropy right).
That said, in a strongly emergent view you can have both. The computational argument for strong emergence is that estimates for the amount of information in the visible universe, which tend to be fairly close to one another, show that the total computing power of the entire universe isn't enough to calculate even relatively simple forms of life. That is, either the universe isn't computable, and it works through processing infinite amounts of information, even in very basic transactions (infinite actualities of precision, with physical events needing inaccessible real numbers to be described perfectly) or higher level structures impose top down causality on micro level interactions and there is true compressibility.
If the latter is true, a position advanced by Paul Davies, Mark Pexton, and Seth Lloyd, among many others, than straight and curved lines both began to exist around the same time, but only exist at certain scales, not absolutely. Conversely, if the reductionist view is true, then lines as such never exist even though continua do and lines across space-time exist as descriptions of real continuous space.
IMO, the pancomputationalist view suffers from trying to force the universe into a box shaped like our current epistemic constraints. I prefer Grisin's view more, applying intuitionist mathematics to physics. I think we're set to see a turn where this idea gains steam and pancomputationalism gets replaced with something new, which might play nicer with conceptions of Copenhagen's complimentarity or even Absolute Idealism, putting the "view from nowhere," to rest. In this sense, lines will exist, but relationally and relative to some scale only.
Infinite division of matter is not possible. Infinite division of lines is.
The former is a physics question, ultimately provable by empirical study.
The latter is a logic question, employing fundamental definitions of geometry and determining what is entailed from the definition.
This idea that lines are more than ideas is where things seem to lead to confusion. We can measure a board and that board will have the property of length that can be clearly identified, but that's not to say length itself exists outside of the things it describes.
A straight line, or a curve?
Climb the mast of a ship in mid ocean and look to the horizon. It appears in a clear day as a very straight line that is continuous with itself as a circle. We live on the surface of a globe, such that the flat surface formed by water is actually imperceptibly curved, but the curvature only becomes apparent at a much greater height than the ship's mast, that is, from space.
Straightness and curvature must arise together, because they get their meaning from the 'line' that we draw between them conceptually.
but see also http://aleph0.clarku.edu/~djoyce/elements/elements.html where it becomes clear that the straight line is defined first, and the circle is derived later as a curve equidistant from a point. and other curves can be defined later still in the elements; an ellipse for example as a curve of equidistant sum from 2 points
I'm impressed by the discussion that has emerged from the initial question. I'm sure that if my observations are merely disruptive, everyone will ignore me.
First, my dim memories of basic geometry are that the system builds up from points to lines (the locus of the points giving the shortest distance between two points), to circles (the locus of points equidistant from a given point. That would be one interpretation of what fundamental means, wouldn't it?
Second, I would suggest that one can explain the relationship between the ideal geometrical objects (points, lines, curves) and physical objects as the limit of our process of developing more and more accurate methods of measuring space, within the limits of Euclidean geometry.
This itself is contested ground. To be sure, that's the most popular take, but in general I don't think most people spend much time considering mathematical foundations. Intuitionists and finitists of various stripes would deny that such infinite division actually exists. I haven't developed strong convictions one way or those other, but I do feel like the general silence re:foundations in how mathematics is taught contributes to problems with mapping mathematics to nature.
Agreed, and more over this length is only clearly identifiable to some level of precision. If you keep trying to make the length measurement more and more precise then eventually different sections of the board will have different lengths because it isn't perfectly smooth. Moreover, predicting how different parts of the board will vary in length will turn out to be a chaotic process with strong susceptibility to initial conditions if we get precise enough, as there will be variance in how long each section is relative to others based on wear in the saw blade, the pulse of the person operating the saw, small changes in atmospheric conditions and moisture, etc. So, complexity and chaos have an epistemically defined element, where how we observe changes how we should describe the system. That's been a big takeaway from my reading as of late (the North Holland Handbook of Philosophy of Science volume on Complex Systems and the Routledge Handbook of Emergence).
Makes sense to me.
But this did leave me with one question. Doesn't a curve require a second dimension that a straight line does not require? Can you have a line curve without reference to either an X and a Y axis, or an implicit reference to two dimensions vis-a-vis how line segments are described relevant to one another? It seems to me like 1 dimensional lines work, but not if they are curved. But then one dimension seems to come prior to the concept of two.
