Universal Map of Mathematics II

F-35 shown obsolete on past post

   The post Universal Map of Mathematics should be read first.
   Now, we will go into bad science fiction.
   Under the mapping, each theorem represents a quasi-vector, its origin if the theorem designators of its predecessors 5. 0103, 4.0658, 3.1057 and it termination the theorem designator 6.0098.  The goal would be to treat that as an actual vector.
   The triple point of the origin means that the theorem space would have to fold into itself for the 3 points to coincide.  This also leads to the conclusion that folds create new theorems and the possibility of arbitrarily creating new theorems by random folds as long as there is no internal contradiction between the source theorems.  `The other result is that an extension of the outer surface of the mathematic topology would also constitute a new theorem.  The analogy is the vectors (1,1,1) and (2,0,1), by inspection the vector (-1,-1,2) is normal, perpendicular to both of them.  If 2 vectors are normal the addition of the product of their respective co-ordinates, x, y, z, will add to 0.  In this case, 1X-1+1X-1+1X2 = -1+-1+2 =0 and 2X-1+0X-1+1X2= -2+0+2=0, the third vector is normal to both of the others.
   If the theorems can be treated as vectors then outer normals to the surface of mathematic topology would automatically constitute new mathematics.   As the diagram above shows, the new vectors exist if a component of them is normal to the surface or if they are entirely normal to the surface.
   For them to be vectors, they must have a well defined magnitude and direction.  Neither is easy to define, or may be possible to define, in this case.

   The designators of the theorems, .06, .09 would have to be given actual spatial meaning in some coherent form such as at right.  Minimally, target areas of mathematical definition would have to be formed, to be fully rigorous there would have to be a specific point indicating each field and a meaning assigned to scaling off between those points.
   There is a related problem of the representation of the theorems themselves.  If the theorem is designated by level and position related to meaning, 5th level, number theory, there could be a need to add another dimension to allow for multiple proofs at that point.  In this additional dimension they could be labeled consecutively out from the indicator point.  But even then, the theorem must connect with multiple other theorems and these connections are probably best handled by a pole in yet another dimension, the sections of the pole corresponding with the different theorem connections and being extended with each additional connection.

   The connections could be treated as rubber strings which stretch form pole to pole.  The location of the conclusion of the joining of 2 or more theorems would have to remain constant under reciprocal relations, the theorem joining 5.0603 with 4.1053 would have to have the same destination as the one joining 4.1053 with 5.0603.   Each would create an angle measured from the connector aimed towards the target space, the corresponding angle and distance would have to land at the same target.
   This leads to another related problem the postulates and assumptions would have to be assigned meaningful locations in an initiating plane.
   I believe the only way to do this is through statistical means.  The initial postulates and assumptions would be used to create the theorems T1.  Those theorems would be assigned descriptors of type, such as number theory,  and a provisional  location point.  By using a large number of theorems the postulates and assumptions would be arranged to minimize the error of theorems going to their designed target point assuming that distances between postulates are treated as vectors and reciprocity is required.  A second statistical pass would then be made using the assigned locations of the postulates and the target points would be adjusted by the outcome of the postulate vectors.  In a few passes going back and forth, a most likely location for the postulates and the target points could be determined.  To go to theorem level T2, the accuracy would decline and may have to be re-rectified.  This would probably have to be repeated at each proof level.  It would assign a meaning to each designator, however tenuous it might be.
   This could then allow for an arbitrary selection of theorem points, an arbtrary fold in theorem space, the ability to approximately predict where the result would be of a theorem combining those results, if the theorem was valid.  Which brings up the joke of the mathematician giving driving directions,"No, I didn't say that this road would take you to where you want to go, what I said was 'If this road goes to where you want then it would be the shortest route there.'"
    The entire process would produce a probability of the theorem's outcome location, which might be useful.  In theory it opens the possibility for machine driven mathematics, computers generating vectors without having to understand the underlying mathematics.