Jesse Mazer wrote:
> IZ wrote:
>
> >And mathematical MWI *would* be in the same happy position *if*
> >it could find a justification for MWI or classical measure.
> >
> >However, in the absence of a satifactory theory of measure,
> >no-once can say that the posit of matter, of material existence
> >is useless. To have material existence is to have non-zero measure,
> >and vice-versa.
>
> Yes, but the point is that almost all of us on this list want to *find* a
> "satisfactory theory of measure" to apply to "everything", so it's a
> strawman to say that it's a prediction of "everything" hypotheses that Harry
> Potter universes should be just as probable as any other.
Wanting to find a measure theory doesn't mean you have
found one, and if you havent found one, it isn't a straw man
to say so.
> Some rough
> proposals for such a theory of measure have been made in this list in the
> past, like the "universal prior" (see
> http://parallel.hpc.unsw.edu.au/rks/docs/occam/node2.html or
> http://www.idsia.ch/~juergen/everything/node4.html ), or my own speculation
> that a theory of consciousness assigning relative and absolute probability
> to observer-moments might have only a single self-consistent solution (see
> http://tinyurl.com/ekz7u or http://tinyurl.com/jnaqb for more on this idea).
>
> >
> > > >You are not going to get anywhere with the
> > > >UDA until you prove mathematical Platonism, and your
> > > >argument for that -- AR as you call it --
> > > >just repeats the same error: the epistemological
> > > >claim that "the truth -alue of '17 is prime is mind-independent"
> > > >is confused with the ontological claim "the number of 17 exists
> > > >separately
> > > >from us in Plato's heaven".
> >
> > > But that is really all that philosophers mean by mathematical platonism,
> > > that mathematical truths are timeless and mind-independent--
> >
> >nope.
> >
> >"Platonists about mathematical objects claim that the theorems of our
> >mathematical theories - sentences like '3 is prime' (a theorem of
> >arithmetic) and 'There are infinitely many transfinite cardinal
> >numbers' (a theorem of set theory) - are literally true and that
> >the only plausible view of such sentences is that they are ABOUT
> >ABSTRACT OBJECTS "
> >
> >(emphasis added)
>
> What do the words "abstract object" mean to you? To me, if propositions
> about numbers have a truth independent of human minds or beliefs, that's
> equivalent to saying they are true statements about abstract objects--how
> could a statement be objectively true yet not be about anything?
By having sense but no reference, for instance.
http://en.wikipedia.org/wiki/Sense_and_reference
The case for mathematical Platonism needs to be made in the first
place; if numbers do not exist at all, the universe, as an existing
thing, cannot be a mathematical structure. (solipsists read: if numbers
are not real, I cannot be mathematical structure). The case for
mathematical Platonism is usually argued on the basis of the objective
nature of mathematical truth. Superficially, it seems persuasive that
objectivity requires objects. However, the basic case for the
objectivity of mathematics is the tendency of mathematicians to agree
about the answers to mathematical problems; this can be explained by
noting that mathematical logic is based on axioms and rules of
inference, and different mathematicians following the same rules will
tend to get the same answers , like different computers running the
same problem.
Your remark is quite telling though. Almost everybody on the list
is making that kind of asumotion with varying degrees of
unconsiousness.
> >
> >http://plato.stanford.edu/entries/platonism/#4.1
> >
> > > this is itself
> > > an ontological claim, not a purely epistemological one.
> >
> >Quite. Did you mean that the other way around ?
>
> No, I was responding to your comment:
>
> >You are not going to get anywhere with the
> >UDA until you prove mathematical Platonism, and your
> >argument for that -- AR as you call it --
> >just repeats the same error: the epistemological
> >claim that "the truth -alue of '17 is prime is mind-independent"
> >is confused with the ontological claim "the number of 17 exists
> >separately
> >from us in Plato's heaven".
>
> Here you seem to be saying that "the truth value of '17 is prime' is
> mind-independent" is a purely "epistemological" claim.
It certainly *could* be, at least. Platonism is *not* the only
philosophy of mathematics!
> What I'm saying is
> that it's necessarily ontological, as are any claims about the objective
> (mind-independent) truth-value of a given proposition.
So you are claiming that mathematical Platonism is not merely
true but *necessarily* true ? That is quite a claim!
>
> >
> > > Few would literally
> > > imagine some alternate dimension called "Plato's heaven" where platonic
> > > forms hang out, and which is somehow able to causally interact with our
> > > brains to produce our ideas about math.
> >
> >Some do. In any case, if numbers don't exist at all -- even
> >platonically --
> >they they cannot even produce the mere appearance of a physical world,
> >as Bruno requires.
>
> But what does "exist" mean in this context? Do you think it makes sense to
> say that there is are objective truths about objects which do not "exist" in
> any sense?
