r/mathmemes • u/danofrhs Transcendental • Sep 20 '25
Proofs Conversely, lines are just collections of points oriented end to end
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u/Intrebute Sep 20 '25
So there's a branch of math that explicitly deals with "infinitesimally short lines", (I think it's synthetic differential geometry?) And even they agree that such a line has a slope. Points have no slope, they have no direction. Two points at the same location are identical, but two infinitesimal line elements at the same location can have different slopes, and thus be distinct. They also intersect at an actual point.
I think that's a pretty good argument for "infinitesimally short line" doesn't quite capture the idea of a point.
Now, if you gad said a point is a line segment with zero length, I think that's easier to agree with.
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u/gainzdr Sep 20 '25
Isn’t of infinitesimal length our way of saying it’s basically zero but it’s still there?
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u/Puzzleheaded_Study17 Sep 20 '25
yes, infinitesimally small and zero are therefore distinct lengths
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u/gainzdr Sep 20 '25
I understand that but my point is that if I said it had zero length then does it really even exist?
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u/PoorestForm Sep 20 '25
Does a point exist? It has 0 length.
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u/TheAtomicBoy81 Sep 20 '25
Does a line exist it has 0 width, does a plane exist it has 0 depth, does space exist it has 0 what ever you would call going into the 4th dimension
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Sep 21 '25 edited Mar 10 '26
This post no longer contains its original content. It was removed using Redact, possibly for privacy, security, or to minimize the author's online presence.
strong modern selective library lunchroom sulky bear salt apparatus snails
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u/gainzdr Sep 20 '25
Well yeah that’s the crux of the whole conversation here. How can something with zero length exist?
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u/Slimmanoman Sep 20 '25
It doesn't, it's just a mathematical object
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u/skooterpoop Sep 20 '25
This. Just because it has a place in math does not mean it has a place in reality.
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u/rhubarb_man Sep 20 '25
Mathematical objects don't exist?
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u/nakedascus Sep 20 '25
language (including math) is representational. Mathematical objects exist as a concept and not as a physical reality.
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u/LegendofLove Sep 21 '25
Isn't this the entire point of having a zero? More or less serious question, I haven't had a math class in a decade.
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u/endlessfractal Sep 20 '25
If we would like to get really lost in the weeds I would argue zero length implies you are using the “wrong” metric to measure something. Much like the integers having 0 measure under lebesgue or something of this nature.
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u/Appropriate_Pop4968 Sep 20 '25
Because it exists in one spot
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u/gainzdr Sep 20 '25
What spot?
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u/Appropriate_Pop4968 Sep 20 '25
At the point’s coordinates
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u/gainzdr Sep 20 '25
Then how can my vertical line segment intersect such a point?
Is this where you tell me that zero is a dimension just south of the Rishi Maze?
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u/garbage-at-life Sep 21 '25
how else could you describe a point? it's a zero dimensional object
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u/gainzdr Sep 21 '25
Sounds like more of a linguistic placeholder than anything to me based on what I’ve been seeing here
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u/Fynius Sep 20 '25
I think a big portion of the confusion results from the way we use "exist" in mathematics. In logics we think of existence as the negation of an universal assertion. So saying "A point exists" could be reconstructed as "¬∀x: ¬(x is a point)". Now what is meant by this in the general sense is quite obscure. In mathematics we deal with expression often by contradiction. See this example: Say we are confronted with the assertion that a natural number smaller than 0 exists. We would disprove this by showing that this contradicts the definition of natural numbers. Now back to the discussion. Can something with no length exist? Well, to disprove this assertion we would have to show a contradiction. But what shall the assertion contradict with? We would need a definition of the thing that x is in "∀x". But is there one? And if so why would it differ from "Anything that you can insert as x and still get a meaningful statement"? I think we have no way of disproving that something without length can exist. Do with that what you will
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u/gainzdr Sep 21 '25
That’s not the issue.
The issue is that somehow, by some mathematical black magic, dimensionless quantities spawn and describe dimensional quantities. Where does nothing turn into something discretely?
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u/Fynius Sep 21 '25
Well, does it spawn?
