Les wrote:
> i haven't seen anyone yet locate your fixed points in actual existing
> processes.
Les,
Consider a much simpler example, from anybody's world: A can with some
liquid. Let the state of the system at $t$ be $x_t$. Now shake the
can. At $t+1$, after the shaking, the system is $x_{t+1}$. Fixed
point theorems say that there'll be at least one point along the
mapping from $x_t$ to $x_{t+1}$ that will be fixed.
Now, you say. But Julio, what do you mean a point in a physical
liquid? Points are dimensionless. Do you mean that a dimensionless
entity has an "objective" existence outside of our minds? And even if
a dimensionless object existed, what if the world is discontinuous at
its most elementary micro-cosmic level, don't theorems predicated on
continuity collapse then?
I'll say, okay Les, I concede: A perfectly dimensionless point can't
exist but in our minds. If the world is not continuous in any
meaningful sense at some level, then there may be a hole right about
where the hypothetical fixed point would be. Etc. But -- but, if the
point of mathematical analysis is *approximation* (Gerretsen & Rau),
and approximation is a *practical human endeavor*, then for certain
*practical* purposes (e.g. inspect the quality of the liquid in a can,
etc.), a molecule can be thought of as a point. They are small
enough. So, for at least one molecule $x_t=x_{t+1}$ will pin a fixed
point.
But Julio, for *other* conceivable practical purposes (something
involving processes at a subatomic level), molecules are very complex
things. Sure. Then let the molecule be a sub-system or element $x^i$
of $x$ and replicate the argument. The state of the molecule at $t$
is $x^i_t$. Shake the molecule every which way. At $t+1$, after the
shaking, there'll be a point defined by $x^i_t=x^i_{t+1}$. Fixed
point.
This is still the early 21st century, how much further would you want to go?
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