Calculus without limits

# Noncommutativity code example

Here is some code illustrating the story. We take the 4 manifold $S^2 \times S^2$ (a favorite manifold of Heinz Hopf) and consider two random functions f,g. Now generate the two manifolds $\{f=0, g=0 \}$ and $\{ g=0,f=0\}$. They are both 2 manifolds. It goes as follows: the sign data of {f,g} are in $\{ (-1,-1),(-1,1),(1,-1),(1,1) \}$ which are 4 points. As noticed in November 2023, I take the 2 dimensional partition complex P = K(1,1,2) which is the kite graph and unique 2-dimensional partition complex of the integer 4. The encoding of the 4 sign data can be done in 6 different ways. We can do so that switching f and g changes the complex to an isomorphic complex with 2 different facets. The two manifolds $\{ f=0, g=0 \}$ and $\{g=0,f=0\}$ are then different. In December 2023, I had illustrated this phenomenon in realizing that the nodal sets $\{ \psi=0 \}$ (Cladni figures) of a complex wave function on the 4 manifold produces a different surface than the nodal set of the conjugate wave $\{ \overline{\psi}=0 \}$. Such strange behavior only appears in the discrete. In the continuum, of course, $\{ \psi=0 \}$ and $\{ \overline{\psi}=0\}$ are the same 2-manifolds (if non-singular, but in the discrete also the singularity issue has disappeared). Similarly also the “coordinates” $f=0, g=0$ are the same than the coordinates $g=0,f=0$. The point I made in the video last Saturday is that all measurements are based on finding intersections of level surfaces (illustrated with the GPS story we always teach in multivariable calculus motivating hyperboloids). We know in quantum mechanics of course that measuring space and momentum coordinates do not commute. But this is a still well understood in the continuum and is the case both in the continuum and the discrete: if we go to a non-commutative algebra of operators, of course, we do not have any more commutativity. That happens already on a space with 2 points where the observables are 2×2 matrices. The transition to the discrete looked at in Sard completely different. There was no postulate to use any non-commutative algebra. Non-commutatiiv models were forced by the founders of quantum mechanics.

[By the way, there are no any data which suggest that space is non-commutative (or discrete) in the very small. It is a speculation, similarly like the Demoktitus speculation who speculated that matter is discrete. Demokritus speculation should be considered flawed because he had based his observations on conclusions which have proven to be wrong (he must have noticed fine sand can not be crushed any more, which is of course wrong of the order by 4-5 orders of magnitudes. Serious experiments what matter and energy is came only more than 2000 years later especially with Lavoisier who started to understand some fundamental conservation principles of chemistry and could base it on experiments. We can in easily crush sand into molecules (by melting and evaporating it) or even atoms and the 20th century has shown 100 years after Lavoisier even beyond on the level of atoms). Modern speculations of the discreteness of space is based on having finite computable approximations like lattice gauge theory that can be used as a numerical method to compute things, say for Monte-Carlo simulations. Any numerical computation is a finite approximation. There is no evidence at all that space is discrete say on a level of $10^{-35}$. Even if it would look discrete on that level (we can currently only reach the size of $10^{-20}$ leaving 15 orders of magnitude to the Planck scale!), it could be that when going to the size of $10^{-100}$ suddenly a perfectly continuum world appears. Speculation about things we can not access with experiments is outside of phytics. It can be done in a mathematical setting. Wittgenstein said it “Worueber wir nicht spechen koennen, darueber muessen wir schweigen”. But one should add that we are allowed to talk about mathematics which is not yet physics. Most mathematical theories probably do not have models realized in the physical world. Why would the Lie group SU(2345) appear in physics? Already SU(5) does not seem and one has tried it in GUT’s. ]

