Re: Arithmetical Realism
Le 12-sept.-06, à 19:20, 1Z a écrit : You have not yet answered my question: what difference are you making between "there exist a prime number in platonia" and "the truth of the proposition asserting the *existence* of a prime number is independent of me, you, and all contingencies" ? "P is true" is not different to "P". That is not the difference I making. All right then. It is an important key point for what will follow. It will help me to represent the modality "True(p)" by just "p"; that is useful because correct machine cannot represent their notion of truth (by Tarski theorem). I'm making a difference between what "exists" means in mathematical sentences and what it means in empiricial sentences (and what it means in fictional contexts...) OK. So with this phrasing, the consequence of the UDA (including either the movie-graph argument, or the use of the "comp-physics" already extracted + OCCAM) can be put in this way: The appearance of "empirical existence" is explain without ontological empirical commitment from the mathematical existence of numbers. Indeed empirical existence, assuming comp, has to be an internal arithmetical modality. The logical case for mathematical Platonism is based on the idea that mathematical statements are true, and make existence claims. Yes. That they are true is not disputed by the anti-Platonist, who must therefore claim that mathematical existence claims are somehow weaker than other existence claims -- perhaps merely metaphorical. But the whole point is that if you take the "yes doctor" idea seriously enough, then "empirical existence" appears to be more metaphorical than mathematical existence. That the the word "exists" means different things in different contexts is easily established. Right. Now a TOE is supposed to explain all those notion of existence and to explain also how they are related. I take the "simple" math existence as primitive, and explain all other notion of existence from it. Perhaps you should wait for it, or peruse in the archive or in my url to see how that works. However, mathematics is not a fiction because it is not a free creation. Mathematicians are constrained by consistency and non-contradiction in a way that authors are not. OK. But after Godel, mathematicians know, (or should know) that the consistency constrained is not enough. Simple example: all sufficiently rich and consistent theory T remains consistent when you add the axiom asserting that T is inconsistent. You get a consistent but unreasonable and incorrect theory. Yes: Godel's second incompleteness result is admittedly amazing. (Incidentally, this approach answers a question about mathematical and empirical truth. The anti-Platonists want sthe two kinds of truth to be different, but also needs them to be related so as to avoid the charge that one class of statement is not true at all. This can be achieved because empirical statements rest on non-contradiction in order to achive correspondence. If an empricial observation fails co correspond to a statemet, there is a contradiction between them. Thus non-contradiciton is a necessary but insufficient justification for truth in empircal statements, but a sufficient one for mathematical statements). Alas no. After Godel's second incompleteness theorem (or Lob extension of it) non-contradiction is insufficient even for the mathematical reality. Any machine/theory can be consistent and false with respect to the intended arithmetical reality. Like Chaitin is aware, even pure arithmetic has some objective "empirical" features. If you agree that the number 0, 1, 2, 3, 4, ... exist (or again, if you prefer, that the truth of the propositions: Ex(x = 0), Ex(x = s(0)), Ex(x = s(s(0))), ... is independent of me), then it can proved that the UD exists. It can be proved also that Peano Arithmetic (PA) can both define the UD and prove that it exists. But again this is just "mathematical existence". You need some reason to assert that mathematical existence is not a mere metaphor implying no real existence, as anti-Platonist mathematicians claim. I do not think that is given by computationalism. It is not given by comp per se. It follows from the UD Argument. Don't hesitate to ask question about any step where you feel not being convinced. Occam does not support conclusions of impossibility. It could be a brute fact that the universe is more complicated than strcitly necessary. You are *trivially* right. This could kill ANY theory. You can say to a string theorist : what about the particles which we have not yet discover and which would behave in a way contradicting the theory. All the facts about mathematical truth and methodology can be established without appeal to the actual existence of mathematical objects. I believe that what you want to say here is this: [All the facts about mathematical truth and methodology can be established without appeal to the empirical (or metaphysical,
Re: Arithmetical Realism
Bruno Marchal wrote: > Le 29-août-06, à 20:45, 1Z a écrit : > > > > > The version of AR that is supported by comp > > only makes a commitment about mind-independent *truth*. The idea > > that the mind-independent truth of mathematical propositions > > entails the mind-independent *existence* of mathematical objects is > > a very contentious and substantive claim. > > > You have not yet answered my question: what difference are you making > between "there exist a prime number in platonia" and "the truth of the > proposition asserting the *existence* of a prime number is independent > of me, you, and all contingencies" ? "P is true" is not different to "P". That is not the difference I making. I'm making a difference between what "exists" means in mathematical sentences and what it means in empiricial sentences (and what it means in fictional contexts...) The logical case for mathematical Platonism is based on the idea that mathematical statements are true, and make existence claims. That they are true is not disputed by the anti-Platonist, who must therefore claim that mathematical existence claims are somehow weaker than other existence claims -- perhaps merely metaphorical. That the the word "exists" means different things in different contexts is easily established. For one thing, this is already conceded by Platonists! Platonists think Platonic existence is eternal, immaterial non-spatial, and so on, unlike the Earthly existence of material bodies. For another, we are already used to contextualising the meaning of "exists". We agree with both: "helicopters exist"; and "helicopters don't exist in Middle Earth". (People who base their entire anti-Platonic philosophy are called fictionalists. However, mathematics is not a fiction because it is not a free creation. Mathematicians are constrained by consistency and non-contradiction in a way that authors are not. Dr Watson's fictional existence is intact despite the fact that he is sometimes called John and sometimes James in Conan Doyle's stories). The epistemic case for mathematical Platonism is be 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. (There is also disagreement about some axioms, such as the Axiom of Choice, and different mathematicians with different attitudes about the AoC will tend to get different answers -- a phenomenon which is easily explained by the formalist view I am taking here). The semantic case for mathematical Platonism is based on the idea that the terms in a mathematical sentence must mean something, and therefore must refer to objects. It can be argued on general linguistic grounds that not all meaning is reference to some kind of object outside the head. Some meaning is sense, some is reference. That establishes the possibility that mathematical terms do not have references. What establishes it is as likely and not merely possible is the obeservation that nothing like empirical investigation is needed to establish the truth of mathematical statements. Mathematical truth is arrived at by a purely conceptual process, which is what would be expected if mathematical meaning were restricted to the Sense, the "in the head" component of meaning. A possible counter argument by the Platonist is that the downgrading of mathematical existence to a mere metaphor is arbitrary. The anti-Platonist must show that a consistent standard is being applied. This it is possible to do; the standard is to take the meaning of existence in the context of a particular proposition to relate to the means of justification of the proposition. Since ordinary statements are confirmed empirically, "exists" means "can be perceived" in that context. Since sufficient grounds for asserting the existence of mathematical objects are that it is does not contradict anything else in mathematics, mathematical existence just amounts to concpetual non-contradictoriness. (Incidentally, this approach answers a question about mathematical and empirical truth. The anti-Platonists want sthe two kinds of truth to be different, but also needs them to be related so as to avoid the charge that one class of statement is not true at all. This can be achieved because empirical statements rest on non-contradiction in order to achive correspondence. If an empricial observation fails co correspond to a statemet, there is a contradiction between them. Thus non-contradiciton is a necessary but insufficient justification for truth in empircal statements, but a sufficient one for mathematical statements).
