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Is 0m=0s=0g? This would seem somewhat logical given the mathematical properties of physical units, but also not necessary.

黄雨伞
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    NO; meters are meters and seconds are seconds. We are measuring different magnitudes. – Mauro ALLEGRANZA Sep 07 '21 at 12:24
  • but you could also say apples are apples and pears are pears, and still 0 apples are the same thing as 0 pears – 黄雨伞 Sep 07 '21 at 12:25
  • I do not like it... :-) I would prefer: "the number of apples (on my desk) is the same as the number of pears (on my desk)". – Mauro ALLEGRANZA Sep 07 '21 at 12:31
  • @MauroALLEGRANZA I would say yes. Zero is zero. The analogy of zero apples and zero oranges is exactly on point. Mathematically, zero is zero. Tell me this. If I showed you zero miles and zero pounds and zero seconds, by what criteria would you distinguish them? – user4894 Sep 07 '21 at 19:22
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    In set theory, all empty sets are identical, but in scalar mathematics, zero values aren’t the same. No apples + a pear = a pear, but 0m + 1s isn’t meaningful any more than 1m+1s. – Frog Sep 07 '21 at 20:28
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    This question belongs on Physics SE. Quantities have units in physics, and equating values with different units is a mistake. – Conifold Sep 07 '21 at 22:58
  • @Frog In a vector space, 0 times any vector is 0, even if the vectors in question are linearly independent. 0e_1 = 0e_2 = 0 for basis vectors e_1 and e_2. If I show you two tables, one containing zero apples and the other containing zero oranges, can you tell them apart? Of course not. – user4894 Sep 08 '21 at 02:19
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    @Conifold This question has two entries on Physics.SE, neither of which settles the matter, though many learned opinions are offered. https://physics.stackexchange.com/questions/286964/should-zero-be-followed-by-units and https://physics.stackexchange.com/questions/174521/is-0-mathrmm-dimensionless Also see Terry Tao's https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/ which is great reading but also does not settle the matter. – user4894 Sep 08 '21 at 04:55
  • @user4894 All the more reason not to rehash it here. – Conifold Sep 08 '21 at 05:07

2 Answers2

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Zero equals zero, whether it be a number of apples, a number of pears, a number of second, or a number of meters.

However, we should not confuse the number of the apples that are on the table with the apples that are on the table. If there are six apples on the table, the number of apples on the table is 6. If there are six pears on the table, the number of pears on the table is 6. So, the number of pears and the number of apples is the same. Yet, six apples is not the same thing as six pears. The same applies to meters and seconds.

However, zero apples have no physical existence, so zero apples are nothing. The same applies to pears. So zero apples is the same thing as zero pears. The same applies to meters and second. The same thing, yes, but still nothing.

Yet, it is possible to have for instance six apples and zero pears. So, having zero pears does not imply having nothing, but having nothing does imply having zero things and therefore zero pears and zero apples.

One way to picture this is to imagine one axis for each quality, with the number of things having one quality indicated by the abscissa corresponding to this quality. At the origin, then, zero things have any quality at all so that nothing is indicated by the origin of the system of axes. Make the best of that.

Speakpigeon
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If one equation equals another at a specific point, there is an equality where they cross. For

s = ut + ½at^2

distance is zero, when time is zero. This is just a reference frame, so it is saying motion will be tracked from the position at time defined as zero. In order to be able to keep track of relationships if you chose a different reference frame, say a different zero position also at the same start time, you need to know which zero is which in order to correctly transform. Consider what is involved in General Relativity, where high speeds & gravity can warp reference frames.

They are not truly equivalent because of dimensional analysis. This keeps track of correct related units, so if you wanted to go to previous position at a given time, you'd know you were using related scales.

From responses to the comments:

Reference frames are crucial. What if one bag has minus five apples, and the other plus five - if you mix them together is it the same as zero pears? Time IS clocks, space IS rulers, & the important thing is transformations - in which we find high speeds cause rotations in spacetime - see this interactive page on relativity, Inside Einstein's Head.

Do zero lots of 10 apples = 0 pears? The idea mathematics gives privileged access to the world doesn't work, rather it's a system of signs we define for our convenience. Zero by one reference is not by another, so to do say a Lorentz transformation you can't just say zero, you have to say the same zero across the calculation - ie explicitly state & be consistent with your reference frame. There is no transcendental zero, it is relative.