And yes, in the abstract sense you can define a curve as one dimensional since a single parameter can describe its points. But you can also fill a square with a line as well. So does it actually make sense to abstract shapes from the space they are embedded in? If we were Flatlanders, curves make sense as objects of perception and intuition, but if we're Pointlanders, the difference between curved and straight doesn't seem like it exists. Then again, I am also not sure why one dimension needs to necessarily come first, it just seems intuitive.
No, it's still a one-dimensional space in itself just as a line is.
We tend to think of dimensions as axes like lines because of Descartes, but it doesn't have to be that way. The curve is exactly as one-dimensional as the line, it just doesn't look to us like it matches up to a dimension.
But --- if you were a point on a curve, could you tell whether you were on a curve or a line? Our intuitions about points are kind of crummy here. It's not the case that in a curve the 'next' point is off-center a little, not least because there is no 'next point'.
I'd like to say you can tell, but I think you probably can't, and that's just as interesting. Lines and curves are equivalent in this way. That they differ can only be seen clearly when you embed them in a larger space, as you suggested, embedding them in a two-dimensional space.
But that two-dimensional space needn't be a plane. If it's a sphere, the surface of a ball, then lines just inherit the constant curvature of the space they're embedded in, so they're "straight" relative to that surface. They're the geodesics. If we embed that sphere in three-dimensional space, then we can talk about the curvature of both.
But it's been literally decades since I did any of this stuff.
We define a line as something that passes through an infinite number of points, where each point has no dimension.
Accepting that two points exist in the world outside the mind, what exactly is the something that passes between them. If this something cannot be seen and doesn't have causal powers, in what sense does it exist ?
Yes, exactly. We have a word for it because it's an interesting case and useful to us. (We also have special names for other curves that are interesting and useful to us, the conic sections, the sigmoid curve, and so on.)
As I was saying to @Count Timothy von Icarus, you can straighten a curve and you can bend a line, so transformations are possible from each to the other. The way to capture that transformability is to see them both as special cases of something more general -- but you can just use curve as already being the more general thing.
There are still curiosities waiting in the wings, like space-filling curves. Like fractals, they can mess with your understanding of dimensionality.
Honestly, if we were talking psychology it would be very tempting to say line comes first. There is a sort of sophistication to the concept of a curve. Lines feel simple, clear. It might hardly matter that children encounter far more curvy and bumpy things than straight and flat things, because cognition is all about simplifying and amplifying. But if we were going to talk developmental psychology, the question would have to be a lot more specific, and the research out there -- and I feel confident there is some -- is probably answering a variety of different versions of your question.
We have some good evidence that specific areas of the visual cortex develop to identify (or perhaps construct) lines out of visual data coming from the optic nerve. You have dedicated areas for vertical lines, horizontal lines, diagonals, etc. If you expose cats to only some types of lines from birth through a critical period of development then they have problems detecting line orientation and object boundaries when exposed to a normal environment.
Of course, if you totally restrict a mammals' access to certain types of sensory data they effectively lose that sense (e.g. sewing an eye shut leads to blindness in that eye if it is done soon after birth, but not in adulthood), which certainly says something about the idea of a priori knowledge in practice. Lack of experience leads to severe brain damage, so maybe it's not a great route to presuppositionless thought. On a similar note, damage to the occipital lobe often causes patients to lose the ability to recall sight, visualize, or dream of sight, whereas damage to the eyes does not impede the ability to visualize.
But even here, it seems like topology, the need to define "edges" of objects, comes first, and our ability to recognize lines and infer 3D shapes from 2D drawings is an ancillary consequence of this. Although arguably, this is only because researchers do what Donald Hoffman accuses them of, and assume that the world is the way vision presents it, and that we just need to explain how evolution got it right for us.
Anyhow, it doesn't seem like one dimension comes before three psychologically, now that I think of it. We don't just experience the world as three dimensional, but our perceptual system maps it out via patterns of neuronal activation that mimic the 3D shape of the thing our nervous system is trying to encode. We are very boundary attuned, which makes it all the more remarkable to discover that discrete physical boundaries are seemingly impossible to define rigorously in most cases if we want a high/fine grained degree of precision.
https://www.frontiersin.org/articles/10.3389/fnhum.2011.00118/full
Good stuff!