Re: Arithmetical Realism
Le 04-sept.-06, à 16:08, 1Z a écrit :
> Arithmetical statements use the word "exists", or the symbolic
> euivalen thereof. However, it is not to be taken literally
> in all contexts.
>
>> No need to add
>> metaphysics at this stage
>
> Yes there is. You need metaphysics to answer the question
> of whether the existence-claims of mathematics shouldbe takne
> literally.
"Metaphysics" is provided through the "yes doctor", which you have no
choice not to take literally.
I mean you would not say "yes" to a doctor who tells you that you will
survive the comp-substitution and then add: don't take this literally.
But even this is strictly speaking suppressed when defining the many
notion of contingency and necessity from the intensional (modal)
variant of the Godel Lob self-referential provability notions.
Recall the UDA and AUDA difference (AUDA = Arithmetical UDA = lobian
interview).
>
>> (nor at any other stage by the way, except
>> the yes doctor, which I prefer to range in "theology" than in
>> "metaphysics").
>
> Is theology better-foudned as a discipline ?
When done by rational theologians, like most of the greek one, it is.
Of course in our civilization "theology" has been appropriated by
"politics" since a long time. Still many Christian theologians have
been "rigorous" or "modest" or "scientific" since, but are generally
put on the margins, if not burned alive or ignored. Today the
aristotelian primary matter hypothesis is defended by the atheist and
the Christian, mainly.
> And Sherlock Holmes lives because Sherlock Holmes lives
> at 221b Baker Street.
Really? Could you give me his phone number please? I will verify.
Come on Peter, this is a diversion which has nothing to do with the
notion of existence of numbers.
You refer to possibly interesting nuances, but those are out of topics
here.
> Arithmetical statements use the word "exists", or the symbolic
> euivalen thereof. However, it is not to be taken literally
> in all contexts.
I don't care. The point is that with comp, the existence of an
electron, or of anything, cannot be taken literally too. The point is
that with comp you can derive from PEANO, why numbers have to believe
in electron, although electron existence is less literal than the
existence of 417.
You keep doing the 1004 fallacy. The question are not metaphysical at
all, and does not address any notion of metaphysical existence in which
I am not interested at all. The point is that the computationalist
hypothesis generates many different notion of existence, and the
interesting thing to do (with respect of explaining quanta, qualia,
where do we come from etc.) consists in finding the relation between
those form of existence, not some intrinsic meaning that they would
have.
Now the simplest notion of existence is the standard interpretation of
"Ex" in first order logic presentation of arithmetic, if only to begin
with. All other notion of existence (psychological, physical,
theological, etc.) are derived from it.
>
>>> Necessary truth
>>> can exist in a world of contingent existence -- providing
>>> all necessary truths in such a world are ontologically non-commital.
>>
>>
>> I don't understand.
>
> If necessary truths don't refer to contingently
> existing things, they cannot be "infected" by their contingency.
I don't understand. A necessary truth could refer to contingently
existing things. If you take (like we will do in the Lobian interview)
"provability B" for "necessity", and consistency D or "possibility",
Godel's second incompleteness theorem is already an example of
necessity about contingencies: it is necessary that if a tautology is
consistent then it is consistent that a falsity is necessary
G proves Dt->DBf, or G* proves B(Dt->DBf). Also B(Ex(x=x)) which is
enough.
>> AR does not ask you for believing in some metaphysical (still less
>> physical) existence of numbers.
>
> Then it does not show the UD exist, and it cannot follow
> that I part of its output.
You should have written: "Then it does not show the UD exists
physically, and it cannot follow
that I am a physical part of its output."
And I agree with you given that I already do not believe you exist
physically in any genuine (applicable) sense of the word (assuming
comp). BTW I have already makes long answer of this, and you did not
reply.
>
>> It ask you to agree that a proposition
>> of the type ExP(x) is true or false independently of any cognitive
>> faculty.
>
> It may well be true. It may well mean nothing more
> than "P(x) is non-contradictory"
No. ExP(x) means that it exist a natural number verifying the property
P.
"P(x) is non-contradictory" is the proposition DP(x),or ~B~P(x), i.e.
~Bew('P(x)') which is a completely different proposition. This one is
even undecidable by *any* lobian machine.
Example: Bf is false but is also non-contradictory for any sound theory
of arithmetic.
Contradictory
>
>> Cognitive abilities are needed to believe o
Re: Arithmetical Realism
Bruno Marchal wrote: > Le 02-sept.-06, à 17:26, 1Z a écrit : > > > > Things don't become necessarily true just > > because someone says so. The truths > > of mathematics may be necessarily true, but > > that does not make AR a s aclaim about > > existence necessarily true. AR as a claim > > about existence is metaphysics, and highly debatable. > > Yes. So let us never do it. Debate is what we are here for. > > Necessary truth doesn't entail necessary existence unless > > the claims in question are claims about existence. > > Exactly. > > > > Whether mathematical truths are about existence is debatable > > and not "necessary". > > > Existential mathematical statement are about existence. And Sherlock Holmes lives because Sherlock Holmes lives at 221b Baker Street. > > Not if AR is only a claim about truth. > > AR is about the truth of arithmetical statements, and among > arithmetical statements, many are existential, so AR makes claim about > the independent truth of existential statements. Arithmetical statements use the word "exists", or the symbolic euivalen thereof. However, it is not to be taken literally in all contexts. > No need to add > metaphysics at this stage Yes there is. You need metaphysics to answer the question of whether the existence-claims of mathematics shouldbe takne literally. > (nor at any other stage by the way, except > the yes doctor, which I prefer to range in "theology" than in > "metaphysics"). Is theology better-foudned as a discipline ? > > Necessary truth > > can exist in a world of contingent existence -- providing > > all necessary truths in such a world are ontologically non-commital. > > > I don't understand. If necessary truths don't refer to contingently existing things, they cannot be "infected" by their contingency. > > As non-Platonists indded take mathematical statements to be. > > AR does not ask you for believing in some metaphysical (still less > physical) existence of numbers. Then it does not show the UD exist, and it cannot follow that I part of its output. > It ask you to agree that a proposition > of the type ExP(x) is true or false independently of any cognitive > faculty. It may well be true. It may well mean nothing more than "P(x) is non-contradictory" > Cognitive abilities are needed to believe or know that ExP(x) > is true (or false), but that's all. Quite. So nothing in the argument can tell me about the nature of my existence. > > That's what White Rabbits are all about. > > > > There is also an apriori argument against Pythagoreanism (=everything > > is numbers). If it is a *contingent* fact that non-mathematical > > entities > > don't exist, > > It is not even a fact. It is an assumption. I already said "if"... > Nobody has proved that > something non mathematical exists, although comp is quite close in > proving this. That isn't the point. The point is the consistency Pythagorean rationalism as a hypothesis. > Indeed comp shows that no first person can be described > mathematically by herself. So *relatively* to a machine first person, > many things will *appear* non mathematical. It is a consequence of > incompleteness + the theaetetical-plotinian definition of knowledge. > > Pythagoreanism cannot be justified by rationalism (=- > > all truths are necessary and apriori). Therefore the > > Pythagorean-ratioanlist > > must believe matter is *impossible*. > > Not impossible. Just useless. The Pythagorean rationalist *must* believe mater is impossible -- the argument becomes inconsistent otherwise. The argument that matter is "useless" is more akin to empiricism than rationalism. > Bruno > > > http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