CriglCragl
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  • If I hand you two empty grocery bags and tell you that one contains zero apples and the other zero oranges, you would agree that the two bags have the exact same contents: zero. Right? Right. How on earth would you distinguish zero meters from zero seconds? It's another empty set weirdness. The set of purple flying elephants is the exact same set as the set of odd numbers divisible by 2. Namely, the empty set. There is only one empty set and there is only one zero. If you claim they are not equivalent, tell me the diagnostic criterion that distinguishes them. – user4894 Sep 07 '21 at 20:21
  • @user4894: I already have, reference frames. What if one bag has minus five apples, and the other plus five - if you mix them together is it the same as zero pears? Time IS clocks, space IS rulers, & the important thing is transformations - in which we find high speeds cause rotations in spacetime https://www.lucify.com/inside-einsteins-head/ – CriglCragl Sep 07 '21 at 20:35
  • Yes. If you have -5 apples and add 5 apples you have zero pears. If this is not clear, you need to think about it. In dimensional analysis, what are units? We take the product of a scalar and a unit. If the scalar is 0, the product is 0. The units no longer matter in the case of the scalar being 0, because 0 times mile is the same as 0 times purple elephant. The result is 0. And frankly if you don't have 0 pears, how many pears do you have? You need to get past your incorrect intuition. 0 times anything is 0. 0 seconds, 0 kilometers, 0 pears, and 0 flying elephants are all 0. They're the same. – user4894 Sep 07 '21 at 20:38
  • You are saying 10 apples = 0 pears? You need to get over the idea mathematics gives privileged access to the world rather than being a system of signs we define for our convenience. Zero by one reference is not by another, so to do say a Lorentz transformation you can't just say zero, you have to say *the same zero across the calculation* - ie explicitly state & be consistent with your reference frame. There is no transcendental zero, it is relative. – CriglCragl Sep 07 '21 at 20:52
  • "You are saying 10 apples = 0 pears?" -- Too dishonest to deserve response. You must know you've lost the argument if you have to stoop to outright lying. – user4894 Sep 07 '21 at 21:32
  • @user4894: Are you talking equations, points in reference frames, or bugging out over the magic of zero? Pick a lane. What do you claim the point of your observation is? It's a meaningless linguistic quibble. And you haven't dealt at all with the transformations issue. Noted. – CriglCragl Sep 07 '21 at 21:39
  • Your response to me was so dishonest and so totally disconnected from anything I wrote that you've demonstrated your insincerity. – user4894 Sep 07 '21 at 21:41
  • Look, here is the deal with "units." Consider a vector space with two basis vectors e1 and e2. For any nonzero scalar k, ke1 ≠ ke2. That's what it means for one second to be different than one meter. But it's an axiom of vector spaces that 0e1 = 0e2 = 0. So 0 seconds and 0 meters are exactly the same, namely 0. That's a mathematical fact, you can't handwave your way around it. Likewise if I have two tables, one with zero apples and another with zero oranges, you can not possibly distinguish them. That's because 0 of anything is 0. The units disappear because multiplying anything by 0 is zero. – user4894 Sep 08 '21 at 00:09
  • @user4894: 'Is the opposite of black white or invisible? Why so?' https://philosophy.stackexchange.com/questions/87223/is-the-opposite-of-black-white-or-invisible-why-so – CriglCragl Dec 04 '21 at 17:44
  • This was all way back in September. I reviewed the comment thread and I agree with my younger self. You're as wrong today as you were in September. Why are you trolling me about a different topic? – user4894 Dec 04 '21 at 19:54
  • Not trolling. Wrong word. You're not trolling. You're stalking. You're demonstrating a degree of emotional disturbance to respond to me in a three month old thread with an entirely unrelated comment. Please have some self-awareness about your apparent obsession with me. And consider the vector space axioms. In R^2, 0e1 = 0e2 = 0. A freshman physics student knows that. Likewise in the free Abelian group generated by a pear and an apple, 0 pears = 0 apples = 0. There is no question or controversy whatsoever. You're just wrong. And, apparently, a little psychologically off. – user4894 Dec 04 '21 at 20:04
  • @user4894: It seems you are unfamiliar with the principles of Socratic dialogue, & agreeing to disagree, or with this being a forum for public debate in which you are not compelled to engage or respond any more than you wish. I only aim to lay out my arguments clearly, I do not require that you agree with me. And I do not agree with you, which you should accept. Using vector jargon does not help your case, as you seem to think. It surprises me you can write so many comments, but not compose your own answer. – CriglCragl Dec 04 '21 at 22:36