Truth-values attach to **sentences**. Those are the only
objects a non-Platonist needs.
> That does not make sense to me.
That's hardly a counterargument. Are you familiar
with the rivals to Platonism ? See below.
> On the other hand, the idea that
> mathematical objects "exist" in the sense of there being objective truths
> about them need not necessarily imply that possible self-aware observers
> within complex mathematically-describable worlds would "exist" in the sense
> that they'd be actual conscious beings whose experiences are just as real as
> yours or mine (although I think it's a lot more elegant to assume they
> would).
I ma saying that not only does mathematical Platonism "not necessarily"
imply consious observers within Platonia , it just doesn't imply
it *at all*. (For heavens' sake, it doesn't even imply
computational *processes*, since Platonia is timeless!)
> But I think it's this question of the consciousness of different
> possible beings within mathematical structures that's the key one in the
> "every universe exists vs. only one universe exists" debate, not whether
> mathematical laws are describing the behavior of "stuff" or whether the
> mathematical relationships between events alone are all there is (how could
> we possibly tell the difference? Believing in 'stuff' as opposed to
> bare-bones mathematical relationships is not something that leads to any
> distinct measurable consequences, so it has no connection to any empirical
> results of science).
As I have stated several times, the maths-only theory leads to
Harry Potter universes. That is its observatioanl consequence.
Substantial theories don't.(They can also support time, consicousness,
etc).
> Jesse
Five Approaches to the Metaphysics of Mathematics
Introduction Why Mathematics Works
Approach 1: Empiricism: The "maths is physics" theory.
Approach 2: Platonism: Objectivity and objects.
Approach 3: Formalism: Mathematics as a game.
Approach 4: Constructivism: How Real Are The Real Numbers?
Approach 5: Quasi Empiricism
Why Mathematics Works
Mathematics is the theoretical exploration of every possible kind of
abstract structure
The world has some kind of structure; it is not chaos or featureless or
void
Therefore, it is likely that at least some of the structures discovered
by mathematicians are applicable to the world
(It would be strange if everything mathematicans came up with was
empirically applicable, and might tend one towards Platonism or
Rationalism, but this is not the case. It might be possible for
mathematicians working completely abstractly and theoretically to fail
to include any empirically useful structures in the finite list of
structures so far achieved, but maths is not divorced from empiricism
and practicallity to that extent -- what I say in (1) is something of
an idealisation).
Empiricism: The "maths is physics" theory.
Mathematical empiricism is undermined by developments since the 19th
century, of forms of mathematics with no obvious physical application,
such an non-Euclidean geometry. I should probably say no obvious
application at the time since non-Euclidean geometery -- or curved
space -- was utilised in Einstein's General Theory of Relativity
subsequent to its development by mathematicians. So M.E. has two
problems: the existence of mathematical structures with no (currently)
obvious physical application, and the fact that the physical
applicability of different areas of mathematics varies with time,
depending on discoveries in physics. Todays mathematical game-playing
may be tomorrow's hard reality. It is also undermined by mathematical
method, the fact that maths is a chalk-and-blackboard (or just
thinking) activity, not a laboratory activity.
Platonism: Objectivity and objects.
Both Platonism and Empiricism share the assumption that mathematical
symbols refer to objects. (And some people feel they have to believe in
Empiricism simply because Platonism is so unacceptable). Platonism gets
its force from noting the robustness and fixity of mathematical truths,
which are often described as "eternal". The reasoning seems to be that
if the truth of a statement is fixed, it must be fixed by something
external to itself. In other words, mathematical truths msut be
discovered, because if they were made they could have be made
differently, and so would not be fixed and eternal. But there is no
reason to think that these two metaphors --"discovering" and "making"--
are the only options. Perhaps the modus operandi of mathematics is
unique; perhaps it combines the fixed objectivity of discovering a
physical fact about the external world whilst being nonetheless an
internal, non-empirical activity. The Platonic thesis seems more
obvious than it should because of an ambiguity in the word "objective".
Objective truths amy be truths about real-world objects. Objective
truths may also be truths that do not depend on the whims or
preferences of the speaker (unlike statements about the best movie of
flavour of ice-cream). Statements that are objective in the first sense
tend to be objective int he second sense, but that does not mean that
all statements that are objective in the second sense need be objective
in the first sense. They may fail to depend on individual whims and
preferences without depending on anything external to the mind.
Formalism: Mathematics as a game.
Both Platonism and Empriricism share the assumption that mathematical
symbols refer to objects. An alternative to both is the theory that
they do not refer at all: this theory is called formalism. For the
formalist, mathematical truths are fixed by the rules of mathematics,
not by external objects. But what fixes the rules of mathematics ?