Le 02-sept.-06, à 17:26, 1Z a écrit : > Things don't become necessarily true just > because someone says so. The truths > of mathematics may be necessarily true, but > that does not make AR a s aclaim about > existence necessarily true. AR as a claim > about existence is metaphysics, and highly debatable. Yes. So let us never do it. > Necessary truth doesn't entail necessary existence unless > the claims in question are claims about existence. Exactly. > Whether mathematical truths are about existence is debatable > and not "necessary". Existential mathematical statement are about existence. > Not if AR is only a claim about truth. AR is about the truth of arithmetical statements, and among arithmetical statements, many are existential, so AR makes claim about the independent truth of existential statements. No need to add metaphysics at this stage (nor at any other stage by the way, except the yes doctor, which I prefer to range in "theology" than in "metaphysics"). > Necessary truth > can exist in a world of contingent existence -- providing > all necessary truths in such a world are ontologically non-commital. I don't understand. > As non-Platonists indded take mathematical statements to be. AR does not ask you for believing in some metaphysical (still less physical) existence of numbers. It ask you to agree that a proposition of the type ExP(x) is true or false independently of any cognitive faculty. Cognitive abilities are needed to believe or know that ExP(x) is true (or false), but that's all. > That's what White Rabbits are all about. > > There is also an apriori argument against Pythagoreanism (=everything > is numbers). If it is a *contingent* fact that non-mathematical > entities > don't exist, It is not even a fact. It is an assumption. Nobody has proved that something non mathematical exists, although comp is quite close in proving this. Indeed comp shows that no first person can be described mathematically by herself. So *relatively* to a machine first person, many things will *appear* non mathematical. It is a consequence of incompleteness + the theaetetical-plotinian definition of knowledge. > Pythagoreanism cannot be justified by rationalism (=- > all truths are necessary and apriori). Therefore the > Pythagorean-ratioanlist > must believe matter is *impossible*. Not impossible. Just useless. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
Le 30-août-06, à 16:37, uv a écrit : > [Bruno's defintiion of Arithmetic Realism I understand to be > " Arithmetical Realism. > All proposition pertaining on natural numbers > with the form Qx Qy Qz Qt Qr ... Qu P(x,y,z,t,r, ...,u) are true > independently > of me. Q represents a universal or existential quantifier, and P > represents a > decidable (recursive) predicate. That is, proposition like the > Fermat-Wiles > theorem, or Goldbach conjecture, or Euclide's infinity of primes > theorem are > either true or false, and this independently of the proposition "Bruno > Marchal > exists". It amounts to accept, for the sake of my argument, that > classical logic is correct in the realm of positive integers. Nothing > more."] Indeed. Good summary, thanks. Third person necessity and contingency will then be defined by the (Sigma1) provability predicate of Godel-Lob, and the n-version persons by intensional (modal) variants of it. Note that Fermat-Wiles, Riemann, Godlbach, Euclide's are all Sigma1. Arithmetical realism bears also on the independence of the truth of Pi1, Sigma2, Pi2, ...SigmaN, PiN ..., sentences, but I have no problem with the lobian machine which have also "realist" analytical beliefs (where we can quantify on sets). Nice example of non P1 or Sigma1 conjectures is given by the famous Syracuse question: http://www.cecm.sfu.ca/organics/papers/lagarias/ Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
Le 02-sept.-06, à 16:03, David Nyman a écrit : > > Bruno Marchal wrote: > >> Please I have never said that primary matter is impossible. Just that >> I >> have no idea what it is, no idea what use can it have, nor any idea >> how >> it could helps to explain quanta or qualia. >> So I am happy that with comp it has necessarily no purpose, and we can >> abandon "weak materialism", i.e. the doctrine of primary matter, like >> the biologist have abandon the vital principle, or like the abandon of >> ether by most physicist. >> But with comp it is shown how to retrieve the appearance of it, by >> taking into account the differences between the notions of n-person >> (and of n-existence) the universal machine cannot avoid. > > Are we not trying to discriminate two possible starting assumptions > here? > > 1) Necessity > 2) Contingency > > Assumption 1 makes no appeal to fundamental contingency, but posits > only 'necessarily true' axioms (e.g. AR). In this sense there could > never be "nothing instead of something" because the 'necessary truth' > of AR is deemed independent of contingency - indeed 'contingency' would > be seen to emerge from it (hence its 'empiricism' Bruno?) All right. > > Assumption 2 posits by contrast the ultimate contingency of 'existence' > - there might indeed have been 'nothing'. The apparent 'necessity' of > AR must consequently be illusory - i.e. AR, CT etc. derive their > 'existence' and characteristics from the prior facts of brute > contingency. > > Under assumption 2, therefore, the semantics of 'bare substrate' boil > down to a fundamental assertion of 'non-relative contingent existence', > and 'primary matter' to 'relative contingent processes/ structures'. > Starting from assumption 2 we could see comp as a schema of relative > contingent process/ structure within which 'primary matter' is deemed > to be 'instantiated', or vice versa (i.e. the 'usual assumption' of > physical instantiation). > > But are assumptions 1 and 2 ineluctably 'theological preferences', or > can we discriminate them empirically? Like you can test the Everett MW from a first person point of view by quantum suicide, you can get first person confirmation of comp by comp suicide. But this is trivial and not so interesting. Now my point is that comp gives enough constraint by itself so that you can derive the physical *law-like* propositions from it, so you can compare them with empiry. So you can get first person plural confirmation (of a purely third person communicable theory). Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
David Nyman wrote: > 1Z wrote: > > > Indeed, but the contingentist doesn't have to regard truth > > as something that exists. > > Fair enough, but even the contingentist needs to express herself > intelligibly without recourse to a constant blizzard of scare quotes. > So she still needs something that FAPP corresponds to 'instantiated > truth', and we can indeed discover such analogs in a contingent world. Finding something that corresponds to instantiated truth -- such as knowledge -- does not make truth contingent. > > That would indicate that logical possibility is a subset > > of physical possibility, which is counterintuitive. That > > is one motivation for sayign that truth (along with other > > abstracta such as numbers) doesn't exist at all. > > Agreed, with the above proviso. > > > No they couldn't, because they do not refer to external > > contingencies ITFP. Where there is no relation, there > > is no variation. Invariance is necessity. > > Well, at the level of metaphor you are correct, but in a strictly > contingentist sense, they implicitly refer to external contingencies, No. They don't refer at all. Maths isn't empirical. > because 'conceptual' contingencies must be instantiated in terms of > those selfsame 'external' ones. Instantiation isn't reference. > IOW, 'reference', 'externality' and the > entire conceptual armamentarium are instantiated in a given contingent > state of affairs if they are instantiated at all. > and consequently are dependent on it for their > 