From: Marc Geddes (m_j_geddes@yahoo.com.au)
Date: Wed Jan 04 2006 - 19:52:10 MST
Yeah, damn friggin right everything you think you know
is wrong. Let me begin to relieve you of some of your
delusions....
---
In other to show where the Eliminative materialists
are going wrong, I will summarize the arguments for
mathematical Platonism. What is mathematical
Platonism? Well it's the idea that mathematical
entities are not just abstract fictions, but have a
real objective existence. Further these entities
cannot be completely reduced to (described in terms
of) material processes. And since (according to
Functionalism) the mind is equivalent to an algorithm
(which is a mathematical entity), if Platonism is
right the Eliminative materialists have to be wrong -
qualia in fact cannot be completely reduced to
material processes.
First, why should we believe in the objective
existence of mathematical entities? Surely, some will
argue, mathematical entities are really just abstract
fictions (or invented languages) we use for describing
what are really material processes. This position is
known as nominalism.
However, there's an argument known as *The argument
from Indispensability*. Certain mathematical theories
(for instance analysis) are indispensable for modern
physics. Physics uses quantifiers which range over
domains that include mathematical entities not in
space and time. Thus, the argument goes; since we
have to accept our best scientific theories of the
world, we should accept that the entities referred to
in our theories really exist.
Now one could try to remove the references to
mathematical entities in scientific theories. For
instance the philosopher Hartry Field (1980) has
proposed this - he suggested trying to remove talk of
real numbers in Newton's theory of gravity and
replacing numbers with space-time points and regions.
But if one tries to do this, one finds that the
theories become enormously unwieldy - mathematical
entities such as numbers are just so *useful* in
science. If there are entities in our theories which
it is very useful to refer to, this provides some
pragmatic grounds for believing in their existence.
The argument at work here is Occam's razor: in
science the general rule of thumb is that simple
explanations are favored over more complex ones.
Since in science references to mathematical entities
simplify scientific theories, the simplest explanation
is that these mathematical entities really exist.
Further it appears that there are some highly abstract
mathematical entities in scientific theories which
*cannot* be replaced with descriptions of material
processes at all. An example of this is 'the quantum
wave function' -in quantum mechanics. The wave
function is necessary for the theory to work, yet the
wave function is wholly abstract. See arguments by
Malament (1982) to the effect that one cannot remove
talk of abstract entities from quantum mechanics.
Since references to abstract entities appears to be
*essential* for QM to work and since QM is such a
successful scientific theory, the simplest possible
explanation is that these abstract entities really do
exist.
Returning to the principle of Ocaam's razor again,
physicist David Deutsch in his book 'The Fabric Of
Reality', uses the principle to establish 'Criterion
for reality'. The idea is that we should regard as
real those postulated entities which, if we tried to
replace them with something else would complicate our
explanations. Deutsch's principle was this:
'If according to the simplest explanation, an entity
is complex and autonomous, then that entity is real.'
('The Fabric Of Reality', Pg 91)
As Detusch points out, mathematical entities do appear
to match the criteria for reality: 'Abstract entities
that are complex and autonomous exist objectively and
are part of the fabric of reality. There exist
logically necessary truths about these entities, and
these comprise the subject-matter of mathematics.'
Professor of mathematics Roger Penrose also neatly
makes the point that mathematicians strongly feel they
are engaged in discovery, not creation, and that
mathematical entities appear to have complex,
autonomous structure not put there by humans:
'The Mandelbrot set provides a striking example. It's
wonderfully elaborate structure was not the invention
of any one person, nor was it the design of a team of
mathematicians. Benoit Mandelbrot himself, the
Polish-American mathematician who first studied the
set, had no real prior conception of the fantastic
elaboration inherent in it...Moreover, the complete
details of the complication of the structure of
Mandelbrot's set cannot really be fully comprehended
by any one of us, nor can it be fully revealed by any
computer. It would seem that this structure is not
just a part of our minds, but it has a reality of its
own.'
In 'Shadows of the Mind', Penrose goes on to make the
very telling point that mathematical theories often
turn out to be useful for science in a manner which
goes far far beyond what the math was originally used
for. The example is given of the mathematics of
Einstein's general theory of relativity:
'In the early years after Einstein's theory was put
forward, there were only a few effects that supported
it and the increase in precision over Newton's scheme
was marginal. However, now, nearly 80 years after the
theory was first produced, its overall precision has
grown to something like *ten million times* greater’
('Shadows of The Mind', Pg 415).
This is not at all what we would expect if the math
was just an invention of the human mind. As Penrose
points out: 'Einstein was not just 'noticing
patterns' in the behavior of physical objects. He was
uncovering a profound mathematical substructure that
was already hidden in the very workings of the world.'
Accepting all this, a skeptic might concede that
mathematical entities have objective existence, but
try to identity them entirely with the material world.
