In my fairly-well-reasoned opinion
to define left
in a mathematical
sense without invoking preexisting conventions
that are typically described using those concepts. The statement
left = up x forward
(where x denotes the cross product, and "up" and "forward" are perpendicular but otherwise arbitrary) doesn't help. The cross product itself is typically defined as a vector operation in a right-handed coordinate system, which is usually defined visually: the y-axis points up, z points out of the page, and x points right. Alternatively, the cross product is defined using the "right hand rule" and a right-handed coordinate system is one where X x Y = Z if X, Y, and Z are the unit vectors. Either the cross product or the unit vectors are defined using the preexisting concepts of left and right.
Elementary particle physics evidently can help - in the standard model there are some things which can distinguish left from right. For more knowledge on this topic than I can furnish see pealco's writeup in Parity and Miles_Dirac's in Chirality.
More basic physics seems as though it should show a difference, but does not. The magnetic field, for instance, would define left and right, if only it was directly empirically observable: if a positive charge moves "up" then the magnetic field in the "forward" region points to the left. But the actual measurable value is not the field but the force on a particle in a field, and that force is always perpendicular to the magnetic field. The convention that says the magnetic field curls right-handedly about the direction of motion of a positive charge dictates that the force on another positive-charged particle is in the direction of the right-handed cross product of the velocity with the magnetic field: if you substituted left for right in this rule, the field direction would change but the force on the second particle would remain the same. The force is coplanar with the velocity of the first particle and therefore doesn't distinguish left from right.