Euclidean geometry assumes that planes are infinitely divisible so there is no end to how many points and lines can be on it. Oddly, Cantor showed that all planes have the same infinity of points if they are truly infinitely divisible. Therefore infinity is defined in an odd sense in mathematics. A sphere within a sphere has the same infinity of area as the one that contains it. They both never end in what can be measured and their areas have an uncountable infinity as their limit. This is a very difficult subject

Thanks Gregory, but I didn't understand the sphere example. Can you please explain it?

Actually a nasty little question (in a good way!). I hope it attracts some of the math heavy hitters. As to a proof such a line exists, I suspect it's just a matter of constructing it. For the rest, I think it devolves to a technical, curated understanding of exactly the infinity in use, what it is, and how it is used.

Well if matter is infinitely divisible than the parts divided go on forever. I don't know how the proof goes but Cantor proved that any size geometric object has the same infinity of parts as any other. This is a very counter intuitive area of study

That's ill-expressed, Gregory. Care to try again? A sphere within another clearly has a smaller area.

...the same infinity of area... — Gregory

Given a point in a plane, how many lines exist such that they do not pass through the point? — anon123

The answer: The number of lines in a plane - the number of lines in that plane that pass through that point = Infinity - Infinity. It's been a long drive, I'm tired. Here, you take the wheels.

First, if you divide something infinitely, infinite parts separate and fall about from each other. Second, Cantor said all geometric objects have the same size infinity of these parts

Perhaps more modern mathematics has improved on Cantor. I've started to think there is no infinite division possible in geometry, but that objects turns back on itself and remains finite in every respect. I wish I new more about the technical aspects of this but I don't

Good. you avoided mention of "area", which was misleading you. Keep going - your point is not yet clear.

You would do well to avoid avoiding the infinite. Rather, take a look at the diagonal argument as a way of understanding how mathematicians tame, rather than avoid, infinity.

A sphere within a sphere has the same infinity of area as the one that contains it. — Gregory

Thanks Gregory, but I didn't understand the sphere example. Can you please explain it? — anon123

Uh yeah, me too. Take the example in two dimensions. You have two concentric circles, one with radius 1 and the other with radius 2. Surely they have different circumferences; and surely if they are spheres they have different surface areas.

Perhaps @Gregory you mean that their points can be placed into one-to-one correspondence, as evidenced by drawing rays from the center, each ray passing through exactly one point of each circle or sphere.

Now the thing is there are infinitely many lines in a plane. Also there are infinitely many lines which pass through a point. So does it mean that no such line exists? Of course not because practically it does. I just need a proof of the same. — anon123

Here's a proof. Take the two-dimensional plane. Put the usual x-y rectangular coordinate system on it. One point's as good as another, so take your point to be the origin (0,0). You want to find lines that don't go through it. How about all the horizontal lines y = k for all nonzero k? That's infinitely many lines that don't go through the origin.

Of course infinitely many lines do go through the origin.

We can in fact make meaningful statements comparing these two quantities. We can in fact show that we can put the lines through the origin into one-to-one correspondence with the lines that don't go through the origin.

It's not much different than noting that we can put all the even numbers into one-to-one correspondence with the odd numbers, even though there are infinitely many of each. Every even number of the form 2n is paired up with an odd number of the form 2n + 1. So we can in fact make meaningful comparisons between two infinite quantities.

Perhaps Gregory you mean that their points can be placed into one-to-one correspondence, as evidenced by drawing rays from the center, each ray passing through exactly one point of each circle or sphere. — fishfry

Am I misreading or do you mean one point on on each circumference or surface?

Perhaps Gregory you mean that their points can be placed into one-to-one correspondence, as evidenced by drawing rays from the center, each ray passing through exactly one point of each circle or sphere.
— fishfry
Am I misreading or do you mean one point on on each circumference or surface? — tim wood

Not sure what you mean. Unless you are interpreting circle and sphere as including their enclosed regions. In math the convention is that a circle is just the points on the circumference, likewise for a sphere. Is that what you meant?

That is, the unit circle is the set of points whose distance is exactly one from the origin. The set of points less than or equal to one is a disk. Not sure what's the corresponding term for a sphere, but the sphere itself is just the points on the surface. It's a 2-dimensional manifold.

I stand instructed.

I stand instructed. — tim wood

That's ok, maybe I'll say something on one of the political threads and then you can throw rocks :-)

ps. Ball. I had to look that up, which means I'm starting to forget things I used to know. A solid sphere, including the enclosed region, is a ball.

So you are saying the cardinality vs density distinction applies in geometry?