In that great Veritasium video about the Ames window there's some talk of the 'constructed world' hypothesis too, that it's exploiting our adaptation to a world full of rectangles which nature would not have provided.
I remember also hearing once long ago that hospitals used to paint the colored guiding paths (how to get to post-op, how to get to labor & delivery, etc.) on the floor because people could keep track of them more intuitively. (Hospitals don't seem to do this much anymore, so maybe the effect is too small.) I remember thinking it was interesting that we might naturally have a firmer grasp of what's above and what's below because those are relatively fixed, whereas left and right are by design constantly shifting.
The boundary-seeking stuff is interesting, because mathematically boundaries are always one dimension down from the dimensionality of the object, but we're also unavoidably talking about projections (onto the eye) so we're dropping a dimension there too. How those go together or work at cross purposes is pretty complicated.
In the Euclidean plane, y=mx+b describes a straight line, in two dimensions, whereas y=b takes that down a step to one dimension.
This is, nevertheless, an entertaining rabbit hole.
Quoting Metaphysician Undercover
Oh dear. Took my eye off the ball. :sad:
Whats the qualitative difference between a straight line in the former & in the latter world? If theres none, how arent you guilty of a false dichotomy?
So, your line isnt of matter, since its infinitely divisible unlike matter. Good to know, as that itself ultimately lets me know where the problem in this whole conversation of ours lied.
A line is not a set of points, a line passes through a set of points
A straight line certainly exists as a concept in the mind. This allows us to talk about them and and use them in science, otherwise this thread wouldn't exist. However, my belief is that straight lines and lines in general don't ontologically exist in a world outside the mind. Therefore, there is a qualitative difference between straight lines in the mind and straight lines outside the mind - straight lines exist as concepts in the mind but don't exist outside the mind.
I assume we both accept that straight lines exist as concepts in the mind. Our concept of a line is something that passes through an infinite number of points, where each point has no dimension. The question is, do straight lines exist outside the mind.
As regards lines outside the mind, if the line passes through a set of points, does the line exist independently of the points it passes through or does the line exist because of the points it passes through.
If situation one, the line exists independently of the points it passes through, then why is the line defined as something that passes through a set of points. Why refer to the points it passes through at all. Of what ontological relevance are the points that the line passes through. A line may happen to pass through points, but why in the first place define a line as something that passes through a set of points.
If situation two, the line exists because of the points it passes through, then the line cannot be independent of the points it passes through, meaning that the line cannot have an ontological existence over and above a set of points, with the consequence that outside the mind there are points but no lines.
Is there any argument that outside the mind a line has an ontological existence independent of the points that it happens to pass through ?
It's not "according to me" but common usage, at least in mathematics.
A line is a curve with constant curvature of 0. It is what mathematicians call the "degenerate case", meaning lines have the property of curvature only in a sort of formal sense. This allows us to unify the cases of wiggly things and straight things as types of a single more general object, and it is a very common thing to do in mathematics.
What I hoped to show was that you can capture in mathematics the intuition that you get a curve by bending a straight line and you get a straight line by straightening out a curve (bending it back). If you're interested in that idea, the bendability of things, then there is a branch of mathematics that gives you a rigorous way of talking about it: topology.
You don't have to treat this as a mathematical question. Mathematics doesn't "own" lines and curves. But it is the result of thousands of years of smart people thinking about these things, so it's there if you want to see how they view it.
A nice metaphysical topic in math. :up:
On Husserl's the Origin of Geometry:
Kinda; though [math] 1/\infty [/math] is only a number within the hyperreals and other such extensions of the reals (like the surreals, to which you linked).
Okay, so here's the thing. You are correct as far as conventional mathematics is concerned. Mathematics has become comfortable with absolute infinity, and this concept allows for the things you speak of.
My stance is rooted in my own philosophy, which I would say is more akin to constructivist mathematics on this point (though correct me if I am wrong). A continuous curve is not, to me, an actual collection of infinitely many points. Instead, it is a collection of finitely many points, between each of which, a boundless number of points could be computed. That is, only potential infinity exists, not absolute infinity. In finite structures exists information capable of endless computation; at no point would there ever be computed an infinite amount, but the computation could be carried on indefinitely nonetheless.
