On Fri, Sep 21, 2012 at 8:40 AM, Rex Allen <rexallen31...@gmail.com> wrote:
> On Tue, Sep 18, 2012 at 11:50 PM, Terren Suydam <terren.suy...@gmail.com>
>> 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.
Can you articulate any important differences between them?
> 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
> Mathematics is a kind of world building. In the imaginative sense.
I am not unsympathetic with this view, given the creativity that goes
into mathematical proofs. However, it falls apart for me when I
consider that an alien civilization is constrained to build the same
worlds if they start from the same logical axioms.
>> 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?
We're using our intelligence and creativity to search a space of
propositions (given a set of axioms) that are either provably true or
false. I would say our intelligence and creativity comes from our
animal nature, evolved as it is to make sense of the world (and each
other) and draw useful inferences that help us survive. I'm not sure
how to answer the question "what does it consist of". Are you asking
how we can act intelligently, how creativity works?
>> 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?
I see. Assuming comp, we are some infinite subset of the trace of the
UD (universal dovetailer), which is a platonic entity. Quarks and
electrons are a part of the physics that emerges from that (the
numbers' dreams)... that's the reversal, where physics emerges from
The question of how we, as "mere" collections of quarks etc. connect
back with Platonia, is answered by CT (Church-Turing Thesis). As we
are universal machines, we can emulate any computation, including the
universal dovetailer (for instance).
>> 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?
The point is that you are constrained in what you can prove starting
from a given set of axioms. You are not constrained in which axioms
you start with - that's where the belief comes in since there is no
way to prove that your axioms are True, except within a more
encompassing logical framework with its own axioms.
> What are beliefs? Why do we have the beliefs that we have? How do we form
> beliefs - what lies behind belief?
Beliefs in the everyday sense are inferences about our experience that
we hold to be true. They help us navigate the world as we experience
it, and make sense of it. Mostly our beliefs are formed by suggestion
from our parents and peers when we are young, and as we learn and grow
we complicate our worldview with new beliefs. There isn't much behind
belief except habituation. Certainly most of us hold onto some beliefs
that are contradicted by facts (particularly the beliefs we hold of
> Can *our* mathematical abilities be reduced to something that is indifferent
> to mathematical truth?
I think if you were doing math in a way that was indifferent to
mathematical truth, you wouldn't be very good at math.
>> Of course, but then what they are doing doesn't relate to the Mandelbrot
> 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?
I think you're just asking if one set of axioms if superior to
another. At the most fundamental level, it is an arbitrary choice.
Practically speaking, there are consequences to that choice which may
make some axioms better than others.
> 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?
Animal reasoning... it would be pretty convenient to have a Platonic
Truth Sense :-) But that's impossible with comp, as oracles are not
> 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?
Any belief that can be articulated can be represented, as the
articulation of that belief is its own representation. Bruno maps
qualia to beliefs that cannot be articulated (shared). I think a more
interesting question is, can all possible subjective experience be
represented using collections of quarks and electrons whose states
change over time in accordance with the laws of physics? Comp says
Perhaps you think it would contradict platonism for an instantiated
being to hold inconsistent beliefs? But in comp, the "numbers'
dreams" are not constrained at all. It's like a flight simulator...
the program running the simulation is utterly determined
(constrained), but the "view from the inside" of the simulation can
entail an infinity of possible scenarios, completely unconstrained in
terms of possibility.
> 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
Yes, and I think that gets at the core of whether you believe in comp
or not (whether you would say yes, doctor). If the idea that all
possible beliefs (including the non-sharable qualia) can be
represented by some computational substrate seems ludicrous then you
should say no to the doctor.
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