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August 1

Drawing a line??

A line, according to geometry, has one dimension, length. However, can we actually draw a line?? Even the finest line that can be drawn is in fact a rectangle whose width is in the millimeter interval. Georgia guy (talk) 14:54, 1 August 2020 (UTC)[reply]

Okay, so you just answered your own question. –Deacon Vorbis (carbon • videos) 15:00, 1 August 2020 (UTC)[reply]

And if I want to know if anything in my answer is wrong. Georgia guy (talk) 15:04, 1 August 2020 (UTC)[reply]

Given an arbitrary drawing, how do you define what is drawn? In principle, you are free to define a drawing of an object to be a line if the width is less than, say 1 mm. Count Iblis (talk) 15:15, 1 August 2020 (UTC)[reply]

Technically, even a rectangle whose length is 5 inches and whose width is 500 micrometers is still a rectangle; its area is 5 times 0.02, or 5/50, or 1/10 of a square inch. Georgia guy (talk) 15:30, 1 August 2020 (UTC)[reply]

Your drawing would have to violate the laws of physics"

"Several experts agreed that real numbers don’t seem to be physically real, and that physicists need a new formalism that doesn’t rely on them. Ahmed Almheiri, a theoretical physicist at the Institute for Advanced Study who studies black holes and quantum gravity, said quantum mechanics “precludes the existence of the continuum.” Quantum math bundles energy and other quantities into packets, which are more like whole numbers rather than a continuum. And infinite numbers get truncated inside black holes. “A black hole may seem to have a continuously infinite number of internal states, but [these get] cut off,” he said, due to quantum gravitational effects. “Real numbers can’t exist, because you can’t hide them inside black holes. Otherwise they’d be able to hide an infinite amount of information.”" Count Iblis (talk) 16:15, 1 August 2020 (UTC)[reply]

Any line physically drawn in the conventional sense of leaving a mark on a piece of paper or such with a pencil or pen must be at least one atom wide – but that will not be enough to make it visible. If you place such a line that is visible to the naked eye under a microscope, you will not see a neat band with sharp edges, but a spotty stripe with fuzzy fractal-like edges. It is a far cry from the neat mathematical abstract notion of a rectangle. If it is true that lengths below the Planck length have no physical meaning, then the laws of physics do not allow an embodiment of the mathematical concept of a line. --Lambiam 18:45, 1 August 2020 (UTC)[reply]

Is this a question on math or metaphysics? Well, summer tends to be slow around here, so sure, I'll play. Plato would have said that a line belongs to the world of ideas and so cannot exist in the real world. (Or rather that the world of ideas is the only real existence and that what we can physically perceive are only approximations of it; we don't need to go quite that far though. See [1] the section on Plato). So while we can draw a representation of a line, a line in the mathematical sense can't be drawn, and saying "draw a line from A to B" is only a figure of speech. Whether the "real world" is an approximation of Plato's ideal world where mathematical concepts actually exist, or mathematical concepts are merely idealized models of what exists in the "real world" (which I believe is the modern point of view), is actually irrelevant for doing mathematics. A mathematical concept is defined in an axiomatic system, where it is either one of a small number of undefined terms or it's defined using those terms. Whether the axiomatic system has anything to do with reality is for physicists to decide; mathematicians make deductions regarding the system using logic, not experiment, and that's carried out independently of what physicists decide about it. --RDBury (talk) 04:10, 2 August 2020 (UTC)[reply]

Some mathematicians might disagree with the notion that mathematical concepts are defined in an axiomatic system; rather, their position is that mathematical objects exist independently of our reasoning about them, and defining them is an issue of describing them in sufficient detail so that they are uniquely determined by the description. The axiomatic system is kind of an afterthought, a tool to help you reason about the mathematical objects. (For all clarity, that is not my position.) --Lambiam 14:49, 2 August 2020 (UTC)[reply]

Very true; the philosophy of mathematics has many schools of thought and fundamental differences of opinion have never been fully worked out. I happened to read a lot of Bertrand Russel when I was a little baby graduate student so the above probably falls into the Logicist camp. The advent of computers has changed the landscape somewhat as well so a pure Logicist philosophy is not as tenable as it once was. --RDBury (talk) 19:07, 2 August 2020 (UTC)[reply]

Of course you can draw a line! It won't look very much like a real geometrical line especially if you take a microscope to it, but you're conveying the concept, which is what drawings are presumed to do. If you draw a house, it also won't look like a house, but you still will have drawn one. 93.136.160.249 (talk) 04:18, 3 August 2020 (UTC)[reply]

Ceci n'est pas une ligne. --Lambiam 08:02, 3 August 2020 (UTC)[reply]

So what is the width of the boundary between a region painted red and one painted blue? It can certainly be seen. → 2A00:23C6:AA08:E500:B166:A7FD:B971:41E9 (talk) 09:30, 3 August 2020 (UTC)[reply]

By now this is no longer a mathematical topic. I could not find photos illustrating this, but I bet that under increased magnification the boundary will start to lose definition, partly because it becomes fractal-like, partly because the colours mix and you cannot tell where the red ends and the blue begins. At distances below 500 nanometer these colours cease to exist, so there is also a natural limit in the order of one micron, however perfectly the regions have been painted. --Lambiam 17:37, 3 August 2020 (UTC)[reply]

Another limit is the size of the molecules making up the pigments. Using other technologies for producing colour, such as stimulated emission, this limit might (theoretically) be passed, but then the next, unsurmountable limit is the atomic spacing in the emitting materials. --Lambiam 14:46, 4 August 2020 (UTC)[reply]