Thus, the Platonic forms of such mathematical objects are not structures of infinite size and resolution, but rather objects of infinite potential. Our interaction with them is through computation, through which they are actualized. Such computation will always represent the curve at some finite resolution, at which the curve will indeed consist of straight lines; despite the fact that a finer resolution could be achieved.
Since absolute infinity, to me, does not exist in any shape or form (not even in my equivalent of the Platonic realm), a continuous curve will only ever be a collection of straight lines, with the potential to consist of even smaller straight lines; but before that potential is actualized, it does not.
This changes what a curve even is. Its substance isn't merely defined by what it is, but also by what it could be. It may sound like I am merely describing mathematics in practice and not in principle; but really, I am making ontic claims here. I think of reality as procedural at every level, and infinity as only ever manifesting in the potential.
But to be perfectly clear, as far as conventional mathematics is concerned, you are correct in this discussion.
:up:
I've come to a similar view. But one of the things I still find weird is this: the infinite, Platonic line, curve, circle, etc. seems to be more compressible than the "actual." That is, a line of infinite length and density can be described very simply. Actualizing it takes computational resources, and we can actualize as much of an approximation of the line as we choose to, increasing the length, increasing the density, etc. But then, when we're done, we're left with an object that is actually harder to describe than the Platonic line, because now we have to specify the finite length and density. The algorithmic entropy has increased.
This is sort of like how removing elements from a set can increase the information needed to describe or construct that set.
Which, I suppose is perhaps not all that weird depending on how we look at it. It means that actuality comes about through a sort of generative, creative force, rather than being something "reduced," from possibility.
You might be interested in some of the ways this sort of thinking gets applied to physics: https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/
As a mathematician who explores the world of complex variables, I agree with you more or less. I have never worked on theory that embodies infinities, treating them like real numbers with an associated arithmetic, only with the notion that there are unbounded processes. When I write a program for constructing a contour in the plane it always depends upon tiny line segments (or vectors) added together.
However, by limiting one's self to these kinds of approximations, limits are ignored, and consequently, ordinary calculus with its integrals (infinite sums) and easy manipulations thereof is out of reach.
But you have a point.
In mathematics, a straight line is a special case of a curve. A curve is a one-dimensional continuum.
All you are doing is demonstrating that what you call "mathematics" is self-contradicting. The "continuum" which you speak of consists of arcs which are created from circles. The circle is two dimensional, and the arc of the circle is two dimensional. yet you now deem it as "one-dimensional" for the sake of constructing your "continuum". Therefore this constructed "continuum" is fundamentally self-contradicting.
Are you saying a straight line is not a continuum? Take a pencil and draw a straight line, then take a pencil and draw a curve that is not a straight line. In both cases you can do so without lifting the pencil. They are both continuum. There is no special difference between the two. A curve is one dimensional.
The curve can be drawn on a two dimensional paper, as can a straight line, but they are both one dimensional.
Mathematics is the least contradictory thing I have come across, everyday intuition is far more contradictory.
What you are doing is exchanging "continuum" with "one-dimensional". Your conclusion "a curve is one dimensional" is based on the unsound premise, that if it is a continuum, then it is one dimensional, i.e. "all continuums are one dimensional". Of course this premise is unsound, as the space-time continuum is a continuum, and it is clearly multidimensional.
Quoting PhilosophyRunner
Mathematics is the most contradictory thing I've come across. That's why I didn't progress far in high school mathematics, and turned toward philosophy instead. My intuition told me not to blindly accept the rules being force-fed into my brain. I ought to understand them first, and only accept what is consistent and coherent. Physics was almost as bad, for self-contradiction. They explained to us that wave motion as activity within a substance, demonstrated this with wave tanks and diagrams as to how the activity is an activity of the particles which make up that substance, then they proceeded to tell us to understand light activity as waves without a substance.
lol, yeah. The modern conception of dimensionality is born out of attempts to resolve all sorts of contradictions. In many cases what we see is definitions agreed upon on a pragmatic basis, because they allow us to do useful things, or because they are the "lesser of two evils." The resolutions often don't come in the form of definitive proofs that change everyone's mind at once, rather taking time to be accepted (and social pressure, I recall Brouwer got thrown out of a prestigious society for intuitionism). The concept of "zero" was itself one of these contentious entities in the West for a considerable period of time.
