When a definition is a close substitute for reality, then it may percolate into common culture with more authority. For example: every citizen can establish the existence of a tax void and Pareto suboptiomal unemployment purely from the logic of the level of gross minimum wages and the official tax statutes - and we don’t need big computers or official bureaus to do some econometrics and then tell us.
Admittedly, there is danger in seductive and seemingly right but wrong definitions. If ‘child’ is defined as ‘irresponsible young human’, then we may be tempted to treat children as such and forget to expect the responsibility that they can handle. But the existence of this danger should not make us close our eyes to the advantages of good definitions.
A side issue concerns our concept of ‘space’. Let us first consider an example of cultural relativism. It appears that different human cultures can have different approaches to one’s orientation in space, and that these approaches are wired into the languages used. [58] Taking a point of reference can be done in three ways: (1) Relative: taking one-self (“the tree is to the left of the house” - seen by me); (2) Absolute: taking the sun (“the tree is to the west of the house”); (3) Intrinsic: taking one of the objects (“the tree is to the back of the house”). If someone is asked to copy a situation in front of him towards a place in the back of him, then there will be a different ‘copy’ depending upon one’s language/culture. If you have a cup of coffee and a pencil in front of you, pick them up, turn yourself around, and recreate the scene, then a Westerner will use relative positions, while an Australian Aboriginal will use absolute positions (and turn the relative positions around). The question now is: while this only concerns the point of reference, can we imagine something similar that affects our concept of space itself ?
I take the position that the human mind apparently is able to conceptualise Euclidean space - and that this actually defines our concept of space. If we take a non-Euclidean geometry - such as a globe - then this still can be imagined to exist within Euclidean space. Pythagoras’s theorem is invalid for triangles drawn on a globe, but to hold that space is a globe would be erroneous - since our definition of space would be Euclidean.
One of the questions often posed is whether the universe - interstellar space - is Euclidean or not. This is a badly posed question. If we define space as Euclidean, then it is another question whether a ray of light follows a straight line or is deflected by gravity.
Barrow (1998:p42-44) provides a troubling quote: [59]
“The most important consequence of the success of Euclidean geometry was that it was believed to describe how the world was. It was neither an approximation nor a human construct. It was part of the absolute truth about things. (…) This confidence was suddenly undermined. Mathematicians discovered that Euclid’s geometry of flat surfaces was not the one and only logically consistent geometry. (…) None had the status of absolute truth. Each was appropriate for describing measurements on a different type of surface, which may or may not exist in reality. With this, the philosophical status of Euclidean geometry was undermined. It could no longer be exhibited as an example of our grasp of absolute truth. (…) These discoveries revealed the difference between mathematics and science.”
This quote is troubling for the following reasons:
1. If we define ‘space’ as Euclidean, then it is an absolute truth. This definition seems to maximise our information power. Other surfaces can be imagined within that space.