This doesn't work. For one thing, as pointed out
earlier, our best scientific theories make explicit
references to wholly abstract entities (for instance
'the quantum wave function') which have no concrete
counter-part at all.
Further more, there exist perfectly good mathematical
facts which be cannot be directly or indirectly
matched to any material facts. A striking example of
this transfinite numbers and infinite sets- here
references are clearly made to infinite entities yet
all available evidence would indicate that all
material entities are finite. Attempting to equate
transfinite numbers or infinite sets with some
nebulous property of a ‘multiverse’ (as some
scientists have attempted to do) is simply begging the
question about what is physical. Nor can infinite
sets be argued away as fictions - they are perfectly
precise and logical mathematics, having the same
'reality' as any other results in mathematics. Greg
Cantor developed a rigorous treatment of transfinite
numbers and later Abraham Robinson and John Conway did
the same for infinitesimals.
In fact one has not even have to consider such
esoteric things as transfinite numbers and quantum
wave functions to see that nominalism (the idea that
abstract entities don't exist) has severe problems.
Logical positivists used to claim that statements
could not be meaningful if they could not be converted
into descriptions in terms of concrete observables.
The positivists were wrong. There are perfectly
meaningful statements in plain every day English that
cannot be converted into concrete descriptions. Here
is one example from the philosophy literature:
Statement: "There are shapes that are never
exemplified"
It can be shown that this statement cannot be
converted into any description of finite length which
refers only to concrete entities. Yet the statement
is clearly meaningful. This shows that abstract
entities do exist.
In fact nominalists cannot speak the words 'No
abstract entities exist' without being in serious
danger of contradicting themselves. For this sentence
itself makes reference to abstract entities. By the
nominalist's own doctrine, the sentence is
meaningless. Indeed the nominalist's cannot think
that he 'saying' anything at all: he is simply making
noises with his mouth! ;)
Semantic considerations provide even more evidence for
believing in the existence of abstract entities. 'The
Fregean argument' is based on the idea that only in
the context of a sentence does a word have meaning.
If a certain expression functions as a singular term
in a sentence, the sentence cannot be meaningful
unless there is an actual real singular entity to
which the term is referring. For instance if ‘2’
functions as a singular term in a sentence, there must
be a real entity '2' to the terms refers.
Finally, let's look at computer science and the mind.
If both Platonism and functionalism is correct and the
mind is a mathematical algorithm, then the mind cannot
be completely reduced to material processes. By the
above arguments, mathematical entities are objectively
real things that exist 'out there' in reality. But
note that many *different* brains (or computers) can
run the *same* algorithm. This suggests that the
mathematical algorithm itself cannot be identical to
the material processes which are enacting a particular
example of it. Penrose again:
'The mind-stuff of strong AI is the logical structure
of an algorithm... the particular physical embodiment
of an algorithm is something totally irrelevant. The
algorithm has some kind of disembodied 'existence'
which is quite apart from any realization of that
algorithm in physical terms' ('The Emperor's New
Mind', Pg 27)
It was an argument similar to this that led to the
demise of the original 'Identity Theory' of mind (a
theory which attempted to identity mental states with
physical processes). Again, the trouble is that many
different brain states could be associated with the
*same* algorithm (or have the same mental states)
which shows that physical processes cannot be
identified with mathematical entities in any simple
way. The weaker 'Token Identity' theories concede
this, but still attempt to equate mental states with
physical processes. Couldn't one simply say that
there's some general high-level properties of physical
matter which can be equated with the algorithm, and
hence dispense with ghostly mathematical entities?
The reason one can't really say this boils down to
Occam's razor and inference to the best explanation
again. Attempting to replace the concept of
'algorithm' with some high level properties of
physical matter is results in descriptions that are
enormously complex and unwieldy. And therefore such
an arbitrary scheme should be rejected, for reasons
explained earlier. Inference to the best explanation
requires that we attempt that mathematical entities
such as 'algorithim's' really do have an objective
existence above and beyond a particular instantiation
in material processes.
To conclude, strong arguments were given to the effect
that mathematical entities are objectively real.
Further, arguments were given that mathematical
entities cannot be completely reduced to descriptions
in terms of material processes. According to the most
popular theories of mind, a mind is equivalent to an
algorithm, which is a mathematical entity. By the
arguments for mathematical Platonism given combined
with the observation that many *different* brains can
run the *same* algorithm, the conclusion must be that
the mathematical algorithm cannot be completely
identified with the material processes occurring in
the brain and therefore (assuming a mind is equivalent
to an algorithm), neither can mental states. It can
further be concluded then, that the position known as
'Eliminative materialism' (the idea that talk of
qualia can be entirely replaced with talk of material
processes) must be false.
--
Cheers!
"Till shade is gone, till water is gone, into the shadow with teeth bared, screaming defiance with the last breath, to spit in Sightblinder’s eye on the last day”
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