This is why I think that the history of mathematics should be taught more at the K-12 level. If nothing else, it's good for students to see how mathematics can be written about, but I think it also helps a certain type of student see the value in math. My problem once getting to algebra in middle school was that math didn't feel rigorous enough. Like, "yeah, you're telling me these are the rules, but why are they the rules? How did we decide on this?"
I was not exchanging one for another, I was stating two separate points. A straight line and a curve are both continuum, as can simply be seen by trying to draw both. In both instances you can draw them continuously with a pencil without needing to stop and start again. This is pretty self evident.
Next point. Both are one dimensional because in both cases, a point on the curve/straight line can be given by a single parameter. I mentioned this only because you brought up "on-dimensional" in your post - this applies to both.
There are no accepted self-contradictory theories in maths. If you are able to show a theory is self-contradictory, then you have disproved that theory beyond all doubt, and it will be binned.
Since a point cannot be straight or curved, what distinguishes a point on a curved line from a point on a straight line? And how does "a single parameter" determine whether the point is a straight line point, a curved line point, a spherical point, a space-time point, or one of an infinite number of other types of points?
Quoting PhilosophyRunner
The skeptic always has the upper hand, because it's not possible to prove anything beyond a doubt. That's just a simple fact of human existence, and human knowledge. We end up accusing the skeptic of being unreasonable, but this does not extinguish the skeptic's doubt.
Now is that a fact?
Maybe, but I'm afraid I can't prove it beyond a doubt.
A little bit of a word game here. In elementary math one can say that the curve in the complex plane, z(t)=cos(t)+isin(2t) depends upon a single parameter, as does the line z(t)=t+2it , but each point on these curves is an ordered pair of real numbers when embedded in the Euclidean plane - a two dimensional vector space.
Quoting ItIsWhatItIs
And the winner is . . . .!? Which would you prefer driving down a dark road at night? :chin:
You know, I read some of the information "about" you on your profile page, wherein, from the first line or so, I think, it reads that you're a retired mathematics professor, & so I'd really like to ask you, if it's no problem, do you believe that you a finite quantity can be made up infinitely of things?
I'll provide an answer to that, if I may. A human being is made up of two distinct types of qualities, material and immaterial (dualism). If you try to quantify all human properties from either one of these categories, you will approach the boundary between these two, immaterial qualities which cannot be counted as material, or material qualities which cannot be counted as immaterial. The use of conventional mathematical axioms will produce the appearance of infinity at the boundary. Therefore trying to quantify all human qualities within one category will produce the appearance of infinity because of the incommensurability between the two distinct types of human qualities.
Sure, since seems to ignore questions intentionally, let's consider what you have to say.
Quoting Metaphysician Undercover
You didn't seem to answer my question, which is/ was a simple yes-or-no one. Your answer here leaves one uncertain as to whether you've said either.
For, you mention the "appearance" of infinity. When using that term, you're invoking the distinction between "appearance" & "reality," yes? As if it only "appears" that way but isn't "really," like Shepard tables differ in size apparently but aren't really? Like, it's only an "illusion"?
Or are you saying that appearance is reality? So that, according to you, the incommensurability between the material & the immaterial is infinite in reality. Yet, even in this latter case, there's a problem to be found, because an incommensurability can only legitimately be represented between what's extended. Now, in your representation of this incommensurability, either the "immaterial" is extended, & therefore it's really not distinct from, & isn't definable as a negation of, the "material," i.e., there's no dualism, or it's not, & thus it can't stand in an incommensurable relation at all, making talk of such a thing absurd.
So, you see, I'm still uncertain as to what your answer can mean. A simple "yes" or "no" helps.
Quoting ItIsWhatItIs
Quoting ItIsWhatItIs
The straight line. There is only one, although it can be placed in various positions in space. Bend it and get curves, of which many exist.
Quoting ItIsWhatItIs
A physical object, no. A mathematical object, yes. The Riemann integral is an infinite sum and is connected to an ordinary derivative, fundamental to virtually all technological advancement in recent years.
Oh that's quite good. I didn't even think of that.
I didn't answer the question, because it didn't make any sense to me. In no way can I understand myself as "a finite quantity". So my answer was meant to show how your proposition, a person as a finite quantity is incoherent.
