Jesse Mazer wrote:
> 1Z wrote:
>
> >
> >Jesse Mazer wrote:
> > > 1Z wrote:
> > >
> >
> > > If a theory can't predict the relative probabilities of X vs. Y, that is
> >not
> > > in any way equivalent to the statement that it predicts X and Y are
> >equally
> > > likely. One is an absence of any prediction, the other is a specific and
> > > definite prediction.
> >
> >IOW, if MMW heories worked, MMW theories would work.
>
> No, that is not a fair paraphrase of what I said. I meant exactly what I
> said I meant--if a hypothesis is not well-defined enough to tell you the
> relative probability of different possibilities, that does not justify the
> claim that the hypothesis predicts each possibility is equally likely. Do
> you agree with this principle or not?
If a hypothesis is not well-defined enough to tell you the
relative probability of different possibilities, the claim
that they are not all the same is unsupported.
> >That isnot really analogous becasue the CC can only have one
> >value at a time.
>
> That difference is irrelevant to my point about probabilities. Again, it is
> *always* unjustified to say that because a theory doesn't predict the
> relative probabilities of different outcomes, that means it predicts they
> are equally likely; it doesn't matter whether or not we are talking about
> the probability in the context of a large ensemble of events (say, the
> probability a certain type of atom will decay in a 1-minute time period,
> where we are repeating the test with a large number of atoms) or in the
> context of a single event.
In the absence of evidence to the contrary , you have to
assume that probabilites are even.
> Anyway, it is quite possible that even if string theory could make
> predictions about the value of the cosmological constant, it would only be a
> probabilistic prediction rather than predicting a single unique value, which
> means that if you are prepared to entertain either the MWI of quantum
> mechanics or "chaotic inflation" where new universes bubble from prior ones
> via inflation, then there might in fact be different "universes" with
> different values of the cosmological constant.
> > > >We can make "sense" of "unicorns have horns", despite
> > > >the lack of reference.
> > >
> > > In this case I would say the reference would be to a certain concept
> >which
> > > humans have collectively defined;
> >
> >No, that's the sense. Sense is in-hte-head , reference
> >is out-of-the-head.
>
> OK, I see. So what if we are talking about a concept in itself, as in "most
> people's concept of a unicorn is that of a horse-like creature with a single
> horn"; would the "concept" itself be a reference?
Only the reference of "a concept" is a concept.
Fictional terms don't have referents. That's why they are unreal.
The point is that you don't need reference for meaning.
> > > I agree, and even a "modal realist" philosopher like David Lewis (see
> > > http://en.wikipedia.org/wiki/David_Lewis_(philosopher) ), who thinks
> >that
> > > propositions about possibilities can only be objectively true or false
> >if we
> > > assume all possible worlds actually exist, would not say that there is
> >any
> > > kind of causal interaction between worlds needed to explain our ability
> >to
> > > reason about them.
> >
> >If we can reason about (for instance)
> >historical what-is without concrete ferefernces is parallel
> >dimensions, we can reason about maths without taking
> >a trip to Plato's heaven.
> But I have already made clear that I *don't* think that we need to refer to
> platonic forms which somehow causally interact with people's brains in our
> explanation of how people reason about math, just like David K Lewis doesn't
> think we need a causal interaction between different possible worlds to
> explain how people reason about possibilities.
So how do we need to refer to them ? Why do we need to refer to them ?
If they are causally inactive, an evil daemon could snap his
fingers and make them vanish. How do we know that hasn't happened
already ?
We can interact with real-world objects, and we must interact it them
in order to confirm the truth of non-mathematical, not-fictional
sentences.
If we had reason to think mathematical sentences were uniform with
empirical sentences, we would be forced to require the existence of
mathematical objects in spite of their lack of a role to play.
But it is the very fact that we do not need experiment or observation
to confirm mathematical sentences that shows they are differnt
from empirical sentences, and different in a way that absolves them
from requiring reference.
> > > The question was to try to help me grasp what you meant by "sense
> >without
> > > reference" and "mind-independent". If it's impossible to come up with
> >any
> > > examples outside of math, that should make you suspicious whether
> > > mathematics really has the strange and marvellous property of there
> >being
> > > objective mind-independent truths about mathematical terms even though
> >they
> > > lack any reference.
> >
> >No it shoudn't. Maths is obviously unique in a number of respects.