Back to the non-commutatiivity I mentioned in the discrete. This is something we can talk about because everybody, can make the measurement and see it. Running computer code is doing an experiment. It is a priori only mathematics and not physics. Mathematical implementations of theories can be considered physical too. It is realized as an idea that can be written down with finitely many symbols and can be programmed with a finite program. It therefore exists. You can run the program bellow and see its output. It is just not necessarily part of the world which has been chosen to host us thinking about it. The physical world could only be a tiny part of what we can explore with our thoughts. In mathematics we can chose our axiom systems as we wish, provided of course that we do not run into an inconsistency. One of the reasons to think in finite mathematics is the possibility that any theory that includes the infinity axiom is fundamentally flawed and inconsistent. That is not impossible and Goedel has shown that we can within such a frame work not even prove consistency. It is a fundamental difficulty. When talking about finite sets only, we live less dangerously.

Here is the code:

Generate[A_]:=If[A=={},{},Sort[Delete[Union[Sort[Flatten[Map[Subsets,A],1]]],1]]];
Whitney[s_]:=Generate[FindClique[s,Infinity,All]];L=Length; Ver[X_]:=Union[Flatten[X]];
sig[x_]:=Signature[x]; nu[A_]:=If[A=={},0,L[NullSpace[A]]];       Rnd:=Random[]-1/2;
F[G_]:=Module[{l=Map[L,G]},If[G=={},{},Table[Sum[If[l[[j]]==k,1,0],{j,L[l]}],{k,Max[l]}]]];
sig[x_,y_]:=If[SubsetQ[x,y]&&(L[x]==L[y]+1),sig[Prepend[y,Complement[x,y][[1]]]]*sig[x],0];
Dirac[G_]:=Module[{f=F[G],b,d,n=L[G]},b=Prepend[Table[Sum[f[[l]],{l,k}],{k,L[f]}],0];
d=Table[sig[G[[i]],G[[j]]],{i,n},{j,n}];{d+Transpose[d],b}];HH[G_]:=MatrixPower[Dirac[G][[1]],2];
Hodge[G_]:=Module[{Q,b,H},{Q,b}=Dirac[G];H=Q.Q;Table[Table[H[[b[[k]]+i,b[[k]]+j]],
{i,b[[k+1]]-b[[k]]},{j,b[[k+1]]-b[[k]]}],{k,L[b]-1}]];
Betti[s_]:=Module[{G},If[GraphQ[s],G=Whitney[s],G=s];Map[nu,Hodge[G]]];
Facets[G_]:=Select[G,(L[#]==Max[Map[L,G]]) &];  Poly[X_]:=PolyhedronData[X,"Skeleton"];
RFunction[G_,P_]:=Module[{},R[x_]:=x->RandomChoice[Range[L[Ver[P]]]];Map[R,Ver[G]]];
RFunction[s_]:=Module[{G},If[GraphQ[s],G=Whitney[s],G=s];R[x_]:=x->Rnd;Map[R,Union[Flatten[G]]]];
AbstractSurface[G_,f_,A_]:=Select[G,(Sum[If[SubsetQ[#/.f,A[[l]]],1,0],{l,L[A]}]>0)&];
Z[n_]:=Partition[Range[n],1]; ZeroJoin[a_]:=If[L[a]==1,Z[a[[1]]],Whitney[CompleteGraph[a]]];
ToGraph[G_]:=UndirectedGraph[n=L[G];Graph[Range[n],
Select[Flatten[Table[k->l,{k,n},{l,k+1,n}],1],(SubsetQ[G[[#[[2]]]],G[[#[[1]]]]])&]]];
S2xS2=Generate[{{1,2,3,4,6},{1,2,3,4,7},{1,2,3,6,9},{1,2,3,7,9},{1,2,4,5,8},{1,2,4,5,9},
{1,2,4,6,8},{1,2,4,7,9},{1,2,5,6,8},{1,2,5,6,9},{1,3,4,6,7},{1,3,5,6,7},{1,3,5,6,9},
{1,3,5,7,10},{1,3,5,9,11},{1,3,5,10,11},{1,3,7,9,10},{1,3,9,10,11},{1,4,5,8,10},
{1,4,5,9,11},{1,4,5,10,11},{1,4,6,7,11},{1,4,6,8,10},{1,4,6,10,11},{1,4,7,9,11},
{1,5,6,7,8},{1,5,7,8,10},{1,6,7,8,11},{1,6,8,10,11},{1,7,8,10,11},{1,7,9,10,11},
{2,3,4,6,8},{2,3,4,7,8},{2,3,5,7,10},{2,3,5,7,11},{2,3,5,10,11},{2,3,6,8,10},{2,3,6,9,10},
{2,3,7,8,11},{2,3,7,9,10},{2,3,8,10,11},{2,4,5,8,9},{2,4,7,8,9},{2,5,6,8,11},{2,5,6,9,10},
{2,5,6,10,11},{2,5,7,8,9},{2,5,7,8,11},{2,5,7,9,10},{2,6,8,10,11},{3,4,6,7,11},{3,4,6,8,10},
{3,4,6,9,10},{3,4,6,9,11},{3,4,7,8,11},{3,4,8,9,10},{3,4,8,9,11},{3,5,6,7,11},{3,5,6,9,11},
{3,8,9,10,11},{4,5,6,9,10},{4,5,6,9,11},{4,5,6,10,11},{4,5,8,9,10},{4,7,8,9,11},{5,6,7,8,11},
{5,7,8,9,10},{7,8,9,10,11}}];   P=ZeroJoin[{1,1,2}];P=P/.{1->3,2->4,3->1,4->2};
G=S2xS2;   V=Ver[P];   FF=Facets[P];   f=RFunction[G];     g=RFunction[G];
fg=Table[f[[k,1]]->(1+(Sign[f[[k,2]]]+1)+(Sign[g[[k,2]]]+1)/2),{k,Length[g]}];
gf=Table[f[[k,1]]->(1+(Sign[f[[k,2]]]+1)/2+(Sign[g[[k,2]]]+1)),{k,Length[g]}];
Hfg=AbstractSurface[G,fg,FF]; Hgf=AbstractSurface[G,gf,FF]; {Betti[Hfg],Betti[Hgf]}

