Journey to the Source of Laughter.
Day 7. The Geometry of Reality. Unobservable Places and Time .
This Section is more technical than the foregoing,you can read quickly and remember only the conclusions C1. and C2.
The philosophical questions like the origin of Life,the real structure of Space-Time ,the existence of some version of a godly power,are usualy considered in some “classical” geometry of space ie a real manifold . In the beginning people viewed everything happening in a Euclidean Geometry that was slowly replaced by an object of differential geometry with the benefit of infinitesimal calculus and allowing more general metrics in calculating distances.The latter seemed necessary in view of relativity theory,where space was interpreted as curved by gravity. In general relativity the mathematical structure of manifolds over the real numbers is used but in a sense the geometry changes in time by certain events involving big gravity changes eg the gravity waves recently detected and predicted by Einstein. In fact the change of geometry also exists on the micro-level ,tiny disturbances created by small gravity effects,too tiny to be actually measurable by the existing tools (and probably will remain so for ages to come).The geometry is therefore at least dynamical and not really fully observable. Of course one may neglect the infinitesimal changes and that will not harm the theory much,so you accept that you can describe local phenomena by choosing local parameters and using real vectorspaces to calculate in.In relativity theory time is used as a parameter and in that theory it can be used as a reversible parameter (running back instead of forward) the laws are independent of that direction. Yet in all our observations time is seemingly irreversible. Now in the definition of time I gave, a parametrization of the totally ordered set of states of the universe,each tiny change represented by a different state,time is not a parameter running through the real numbers. It can be ,but that is then an extra axioma put on the structure of the universe (not founded on any observaton or experiment!) ,so we do not need to do that for the moment.The noncommutativity of certain operators associated to different observations of a particle,yielded the incertainty principle (and the passing to quantum description of the micro-world via probability). The fact of representing some quantities to be measured as operators on some Hilbert space is assuming they are vectors in the geometry of space ! However if that geometry is not assumed to be a classical one (even locally) ,but for example a non-commutative geometry then the operator formalism does not apply , in fact if there would be given some operator “version” of the quantities then that would automatically be a non-commutative one. If one could measure in the non-commutative geometry then one could measure at the same time things one could not do in the classical geometry, however it may not be possible to measure in the non-commutative geometry by using real or complex numbers !
I will be short about the very technical point about non-commutative geometry,there is a concept of a non-commutative toplogy on a set of places (places do not have to be sets of points !) ,the essential thing is that for a topology to be non-commutative it is necessary and sufficient to have open sets with non-trivial self intersection (this of course can happen when they are not given as sets of points in some space!); Such places are non-observable for geometric reasons,because everything we observe we embed as sets of points in some space.You can thus think of a non-commutative place as a mathematical point in the observed space (a point cannot be observed,it has no size),yet an object can be at such place,hence non-observable for geometric reasons,not for reasons of not good enough tools!)The observable sets form a commutative geometry and we can for example assume thet this commutative shadow is classical space-time. Then we can see the non-commutative geometry of reality as the micro-process of the usual space time (the inclusion relation being the partial order on the commutative shadow defined by taking for aspects of an observable set those non-obervable ones containing the observable one as the largest observable subset,the micro relation is the partial order defined on the non-commutative geometry.Hence this model of non-commutative space-time also fits the formalism of a learning process on the ingredients “the space-time” with microstructure defined by aspects as just explained! In conclusion we may view the non-commutative space as usual space-time with at each set a number (a priori not necessarily finite) of mathematical points (ie not observable in space) representing non-commutative places; In the latter there is no distance defined and also no time observed! So also it is not defined what the speed of a particle is in a non-commutative place,it is non-defined but we might think it is infinite,mobility is immediate then there (agreeing with representing the place by amathematical point).
We may cut the story short by assuming there are unobserved places and unobserved time in the non-commutative geometry of reality and in the underlying real geometry only observable places and observed time appears!
You could argue that our abstract world consisting of meanings is in fact existing in the non-commutative world,meanings are not in an observable place and even if their construction takes time they themselves do not exist in time. There is no contradiction in doing this but I will not do it as there is no need for it and it makes a rather strange connection between reality and our abstract world ! It is an interesting idea though,it would make free will into some unobservale object in the geometry of reality !
After this technical section,the conclusion is that we need to exend the definition of EXISTING in the reality,because objects that cannot be observed ,for some period of time,(eventually for ever,we do not know for how long) may exist in non-commutative reality. So if the geometry of reality is a non-commutative one,and why would it not be ( ?!), the existence of some real objects may be( forever) unobservable.
C1.This is obviously important in dicussing the existence of things like soul,god,superpower,cosmic awarenes,yet it will not allow arbitrary assumptions because of the following approximating observations.The effect of some object existing in a non-observable place on near-by observable places can give away the existence of that object without actually observing it,eg gravity effects (can dark matter be explained this way?).
If this section was too technical you just have to remember that the notions of existing in reality and being observable need not be related ! That is important ,everything depends on the assumption on which geometry you.want to describe reality by.That in the example I now explained a little the whole geometric picture can completely be seen as a learning process on the classical space-time is important from the cognitive interpretation point of view. Remember the definition of understanding !
C2.To understand the usual space-time process you have to have knowledge about the micro-process ,hence about the non-commutative deformation of it!