> >That is why there is such a subject as philosophy-of-mathematics.
> That's pretty vague--unique in what respects? Does uniqueness in these other
> respects somehow justify the belief that it is unique in the respect of
> involving both sense-without-reference and mind-independence?
Yes. Maths is apriori. It doesn't require experiment or
observation. Aprioriness is explained by analycity.
An analytical sentences contains in its meaning everything
necessary to determine its truth-value. Analycity is explained
by sense. Analycity requires a kind of meaning that is
in-the-head, in addition to reference.
Analycity and aprioriness explain objectivity and necessity. if
mathematical sentence doesn't require anything outside
itself to arrive at its truth value, then its truth-value
is not going to vary between times, places and persons.
> > > If you really believe this, you should at least be able
> > > to give an argument about *why* math is different from every other
> >domain in
> > > this respect.
> >
> >
> >It is on a deeper level of abstraction.
>
> That doesn't remotely resemble an argument--can you define precisely what
> "deeper level of abstraction" means, and why "deepness of abstraction"
> should be in any way related to statements that lack sense but are
> objectively true?
Don't you mean "lack reference but are objectively true" ?
This is hardly contentious. To lack references is (on my
view) to lack concrete references. To lack concrete
references is to be abstract. (Even Platonists
think mathemtical statements have *abstract* references,
but that leads to the same conclusion).
> > >The case for mathematical Platonism needs to be made in the first
> >If you are going to claim we are already inside Plato's
> >heaven, as many on the list do, you are already dealing
> >with a stronger definition of ontology than that,
> No I'm not. In terms of the notion that "existence" just means the entities
> you'd need to refer to in an exhaustive list of all objective truths, the
> idea that "we are already inside Plato's heaven" could be understood to mean
> that as long as you include all the true statements that deal with the
> mathematical question of how a theoretical universe with our laws of nature
> and initial conditions would behave, then your exhaustive list of truths
> doesn't need to include any additional non-mathematical truths which tell
> you this particular mathematical description is actually "physically
> instantiated" or something like that.
You don't need to refer to any entities to establish mathematical
truths.
You do need to refer to entities to establish which particular
mathematics
is exemplified by the physics in this universe. So your list
of truths is going to be partnered with a much shorter list
of necessary references.
Even if you insist that every truth must have a reference, even if it
is causally, and therefore epistemolgocially, idle, that just leaves
you,
as usual, with the HP problem.
> Of course, I'm not sure I believe myself that the exhaustive list of true
> statements would include *nothing* but mathematical statements--I think it
> might also have to include some truths about consciousness and qualia.
If they are a different kind of truth (presumably contingent),
wouldn't they require a different kind of instantiation (presumably
physical) ?
> In
> this sense, it is conceivable that the exhaustive list of truths would tell
> you that only some mathematically-describable worlds containing intelligent
> life would actually lead to conscious experiences, while others wouldn't--in
> this sense only some (or one) possible world would be "real". I think it's a
> lot simpler to just assume all mathematically-possible observers are equally
> real with respect to consciousness, but I can't prove it obviously.
Mathematics is no good at all in telling us
which mathemtical structures would be consciouss,
whether all would be, whether none would be,
or what consciousness is.
Nevertheless, consciousness exists.
So there is at leat one non-mathematical fact!
> Still, I
> have no idea what it would mean to say that some mathematically possible
> worlds are *physically* real if "physical" was meant as anything other than
> a shorthand way for talking about consciousness.
If mathematical reality is constituted by being the reference
of a mathematical statement, then physical reality is constituted
by being the reference of an objective but non-mathematical
statement. You have conceded that such truths could exist.
Or, to take my usual line,
1) You are physically real.
2) Physically real things can interact with you
3) Physically unreal things can't.
(Of course, if everything is mathematical, and if mathematical
entities are casually idle, as you have said, nothing can interact
with anything).
> > > Incidentally, does your definition of "exists" mean that you don't think
> > > anything exists beyond the boundaries of the observable universe
> >
> >That depends on what the "able" means in "observable".