The first surface of co-dimension 2 in the 4-manifold is a 2-torus. The second is a sphere. The first is {f=0,g=0}, the second is {g=0,f=0}. The code above generates random cases. For the picture, I used the following functions f,g leading to the functions called fg,gf from the 4-manifold to $\{1,2,3,4\}$ representing the possible sign cases for f and g. By the way, if you run the code, it does (while rarely) happen that the level surface is empty. This does not happen if we look at a level surface of one function (if the function takes both negative and positive values) but it can happen already with a level surface of two functions. For larger host manifolds, the probability to hit an empty surface is so small that we have almost zero chance to encounter it when looking at random functions.

(* For the picture, I had the following functions  *)
P={{3},{4},{1},{2},{3,4},{3,1},{3,2},{4,1},{4,2},{3,4,1},{3,4,2}};
f={1->-0.25,2->-0.05,3->0.35,4->0.25,5->0.16,6->-0.04,7->0.30,8->0.07,9->-0.39,10->-0.29,11->-0.15};
g={1->0.30,2->0.01,3->-0.14,4->-0.14,5->-0.27,6->0.15,7->-0.002,8->0.01,9->-0.29,10->-0.13,11->-0.10};
fg={1->2,2->2,3->3,4->3,5->3,6->2,7->3,8->4,9->1,10->1,11->1};
gf={1->3,2->3,3->2,4->2,5->2,6->3,7->2,8->4,9->1,10->1,11->1};
Hfg=AbstractSurface[G,fg,FF]; Hgf=AbstractSurface[G,gf,FF]; {Betti[Hfg],Betti[Hgf]}
{GraphPlot3D[ToGraph[Hfg]],GraphPlot3D[ToGraph[Hfg]]}