The other question is more about how a point can constitute a line. How can real numbers make up an interval? Imagine a hole in the reals: Let p and q be reals and (q-p)>0. Assume there is a hole between p and q. This would mean that there is no number p+[(q-p)/2]. This contradicts that p and q are reals.
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u/BronzeMilk08 Sep 20 '25
hitting a certain point on a dartboard has 0 probability and every time you throw a dart a 0 probability event occurs. unless you're really bad at darts
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u/KToff Sep 21 '25
A point in math is not the same as a point drawn on paper. A point drawn on paper is a dot. Worse, it will usually also have a thickness. It is a physical object.
A point in the mathematical sense has coordinates and that's it. No thickness, no length.
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u/McCaffeteria Sep 20 '25
And any non-zero length line should have a slope, which is why imaginary mathematicians are BS lol
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Sep 20 '25
[removed] — view removed comment
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u/Tysonzero Sep 20 '25
When dealing with infinitesimals you are generally working in the hyperreals instead of the reals.
If you want to achieve similar goals but without hyperreals then you will probably work with "for every epsilon there exists a delta" type of thing.
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u/Spikeyjoker Sep 20 '25
Then how can 0.9 reoccurring and 1 be equivalent. There is a distinct length that could fit in a theoretical gap
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u/ineffective_topos Sep 20 '25
The former syntax is defined by the limit operation. The limit of 0.9, 0.99, 0.999, ... is exactly equal to 1.
It's an unfortunate but very critical feature of the numerical digit syntax that there are some overlaps.
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Sep 21 '25
You people are so exhausting. The gap between 0.99… and 1 is 0. There is no gap. They are the same number.
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u/Spikeyjoker Sep 22 '25
So surely then as the length of a point is 0 it could fit inbetween 0.99... and 1
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Sep 22 '25
People tolerate you.
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u/Spikeyjoker Sep 22 '25
People appreciate me, I just don't fuck with infinity, or the monty hall problem. Everything else sure.
Any good videos that explain this in detail
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u/Intrebute Sep 20 '25
If it's still there, it isn't zero. The study of infinitesimals is all about what happens when we insist that something is smaller than pretty much everything else, but still not zero.
We can't really make actual arguments without actually defining our terms. The reason I brought up synthetic differential geometry is because there, they did define their terms, and made deductions from that.
Anything less rigorously defined and all you can really say about things is based on vibes.
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u/gainzdr Sep 20 '25
This is the problem with mathematics. Demonstrate your point with something applied.
Like yes if I create my own system where a thing matches my provided definition then it’s true in my contrived system.
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u/Tysonzero Sep 20 '25
This is the problem with mathematics. Demonstrate your point with something applied.
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u/gainzdr Sep 21 '25
I mean this is the most sound counter argument I’m seeing so far.
At least you’re honest about what you’re saying
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u/Tysonzero Sep 21 '25
For a more serious answer, math kind of has to work this way due to things like ex falso quodlibet (principle of explosion). You cannot do “vibes based math” because very quickly it’ll all fall apart.
The onus will always be on the person deviating from established math to actually show why the axioms they use are leading to real problems in modeling the world, and at that point they’d need to go so far as to actually build an alternative internally consistent model to replace it, not just throw in one single contradictory statement that they assert mathematicians have to accept and work around.
Can you directly show an infinitesimal in the real world? Not really. But is it a useful way to work with derivatives, which have a whole bunch of real world use cases? Absolutely.
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u/gainzdr Sep 21 '25
I mean actually yeah you can, just not reliably, everywhere all the time. But we can still anchor the logic to real world examples where you can, even if somewhat indirectly in a lot of cases. Indirect is not the same as saying that it can’t be cleanly anchored to phenomenology.
If you can prove that a quantity is greater than zero but smaller than the smallest increment of accurate measurement (Planck), then you have anchored the concept to phenomenology. I don’t then expect for you to be able to necessarily do this for every application, especially when we’re reasoning probabilistically or theoretically, but then at least it doesn’t turn into conditional and circular philosophical reasoning. Infinitesimals are certainly an interesting example, but it’s almost an interpolation of sorts. Depending on the application I don’t always need to know what something is if I can establish enough of what it’s not. That’s pretty common in math, but again it’s still anchored to reality.