'logic'. Clearly not, since we are able to concive physically impossible worlds. The virtual "logic" isn't determined by physics. A computer running on real phsyics can simulate a world where graivity is an inverse cube law. > Were these contingencies different, white rabbits might become > quite mundane. Yes. It is logically possible for what is physically (im)possible to have been different. Physical possibillity is a subset of logical possibility. Logical possibility isn't determined by physical possibility. > > You seem to be intent on defining truth in > > the most baggy way possible. > > Yes, but I'm just trying to point out that we can pragmatically deploy > a variety of means to establish agreement to some level of accuracy > without having to believe in the 'transcendent existence' of truth. That is tangential to the discussion. The point is that anti-Plaotonists can agree with Platonists 100% about the mind-independence of mathemaical trth, whilst agreeing 0% about the mind-independent existence of mathematical objects."Transcendent" truth does not have to be sacrificed to ontological contingency. > > > In this > > > view, 'conceptual existence' is just the instantiated existence of a > > > concept. > > > > What has that got to do with truth ? > > Well, the existence of truth is just the instantiated existence of a > truth, in the contingentist view. Actually, I don't really want to push > all this too far. FAPP the distinctions you make are valid, and I'd > much rather agree to deploy a metaphorical sense of the 'existence' of > truth rather than chase about looking for its multifarious > contingentist instantiations. I was originally trying to contrast the > contingent vs. necessary ontic assumptions that seemed to me implicit > in your dialogue with Bruno. As it happens, my own preference lies on > the side of contingency. OK. > Conceptual > > > 'existence' is simply the sum of the instantiations of all (agreed) > > > instances of a concept - IOW they're all apples if we agree they are. > > > Any other view is surely already 'Platonic'? > > > > Nope. > > Why isn't it? Do you mean that we can ascribe metaphorical 'existence' > to a conceptual framework that transcends any or all particular > instantiated examples, without ascribing literal existence to it? In > this case, as with 'truth', I would concur. Mathematical "existence" operates under constraints of logical coherence, non-contradicition, consistency. It is not just a case of conceiving something. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
1Z wrote: > Indeed, but the contingentist doesn't have to regard truth > as something that exists. Fair enough, but even the contingentist needs to express herself intelligibly without recourse to a constant blizzard of scare quotes. So she still needs something that FAPP corresponds to 'instantiated truth', and we can indeed discover such analogs in a contingent world. > That would indicate that logical possibility is a subset > of physical possibility, which is counterintuitive. That > is one motivation for sayign that truth (along with other > abstracta such as numbers) doesn't exist at all. Agreed, with the above proviso. > No they couldn't, because they do not refer to external > contingencies ITFP. Where there is no relation, there > is no variation. Invariance is necessity. Well, at the level of metaphor you are correct, but in a strictly contingentist sense, they implicitly refer to external contingencies, because 'conceptual' contingencies must be instantiated in terms of those selfsame 'external' ones. IOW, 'reference', 'externality' and the entire conceptual armamentarium are instantiated in a given contingent state of affairs and consequently are dependent on it for their 'logic'. Were these contingencies different, white rabbits might become quite mundane. > You seem to be intent on defining truth in > the most baggy way possible. Yes, but I'm just trying to point out that we can pragmatically deploy a variety of means to establish agreement to some level of accuracy without having to believe in the 'transcendent existence' of truth. > > In this > > view, 'conceptual existence' is just the instantiated existence of a > > concept. > > What has that got to do with truth ? Well, the existence of truth is just the instantiated existence of a truth, in the contingentist view. Actually, I don't really want to push all this too far. FAPP the distinctions you make are valid, and I'd much rather agree to deploy a metaphorical sense of the 'existence' of truth rather than chase about looking for its multifarious contingentist instantiations. I was originally trying to contrast the contingent vs. necessary ontic assumptions that seemed to me implicit in your dialogue with Bruno. As it happens, my own preference lies on the side of contingency. Conceptual > > 'existence' is simply the sum of the instantiations of all (agreed) > > instances of a concept - IOW they're all apples if we agree they are. > > Any other view is surely already 'Platonic'? > > Nope. Why isn't it? Do you mean that we can ascribe metaphorical 'existence' to a conceptual framework that transcends any or all particular instantiated examples, without ascribing literal existence to it? In this case, as with 'truth', I would concur. David > David Nyman wrote: > > 1Z wrote: > > > > > Why should the *truth* of a statement be dependent on > > > the *existence* of an instance of it > > > What I mean is that - for a 'thoroughgoing contingentist' - > > 'statements', 'concepts', 'truths', 'referents' and anything else > > whatsoever can exist solely in virtue of their actual contingent > > instantiation (i.e. there literally isn't any other sort of > > 'existence'). > > Indeed, but the contingentist doesn't have to regard truth > as something that exists. > > Within such a world-view, even apparently inescapable > > logical truths are 'necessary' only within a relational system > > instantiated solely in terms of a contingent world. > > That would indicate that logical possibility is a subset > of physical possibility, which is counterintuitive. That > is one motivation for sayign that truth (along with other > abstracta such as numbers) doesn't exist at all. > > > They cannot > > 'transcend' present contingencies, and under different contingencies > > (about which we can know nothing) they could be different. > > No they couldn't, because they do not refer to external > contingencies ITFP. Where there is no relation, there > is no variation. Invariance is necessity. > > > This > > establishes an 'epistemic horizon' for a contingent world. > > > > > > What does instantiation have to do with truth ? > > > > Everything. 'Truth' in contingent terms is (very loosely) something > > like: > > > > 1) dispositions to believe that certain statements correspond with > > putative sets of 'facts'. > > That is belief, not truth. > > > 2) sets of 'facts' > > Facts exist. Statements are true. Which do you mean ? > > > 3) logical/ empirical processes of judgement > > What is judged may be true, since it > may be a statement or proposition. > > Processes of judgement are neither true nor false. > > > 4) conclusions as to truths asserted > > Defining truth in terms of truth. > > > 5) behaviour consequent on this > > Behaviour is neither true nr false. It is not a > statement or proposition. > > > 6) etc. > > You seem to be intent on defining truth in > the most baggy way possible. > > > If any element of this - from soup to nuts - fails to be ins
Re: Arithmetical Realism