Quoting ItIsWhatItIs
All cases of infinity are appearance only. For example, that the natural numbers might continue infinitely is just an appearance. The same is the case for any instance when it looks like something might continue infinitely, it is always an appearance. It only appears that way.
But appearances are just as real as anything else, that's why we have real appearances. So I am not attempting to distinguish between appearance and reality.
... & therefore, according to you, a "mathematical" object isn't a "physical" object. Okay, thus, 'tis Platonism for you, mister professor. Got it.
Quoting Metaphysician Undercover
My question wasn't: can you understand yourself as a finite quantity? So, for you to say that there's no way that you can, is irrelevant & inconsequential. Moreover, I never propositioned what you claim that I did with my question. You're going somewhere else with this, bro.
Quoting Metaphysician Undercover
So, according to you, infinity appears to us. Okay, got it.
Yes but that doesn't affect the dimensions of the curve. You can similarly embed the curve in 3,4,5... dimensions, however the embedded curve is still one dimensional. The additional coordinates can all be derived from a single parameter, for example t using (sin(t), cos(t)).
This question makes no sense to me. It is like asking - Since a handle cannot be a door, what distinguishes a handle on a door from a handle on a window? Pretty obvious answer, isn't it - a handle on a door is on a door, and a handle on a window is on a window.
Similarly a point on a curve is on a curve, and a point on a straight line is on a straight line (which is a type of curve as I have explained previously).
A point is an exact location in space.
See my answer in the post above.
For those fascinated by dimensions of curves: How can a curve be one-dimensional?
Ok, by your analogy, the location of "the handle" is what distinguishes whether it is on a door, or on a window. Now the problem is to justify calling both of these "a handle". I mean, you could randomly pick any part of a window, and call it a handle, or any part of a door, and call it a handle, and say that's a handle on a window, and that's a handle on a door, but then "handle" loses its meaning. You allow "handle" to refer to anything, in your effort to say that there is a handle on a window, and a handle on a door, there could be also a handle on the ceiling, and a handle on the floor.
So we must define "handle", in order to make the judgement that there is a handle on the window and a handle on the door. And "handle" is defined by its function. Now, let's assume a point. Can we define "a point", so as to justify the claim that there is a point on a straight line, and also a point on a curved line? This requires showing how the point would relate to a straight line, and how it would relate to a curved line, like we show through definition, that the handle has the function of being something we grasp, and hold on to, and this is how we justify that there is a handle on a door, as well as on a window.
Quoting jgill
The explanation on that page, where the person describes the top half, and the bottom half of a circle, is not acceptable, because the top and bottom both have the exact same description. In reality, the top half and the bottom half of a circle are curved in opposite directions, so they cannot have the same description.
A point is an exact location in space. There is a location in space on both a curve and a straight line (more than one location).
For an intuitive approximation, draw a straight line. Now put a pin on that straight line, that is a point on the straight line. Do the same for a curve, that is a point on a curve.
The entire circle can be given by one parametric variable t, where:
x=rcos(t)
y=rsin(t)
Curved and straight lines are spatial concepts. they do not have any particular "location in space". So if a point is "an exact location in space", the point does not exist on a line, neither curved nor straight.
Quoting PhilosophyRunner
You said a point "is an exact location". Obviously "an intuitive approximation" cannot serve as an example of an exact point, because "exact" and "approximation" have incompatible meanings.
Quoting PhilosophyRunner
It seems to me, that "cos" and "sin" are not one, but two parameters.
Spatial - relating to or occupying space. Your two sentences contradict each other.
Neither "sin" nor "cos" are parameters. "t" is the parameter.
In the analogy, the point would be an approximate location. But it is an analogy, not meant to be perfect. That a straight line is a curve is formally correct in math, and also intuitively obvious.
Read your own definition PR, it says relating to "or" occupying space. Obviously, in the context above, "spatial concepts" means "relating to" rather than occupying space. And there is no such contradiction. Spatial concepts relate to space, they do not occupy space.
Quoting PhilosophyRunner
Ok, my mistake, I was careless in my choice of words. Let's just say that sin and cos are two distinct aspects, and "dimension" is defined as an aspect. Therefore there is two dimensions, regardless of how many "parameters" there are.
