One of the most important questions in both physics and mathematics is: How do we measure things? In mathematics this is mostly settled. We have a number system which is good enough for all of our needs. It allows us to assign numerical representation to distance and it is also easy to do algebraic calculation with it. Also, on a graph it doesn't really matter what the numbers stay for - they can refer to meters or to microns (or to bytes). It begins to matter only when we what to actually understand what the graph says us. That is, when we want to use mathematics to describe not only itself but also the world around us.

Describing the physical world is the main goal of physics. So lets look how physics deals with our question.

Currently we have the M.K.S system - meter, kg, second. It is a system that works very well. But is it natural?

Surprisingly this units are not only not natural, but they are not even well defined. The meter for example, was supposed to be 1/40000 of the earth equator. It turned out that the equator length was not calculated correctly, so the length of the meter is not related to it.

For a long time there was a special metal bar 1 meter long, that was kept at a constant temperature. It was called "meter" - and it is all the connection to the physical world that this system has and ever had. The meter is not well defined because it is impossible to build perfect replicas of this metal bar, they will all be slightly different in size and shape. Currently it is defined as the distance traveled by light in 1/299,792,458 of a second.

This is also true for kg. Initially it was defined as the 1/1000 of the weight that one cubic meter of water had at sea level. Well this weight is hardly a natural constant. Also it cannot be natural because it was defined using meters, and they are not natural.

The second is also not natural. Initially the division of the day into 24 hours and then into 60 minutes and 60 seconds was due to the fact that this was the easiest relation to build the watch for. You can read more about measuring time in my post about calendars.

In other ares of physics, we have other units. But most of them are also unnatural. An example on a natural unit used in physics is kelvin. The kelvin temperature scale is defined by pure reason based on motion of mater. Also since Kelvin can be converted to Celsius by simple addition of 273, Celsius is also a natural unit. the only difference between the two is that Celsius has a different point denoted by zero.

The reason why we have so many unnatural units is twofold. Firstly, when those units where invented natural units were unknown. Secondly many natural units are two small. For example the electron charge can be considered a natural unit of charge. But it is only 1.6021765 × 10^{−19} coulomb. It is simply too small to be used.

The Geometric McKay Correspondence (Part 1)

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