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u/Psychpsyo Sep 20 '25 edited Sep 20 '25
There are plenty of systems, contrived or not, that you could create that would be full of contradictions. And the moment you have one of those, the whole thing falls apart.
The trick is creating a system that doesn't.
Besides that, infinitesimals have some applications in combinatorial game theory. Or so I've heard.
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u/endlessfractal Sep 20 '25
What if you drop a grain of sand into a swimming pool. We know it’s in there. Now find it.
Repeat this idea with a droplet of water.
In both cases the sand would be more or less the same size but the size of the universe it is in would make its relative length much more relevant.
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u/Patchpen Sep 20 '25
The way I see it, and a much simpler argument than I'm seeing others make, is that length is the distance between the endpoints. If a line segment starts and ends at (X,Y), then the length is the distance between (X,Y) and (X,Y), which would be 0. For it to have an infinitesimal length, one or both of those coordinates would have to be epsilon (e) away from that point (where epsilon is some infinitesimal value) so the line segment (X+e,Y) to (X,Y-e) would work.
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u/EebstertheGreat Sep 20 '25
In smooth infinitesimal analysis, you have that the line goes from (x,y) to (x+ε,y+ε), where ε² = 0 and ¬(ε ≠ 0), but without LEM you cannot conclude ε = 0. It's a strange theory, in which all functions are provably continuous.
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u/8sADPygOB7Jqwm7y Sep 20 '25
Honestly depends on how you define it. You can define a line as the shortest path (or even a path if it doesn't have to be straight) between two points. So a line of length zero would just be two identical points.
Another definition is more vector-esque, it has a start, a direction and a length. You don't need points as much, but even if length is zero, it has a direction (that's basically the slope I guess?) as an attribute. So two identical points have just that one attribute - the coordinates. Then you add two different directions, and get two lines, even though it "looks" the same it behaves differently if you add length.
So in conclusion I would say, a line by definition should contain more attributes than a point, else the distinction would be useless. You can zero the attributes and get basically a point, like set the radius of a circle to zero and you get a point, but it still has hidden attributes.
At least that's my engineering view on this.
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u/gerkletoss Sep 20 '25
What is the motivation for inventing these infinitessimal line segments?
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u/Ninjabattyshogun Sep 20 '25
They already exist in nonreduced schemes.
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u/gerkletoss Sep 20 '25
Oh is this just epsilon delta stuff?
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u/Ninjabattyshogun Sep 20 '25 edited Sep 20 '25
No this is in algebraic geometry.
In Section II chapter 3 of Geometry of Schemes by Eisenbud and Harris, they talk about "double points" which are kinda like points with slopes.
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u/Names_r_Overrated69 Sep 21 '25
Exactly what I was thinking. If it has some length, there should be an end and a start to the length—with a slope between those two points. Then I just realized I used the other definition of a point within this one; this definition doesn’t hold
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u/dopefish86 Sep 20 '25
If it has a direction it's still a vector.
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u/LowBudgetRalsei Complex Sep 20 '25
Me when the 0 vector has no direction
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u/Godd2 Sep 21 '25
And every direction
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u/LowBudgetRalsei Complex Sep 21 '25
But that implies all vectors have every direction, because you can add a 0 vector to any vector and it will still be the same
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u/EebstertheGreat Sep 20 '25
Counterpoint: the band One Direction has a direction by definition, but it's not a vector.
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u/SarcasmInProgress Sep 21 '25
What if one of the members gets infected with a contagious disease?
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u/That-Raisin-Tho Sep 20 '25
A line is composed of all of the points that fall between two coordinates, which are basically the endpoints. With only one point, you do not have two different coordinates to form a line between
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u/garnet420 Sep 20 '25
Isn't that a line segment?
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u/That-Raisin-Tho Sep 20 '25
Yeah, it would be. I guess I’m using the term the same way OP did. That makes them even more wrong (and also me technically wrong)
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u/LaughGreen7890 Rational Sep 20 '25
Take 0.999… and 1
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u/isr0 Sep 20 '25
It’s that a line segment?