And this spatial concept has conceptual points on it.
Sorry, I can't make sense of your statement.
The concept of a line is referencing something in space, even though the concept itself is not in space. That referent has points on it.
Just like a concept of a strawberry does not occupy space, but a strawberry does occupy space. If we discuss a conceptual strawberry this conceptualization will include it being in space.
Similarly if we discuss a conceptual line or curve, this conceptualization includes it being in space (at least 1 dimension, but can be embedded in more).
Thanks for recognizing and admitting the fault of your previous reply. However, I believe there is still a fault here. The concept of a line, does not itself reference any particular thing, or place in space. When mathematics is applied to measure something, or represent something, such as a line on a map or plan, then there is something in space referenced.
This is the difference between the universal and the particular. The concept itself is known as a universal, there is no particular which it represents, because it can be applied to a vast multitude of things in space, yet it does not actually reference any particular one of them. The particular thing in space only gets referenced by the concept according to the specifications of the application.
Quoting PhilosophyRunner
According to what I stated above then, we need to distinguish between the concept itself, which is the universal, referring to not any particular object in space, but potentially referring to a vast multitude, and the specifications for its application. Do you agree that such a distinction ought to be made, the distinction between the concept itself, and the specifications of how the concept ought to be applied to particular things which we understand as existing in space?
I propose that this distinction is important and necessary because the concept indicates a very specific aspect of existence in space. So to apply the concept to things actually existing in space we need a formula to determine that aspect which the concept will be properly applied to.
While the universal concept does not exist in space, included within that universal concept is space. This is the thing that I tried to clumsily explain in one line yesterday, and I can't quite find elegant words for it.
You cannot have a concept of a line that is removed from space - such a thing would be a concept of something other than a line.
Just like the concept of a strawberry includes space in that concept. If it didn't it would be a concept of a something that had no height or width or length, which is not a concept of a strawberry at all.
Another way I could say it is as follows. Close your eyes and have a mental picture of a curve. That mental picture you have is one of a curve and one of space as well. You can't have a mental picture of a curve without that mental picture including space (at least 1 dimension). Your criticism is that the mental picture does not exist in space. No it doesn't, but within that mental picture the curve is in space (at least 1 dimension).
Now to give a contrasting example of a completely different concept - the concept of a number does not require space to be included in that concept. You can correctly challenge me if I were to talk about a point on the number 3. But a concept of a line does require space to be included in it, otherwise it is not a concept of a line.
You could draw a line to represent the concept of line. And, I suppose one could have a mental picture of a line. But how does that drawn picture, or the mental picture for that matter, include space? This makes no sense to me. When I see a line drawn on a piece of paper, in no way do I see the concept of space there. Nor do I see one dimension of space. So your statement "You can't have a mental picture of a curve without that mental picture including space (at least 1 dimension)", not only appears false, but it doesn't make any sense to me.
Anyway, none of this seems very relevant. The issue was, how are we going to relate "a point", which indicates an exact location in space, to a straight line and to a curved line.
Two infinitesimals can make a segment. 3 can make a curve ib
Well, two distinct points can be interpreted as end points of a line segment. The arithmetic of infinitesimals is different. As part of the hyperreals they do not form a metric space, so there is no "distance" between them. But they are complicated and beyond my realm.
Do infinitesimals have shape?
To my knowledge, no. But new aspects of mathematics seem to open up daily. For me, for a and b = infinitesimals, a+b=a, a-b=0. But I have avoided non-standard analysis, as do most in classical complex variables.
If i can pick your brain, a line segment seems to be convergent (because it has two end points) but it seems also to be divergent (because the segment is groudless by being composed of spaceless points). Berkeley said something about "ghosts of departed points" (with respect to infinitesimals). It appears to me that all geometry is infinite and finite somewhat like a content and form respectively
The line on a piece of paper is a one dimensional entity (the line) embedded in a two dimensional entity (the paper). So most definitely there is space in that concept.
Imagine a paper without space. A no dimensional paper. You can't - such a thing is not a paper, and you can't draw a line on that thing.
I wish to add that mathematical space does not have to be the 3 dimensional space you or I perceive. You can have, for example, a 100 dimensional space, with a 1 dimensional line embedded in it. You can have an space with any positive integer dimension.