>
> Well, put it this way--as long as the expansion of the universe is not
> slowing to zero in the limit as time goes to infinity (and observational
> evidence suggests the rate of expansion is actually accelerating), then any
> event beyond the boundaries of the observable universe at this moment of
> cosmological time will never, ever be able to have any causal influence on
> us, because the space between us and the point where the event occurred will
> always be expanding faster than light from the event can close the gap
> between it and us. So unless relativity is wrong and it's possible to travel
> faster than light, you would not in principle be "able" to observe anything
> happening beyond the observable universe, at any point in the future. So do
> you think that anything "exists" beyond the observable universe? And what if
> you and I are in different galaxies, so that the boundaries of the
> "observable universe" are slightly different for each of us? Do you think
> that existence is relative to the observer, rather than being objective and
> universal?
It's all causally connected providing it starts from the BB.
> > > Do you think there is any sense in which your projection could be
> > > objectively wrong, even if you believe it is correct?
> >
> >It could be wrong. So it doesn't necessarily
> >deliver objectivity, Hiwever it allows us to understand
> >what objectivity is.
>
> But again, to have a notion of objectivity you must have at least an ideal
> of an error-free theorem-producer
Objective doesn't mean "infallible", it means "not subjective",
ie not varying with personal opinions and prejudices.
> >Formalism is a bit iffy. Apart from that, they all do.
>
> Are you sure? Does empiricism say there are objective mathematical truths
> separate from whatever empirical "observations" we happen to make, for
> example?
No, it says there are objective truths
which are not separate from empircal investigation.
Empirical truths are
mind-independent too.
> > > You are still misunderstanding, of course MP is an ontological thesis,
> >where
> > > do you think I was arguing otherwise? What I'm saying is that any
> >statement
> > > of the form "there is a mind-independent truth about X" is an
> >ontological
> > > statement, by necessity.
> >
> >It is obviously epistemoligcal, It deals with truth, not being.
>
> But as I pointed out, it is common among philosophers to define "being" in
> terms the set of all possible objective truths.
I think you'll find that's "all true facts" or "all obtaining
states-of-affairs".
> I certainly don't think many
> philosophers would define "being" the way you do in terms of ability to
> causally interact with us.
Are there any philosophers who think non-existent things can interact
with them ?
> > > It is not a "necessity" to believe that statements
> > > about math are ontological ones, though, because you are free to deny
> >that
> > > there is any mind-independent truth about them (in which case you are
> > > obviously not a mathematical platonist).
> >
> >I believe there is mind-independent truth about them AND deny they are
> >ontological
>
> Then I think you are defining "ontological" in a different way from most
> philosophers, or at least most analytic philosophers of the 20th century
> (the 'continental' philosophers have their own way of thinking about
> ontology, but then continental philosophers hardly ever deal with philosophy
> of math).
Hmmm. But if you were extensively versed in analytical
philosophy, would you need readings from St Frege and St Carnap on the
subjects
of Sense and Analycity ?
> >How can it be ontological when it says nothing about being, existence,
> >etc ?
> >
> >Bearing in mind that *my* definition of existence entails the
> >possibility
> >of causal interaction...
>
> I acknowledge that with *your* definition of existence, mathematical objects
> do not exist. But I think you're using a pretty idiosyncratic definition
> that would be miles away from how most analytic philosophers would define
> existence, in particular those who work with philosophy of math.
All Analyticals are Platonists ?
> > > Yes, but that's because my notion of "existence" is simply a shorthand
> >for
> > > an element of reality about which there exist objective truths.
> >
> >Oh come on, that's like defining God as a necessarily
> >existing being. You need to *show* that truth
> >implies existence, not just assert it.
>
> You're acting as though we have some mutually agreed-upon prior definition
> of "existence" which is separate from objective truth,
Dictionaries do not define "truth" and "existence" as synonyms.
> and that I am trying
> to make an *argument* that objective truth implies existence in this sense.
> But that's not what I'm doing at all--I am *defining* the word "existence"
> in terms of objective truth, not making an argument.
That is exactly what I am complaining about.
> I agree that if
> existence is defined in your way instead of my way, then objective truth
> doesn't imply existence, but then I think you're using a very weird
> definition which is different from how virtually any analytic philosopher
> would define it,
As truth ?
1) existence is truth, truth existence
2) some false statements exist
3) therefore, from (1), false statements are true
By reductio, (1) is false.
> and which leads to internal problems like the issue of
> whether anything "exists" beyond the boundaries of the observable universe.
> > > But that would make "existence" local too, rather than objective. My
> >light
> > > cones are different from yours, so if you want to say that the past is
> > > "real" in a sense that the future is not, that would make the reality of
> > > events different for each observer.