David Nyman wrote: > 1Z wrote: > > > Why should the *truth* of a statement be dependent on > > the *existence* of an instance of it > What I mean is that - for a 'thoroughgoing contingentist' - > 'statements', 'concepts', 'truths', 'referents' and anything else > whatsoever can exist solely in virtue of their actual contingent > instantiation (i.e. there literally isn't any other sort of > 'existence'). Indeed, but the contingentist doesn't have to regard truth as something that exists. > Within such a world-view, even apparently inescapable > logical truths are 'necessary' only within a relational system > instantiated solely in terms of a contingent world. That would indicate that logical possibility is a subset of physical possibility, which is counterintuitive. That is one motivation for sayign that truth (along with other abstracta such as numbers) doesn't exist at all. > They cannot > 'transcend' present contingencies, and under different contingencies > (about which we can know nothing) they could be different. No they couldn't, because they do not refer to external contingencies ITFP. Where there is no relation, there is no variation. Invariance is necessity. > This > establishes an 'epistemic horizon' for a contingent world. > > What does instantiation have to do with truth ? > > Everything. 'Truth' in contingent terms is (very loosely) something > like: > > 1) dispositions to believe that certain statements correspond with > putative sets of 'facts'. That is belief, not truth. > 2) sets of 'facts' Facts exist. Statements are true. Which do you mean ? > 3) logical/ empirical processes of judgement What is judged may be true, since it may be a statement or proposition. Processes of judgement are neither true nor false. > 4) conclusions as to truths asserted Defining truth in terms of truth. > 5) behaviour consequent on this Behaviour is neither true nr false. It is not a statement or proposition. > 6) etc. You seem to be intent on defining truth in the most baggy way possible. > If any element of this - from soup to nuts - fails to be instantiated > in some form it cannot exist in a purely contingent world. Hardly anything in your list actually has anythig to do with truth. The possible exception is (2), "facts". But "fact" is a notoriously[*] ambiguous word. [*] Not notoriously *enough* , though. > In this > view, 'conceptual existence' is just the instantiated existence of a > concept. What has that got to do with truth ? > AFAICS any other view would have to assert some sort of > transcendent 'conceptual existence' that subsumes 'contingent > existence'. No, because truth and existence are different. Thus, a proposition can both exist contingently and have a necessary truth-value. > > Logical possibility is defined in terms of contradiciton. > > Why should it turn out to be nonetheless dependent > > on instantiation ? > > Because 'contradiction' itself depends on instantiation. No it doesn't. > A statement is > 'contradictory' because its referent is impossible to instantiate under > present contingencies. No, it is contradictory becuase it contains a clause of the form [A & ~A] (A and not-A). Contradiciton is a formal, logical property. > In this world-view, answering such questions is > easy - *everything* depends on such instantiation. Conceptual > 'existence' is simply the sum of the instantiations of all (agreed) > instances of a concept - IOW they're all apples if we agree they are. > Any other view is surely already 'Platonic'? Nope. > > I don't see why. You just seem to be treating > > truth and existence as interchangeable, which > > begs the questions AFAICS. > > No, I'm saying (above) that 'truth' in a contingent world can only be > *derived* from present contingencies. It can also be derived from the interrelation of concepts. > By this token, truth in any > 'transcendent' sense Could you specify a "transcendent sense" ? > is either impossible (if one believes in a > contingent world), or alternatively *must* be a de facto 'existence' > claim that rules out 'primary contingency' - i.e. the world 'in the > sense that I exist' is supposed to emerge from 'necessity'. I couldn't make sense of that. Necessity is an abstract logical property, not a thing. > So I'm > agreeing with you (I think) in your contention that 'number theology', > to be ontically coherent, must be an existence claim for a priori truth > in this 'strong' sense. Platonists feel they must reify the supposed referents of necessarily true statements in order to explain their necessity. Number theologians only need to reify numbers. I have no idea why you are so keen on reifying truth. > > > To be coherent AFAICS one would need to be making > > > ontic claims for 'necessary truth' that would constrain 'contingent > > > possibility'. > > > > I have no idea what you mean by that. Why would a claim about > > necessary truth be ontic rather than epistemic, for instance ? > >
Re: Arithmetical Realism
1Z wrote: > Why should the *truth* of a statement be dependent on > the *existence* of an instance of it ? What I mean is that - for a 'thoroughgoing contingentist' - 'statements', 'concepts', 'truths', 'referents' and anything else whatsoever can exist solely in virtue of their actual contingent instantiation (i.e. there literally isn't any other sort of 'existence'). Within such a world-view, even apparently inescapable logical truths are 'necessary' only within a relational system instantiated solely in terms of a contingent world. They cannot 'transcend' present contingencies, and under different contingencies (about which we can know nothing) they could be different. This establishes an 'epistemic horizon' for a contingent world. > What does instantiation have to do with truth ? Everything. 'Truth' in contingent terms is (very loosely) something like: 1) dispositions to believe that certain statements correspond with putative sets of 'facts'. 2) sets of 'facts' 3) logical/ empirical processes of judgement 4) conclusions as to truths asserted 5) behaviour consequent on this 6) etc. If any element of this - from soup to nuts - fails to be instantiated in some form it cannot exist in a purely contingent world. In this view, 'conceptual existence' is just the instantiated existence of a concept. AFAICS any other view would have to assert some sort of transcendent 'conceptual existence' that subsumes 'contingent existence'. > Logical possibility is defined in terms of contradiciton. > Why should it turn out to be nonetheless dependent > on instantiation ? Because 'contradiction' itself depends on instantiation. A statement is 'contradictory' because its referent is impossible to instantiate under present contingencies. In this world-view, answering such questions is easy - *everything* depends on such instantiation. Conceptual 'existence' is simply the sum of the instantiations of all (agreed) instances of a concept - IOW they're all apples if we agree they are. Any other view is surely already 'Platonic'? > I don't see why. You just seem to be treating > truth and existence as interchangeable, which > begs the questions AFAICS. No, I'm saying (above) that 'truth' in a contingent world can only be *derived* from present contingencies. By this token, truth in any 'transcendent' sense is either impossible (if one believes in a contingent world), or alternatively *must* be a de facto 'existence' claim that rules out 'primary contingency' - i.e. the world 'in the sense that I exist' is supposed to emerge from 'necessity'. So I'm agreeing with you (I think) in your contention that 'number theology', to be ontically coherent, must be an existence claim for a priori truth in this 'strong' sense. > > To be coherent AFAICS one would need to be making > > ontic claims for 'necessary truth' that would constrain 'contingent > > possibility'. > > I have no idea what you mean by that. Why would a claim about > necessary truth be ontic rather than epistemic, for instance ? For the reasons you yourself have argued - i.e. that claims based on 'Platonic numbers' must be regarded as ontic in a strong sense if they are supposed to account for a world that exists 'in the sense that I exist'. Epistemic claims would then follow from this. David > David Nyman wrote: > > > 1Z wrote: > > > > > Statements, concepts and beliefs must > > > be contingently instantiated. That doesn't > > > mean that their truths-values are logially > > > contingent. > > > > > > > I'm not sure that in a world of strictly contingent existence one can > > establish a 'logical necessity' that is independent of 'contingent > > instantiation' > > Why should the *truth* of a statement be dependent on > the *existence* of an instance of it ? > > Moreover, the necessary truth of mathematical statements > follows from their lack or real referents:- > > > Mathematical statements > are necessarily true because there are no possible circumstances > that make them false; there are no possible circumstances that > would make them false because they do not refer to anything > external. This is much simpler than the Platonist > alternative that mathematical statements: > > 1) have referents > which are > 2) unchanging and eternal, unlike anything anyone has actually seen > and thereby > 3) explain the necessity (invariance) of mathematical statements > without > 4) performing any other role -- they are not involved in > mathematical proof. > > > > and thus escapes restriction to 'necessary under certain > > contingencies' (even if these are equivalent to 'any that I can > > imagine'). > > > > If one is going to be a 'contingentist', then one might as > > well be a thoroughgoing one. > > > > > But physical possibility is a subset > > > of logical possibility, so the physical > > > systems can't do anything its abstract counterpart > > > cannot do, so what is true of the abstract system > > > is true of any phsycial systems that really instantiates it. >
