[FOM] Removing deep pathology
Frank Waaldijk
fwaaldijk at gmail.com
Mon Aug 24 07:53:18 EDT 2015
Just like in medicine (`pathology' is after all a medical term), anyone
wishing to remove pathology should consider what caused the pathology in
the first place.
Let´s look at Harvey Friedman's example: total real functions f for which
f(x+y)=f(x)+f(y), and yet not of the type f(x)=cx (for some constant c).
Obviously, such a function f is discontinuous. A discontinuous total real
function only exists if one believes in PEM (principle of excluded middle)
for infinite sets. Added to this we also use the axiom of choice AC to
conclude that such functions f must exist (even though we cannot describe
them in a meaningful way, as Harvey points out).
It seems to me that if one wants both PEM and AC, then one cannot avoid
what Harvey calls `deep pathology'. On the other hand, like Hendrik Boom
pointed out, in constructive mathematics you will not find such `deep
pathology'. Precisely because of the insistence that everything that you
claim to exist has to have a meaningful description (read: construction).
The question then becomes: does the liver patient wish to stop drinking?
:-) My advocacy for constructive mathematics is known, so I leave it at
that.
The example of fractals that Martin Davis gives, interests me more inasfar
as it touches on the same phenomenon of `meaningful' but on a `lower'
level: the question of whether we can really describe infinite objects,
even if the description is recursive. This is of course the driving
question behind (ultra)finitism.
In this vein, to illustrate the problem that (ultra)finitism tries to
tackle I thought of a rephrasing (strengthening?) of Zeno's paradox in the
following way, using a fractal.
Consider Achilles who is at the origin (0 on the real line), and he wishes
to reach the Turtle who is at point 1, and doesn't move at all. Now we
impose a recursive (fractal) rule for Achilles to follow, called Funny-Walk
(FW): in order to FW-reach 1 Achilles must first FW-reach 2/3, then
FW-redouble to 1/3, and then FW-proceed to 1.
Taking Cauchy-limits, this procedure in facts describes a continuous
function FW from [0,1] to [0,1] such that (indeed) FW(0)=0 and FW(1)=1, so
we could say: Achilles reaches the Turtle. (FW(1/3)=2/3, FW(2/3)=1/3, and
each new 3^⁻(n)-interval is treated in the same manner, the function's
graph is a fractal).
There is only one problem (the paradox): Achilles has covered infinite
distance. This because in each recursive step, the distance that Achilles
has walked increases with a factor 5/3.
So, I do not entirely agree that fractals are unproblematic. In fact I
think infinity always poses really difficult problems. I recently came
across Edward Nelson's internal set theory (IST), and I wonder if anyone is
carrying on with this? I really like his idea that in every infinite set
there is a `vague' element.
In the hope to have contributed something to the discussion,
best wishes to all,
Frank Waaldijk
www.fwaaldijk.nl/mathematics.html
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