> >
> >Not if ptoetnial causal interaction is allowed to run in both
> >directions. There
> >is chain of ackward causes linking me to the BB; running forward again,
> >another chain connects to events outside my light-cone.
>
> Yes, but as I pointed out above, events happening at the current
> cosmological time outside the current observable universe cannot ever have
> an effect on anything in your future light cone, nor can they have been
> affected by any event in your past light cone, assuming the expansion of
> space is not slowing to zero.
As I say, I allow causality to work both ways.
> > > >http://www.geocities.com/peterdjones/met_time2.html
> > >
> > > His argument simply assumes that a moment can "become existent", without
> > > addressing this question of whether we need a second time dimension to
> >make
> > > sense of this
> >
> >We don't, since nothing changes once it has come into existence.
>
> I don't understand how this response addresses the issue of needing a second
> time dimension.
The regress of time dimension comes in when you require that
moments in time themselves change as time progresses.
'(7) The argument that the A series cannot exist is as follows: On the
one hand, (a) past, present, and future are
incompatible determinations (whether relations or qualities) of an
event; if any event is present, it cannot be past,
nor be future. On the other hand, (b) every event has them all; if any
event is past, it has been present and future.
These two, (a) and (b), are simply inconsistent, and therefore the A
series, and consequently change and time, cannot
exist. (468) Notice that we cannot presuppose the existence of time, in
order to evade this argument.
(8) An obvious objection against this argument is that (a) or (b), or
both, involves equivocation in terms of verb-forms for
tense; the determinations as past, present, and future are only
incompatible when they are simultaneous, and there should be no
contradiction when an event has all of them successively. However, this
objection involves a vicious circle, because it presupposes the
existence of time, in that those determinations are supposed to be
taking place in time. (468)
(9) Thus the previous conclusion cannot be evaded. Since the notions of
A series and of time contain a contradiction, they cannot
be applied to reality, and hence time is unreal. (470-471) It must be
remembered that this whole argument is taking place
within the context of attributing time to the reality itself, not to
subjective consciousness. (471-472) '
http://www.bun.kyoto-u.ac.jp/~suchii/mctaggart.html
My basic point is that it is only contradictory for a thing (in this
case an event, or moment-of-time)
to be variously past, present and future if those are
<em>intrinsic</em> properties. There
is no contradiction in having different relational properties, scuha s
being both a mother
and a daughter!
> Our notion of spatial movement depends on the idea of a time
> dimension for something to have moved from one location in space at an
> earlier time to another location at a later time; if you believe the present
> is "moving forward" along our universe's time dimension, that requires some
> sort of meta-time for it to make sense in the same way (as in, 'at an
> earlier point in meta-time, the present moment was 1985; but later in
> meta-time, the present moment had moved forward to 2006').
I don't believe the present is "moving forward" along our universe's
time dimension.
I think new present states are popping into existence
and remaining unchanged thereafter.
http://www.geocities.com/peterdjones/diagrams/time_A_series2.jpg
http://www.geocities.com/peterdjones/met_time2.html
> And then you have
> the same issue with whether you take an A-series view or a B-series view of
> meta-time, and if you want to adopt an A-series view you'd have to introduce
> meta-meta-time to make sense of that, and so forth.
It's all B-series, as I have said.
> > > . And as he admits, he is "assuming that such a thing as
> > > becoming is possible without describing or explaining it".
> >
> >It is philoosphically respectable to regard tiem as fundamental.
> >something has to be fundamental.
>
> It's "philosophically respectable" because it has such a long history, but I
> can think of a number of similar ideas with a long history that don't seem
> in any way coherent to me, like the notion of "free will" as distinct from
> determinism or randomness or any combination of the two.
Well I have an argument for that too...
In any case, it is a question of showing a contradiction,
not of things "seeming incoherent".
> >
> > > >Mathematical Platonism also doesn't (obviously) have the resources
> > > >to keep "worlds" separate.
> > >
> > > Sure it does. Different Turing machine programs are mathematical
> >objects,
> > > no?
> >
> >Different substrings within TM programmes can be identical, no ?
>
> Sure, but you still aren't explaining how this suggests beings in one
> program would have knowledge of programs different from their own.