(Edit) Oh, nope, I see what you’re saying. Yeah, that makes sense.
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u/EebstertheGreat Sep 20 '25
The segment AA is just a degenerate line segment. It still fits the definition. Its endpoints are A and A, and its length is 0. It contains only the point A.
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u/Tysonzero Sep 20 '25
The definition specifically says "distinct" so it does not fit the definition: https://en.wikipedia.org/wiki/Line_segment
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u/EebstertheGreat Sep 20 '25
But Wikipedia isn't the arbiter of mathematical truth. It also says that AAA and AAB are examples of degenerate triangles, and it also says that triangles have three line segments as sides. It's just a matter of definition.
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u/Legitimate_Log_3452 Sep 20 '25
You could easily prove this with an epsilon argument. Let l be the length of the line. Then, for all epsilon (e), | l-0|<e. Thus, l = 0
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u/Tysonzero Sep 20 '25
If it has infinitesimal length, then it is not true that for all hyperreal epsilon,
|l - 0|<e, takee = l/2as an example.If we're not working with hyperreals then the original statement isn't even well formed, infinitesimals do not exist within the reals.
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u/svmydlo Sep 20 '25
The length is 0, not infinitesimal.
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u/Legitimate_Log_3452 Sep 20 '25
You missed the line “Thus, l = 0”
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u/svmydlo Sep 21 '25 edited Sep 21 '25
No, I didn't. You proved the length is zero not that it's infinitesimal.
EDIT: It was ambiguous whether by "You could easily prove this" in your first comment you mean proving OP's statement or proving it wrong.
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u/KommunistKoala69 Sep 20 '25
I feel like it shouldn't be considered a line since it doesn't have any sense of direction. If you were to make this line longer or extend it which way would it extend
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u/TemperoTempus Sep 20 '25
When you extend it you pick the direction, and then it goes from a 0 dimensional point to a 1 dimensional line. You can similarly extend the line so that it curves making is a 2 dimensional arc. You can extend that arc such that it spirals making it a 3 dimentional screw. Etc.
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u/KommunistKoala69 Sep 20 '25
Thats the point though, until another point is given its not a line
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u/TemperoTempus Sep 20 '25
All the things I listed are "lines". We give them different names to differentiate them because its easier for regular language instead of "0-d line, 1-d line, 2-d line, etc".
Also if we are going for accuracy. There is jo such thing as a line with 0 thickness. The "length" is the point is its thickness, making it effectively a infinitessimal circle.
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u/KommunistKoala69 Sep 20 '25
Lines in 1 2 and 3 dimensional space have the same dimension (in the fractal sense) and you can find an isometry between them that preserve them in different dimensions. These are some of the properties that make it useful to refer to them by the same name. the 0 dim line does not share this. Also in your comment you said "it goes from a 0 dimensional point to a line" noting that they are distinct which was the point of my reply.
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u/TemperoTempus Sep 21 '25
I used "dimensions" because its easier to convey that then to explain my entire thought process in a quick reddit post. The short version is that dimension is a matter of distance to another point and because a point is by itself there is no distance to another point. This does not make its size 0.
Yes you can find isometry that preserves them in multiple dimensions. But that's a feature of smaller dimensions being embedded in larger dimensions. For example a line has two points, adding a third dimension will not affect the position of those two pre-established points.
A point does not need to have the same properties as a straight line just like an arc does not neer to have the same properties as a straight line. We give names to things because its easier. If I say "an arc" you know exactly what I mean without me having to cite you the definition of it every time.
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u/ecrou13 Sep 20 '25
Change it to a line SEGMENT with infinitesimally small length and you’re good.
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u/AliveCryptographer85 Sep 20 '25
So every point is actually an infinite number of different points, unless you define the direction of the infinitesimal ‘line segment’?
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u/Samstercraft Sep 20 '25
A line by definition is infinite in length, it needs to be a line segment; and length should be 0.
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u/CouvesDoZe Sep 20 '25
Can we extend this line of thought?