Re: Arithmetical Realism
David Nyman wrote: > 1Z wrote: > > > Statements, concepts and beliefs must > > be contingently instantiated. That doesn't > > mean that their truths-values are logially > > contingent. > > > > I'm not sure that in a world of strictly contingent existence one can > establish a 'logical necessity' that is independent of 'contingent > instantiation' Why should the *truth* of a statement be dependent on the *existence* of an instance of it ? Moreover, the necessary truth of mathematical statements follows from their lack or real referents:- Mathematical statements are necessarily true because there are no possible circumstances that make them false; there are no possible circumstances that would make them false because they do not refer to anything external. This is much simpler than the Platonist alternative that mathematical statements: 1) have referents which are 2) unchanging and eternal, unlike anything anyone has actually seen and thereby 3) explain the necessity (invariance) of mathematical statements without 4) performing any other role -- they are not involved in mathematical proof. > and thus escapes restriction to 'necessary under certain > contingencies' (even if these are equivalent to 'any that I can > imagine'). If one is going to be a 'contingentist', then one might as > well be a thoroughgoing one. > > > But physical possibility is a subset > > of logical possibility, so the physical > > systems can't do anything its abstract counterpart > > cannot do, so what is true of the abstract system > > is true of any phsycial systems that really instantiates it. > > I agree. However what I'm saying is that in a world of contingent > existence *everything* is contingently instantiated. What does instantiation have to do with truth ? > Consequently, > neither 'physical possibility' nor 'logical possibility' can escape > dependency on such instantiation. Logical possibility is defined in terms of contradiciton. Why should it turn out to be nonetheless dependent on instantiation ? > In a world of contingent existence > the elevation of any 'necessary truth' above contingency is dubious and > possibly incoherent. I don't see why. You just seem to be treating truth and existence as interchangeable, which begs the questions AFAICS. > To be coherent AFAICS one would need to be making > ontic claims for 'necessary truth' that would constrain 'contingent > possibility'. I have no idea what you mean by that. Why would a claim about necessary truth be ontic rather than epistemic, for instance ? --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
1Z wrote: > Statements, concepts and beliefs must > be contingently instantiated. That doesn't > mean that their truths-values are logially > contingent. > I'm not sure that in a world of strictly contingent existence one can establish a 'logical necessity' that is independent of 'contingent instantiation' and thus escapes restriction to 'necessary under certain contingencies' (even if these are equivalent to 'any that I can imagine'). If one is going to be a 'contingentist', then one might as well be a thoroughgoing one. > But physical possibility is a subset > of logical possibility, so the physical > systems can't do anything its abstract counterpart > cannot do, so what is true of the abstract system > is true of any phsycial systems that really instantiates it. I agree. However what I'm saying is that in a world of contingent existence *everything* is contingently instantiated. Consequently, neither 'physical possibility' nor 'logical possibility' can escape dependency on such instantiation. In a world of contingent existence the elevation of any 'necessary truth' above contingency is dubious and possibly incoherent. To be coherent AFAICS one would need to be making ontic claims for 'necessary truth' that would constrain 'contingent possibility'. David > David Nyman wrote: > > > 1Z wrote: > > > > > Necessary truth doesn't entail necessary existence unless > > > the claims in question are claims about existence. > > > > If one claims (which I don't BTW) that something is 'necessarily true' > > *independent of contingent existence* then I think for this to be in > > any way coherent, one must be making some sort of existence claim for > > 'necessary truth'. > > But only the sort of abstract "exisence" that > numbers have in the first place, which is > not genuine existence at all for anti-Platonists. > > > By contrast, within contingent existence, some > > things may seem 'necessarily true', but this truth can only be derived > > from aspects of contingency (i.e. in virtue of the concept and its > > referents being contingently instantiated). > > What things ? Are they really necessarily true, > or only seemingly so ? > > > > Not if AR is only a claim about truth. Necessary truth > > > can exist in a world of contingent existence -- providing > > > all necessary truths in such a world are ontologically non-commital. > > > As non-Platonists indded take mathematical statements to be. > > > > I agree insofar as you mean what I'm saying above: i.e. the 'existence' > > of 'necessary truth' in a world of contingent existence must itself be > > 'contingently instantiated'. > > Statements, concepts and beliefs must > be contingently instantiated. That doesn't > mean that their truths-values are logially > contingent. > > > 'Necessity' in this sense is restricted to > > 'necessary under ceratin contingencies'. In a world of contingent > > existence the behaviour of a logical system must reduce ultimately to > > the behaviour of a contingently instantiated system. > > But physical possibility is a subset > of logical possibility, so the physical > systems can't do anything its abstract counterpart > cannot do, so what is true of the abstract system > is true of any phsycial systems that really instantiates it. > > > > There is also an apriori argument against Pythagoreanism (=everything > > > is numbers). If it is a *contingent* fact that non-mathematical > > > entities > > > don't exist, Pythagoreanism cannot be justified by rationalism (=- > > > all truths are necessary and apriori). Therefore the > > > Pythagorean-ratioanlist > > > must believe matter is *impossible*. > > > > Yes, I agree. That's what I mean about the 'existence' claim of > > 'necessary truth' - since it rules out 'contingent instantiation', it > > must replace it with 'necessary instantiation', or be incoherent as to > > ontology. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