> Obviously
> if you had two simulations which behaved identically in some region of
> simulated space and time, then simulated observers within these regions
> would have identical experiences, but they are still both experiencing
> things which happen in their own simulation, they have no knowedge of events
> which don't correspond to anything in their simulation, which is what you
> seemed to be suggesting earlier when you claimed that if the "everything"
> idea was right we should be able to observe events in alternate worlds which
> didn't happen in our own world.
The set of every mathematical possibility includes possibilites where
the internal thoughts, memories etc are completely out of
sync with what is going on externally (and/or with each other).
Platonia is NOT a Matrix theory. In the Matrix , one world
is simulated, in Platonia every mathematical possibility is.
Most mathematical possibilities make no sense physically.
> >Bearing in mind that the physical universe provides us with a
> >mechanism --
> >spatio-temporal location -- which allows identical things to be kept
> >separate,
> >and which doesn't exit in Platonia.
>
> But we're not talking about keeping identical things separate, we're talking
> about keeping *different* things separate, namely different worlds within
> the "everything" (like a lawlike universe vs. a universe that obeys one set
> of laws up to a given moment but then switches over to totally different
> 'Harry Potter' laws after that moment).
The two universes you mention are identical up to a point. They both
might include verions of me.
(I notice that time seems to have come bac into the picture, BTW)
> > > If you run a particular Turing machine program which contains
> > > intelligent beings, will they somehow have psychic knowledge of what's
> > > happening in other distinct programs? Obviously not, we could run the
> > > program on a real computer
> >
> >uh-uh! Real programmes are run at distinct spatio-temporal
> >locations. They don't exist in Platonia.
>
> But the programs themselves are distinct, just like the integers 8 and 9 are
> distinct. Are you saying that we can have no concept of 8 and 9 being
> different numbers without referring to specific collections of 8 objects and
> 9 objects at different spatio-temporal locations? If you're not saying this,
> what makes programs any different from integers? After all, you can make a
> one-to-one mapping between the set of all integers and the set of all turing
> machine programs.
But substrings in the programmes will be identical, as I have said.
The "running on different machines" analogy doesn't work, since
space is supposed to emerge form programmes in Platonia.
> >
> > > and see that the beings have no such mysterious
> > > knowledge, and barring errors the ideal "Platonic" program should have
> >the
> > > same output as the 'physical' instantiation
> >
> >That is beside the point. The question is
> >waht it woould feel like to be in the programme.
>
> Unless you are an extreme mind/body dualist, the feelings and thoughts of a
> being in a computer program should correspond to the type of
> information-processing going on in his simulated brain,
Not quite. Physicalists can reject computationalism. Feelings
(particularly) might supervene on non-information-processing
physics.
> and should generally
> correspond to what the simulated being claims to be experiencing within the
> simulation.
It's very possible that they actually wouldn't.
> If he doesn't say he's experiencing an infinite number of
> universes at once when you examine the program's output, then you can
> probably assume he isn't.
That is entirely academic, since no Platonic simulation exists.
> Of course there is still the issue of whether he's
> actually "experiencing" anything at all, or whether he's a philosophical
> zombie with no qualia at all (see
> http://en.wikipedia.org/wiki/Philosophical_zombie ).
Quite.
> > > ('physical' from the perspective
> > > of the most fundamental laws of our universe, which could itself be a
> > > program running in a bigger universe or in 'Platonia').
> >
> >The laws of physics, then, are "in" the programme -- not vice--versa.
>
> I don't understand this comment. What would it mean for the program to be in
> the laws of physics?
It means a phsyical computer, like the one in front
of you, in a physical universe.
> And what relation does this have to the context of my
> claim that 'barring errors the ideal "Platonic" program should have the same
> output as the 'physical' instantiation'?
We just have to hope or assume the programme isn't a zombie.
> > > >All mathematical multiverse theories have the implication that
> > > >I have many identical counterparts.
> > >
> > > "Identical" only to the extent they are experiencing the exact same
> >things
> > > you are.
> >
> >Nope. Also identical in that they share all my memories up
> >until time T , when things turn Harry Potter.
>
> OK, but the point is that the version of you in the universe where the laws
> of nature *didn't* change after time T is no longer identical to the version
> of you in the universe where they did,
But he would still identify himself as me. What entitles me
to claim to be the real me, when we have 99.99% of our memories in
common ?
> and there's no reason that you in the
> non-Harry-Potter universe would have any awareness of what he's experiencing
> at this moment.
