On 9/21/2012 5:40 AM, Rex Allen wrote:
On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam <terren.suy...@gmail.com <mailto:terren.suy...@gmail.com>> wrote:


    On Tue, Sep 18, 2012 at 10:19 PM, Rex Allen <rexallen31...@gmail.com
    <mailto:rexallen31...@gmail.com>> wrote:
    >
    > On Mon, Sep 17, 2012 at 1:36 PM, Terren Suydam <terren.suy...@gmail.com
    <mailto:terren.suy...@gmail.com>>
    > 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.

    I'm curious about what a plausible "fictionalist" account of the
    Mandelbrot set could be. Is fictionalism the same as constructivism,
    or the idea that knowledge doesn't exist outside of a mind?


I lean towards a strong form of fictionalism - which says that there are few important differences between mathematics and literary fiction.

So - I could give a detailed answer - but I think I'd rather give a sketchy answer at this point.

I would say that mathematics is just very tightly plotted fiction where so many details of the story are known up front that the plot can only progress in very specific ways if it is to remain consistent and believable to the "reader".

Mathematics is a kind of world building.  In the imaginative sense.



    > 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?
    >
    > How is it that we are able to reliably know things about Platonia?

    I think just doing logic and math - starting from axioms and proving
    things from them - is interacting with the Platonic realm.


But how is it that we humans do that? This is my main question. What exactly are we doing when we start from axioms and prove things from them? Where does this ability come from? What does it consist of?



    > I would have thought that quarks and electrons from which we appear to be
    > constituted would be indifferent to truth.
    >
    > Which would fit with the fact that I seem to make a lot of mistakes.
    >
    > But you think otherwise?

    I didn't understand the above... what do quarks and electrons have to
    do with arithmetical platonism?


Are we not composed from quarks and electrons? If so - then how do "mere" collections of quarks and electrons connect with platonic truths?

By chance? Are we just fortunate that the initial conditions and causal laws of the universe are such that our quarks and electrons take forms that mirror Platonic Truths?


    >>
    >> 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.

    I only meant that all possible intelligences that start from a
    mathematics that includes addition, multiplication, and complex
    numbers will find that if they iterate the function z' = z^2 + c, they
    will find that some orbits become periodic or settle on a point, and
    some escape to infinity. If they draw a graph of which orbits don't
    escape, they will draw the Mandelbrot Set. All possible intelligences
    that undertake that procedure will draw the same shape... and this
    seems like discovery, not creation.


It seems like a tautology to me. If you do what I do and believe what I believe then you will be a lot like me...?

Is there anything to mathematics other than belief?

What are beliefs? Why do we have the beliefs that we have? How do we form beliefs - what lies behind belief?

Can *our* mathematical abilities be reduced to something that is indifferent to mathematical truth?



    > Could there be intelligences who start from vastly difference premises, 
and
    > use vastly different rules of inference, and draw vastly different
    > conclusions?

    Of course, but then what they are doing doesn't relate to the Mandelbrot 
Set.


However - they might *believe* their creations to be just as significant and universal as you consider the Mandelbrot Set to be - mightened they?

What would make them wrong in their belief but you right in yours?



    >
    > 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?

    I don't think so... based on my understanding of how mad "maddest of
    the mad" can get.

    > 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?

    I'm guessing you meant to say "and if there is such a belief...".  I'm
    having a tough time understanding where you're going with this... it
    seems like an interesting line of questions, but I have no idea how it
    relates to what we were discussing.


So - you spotted the ambiguity in that sentence. What kind of reasoning did you use to do that? Did you use your Platonic Truth Sense? Or some more ordinary, more "animal" reasoning?

The sentence was garbled, I admit. What I was trying to get at is: if all beliefs are possible, then can all beliefs, even the most bizarre and alien, be represented using collections of quarks and electrons whose states change over time in accordance with the laws of physics?

Put a slightly different way: Are all possible beliefs possible in our universe using our physics? Or would some beliefs require some other representational substrate capable of state changes requiring some more alien physics?

This sort of opens up the question of exactly what is involved in 
"representation".

Rex

I think you would need to have some definition of 'belief' in order to address such questions. If you simply mean an expression, "I believe X." then certainly any value of X, even "y and not-y" is possible. Generally we suppose belief includes at least the ability to clearly imagine X and describe it - which rules out self contradictions like "y and not-y" - and make some inferences from it. I'd say real belief entails at least the possibility of acting on the belief.

Brent

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