Recently seen by a colleague as a True/False question in homework, I'd be interested in your answers. Please respond to the poll, express opinions in replies, and boost for reach.

A line is parallel to itself:

@ColinTheMathmo True, which makes “being parallel” an equivalence relation among lines?

@ColinTheMathmo Weighed in on the other site; watching this one with interest. I have motivations for my answer but am saving them for after the vote.

I say "True" provided the definition is the slope is the same. The other concept that the lines don't touch each other is very weak and does not hold true for separate parallel lines in non Euclidean geometries.

@ColinTheMathmo My instinct (and vote) was false, since a line shares points with itself and parallel lines do not. But then I thought I want parallel to be an equivalence relation, so now I'm thinking true.

@olligobber @ColinTheMathmo I had an instinctive preference for 'true' and was trying to think why, and the mention of equivalence relations crystallized it for me. A line paralleling itself feels of a kind with the idea that an angle is congruent to itself.

(But I don't do serious geometry and there may be good reasons to not want parallelism to be an equivalence relation.)

@nebusj @olligobber @ColinTheMathmo Really it's a matter of definition but my preference is for false. Two lines can cross (one real point of intersection), be parallel (zero real points of intersection or one infinite point) or coincide (infinitely many points of intersection). I think we should keep those three cases separate, just like for integers we keep separate units (0 prime factors), primes (1 prime factor), composites (>1 but finitely many prime factors), and zero (all the factors).

@11011110 The question then is whether that's a good definition. It means that if A is parallel to B, and B is parallel to C, you can't necessarily say that A is parallel to C.

That feels ... unfortunate.

CC: @nebusj @olligobber

@11011110 But Euclidean and Hyperbolic geometries are different, and in Euclidean geometry it's "most true". So allowing it to be "always" true just solidifies a difference that's already there, and creates a consistency.

CC: @nebusj @olligobber

@ColinTheMathmo @11011110 @nebusj @olligobber
In another direction, in 3(+) dimensions, I think it's common for "parallel" to mean "in the same direction", in contrast to "intersecting". Now those two pieces are completely independent, so there are four types of lines: skew, intersecting but not identical, parallel but not identical, identical.

@bmreiniger That's true. The question remains as to whether the best/cleanest/elegantest underlying idea "parallel including coincident", or is the "best" underlying idea "parallel, not including coincident".

In some sense it really doesn't matter, but the thought processes in this non-important example may then prove to be useful and important in developing the concept of "taste" in definitions, etc.

CC: @11011110 @nebusj @olligobber

My understanding is that you need other line to compare to first to say if they are parallel or not. Statement just doesn't sound good to me, maybe I'm wrong

@ColinTheMathmo Yes if "parallel to" is defined as "have the same slope" or and it's Euclidean geometry. No if it is defined as "have no points in common.

@ColinTheMathmo @grainloom I’d say true if you define it as parallel at distance=0

Since True is the obvious answer, and so that is what I checked, I will also assume that the "correct" answer is false and that @ColinTheMathmo will explain why...otherwise, no reason for the poll...

It's interesting to see the majority of people from the fediverse I can see getting this wrong

@ColinTheMathmo Definition #1: Lines with the same slope are parallel. True.

Definition #2: Lines that never intersect when extended infinitely are parallel. False.

@ColinTheMathmo False; by definition, parallel lines cannot intersect anywhere along their lengths. A line that is "parallel with itself" cannot exist, as the "other line" (such as it is) intersects with itself at every point. The question is meaningless in the same way as asking if dividing infinity by infinity yields 1; but, since a boolean answer had to be given . . .

@vertigo Your definition is not the only option.

It's interesting that this was set as a homework question for a child, it's created an interesting discussion among several working mathematicians.

@ColinTheMathmo To be clearn, I didn't say it was. You asked for an answer to the question, and a rationale behind the answer. I provided one such answer to each.

This is why I clarified with the question being meaningless as queried.

@vertigo Oh, indeed. My comment wasn't a criticism, it was simply observing that there is a lot going on, much of it unrealised by most, and generally hidden from students.

@ColinTheMathmo @vertigo
It's always fun to have a go at Euclid, but did the working mathematicians say anything interesting? (In my working definition, a thing might be interesting because of the presence or absence of novelty, aesthetics, or utility)

@vertigo @ColinTheMathmo that was my reasoning as well, but i was never strong on relations


!@#$% lazy question, perhaps with no pedagogical function.

@seachanged As stated, in that context, agreed. It has, however, sparked a fantastic conversation elsewhere with potentially huge pedagogical value.



To be more clear, I was not saying that *your* question was lazy, or of no pedagogical importance. I love the graph that you've built.

@ColinTheMathmo I'd say that the topological definition (A ∩ B = ø, thus false) is more general/suitable than the analytic one (lines {a+b*t, c+d*t}, parallel iff b=d, thus 'true')
, but, my intuition is still on the fence about it

@ColinTheMathmo that depends of the truthness of zero being a even number

@ColinTheMathmo in short, our decimal system is suited for space-distance related modelling and zero is the reference/starting point. If one acknowledges zero as a neutral number that is neither odd nor even, it means the line is not paralell to itself. If zero is even, the line is paralell to itself

@dani I feel like you might be mixing up the question of "odd vs even" with the question of "positive vs negative". Different groups take different conventions about whether zero is positive, negative, both, or neither, but everyone I know of in mathematics agrees that zero is even.

@ColinTheMathmo for me, zero is neither odd nor even. It seems I like to come up with different answers :)

@dani To claim that zero is neither odd nor even is definitely not mainstream. To help me understand your thinking, let me ask a few questions:

* Is zero a number? (Some people say zero isn't a number at all).

* Is 6 an even number?

* Is 2 an even number?

* Is -2 an even number?

* What is your definition of an even number?

@ColinTheMathmo I enjoy your interest. For me, odd and even should be 3-fold instead of 2-fold. It could be achieved by adding a 3rd class, such as neutral.
Maintaining the 2-fold, the zero for me, is: out off the 2 classes; or gives an error; or is impossible to answer.
Answering your questions:
* That is a great question. In the decimal system, which is in scope, zero is a number
* 6 is even
* 2 is even
* -2 is even
* even number is a number which the modulus on the 2 ( n % 2 ), gives zero

@dani In every other instance, adding two even numbers gives and even number. On that basis, (-2) plus (2) should give an even number, which implies that 0 would be even.

And consider (n%2) when n=0 ... the answer is 0. That implies that 0 is even.

0 is between two odd numbers, that implies that 0 is even.

Do you still think that 0 is neither even nor odd? If it's *not* even then you can have two even numbers which add to a result that is *not* even. That seems unhelpful.

@ColinTheMathmo regarding my decimal system answer, you can replace my wording with: odd -> negative . even -> positive . the logic of the answer maintains, with such replacement.

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