It's mathematically possible for both sets of memories to splice
up again. You keep assuming Platonia has sensible laws of physics.
But physical laws are a constraint on All Mathematical Possibilities.
> > > There's no reason to think that counterparts basically similar to
> > > you but having different experiences (say, of a hippogriff flying
> >through
> > > the window) would have some sort of psychic knowledge of each other.
> >
> >They would share my memories, identify themselves as me,
> >and so on. There would be counterparts which have my memories
> >up to time T, then an outpurst of HP, then normal memories from time
> >T+1 onwards.
>
> Yes, I agree. But again, the version of you in the universe with no HP
> events would not have any awareness of them.
And the other version would. Yet my memories are consistent.
That can only be a coincidence , if I am in Platonia.
> > > > > 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;
> > > >
> > > >which is not how eeverybody else defines it.
> > >
> > > If not, then perhaps that's just because they don't in fact think that
> > > "formalism" means believing there's an objective truth about whether a
> >given
> > > statement is derivable from a given set of axioms.
> >
> >It's because they do think Platonism means numbers exist in some sense.
>
> Only in the sense that they must be included in an exhaustive list of
> objective truths about reality. Again, my understanding is that this
> basically is how most analytic philosophers would *define* existence,
> including philosophers studying philosophy-of-math.
That would be question-begging.
> > > If you are claiming that
> > > "everybody" does think that formalists believe this, yet they are still
> >not
> > > considered "Platonists" in any way, I'd like to see some evidence for
> >this
> > > claim.
> >
> >Formalism is alaways cited as a different position to Platonism
> >
> >http://en.wikipedia.org/wiki/Philosophy_of_mathematics
>
> I didn't just ask for evidence that formalists and Platonists are different,
> I asked for evidence that there is anyone who considers them completely
> different but who *also* understands "formalists" to be acknowledging that
> there are objective mind-independent truths about whether a given formal
> system produces a given theorem (as opposed to just understanding
> 'formalists' to be saying that mathematics is a game we humans play with
> deriving theorems from axiomatic systems, without making any claims that
> there is an objective truth about whether our derivations are 'correct' or
> not).
Whatever. If formalism is true, Platonism is false.
> > > >But if any non-Platonic hteory is correct, truths do not
> > > >need to refer.
> > >
> > > Can you provide a quote or citation for the idea that any philosophers
> >of
> > > math subscribe to a view where there are objective truths about
> >mathematical
> > > objects yet the statements do not refer?
> >
> >
> >
> >"Logicism is the thesis that mathematics is reducible to logic, and
> >hence nothing but a part of logic (Carnap 1931/1883, 41). Logicists
> >hold that mathematics can be known a priori, but suggest that our
> >knowledge of mathematics is just part of our knowledge of logic in
> >general, and is thus analytic, not requiring any special faculty of
> >mathematical intuition"
> >
> >
> >http://en.wikipedia.org/wiki/Philosophy_of_mathematics
> OK, but are you assuming that purely logical statements don't refer?
Yes. "Analytic".
> As I said earlier, this criticism would only make sense if we had some prior
> notion of "existence", and I was trying to make the argument that objective
> truth necessarily implies this prior notion. But I'm not, I'm just defining
> the word "existence" in terms of objective truths, because I can't think of
> any way to define the word that seems coherent and which doesn't lead to
> bizarre conclusions like "nothing exists beyond the boundaries of the
> observable universe" (except possibly for the notion of equating existence
> with consciousness, which might make sense and would potentially allow for
> mathematical structures containing intelligent observers which nevertheless
> don't 'exist'). I agree that if we define "existence" in your way,
> mathematical objects do not exist. I just don't think your way of defining
> it is how most philosophers would define it, particularly the analytic
> philosophers who study philosophy of mathematics;
OTOH, they don't all define it your way,
or they would all be Platonists.
> most of them would define
> it in a way more similar to mine, I think. This is a sociological question
> rather than a philosophical one, I suppose we'd have to do a poll or
> something to be sure.
No, it can be settled. Empirically, you are conscious.
I am conscious. Those are not mathematical
facts. Therefore there are non-mathematical
truths. Therefore, there is non-mathematical
existence, by your definition.
Moreover, nothing beyond non-mathematical existence
is needed to explain mathematical truth, since it
is analytic.
> Jesse
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Received on Mon Jul 17 2006 - 16:30:13 PDT
