On Sep 18, 2012, at 9:19 PM, Rex Allen <[email protected]> wrote:
On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam <[email protected]
> wrote:
Rex,
Do you have a non-platonist explanation for the "discovery" of the
Mandelbrot set and the infinite complexity therein?
I find fictionalism to be the most plausible view of mathematics,
with all that implies for the Mandelbrot set.
But ;et me turn the question around on you, if I can:
Do you have an explanation for how we "discover" mathematical
objects and otherwise interact with the Platonic realm?
We study and create theories about objects in the mathematical realm
just as we study and create theories about objects in the physical
realm.
It's not much different from how we develop theories about other
things we cannot interact with: the early universe, the cores of
stars, the insides of black holes, etc.
We test these theories by following their implications and seeing if
they lead to contridictions with other, more established, facts.
Just as with physical theories, we ocasionally find that we need to
throw out the old set of theories (or axioms) for a new set which has
greater explanatory power.
How is it that we are able to reliably know things about Platonia?
The very idea of knowing implies a differentiation between true and
false.
This leads quite directly to boolean algebra. Boolean algebra leads
to concepts of numbers. (e.g., even numbers of not operators cancel
out, so counting them becomes an issue). Once you get counting and
numbers, you get the uncapturable infinite truths concerning them, and
infinite hierarchies if ever more powerful consistent theories.
Nearly any intelligent civilization that notices a partition between
true and false will eventyally get here.
I would have thought that quarks and electrons from which we appear
to be constituted would be indifferent to truth.
The unreasonable effectiveness of math in the physical sciences is yet
further support if Platonism. If this, and seemingly infinite
physical universes exist, and they are mathematical structures, why
can't others exist?
Which would fit with the fact that I seem to make a lot of mistakes.
But you think otherwise?
We are imperfect beings.
Jason
How can you make
sense of that in terms of the constructivist point of view that you
are (I think) compelled to take if you argue against arithmetical
platonism? It seems obvious that all possible intelligences would
discover the same forms of the Mandelbrot so long as they iterated on
z' = z^2 + c, but maybe I am missing the point of your argument.
I will agree with you that all intelligences that start from the
same premises as you, and follow the same rules as inference as you,
will also draw the same conclusions about the Mandelbrot set as you
do.
However - I do not agree with you that this amenable group exhausts
the set of all *possible* intelligences.
Could there be intelligences who start from vastly difference
premises, and use vastly different rules of inference, and draw
vastly different conclusions?
If not - what makes them impossible intelligences?
=*=
What are the limits of belief, do you think? Is there any belief
that is so preposterous that even the maddest of the mad could not
believe such a thing?
And if there is no such belief - then is it conceivable that quarks
and electrons could configure themselves in such a way as to *cause*
a being who holds such beliefs to come into existence?
And if this is beyond the capacity of quarks and electrons, does it
seem possible that there might be some other form of matter with
more exotic properties that might be up to the task?
And if not - why not?
Rex
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