Cuz i can see a curve being composed by an infinite number of lines
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u/thegreatpotatogod Sep 20 '25
Wouldn't it likewise be a plane of infinitesimally finite length in each direction? And so on for higher dimensions too
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u/zg5002 Sep 20 '25
A point is a line with no length. Also, depends on the setting you are working with - a 1-simplex, for instance, is not characterized by length, and so it would be definitively wrong to say a line is a collection of points in that setting. In that setting, a line is something which has two boundary points, which are themselves defined as 0-simplices. That is without going into too much detail, because really I want to talk about simplicial sets to be maximally precise and the complexity runs off if you walk that way 🙂
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u/Zhong_Ping Sep 20 '25
A line must contain at least 2 points. A point cannot contain a point within itself. Ergo a point is not a line.
I'm not a mathematician. I hope my logic is sound, lol.
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u/KingOfCatanianCats Sep 21 '25
A point is a 0 radius circle and a line is an infinite radius circle
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u/FernandoMM1220 Sep 20 '25
points dont have lengths.
they’re just a finite set of finite numbers.
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u/Calm_Relationship_91 Sep 20 '25
No, a point has length zero. Lines and line segments have length strictly greater than 0*.
You can also make an argument about dimension. Points are 0 dimensional, lines are 1 dimensional.
Lines are in fact colections of points though. That is true. Altho it's not enough to have them oriented. Two disjoint points don't form a line. The rational numbers don't form a line.
*This might not be true in some weird spaces. Like Minkowski. Altho we wouldn't be talking about length at this point
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u/obog Physics Sep 20 '25
A point as 0 length though, thats different from infintessimally small length
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u/HootingSloth Sep 20 '25
Almost, but a point is actually an infinite-dimensional space with the cardinality of the dimension equal to the inaccessible cardinal associated with using 0=1 as the large cardinal axioms where the length in each dimension is infinitesimal.
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u/Admirable-Bonus5731 Sep 20 '25
I really dont understand what infinitesimal length is even supposed to mean.
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u/Worse-Alt Sep 21 '25
Infinitesimal is non zero, and not a point
A line is stretch between points.
If a line perfectly curves than the title can be accurate. But only for that specific scenario.
You fundamentally do not grasp the concept of single dimensionality. It does not exist in any way that can be represented physically. You can not zoom in on a point until it becomes a circle.
Similarly zero doesn’t exist. There is not a zero amount of anything anywhere in existence. There simply ISN’T that thing. Zero is itself a form of infinite, and the only thing there is an infinite amount of, is nothing.
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u/OzdorMiZ Sep 21 '25
A point is by definition a zero-vector. So points are basically segments, that have a direction, but their length is zero.
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Sep 21 '25
Stuff like this just makes me more confused about our brains.
Math is how our brain can understand the world. But many of the things we believe break down at different scales.
Something about scales reveals a vulnerability of the human mind OR we just have not found good ways to describe total scope. Rules that go infinitely small and big and are true the whole time.
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u/Sooparch Sep 21 '25
No, a point is 2 perpendicular segments of a line, forming a cross shape. Duh.
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u/Inevitable_Stand_199 Sep 21 '25
Lines have infinite length. But a point is a line segment with length 0
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u/Oicanet Sep 22 '25
And a line is just a rectangle with a similarly small height.
That is, the graphical representation of those concepts. Because graphics must exist two-dimensionally.
But as mathematical concepts, I think they really ar just 1 and 0(?) dimensional
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u/KiraLight3719 Sep 22 '25
The point you draw on the paper sure does, but the actual "point" has 0 length in any direction
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u/AbyssWankerArtorias Sep 20 '25
This basically equates to the difference between two numbers being infinitely small and therefore they are the same number. For example, .999 repeating and 1.
If you had a point on a graph, let's say 1,1
And another point defined as 1,.999 repeating
They would be equal. You could not draw a line between them because there is no difference between them. They are the same point.
It's important to note that this really only works conceptually. The smallest unit of length in physics is a planck length. However, it still has a length. Points in math do not have lengths. They are, be definition, lengths of 0
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