= symbol has 3 meanings in computers. Often languages express these with 3 syntax: =,==,:=
(a) 1+2=3 is the Mathematical "scales". This originated from market place scales and it says the left and right balance. Algebra works on the basis that if you do the same thing to both sides of a scale you know the scale will still balance. If you have 1Kg of rice and 1Kg or gold and you do the same material action to the rice and the gold the mass will remain the same on each side. The scale will balance. This is the origin of all mathematics. Indeed Greeks could only think in terms of ratios like this, the idea of abstract numbers by themselves began later with the Arabs after the Roman Empire.
(b) 1+2 == 3 is the Logical equality. We accept the statement 1+2 == 3 because indeed 2 and 1 on the left of the scales is the same weight as 3 on the right. Unlike with mathematics symbol there is the super powerful negation logical operator so we can produce statements like 1+2 != 4 (that is 1+2 == not(4) ). Indeed "1+2 is not the same size as 4" is a true statement.
(c) X := 1+2 this simply gives a new name to 1+2. In this case X. Its says that X can now be replaced with 1+2. Using (a) above to equate 1+2 = 3, and expressing in the logical form of (b) we can say X==3
Now in Logic the obsession is with contradictions. Using the above 3 distinctions. If we define X := 4 then 1+2 == X is false. If however we now define '4' to mean 4 := 1+2 then 1+2 == 4 is true. But this now means 2 + 2 == 4 is false because we've redefined what '4' means.
So in fact a "contradiction" is impossible if we just accept the contradiction as a definition.
We are talking to a witness of a murder. The witness says they heard a scream and then a tall man ran out of the building and got into the red car. Then they say a short man got out of the car. Now this means either its the same man how was tall but is now short or there was another man in the car. We ask the witness was there another man in the car. They say no. So lets define the tall man as "John". So John is tall. John got into the car. John now gets out of the car. But John is short. John cannot be tall and short so we have a contradiction. However what if the witness' has bad eye sight. So we can quite happily say that John appeared tall and then John appeared short and no contradiction. But lets get crazy. There is no explanation. The story starts with Tall John and end with Short John. Don't we then just make 2 separate witness statements. They must be separate because they contradict, but within their own world's they are fine.
Now this is the actual problem with contradiction. Every time we contradict we fracture reality. I forget the Analytic Philosophers involved but every text defines (3 above) an "ontology." And if it starts redefining this like John is now defined as short we have a different ontology and a different narrative.
In the end it may turn out that the witness' first statement about a Tall man leaving the building fits in with other statements. Perhaps another witness says they say a Tall Man running around the corner. The story of a Tall man getting into a car, and a short man getting out may never be useful or explained and never fits in with anything else from the investigation. A different world that the investigation can just discard.
I once called this idea "harmonic structuralism." A narrative could be judged simply by its size and usefulness. There was no "truth" but rather usefulness. Most people can see and most people on a rotating globe can see the sun at some point in their lives. There must be unfortunate Eskimo babies who are born and die within a few months during the Arctic winter who never see the Sun. But for most people the story of the Sun fits in well with their lives without contradicting too much. So its a popular "truth".
Now "dynamic systems." A dynamic system is not like X:= 1+2 because it is recursive. A very famous example is X:= k X (1-X) studied by Prof Robert May in relation to population growth constrained by resources.
Unless X is the "fixed point" it is a contradiction. So if k := 2 then X:= 0.5 means that X == k X (1-X) is true. Indeed with X := 0.5 the equation gives the result 0.5.
However for X := 0.75 then the equation gives the result 3/8. Now if we ignore the contradiction like above and redefine X:=3/8 we can do the calculation again and keep going.
Perhaps we need a 4th definition of equal to mean "keep redefining"
X =: k X (1-X) means ignore any contradiction and keep redefining.
Now for X =: 2 X (1-X) where X := 0.75 after 5 interations X == 0.5 is true. The result is the fixed point. Indeed in stable dynamic systems the result of either the fixed point or infinity. Mandelbrot Set most famously plotted the stability of the dynamic system X =: X^2 + C in the complex domain to get this picture.
the speed of convergence should be a second result of such statements. So the full statement for dynamic systems should be:
X =: 2 X (1-X) == (0.5,5) where X := 0.75 and
X =: X^2 + C == (inf,inf) where X := 0, C := 3+4i
But what of:
X =: 3X (1-X) where X := 0.75
This ends up cycling between two values of X := {0.673..,0.659..}
And
X =: 3.6 X (1-X) where X := 0.75
This definition never settles down. It keeps changing forever. It is what is called Chaotic.
This constant motion or "struggling" of some dynamic systems to find a stable equilibrium is analogous to the struggling in other systems to deal with contradictions. X =: f(X) is not happy constantly changing in the hope of finding a value which solves the equality and balances the scales. But sometimes like a Greek Tragedy the trajectory conspires against the system to ensure it never gets to that value (even if that value is finite).
Can we define Infinity as the solution to some dynamic systems.
X =: 1 + X only has a solution at X := infinity. That famous conundrum of the infinite hotel with infinite guests staying there. Nevertheless if everyone moved to the next room then Room 1 would become available. In fact isn't this analogous to what Cantor did with his Diagonalisation argument?
Dynamic systems however are constrained and you can't split the ontology in order to relieve the problem of contradiction. It is this problem that provides them their "energy." I wonder if we could define some quantity E for dynamic systems to measure the "movement" in values as they seek resolution.
On the subject of resolving contradictions Hegelian dialectics utilises exactly this "energy" to trigger Aufhebung and context changes that provide new perspectives that resolve apparent contradictions. It is this energy that motivates History itself. The resulting perspective is one where the contradictions between individual and group are satisfied, captured in the maxim "the I that is we, and the We that is I". Marx famously took up this idea to argue that the resolution to the contradictions in society was the state of Communism where individuals and society becomes stable. Its a sad reflection on Post Modernity that rather than solve these contradictions it seems the energy has just been bled out.