I think there are too many problems with abstract geometry in this regard. I have always wonder if I could go off in one direction in space forever until a friend told me I would simply return to where I started. This blow my mind. Then recently I realized this applies too, perhaps, to the infinite divisibility thing. If I divide an object too much I will come back to the surface area. This makes sense to me as a looped curve

So you are saying the cardinality vs density distinction applies in geometry? — Gregory

I'm not sure how you got that from anything I wrote. I pointed out that there's a one-to-one correspondence between the points of any two circles, and that your use of the word area was off the mark.

I was trying to think if there was a density or measure argument regarding the set of lines through and not through a point, but I couldn't think of anything useful.

ps -- There is a dimension argument. The lines through a point are parameterized by a single real number, the slope of the line through the point. The lines not through the point require two points to specify their slope and y-intercept.

I think there are too many problems with abstract geometry in this regard. I have always wonder if I could go off in one direction in space forever until a friend told me I would simply return to where I started. — Gregory

That's a question of physics, not math. But it's a good question. You'd laugh at someone with a fat ass, then realize it's yourself. :-) Or rear end your own car and send your insurance company into an infinite loop.

This blow my mind. Then recently I realized this applies too, perhaps, to the infinite divisibility thing. If I divide an object too much I will come back to the surface area. This makes sense to me as a looped curve — Gregory

I don't think I can visualize that at all. If you take the unit interval and divide it into lengths 1/2, 1/4, 1/8, 1/16, etc., you don't get back to the unit interval. On the other hand, each subinterval is topologically the same as the original one. So if you're an ant on the real line, you have no way to know the scale of the points around you unless they're marked. In fact any open interval (an interval that omits its endpoints) is topologically equivalent to the entire infinitely long real line. I've always thought that's kind of interesting. Even if the universe is infinite in extent, it might really be finite, and the laws of physics just make you slow down as you reach the boundary. Mathematically it would amount to the same thing.

Well as I've mentioned to you before, I don't see how you can slice through a geometric object forever, creating infinite little pieces that keep dividing, while the whole remains finite

I see a point\infinitesimal as like a hole with a duplicate hole in it that 1) is the first one 2) but reverts back the the whole making the formation of the single finite object. Otherwise we have something finite made of infinite parts

"Given a point in a plane, how many lines exist such that they do not pass through the point?" — anon123

What's a line in a 2 dimensional plane?

A line is a set of pairs of points (x,y) such that y=mx+c, c is its intercept with the y axis and m is its gradient.

So let's pick some point in the plane, (u,v) - what defines a line that does not go through the point (u,v)? Well, it's a set of pairs that satisfies a line equation above such that (u,v) is not an element of the set.

That can be translated into an equation, a line does not intersect the point (u,v) if (u,v) is not a solution to its line equation. IE if (y-v) != m(x-u) for all y and x in the line equation. How many ways could that happen?

To get at that, let's consider an example, the point (u,v)=(1,1) isn't on the line y=2x+3, what would happen if we subbed it in? We'd get 1 on the left and 5 on the right, so the discrepancy between the left and the right would be 4. With that we could write that 1=2*1+3-4, subbing the point into y=mx+c and noting the discrepancy -4 on the right hand side.

What that reveals is that there's another way of thinking about the problem, we could make the following construction:

(1) A point (u,v)
(2) A line equation y=mx+c
(3) A discrepancy d such that v=mu+c+d

Notice that (u,v) will lay on the line y=mx+c if and only if d=0.

For a given point (u,v), can you find a discrepancy such that v=mu+c+d where d>0? Yes, the line equation y=mx+(c+d) generates such a point. Consider the point (1,1) and the discrepancy 3, That means we'd need to have an m and a c such that 1=m*1+c+3, which is one equation with two unknowns, so you can an infinite family of solutions m+c=-2.

And that was for a single discrepancy value.

Put all that together, then to have a a recipe for a line which doesn't contain a point, you pick a point, then you pick a discrepancy (say there are n discrepancies), then you have an infinite family of solutions for that discrepancy.

How many is that exactly? There's a discrepancy for every nonzero real number, and there's a line of solutions for every discrepancy, so that's the cardinality of the continuum squared. Which is the cardinality of the continuum again by the rules of transfinite arithmetic.

So there are as many lines satisfying your question as there are points in the plane, or points in your original line. Just infinity things.

Well as I've mentioned to you before, I don't see how you can slice through a geometric object forever, creating infinite little pieces that keep dividing, while the whole remains finite — Gregory

Suppose I cut a circular disk (that includes the boundary and the enclosed region) in half. How many pieces do you have? 2. Finite or infinite? Finite.

Cut the halves in half. Total pieces, 4. Finite. Cut them in half. Pieces 8, finite. Again. Pieces 16, finite.

You can always cut the pieces in half, and you always have finitely many pieces.

At no time is there ever an infinite number of pieces. Your intuition is fooling you. There are always finitely many pieces, whose area adds up to the size of the original circle.