David Nyman wrote: > 1Z wrote: > > > Necessary truth doesn't entail necessary existence unless > > the claims in question are claims about existence. > > If one claims (which I don't BTW) that something is 'necessarily true' > *independent of contingent existence* then I think for this to be in > any way coherent, one must be making some sort of existence claim for > 'necessary truth'. But only the sort of abstract "exisence" that numbers have in the first place, which is not genuine existence at all for anti-Platonists. > By contrast, within contingent existence, some > things may seem 'necessarily true', but this truth can only be derived > from aspects of contingency (i.e. in virtue of the concept and its > referents being contingently instantiated). What things ? Are they really necessarily true, or only seemingly so ? > > Not if AR is only a claim about truth. Necessary truth > > can exist in a world of contingent existence -- providing > > all necessary truths in such a world are ontologically non-commital. > > As non-Platonists indded take mathematical statements to be. > > I agree insofar as you mean what I'm saying above: i.e. the 'existence' > of 'necessary truth' in a world of contingent existence must itself be > 'contingently instantiated'. Statements, concepts and beliefs must be contingently instantiated. That doesn't mean that their truths-values are logially contingent. > 'Necessity' in this sense is restricted to > 'necessary under ceratin contingencies'. In a world of contingent > existence the behaviour of a logical system must reduce ultimately to > the behaviour of a contingently instantiated system. But physical possibility is a subset of logical possibility, so the physical systems can't do anything its abstract counterpart cannot do, so what is true of the abstract system is true of any phsycial systems that really instantiates it. > > There is also an apriori argument against Pythagoreanism (=everything > > is numbers). If it is a *contingent* fact that non-mathematical > > entities > > don't exist, Pythagoreanism cannot be justified by rationalism (=- > > all truths are necessary and apriori). Therefore the > > Pythagorean-ratioanlist > > must believe matter is *impossible*. > > Yes, I agree. That's what I mean about the 'existence' claim of > 'necessary truth' - since it rules out 'contingent instantiation', it > must replace it with 'necessary instantiation', or be incoherent as to > ontology. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
1Z wrote: > Necessary truth doesn't entail necessary existence unless > the claims in question are claims about existence. If one claims (which I don't BTW) that something is 'necessarily true' *independent of contingent existence* then I think for this to be in any way coherent, one must be making some sort of existence claim for 'necessary truth'. By contrast, within contingent existence, some things may seem 'necessarily true', but this truth can only be derived from aspects of contingency (i.e. in virtue of the concept and its referents being contingently instantiated). > Not if AR is only a claim about truth. Necessary truth > can exist in a world of contingent existence -- providing > all necessary truths in such a world are ontologically non-commital. > As non-Platonists indded take mathematical statements to be. I agree insofar as you mean what I'm saying above: i.e. the 'existence' of 'necessary truth' in a world of contingent existence must itself be 'contingently instantiated'. 'Necessity' in this sense is restricted to 'necessary under ceratin contingencies'. In a world of contingent existence the behaviour of a logical system must reduce ultimately to the behaviour of a contingently instantiated system. > There is also an apriori argument against Pythagoreanism (=everything > is numbers). If it is a *contingent* fact that non-mathematical > entities > don't exist, Pythagoreanism cannot be justified by rationalism (=- > all truths are necessary and apriori). Therefore the > Pythagorean-ratioanlist > must believe matter is *impossible*. Yes, I agree. That's what I mean about the 'existence' claim of 'necessary truth' - since it rules out 'contingent instantiation', it must replace it with 'necessary instantiation', or be incoherent as to ontology. David > David Nyman wrote: > > Bruno Marchal wrote: > > > > > Please I have never said that primary matter is impossible. Just that I > > > have no idea what it is, no idea what use can it have, nor any idea how > > > it could helps to explain quanta or qualia. > > > So I am happy that with comp it has necessarily no purpose, and we can > > > abandon "weak materialism", i.e. the doctrine of primary matter, like > > > the biologist have abandon the vital principle, or like the abandon of > > > ether by most physicist. > > > But with comp it is shown how to retrieve the appearance of it, by > > > taking into account the differences between the notions of n-person > > > (and of n-existence) the universal machine cannot avoid. > > > > Are we not trying to discriminate two possible starting assumptions > > here? > > > > 1) Necessity > > 2) Contingency > > > > Assumption 1 makes no appeal to fundamental contingency, but posits > > only 'necessarily true' axioms (e.g. AR). > > Things don't become necessarily true just > because someone says so. The truths > of mathematics may be necessarily true, but > that does not make AR a s aclaim about > existence necessarily true. AR as a claim > about existence is metaphysics, and highly debatable. > > > In this sense there could > > never be "nothing instead of something" because the 'necessary truth' > > of AR is deemed independent of contingency - indeed 'contingency' would > > be seen to emerge from it (hence its 'empiricism' Bruno?) > > Necessary truth doesn't entail necessary existence unless > the claims in question are claims about existence. > > Whether mathematical truths are about existence is debatable > and not "necessary". > > > Assumption 2 posits by contrast the ultimate contingency of 'existence' > > - there might indeed have been 'nothing'. The apparent 'necessity' of > > AR must consequently be illusory > > Not if AR is only a claim about truth. Necessary truth > can exist in a world of contingent existence -- providing > all necessary truths in such a world are ontologically non-commital. > As non-Platonists indded take mathematical statements to be. > > > - i.e. AR, CT etc. derive their > > 'existence' and characteristics from the prior facts of brute > > contingency. > > > > Under assumption 2, therefore, the semantics of 'bare substrate' boil > > down to a fundamental assertion of 'non-relative contingent existence', > > and 'primary matter' to 'relative contingent processes/ structures'. > > Starting from assumption 2 we could see comp as a schema of relative > > contingent process/ structure within which 'primary matter' is deemed > > to be 'instantiated', or vice versa (i.e. the 'usual assumption' of > > physical instantiation). > > > > But are assumptions 1 and 2 ineluctably 'theological preferences', or > > can we discriminate them empirically? > > That's what White Rabbits are all about. > > There is also an apriori argument against Pythagoreanism (=everything > is numbers). If it is a *contingent* fact that non-mathematical > entities > don't exist, Pythagoreanism cannot be justified by rationalism (=- > all truths are necessary and apriori). Therefore the > Pythagorean-ratioanl
Re: Arithmetical Realism
David Nyman wrote: > Bruno Marchal wrote: > > > Please I have never said that primary matter is impossible. Just that I > > have no idea what it is, no idea what use can it have, nor any idea how > > it could helps to explain quanta or qualia. > > So I am happy that with comp it has necessarily no purpose, and we can > > abandon "weak materialism", i.e. the doctrine of primary matter, like > > the biologist have abandon the vital principle, or like the abandon of > > ether by most physicist. > > But with comp it is shown how to retrieve the appearance of it, by > > taking into account the differences between the notions of n-person > > (and of n-existence) the universal machine cannot avoid. > > Are we not trying to discriminate two possible starting assumptions > here? > > 1) Necessity > 2) Contingency > > Assumption 1 makes no appeal to fundamental contingency, but posits > only 'necessarily true' axioms (e.g. AR). Things don't become necessarily true just because someone says so. The truths of mathematics may be necessarily true, but that does not make AR a s aclaim about existence necessarily true. AR as a claim about existence is metaphysics, and highly debatable. > In this sense there could > never be "nothing instead of something" because the 'necessary truth' > of AR is deemed independent of contingency - indeed 'contingency' would > be seen to emerge from it (hence its 'empiricism' Bruno?) Necessary truth doesn't entail necessary existence unless the claims in question are claims about existence. Whether mathematical truths are about existence is