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Imagine, a mathematician from another galaxy lands on the earth. Is there a way we can explain to him what is "counterclockwise" without showing him a picture?

Things like Green's formula, Stokes formula etc. do not work - if in the beginning we choose to use "left handed" $x, y, z$-coordinate axes and for any vector field ${\bf F}$ define $\text{curl}{\bf F}$ as usual, the forms of these formulas remain the same. So is there a math theory that can make an absolute distinction between left handed and right handed coordinate systems?

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    $\begingroup$ Choosing "left-handed" or "right-handed" coordinate is essentially choosing an orientation of a manifold (and in turn a volume form). When you choose the left-handed coordinate system on $\Bbb R^3$, the volume form in Stokes theorem etc. is the negative of the usual volume form (ie. the one in right-handed coordinates). So, by choosing right-handed coordinate system on $\Bbb R^3$, we are just choosing an orientation of $\Bbb R^3$. Note that, an 'orientation' of $\Bbb R^3$ (or more generally, $\Bbb R^n$) is just a choice of an ordered basis of $\Bbb R^3$. $\endgroup$
    – feynhat
    May 22 at 2:18
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    $\begingroup$ We can say to your mathematician from another galaxy, that, "We have a preferred orientation (ie. choice of an ordered basis) of our space, which we call the right-handed system." $\endgroup$
    – feynhat
    May 22 at 2:21
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    $\begingroup$ Where did these tags come from? Seriously. $\endgroup$
    – Ted Shifrin
    May 22 at 3:31
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    $\begingroup$ It doesn't matter what galaxy the mathematician comes from; mathematical descriptors like "right handed" come from definitions, not observations. They may use different definitions than you (i.e. they may use the word "right" to refer to the opposite orientation of $\mathbb{R}^3$ than you do), but if these definitions are brought into agreement, there shouldn't be any issue. $\endgroup$
    – Kajelad
    May 22 at 3:41
  • $\begingroup$ I expect you'll enjoy some edition of Martin Gardner's The Ambidextrous Universe. $\endgroup$
    – Andrew D. Hwang
    May 22 at 11:32

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This is actually a question about physics, not about mathematics. Mathematics has no problem defining “the plane” as $\mathbb{R}^2$ and defining clockwise and counterclockwise on “the plane”. The problem is when you want “the plane” in mathematics to correspond to an actual plane in the real world - the same applies to 3D space.

It turns out that the laws of physics do not respect reflection symmetry (also known as parity symmetry or P-symmetry). That is, there are actually laws of physics whose equations depend on whether a left- or right-handed coordinate system is chosen. This is not at all obvious; the rules of Newtonian mechanics don’t indicate such a thing occurs, nor do the rules of relativity. But if your mathematician were a physicist, you could describe certain physical experiments related to particle decay they could perform to determine whether they’re using a left- or right-handed coordinate system.

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    $\begingroup$ From what I've read, parity violation is enough to distinguish left from right if the aliens are made of matter, rather than antimatter. But if the aliens are made of antimatter, they'll get left and right backwards. That works in OP's scenario, in which an alien lands on Earth and doesn't blow up. For general communication with aliens, you can apparently use CP-violation to tell if you're made of matter or antimatter, and, once you know that, you can use parity violation to distinguish left from right. See the answer physics.stackexchange.com/a/12970/257059 for more information. $\endgroup$
    – Mitchell Spector
    May 22 at 3:00
  • $\begingroup$ Yes ... But is there any pure math constructions that can make this distinction? Do we have this kind of math that physicists use (on weak interactions, etc.)? $\endgroup$
    – Yuval
    May 22 at 3:20
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    $\begingroup$ @Yuval In mathematics, we just know things like that $\hat \imath \times \hat \jmath = \hat k$. This doesn't give us any "right-handed" or "left-handed" distinction until we try to associate it with our physical world of three spatial dimensions in one of two possible orientations. $\endgroup$
    – aschepler
    May 22 at 4:09
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    $\begingroup$ @Yuval You are missing the point. If we are talking about the mathematical line $\mathbb{R}$, I can define the statement “$a$ is to the left of $b$” to mean $a < b$. But if I show you a line in space and ask you which way is “left”, mathematics alone cannot answer that question. We would have to use physics to determine whether there is some property about going one way along the line that’s different than going another (which, in general, there isn’t). But there is a physical difference between the left-hand and right-hand rule (as defined by an actual human’s hands). $\endgroup$
    – Mark Saving
    May 22 at 5:07

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