The next step is to understand what is the replacement of the Riemannian paradigm for noncommutative spaces. To prepare for that, and using the excuse of the summer holidays, let me first tell the story of the change of paradigm that already took place in the metric system with the replacement of the concrete "mètre étalon" by a spectral unit of measurement.
The notion of geometry is intimately tied up with the measurement of length. In the real world such measurement depends on the chosen system of units and the story of the most commonly used system--the metric system--illustrates the difficulties attached to reaching some agreement on a physical unit of length which would unify the previous numerous existing choices. As is well known, the United States are one of the few countries that are not using the metric system and this lack of uniformity in the choice of a unit of length became painfully obvious when it entailed the loss of a probe worth 125 million dollars just because two different teams of engineers had used the two different units (the foot and the metric system).
In 1791 the French Academy of Sciences agreed on the definition of the unit of length in the metric system, the "mètre", as being the ten millionth part of the quarter of the meridian of the earth. The idea was to measure the length of the arc of the meridian from Barcelone to Dunkerque while the corresponding angle (approximately 9.5°) was determined using the measurement of latitude from reference stars. In a way this was just a refinement of what Eratosthenes had done in Egypt, 250 years BC, to measure the size of the earth (with a precision of 0.4 %).
Thus in 1792 two expeditions were sent to measure this arc of the meridian, one for the Northern portion was led by Delambre and the other for the southern portion was led by Mechain. Both of them were astronomers who were using a new instrument for measuring angles, invented by Borda, a French physicist. The method they used is the method of triangulation and of concrete measurement of the "base" of one triangle. It took them a long time to perform their measurements and it was a risky enterprize. At the beginning of the revolution, France entered in a war with Spain. Just try to imagine how difficult it is to explain that you are trying to define a universal unit of length when you are arrested at the top of a mountain with very precise optical instruments allowing you to follow all the movements of the troops in the surrounding.
Both Delambre and Mechain were trying to reach the utmost precision in their measurements and an important part of the delay came from the fact that this reached an obsessive level in the case of Mechain. In fact when he measured the latitude of Barcelone he did it from two different close by locations, but found contradictory results which were discordant by 3.5 seconds of arc. Pressed to give his result he chose to hide this discrepancy just to "save the face" which is the wrong attitude for a Scientist. Chased from Spain by the war with France he had no second chance to understand the origin of the discrepancy and had to fiddle a little bit with his results to present them to the International Commission which met in Paris in 1799 to collect the results of Delambre and Mechain and compute the "mètre" from them. Since he was an honest man obsessed by precision, the above discrepancy kept haunting him and he obtained from the Academy to lead another expedition a few years later to triangulate further into Spain. He went and died from malaria in Valencia. After his death, his notebooks were analysed by Delambre who found the discrepancy in the measurements of the latitude of Barcelone but could not explain it. The explanation was found 25 years after the death of Mechain by a young astronomer by the name of Nicollet, who was a student of Laplace. Mechain had done in both of the sites he had chosen in Barcelone (Mont Jouy and Fontana del Oro) a number of measurements of latitude using several reference stars. Then he had simply taken the average of his measurements in each place. Mechain knew very well that refraction distorts the path of light rays which creates an uncertainty when you use reference stars that are close to the horizon. But he considered that the average result would wipe out this problem. What Nicollet did was to ponder the average to eliminate the uncertainty created by refraction and, using the measurements of Mechain, he obtained a remarkable agreement (0.4 seconds ie a few meters) between the latitudes measured from Mont Jouy and Fontana del Oro. In other words Mechain had made no mistake in his measurements and could have understood by pure thought what was wrong in his computation. I recommend the book of Ken Adler
for a nice account of the full story of the two expeditions.
In any case in the meantime the International commission had taken the results from the two expeditions and computed the length of the ten millionth part of the quarter of the meridian using them. Moreover a concrete platinum bar with approximately that length was then realized and was taken as the definition of the unit of length in the metric system. With this unit the actual length of the quarter of meridian turns out to be 10 002 290 rather than the aimed for 10 000 000 but this is no longer relevant.
In fact in 1889 the reference became another specific metal bar (of platinum and iridium) called "mètre étalon", which was deposited near Paris in the pavillon de Breteuil. This definition held until 1960.
Already in 1927, at the seventh conference on the metric system, in order to take into account the inevitable natural variations of the concrete called "mètre étalon", the idea emerged to compare it with a reference wave length (the red line of Cadmium).
Around 1960 the reference to the called "mètre étalon" was finally abandoned and a new definition of the unit of length in the metric system (the "mètre) was adopted as 1650763.73 times the wave length of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the Krypton 86Kr.
In 1967 the second was defined as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of Caesium-133. Finally in 1983 the "mètre" was defined as the distance traveled by light in 1/299792458 second. In fact the speed of light is just a conversion factor and to define the "mètre" one gives it the specific value of c= 299792458 m/s. In other words the "mètre" is defined as a certain fraction 9192631770/299792458~ 30.6633... of the wave length of the radiation coming from the transition between the above hyperfine levels of the Caesium atom.
The advantages of the new standard of length are many. First by not being tied up with any specific location, it is in fact available anywhere where Caesium is. The choice of Caesium as opposed to Helium or Hydrogen which are much more common in the universe is of course still debatable, and it is quite possible that a new standard will soon be adopted involving spectral lines of Hydrogen instead of Caesium. See this paper of Bordé
for an update.
While it would be difficult to communicate our standard of length with other extra terrestrial civilizations if they had to make measurements of the earth (such as its size) the spectral definition can easily be encoded in a probe and sent out. In fact spectral patterns provide a perfect "signature" of chemicals, and a universal information available anywhere where these chemicals can be found, so that the wave length of a specific line is a perfectly acceptable unit, while if you start thinking a bit you will find out that we would be unable to just tell where the earth is
in the universe... Coordinates ? yes but whith respect to which system? One possibility would be to give the sequence of redshifts to nearby galaxies, and in a more refined manner to nearby stars but it would be quite difficult to be sure that this would single out a definite place.