Duality has become fundamental to our application of measurement.
The very special nature of duality excites interest as does triplet , quadruplets etc..Duality is ur simplest notion that 2 things are the "same"(dual) and our most direct inference when fitting 2 things together, abd or comparing 2 things. In fact we donot need to conclude on the basis of 2 instances, but w naturally do. However induction is concluding that some thing is the "same"(multiplet) based on multiple instances. Euclid chose to develop his system of measurement and comparison on the god Isis thus we use duality for most of our reasoning. And this was not just Euclid's choice. It appears the most of mankind prefers this type of dual reasoning and comparison and measurement.
Without understanding that Euclid starts with the circular disk and the periphery of that disk, one misses the importance of the dual. As i have explained before , from a fixed veinlike branch, used as a compass pair, a circle can be constructed, and from this circle a straight line called a diameter can be constructed point by point using different pairs of compasses from the same 2 points on the circle periphery.. Thus Euclids definition that a god line has all dual points in it.
how many take for granted that the radius of a circle divides the circumference into 6 equal arcs? This is challenging to prove using duality and dual points, because the common inference that if 2 things coincide they are dual allows duality to be attributed by coincidence of points or magnitudes pointwise, not by what appears to happen. Thus i can construct any number of six marks on a periphery but i cannot say that the 6th coincides with the starting point precisely or necessarily. In fact quite often, when doing this precise construction they do not, and so i fudge it. Why? Because i was taught that they should. Therefore i justify my action by saying i have introduced some innacuracy somewhere, or the compass point slipped or paralla etc. However, even formally i cannot "prove " this relationship to be the case, as i can "prove" that duals can cut the periphery into 4 arcs, because i can directly measure or apply the dual to the periphery even formally. However this does rely on the notion of parallel lines and thales theorem and circle arcs.
We cannot even accept that vertically opposite angles are equal(dual) without proof. Thus these fundamental relationships may well have been defined, arrived at inductively, shown to be consistent through generations of consistency, tautologically assigned. The reality may be so close that we cannot measure it without quantum interference!Nevertheless it is not formally proved but defined away.
For example, by use of Neusis Archimedes was able to trisect an angle with a "ruler" and a compass. In fact by formal logic he deduced the form and then transferred the measurement to a straight edge by is compass or dividers, and then used the straight edge to settle the final positioning which he was not able to do by compass alone because it require parallel lines.
This kind of ingenuity is fantastic for a pragmatist, but it is of course suspect to subjective skill and intuition. The formal and the real divide at just these points of conflict. What we may formally assign and imagine may not turn out to be achievable with the instruments we plan o use, or our formal systems may be incoherent with our "real" systems.
To prove collinearity and coincidence are other such problems. The problem arises due to defining dual tiple etc as a "measurement" not an assignment or attribute. Thus if we cannot make the measurement w cannot say the thing is dual.
How can we define dual, triple in the first place then? Logically we can take the naivities and use them to define a complex, and then we can use the complex tautologically to define the naiveties. Thus it is an iterative tautology which works in definition but without the prior naive basis it will not be consistent. This is what happens when we try to prove collinearity and coincidence and that the radius cuts exactly 6 times along the periphery of a circle ,
Sometimes we have to accept the naive sensory information and run with it, and of course that means aour forma; models have to do the same.
Euclid allows duality to be determined, inferred or understood by fitting things on top of each other. Thus while he does not propose sophisticated measurement, he allows proportional measurement using Eudoxus trichotomy, les, duyal, greater.
This fitting is achieved by movement of the form which presumably is held rigid; and such movements are rotation, folding or flipping and sliding along straight lines. These are of course the affine symmetries which underpin Euclids notion of dual. If these movements are not of the rigid sort then dual cannot be justified as a judgement of sameness.
There are other judgement of sameness which relate to magnification and thus are called similarity. Similariy thus exists for all "scales" which is a way of saying that the duality scheme that has been accepted and fractalise to all parts and multiples, provides a system of sizes to fit each situation. Theses sizes are chosen by identifying them as a scale for what particular purpose one has in mind. It is therefore paramunt that a scale is based on a rigid set of relationships, and such a set can hardly be found in nature, but we do try. Eventually we have to admit that it is our formal schemes on which our measurements rely, particularly when we realise that some measurement standards are beyond human ability, and some machine or other tool is relied upon to provide data forour formal calculation of a "value" or standard.
Many times the tautology is underplayed, but as far as standards go we must always bear in mind the essential tautological nature of any standard. This precisely makes sense only when we allow for implicit , iterative definitions of standards, thus allowing us to asymptotically approach a level of ideal values and behaviours.
Using this freedom to do affine motions we can formally demonstrate that the radius does cut the periphery of a circle in just 6 points, provided we have a notion of collinearity, and or a notion of a unique straight(dual pointed) line.
This last bit is the fudge, but a reasonable and pragmatic one: if i can construct a dually pointed line throufg 2 given points, then there is only one dually pointed line from the 2 points used to define the dual points of the line. However there are many sets of 2 points either side of the dually pointed line that could be used to produce a dual pointed line which going through the same two points will be collinear.