IZ wrote:
>
>
>
>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.
>
But it is a straw man to say "everything-theories makes the prediction that
Harry Potter universes should be just as likely as lawlike ones", because in
fact they do *not* make that definite prediction. If you had just said
something like, "everything theories do not yet have any rigourous proof
that Harry Potter universes should be less likely than lawlike ones" I
wouldn't object.
> > 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 sense/reference distinction is about the possibility of our having
multiple mentally distinct terms which map to the same real-world
object...but what would "sense but no reference" mean? A term that is
completely meaningless, like a round square? I don't see how there can be an
objective, mind-independent truth about a term that doesn't refer to any
coherent object or possibility. Can you think of any statements outside of
math or logic that you would say have "sense but no reference" but also have
a mind-independent truth value?
>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.
Again, what does "exist" mean for you?
>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.
"Tend to", although occasionally they can make mistakes. For the answer to
be really objective, you need to refer to some sort of ideal mathematician
or computer following certain rules, but that is just another form of
Platonism.
>
> > >
> > >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!
I think it's the only philosophy of mathematics that says that mathematical
statements have a *mind-independent* truth-value, though.
>
> > 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!
No, you misunderstood. I'm saying that *if* you believe that mathematical
statements have a mind-independent truth-value, that is necessarily is
equivalent to what I understand "mathematical Platonism" to mean. Of course,
you may not in fact believe that mathematical statements have any such
mind-independent truth-value.
>
> >
> > >
> > > > 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!)
Most physicists today take a "spacetime" view of the universe in which the
notion of a global objective past, present and future is meaningless (for
any given event, it is of course true that everything in its future light
cone objectively lies in its future and everything in its past light cone
objectively lies in its past, but there is no objective truth about whether
events not in either light cone lie in the first event's past, future, or
present). Philosophically, I don't think the notion of time "really moving"
is even coherent--how could the present "move" without introducing a second
time dimension, for example? Are you familiar with McTaggart's distinction
between the A-series and the B-series view of time? Are you arguing for the
A-series here? If so I think few physicists would agree--see for example
http://tinyurl.com/nesh7
>
> > 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.
It leads to them, but it doesn't necessarily lead to the idea that they have
a high probability. Again, you would need to define a measure on
universes/observer-moments to make any predictions about this one way or
another--until then these "everything" ideas are simply incomplete
hypotheses, a bit like string theory (which also has not been developed to
the point where it can make definite predictions on many physical issues,
although I certainly think string theory is much, much closer to being a
real predictive theory than the very speculative everything-ideas).
>That is its observatioanl consequence.
It's not an observational consequence if you don't happen to be in one of
the Harry Potter worlds! No multiverse theory predicts that observers should
have an omniscient view of all universes, they only see the one they are
living in.
>
>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
Is Peter D Jones a philosopher? I have my doubts that philosophers of
mathematics would see the categories described here as mutually exclusive.
For example, a formalist, to the extent he believes there is an objective
truth about whether certain statements are derivable from a set of axioms
and rules of inference, is just a species of platonist as I would define it;
the objective truth necessarily involves an "ideal" case of following the
rules without any possibility of error, not any specific mathematician or
computer which may slip up in deriving new propositions from the axioms. In
this case the ideal axiomatic system is the "abstract object" which there
are objective truths about, since the truths cannot refer to any specific
attempt to implement the system in the real world. Some formalists may not
think in this way, but in this case they do not really believe there are
objective mind-independent truths about axiomatic systems.
Jesse
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Received on Tue Jul 11 2006 - 19:11:39 PDT