I mentally divide objects though. I don't think if this is intuition or imagination, but flexing the segment infinitely than bringing it back like a slinky to the finite aspect of the segment, which certainly seems to require an extra thought in logic in order to accomplish because the slinky would stretch to infinity in both directions

I mentally divide objects though. I don't think if this is intuition or imagination, but flexing the segment infinitely than bringing it back like a slinky to the finite aspect of the segment, which certainly seems to require an extra thought in logic in order to accomplish because the slinky would stretch to infinity in both directions — Gregory

Do you see that no matter how many times you cut an object and its subparts in half, there are always finitely many parts? And that their area always sums up to exactly the original size?

Stretching a slinky is a little different. I take this as your noting that we can stretch the unit interval (0,1) to the entire real line? It's interesting that we can do that. The tangent function from high school trigonometry class does that very nicely, it maps the interval (-pi/2, pi/2) to the real line. The key point is that length is not a topological property. We can stretch things, even to infinity, while preserving all topological relationships.

Can we visualize this? Yes. Imagine a straight line through the origin of the x-y plane. Its slope at any point is actually the tangent of the angle it makes with the positive x-axis. When the line is nearly vertical going from lower left to upper right, its slope gets larger and larger, approaching infinity. When the line is nearly vertical going from upper left to lower right, its slope is smaller and smaller, approaching minus infinity. So as the line sweeps out an arc from going straight down to straight up. it corresponds every angle from -pi/2 to -pi/2, exclusive of the endpoints. to every real number. Literally easy to see.

Or just take the function 1/x. That maps (0,1) to (0, infinity). In this case we're only stretching half the slinky to infinity. Mathematically it's trivial, 1/x is one of the basic high school functions.

I may not be understanding the point you're making. The math is at high school level, it's very simple. Topological stretching does not preserve length, so we can stretch things arbitrarily, even to infinity.

Of course this is math, not physics. Nobody knows if these infinite stretching operations have physical meaning. You can't arbitrarily divide or stretch physical objects.

I get what you are saying. However a point has no dimensions, so how can it have any relation to space except as a limit. An infinitesimal is what comes closest to zero but that's an infinite region and is infinite. Infinite infinities equal finite space? How?

I get what you are saying. However a point has no dimensions, so how can it have any relation to space except as a limit. — Gregory

Do you mean physical space, as in the real world? No relation at all. Dimensionless points are purely a mathematical abstraction. For example a point on the x-y plane that we call, say, (2,3), meaning the point at x = 2 and y = 3, has mathematical existence as a pair of real numbers. But there are no dimensionless points in the real world. They're purely conceptual mathematical entities. Physicists find them useful when building mathematical models of the world, but they are not real (as far as we know, of course). They're only mathematical. Does that make sense?

If it helps to think about it this way, math is a complete, total fiction that just happens to be useful to people. But that's not unusual. The novel Moby Dick is a work of fiction, but it teaches us not to follow our obsessions to our doom. So works of fiction can be highly useful. Sometimes it's helpful to think about math that way. I'm not necessarily saying math IS this way, but that it often helps to THINK about it this way.

So: Math is fiction; fiction can sometimes be useful, so we need not believe that math is true in order to use it. Think Moby Dick.

I
An infinitesimal is what comes closest to zero — Gregory

There are no infinitesimals in the real numbers. We can work out mathematical systems that have infinitesimals, but I'm not sure how that helps us in the present conversation. If there are no dimensionless points in the real world, it's likewise unlikely that there are infinitesimals. I think that if you would separate math and physics in your mind it would be helpful.

but that's an infinite region and is infinite. — Gregory

Infinitesimals are not infinite. I'm not sure what you're saying.

Infinite infinities equal finite space? How? — Gregory

If you're asking how infinitely many dimensionless points can make up a region that has size, even mathematically, it's just the way it is. Nobody knows the answer to that.For example the unit interval consisting of all the real numbers between 0 and 1 has length 1, but it's made up of uncountably many dimensionless points. Just how it is. The concept goes back to Euclid., , although I don't think he talked about infinity. There's no explanation for how it works, if that's what you mean.

That helps me. Thanks :)

That helps me. Thanks — Gregory

Glad I could help. You're welcome :-)

Consider a Koch snowflake.... finite area, infinite circumference.

"Infinite infinities equal finite space? How?"

Integral calculus is how. I think 3blue1brown's 'essence of calculus' is a really good exploration of the intuitions and the intuitive objections and shocking weirdness of calculus. https://www.youtube.com/watch?v=WUvTyaaNkzM . "My iidea is for you to come away thinking you could have invented calculus yourself..."

Must be a slow thread but this is the clearest answer so far.

For those who like geometry puzzles, try Xsection. It starts off really easy but quickly becomes quite difficult. I'm proud to say I managed to complete it after 6 months and now in hindsight it has become really easy. So it trains your geometric thinking and imagination too.

You're a good teacher. Thanks.