Formalism suggests that mathematics is a meaningless game, and the
rules can be defined any way we like. Yet mathematicians in practice
are careful about the selection of axioms, not arbitrary. So do the
rules and axioms of mathematics mean anything or not ?
The reader may or may not have noticed that I have been talking about
mathematical symbols "referring" to things rather than "meaning"
things. This eliptically refers to a distinction between two different
kinds or shades of meaning made by Frege. "Reference" is the
external-world object a symbol is "about". "Sense" is the kind of
meaning a symbol has even if does not have a reference. Thus statements
about unicorns or the bald King of France have Sense but not Reference.
Thus it is possible for mathematical statments to have a sense, and
therefore a meaning, beyond the formal rules and defintions, but
stopping short of external objects (referents), whether physical or
Platonic. This position retains the negative claim of Formalism, that
mathematical symbols don't refer to objects, and thus avoids the
pitfalls of both Platonism and Empiricism. Howeverm it allows that
mathematical symbols can have meanings of an in-the-head kind and thus
explains the non-arbitrary nature of the choice of axioms; they are not
arbitrary because they must correspond to the mathematician's intuition
-- her "sense" -- of what a real number or a set is.
So far we have been assuming that the same answer must apply uniformly
to all mathematical statmentents and symbols: they all refer or none
do. There is a fourth option: divide and conquer -- some refer and
others don't.
Constructivism: How Real Are The Real Numbers?
Mathematical Platonism is the view that mathematical objects exist in
some sense (not necessarily the sense that physical objects exist),
irrespective of our ability to display or prove them.
Constructivism is an opposing view, according to which only objects
which can be explicitly shown or proven exist.
Some constructivists believe in constructivism as an end in itself.
Others use it as a means to an end,namely the elimination of
infinities.
Real numbers (numbers which cannot be expressed in a finite number of
digits) are a particular problem for both kinds of constructivist. For
the first, because most real numbers cannot be shown. Some are well
known, such as pi and e (the base of natural logarithms). These can be
generated by an algorithm; a sort of mathematical machine, where if you
keep turning the handle, it keeps churning out digits. Of course, real
numbers are not a finite string of digits, so the process never
finishes. Nonetheless, the fact that the algorithm itself is finite
gives us a handle on the real number, an ability to grasp it. However,
the real number line is dense, so between any of these well known real
numbers are infinities of intransigent real numbers, whose digits form
no pattern, and which therefore cannot be set out, either as a string
of digits, or as an algorithm.
Or, at least, so the Platonists would have it. What alternatives are
available for the constructuvist ?
1. All numbers are equally (ontologically) real.
2. The accessible real numbers (the transcendental numbers) are
(ontologically) real, and the other (mathematically) "real" numbers are
not.
3. No (mathematically) real number is (ontologically) real.
1. is the position constructivists are trying to get away from. 3 is
hardly tenable, since some irrational numbers, such as pi, *can* be
constructed. And (2) contradicts the idea of density -- it suggests the
real number line has gaps. Moreover, something like the requirment of
denstity can be asserted without making any strong ontological
commitments towards the reality of the real number line.
Quasi Empiricism
A number of recent developments in mathematics, such as the increased
use of computers to assist proof, and doubts about the correct choice
of basic axioms, have given rise to a view called quasi-empiricism.
This challenges the traditional idea of mathematical truth as eternal
and discoverable apriori. According to quasi-empiricists the use of a
computer to perform a proof is a form of experiment. But it remains the
case that any mathematical problem that can in principle be solved by
shutting you eye and thinking. Computers are used because mathematians
do not have infinite mental resources; they are an aid. Contrast this
with traditonal sciences like chemistry or biology, where real-world
objects have to be studied, and would still have to be studied by
super-scientitists with an IQ of a million. In genuinely emprical
sciences, experimentation and observation are used to gain information.
In mathematics the information of the solution to a problem is always
latent in the starting-point, the basic axioms and the formulation of
the problem. The process of thinking through a problem simply makes
this latent information explicit. (I say simply, but of ocurse it is
often very non-trivial). The use of a computer externalises this
process. The computer may be outside the mathematician's head but all
the information that comes out of it is information that went into it.
Mathematics is in that sense still apriori.
Having said that, the quasi-empricist still has some points about the
modern style of mathematics. Axioms look less like eternal truths and
mroe like hypotheses which are used for a while but may eventualy be
discarded if they prove problematical, like the role of scientific
hypotheses in Popper's philosophy.
Thus mathematics has some of the look and feel of empirical science
without being empricial in the most essential sense -- that of needing
an input of inormation from outside the head."Quasi" indeed!
Peter D Jones 5/10/05-26/1/06
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Received on Tue Jul 11 2006 - 17:51:00 PDT