debatable and not "necessary". > Assumption 2 posits by contrast the ultimate contingency of 'existence' > - there might indeed have been 'nothing'. The apparent 'necessity' of > AR must consequently be illusory Not if AR is only a claim about truth. Necessary truth can exist in a world of contingent existence -- providing all necessary truths in such a world are ontologically non-commital. As non-Platonists indded take mathematical statements to be. > - i.e. AR, CT etc. derive their > 'existence' and characteristics from the prior facts of brute > contingency. > > Under assumption 2, therefore, the semantics of 'bare substrate' boil > down to a fundamental assertion of 'non-relative contingent existence', > and 'primary matter' to 'relative contingent processes/ structures'. > Starting from assumption 2 we could see comp as a schema of relative > contingent process/ structure within which 'primary matter' is deemed > to be 'instantiated', or vice versa (i.e. the 'usual assumption' of > physical instantiation). > > But are assumptions 1 and 2 ineluctably 'theological preferences', or > can we discriminate them empirically? That's what White Rabbits are all about. There is also an apriori argument against Pythagoreanism (=everything is numbers). If it is a *contingent* fact that non-mathematical entities don't exist, Pythagoreanism cannot be justified by rationalism (=- all truths are necessary and apriori). Therefore the Pythagorean-ratioanlist must believe matter is *impossible*. (BTW, empiricists can accept *some* apriori arguments). --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
Bruno Marchal wrote: > Please I have never said that primary matter is impossible. Just that I > have no idea what it is, no idea what use can it have, nor any idea how > it could helps to explain quanta or qualia. > So I am happy that with comp it has necessarily no purpose, and we can > abandon "weak materialism", i.e. the doctrine of primary matter, like > the biologist have abandon the vital principle, or like the abandon of > ether by most physicist. > But with comp it is shown how to retrieve the appearance of it, by > taking into account the differences between the notions of n-person > (and of n-existence) the universal machine cannot avoid. Are we not trying to discriminate two possible starting assumptions here? 1) Necessity 2) Contingency Assumption 1 makes no appeal to fundamental contingency, but posits only 'necessarily true' axioms (e.g. AR). In this sense there could never be "nothing instead of something" because the 'necessary truth' of AR is deemed independent of contingency - indeed 'contingency' would be seen to emerge from it (hence its 'empiricism' Bruno?) Assumption 2 posits by contrast the ultimate contingency of 'existence' - there might indeed have been 'nothing'. The apparent 'necessity' of AR must consequently be illusory - i.e. AR, CT etc. derive their 'existence' and characteristics from the prior facts of brute contingency. Under assumption 2, therefore, the semantics of 'bare substrate' boil down to a fundamental assertion of 'non-relative contingent existence', and 'primary matter' to 'relative contingent processes/ structures'. Starting from assumption 2 we could see comp as a schema of relative contingent process/ structure within which 'primary matter' is deemed to be 'instantiated', or vice versa (i.e. the 'usual assumption' of physical instantiation). But are assumptions 1 and 2 ineluctably 'theological preferences', or can we discriminate them empirically? David > Le 31-août-06, à 22:20, 1Z a écrit : > > > > > > > Bruno Marchal wrote: > > > >> Le 29-août-06, à 20:45, 1Z a écrit : > >> > >> > >> > >>> The version of AR that is supported by comp > >>> only makes a commitment about mind-independent *truth*. The idea > >>> that the mind-independent truth of mathematical propositions > >>> entails the mind-independent *existence* of mathematical objects is > >>> a very contentious and substantive claim. > >> > >> > >> You have not yet answered my question: what difference are you making > >> between "there exist a prime number in platonia" and "the truth of the > >> proposition asserting the *existence* of a prime number is independent > >> of me, you, and all contingencies" ? > > > > "P is true" is not different to "P". That is not the difference I > > making. > > > I am glad to hear this. > > > > > > I'm making a difference between what "exists" means in mathematical > > sentences and what it means in empiricial sentences (and what it means > > in fictional contexts...) > > > Of course I do that difference too! Each hypostase has its own notion > of existence. > When I say that a number exists, it is in the usual sense of a realist > arithmetician. > But physical existence is a completely different things having a logic > of its own. The UDA shows that the logic of the physical propositions > should emerge from the logic of what will be true in all accessible > worlds. The world correspond to the relative consistent extension and > are eventually characterized by the discourse which remain invariant > through world-transition, themselves eventually given by the interview > of the lobian machine. > I am certainly not identifying many different notion of existence, on > the contrary. Recall perhaps that each hypostase (that is "notion of > person") defines some "canonical" Kripke "multiverses". > Perhaps search on "Kripke" in the archive, but I guess we will go back > to this at some point. > > > > > > > > The logical case for mathematical Platonism is based on the idea > > that mathematical statements are true, and make existence claims. > > That they are true is not disputed by the anti-Platonist, who > > must therefore claim that mathematical existence claims are somehow > > weaker than other existence claims -- perhaps merely metaphorical. > > That the the word "exists" means different things in different contexts > > is easily established. > > > > > > > > > > > > > (Incidentally, this approach answers a question about mathematical and > > empirical > > truth. The anti-Platonists want sthe two kinds of truth to be > > different, but > > also needs them to be related so as to avoid the charge that one class > > of > > statement is not true at all. This can be achieved because empirical > > statements rest on non-contradiction in order to achive correspondence. > > If an empricial observation fails co correspond to a statemet, there > > is a contradiction between them. Thus non-contradiciton is a necessary > > but insufficient justification for tru
Re: Arithmetical Realism
Le 31-août-06, à 22:20, 1Z a écrit : > > > Bruno Marchal wrote: > >> Le 29-août-06, à 20:45, 1Z a écrit : >> >> >> >>> The version of AR that is supported by comp >>> only makes a commitment about mind-independent *truth*. The idea >>> that the mind-independent truth of mathematical propositions >>> entails the mind-independent *existence* of mathematical objects is >>> a very contentious and substantive claim. >> >> >> You have not yet answered my question: what difference are you making >> between "there exist a prime number in platonia" and "the truth of the >> proposition asserting the *existence* of a prime number is independent >> of me, you, and all contingencies" ? > > "P is true" is not different to "P". That is not the difference I > making. I am glad to hear this. > > I'm making a difference between what "exists" means in mathematical > sentences and what it means in empiricial sentences (and what it means > in fictional contexts...) Of course I do that difference too! Each hypostase has its own notion of existence. When I say that a number exists, it is in the usual sense of a realist arithmetician. But physical existence is a completely different things having a logic of its own. The UDA shows that the logic of the physical propositions should emerge from the logic of what will be true in all accessible worlds. The world correspond to the relative consistent extension and are eventually characterized by the discourse which remain invariant through world-transition, themselves eventually given by the interview of the lobian machine. I am certainly not identifying many different notion of existence, on the contrary. Recall perhaps that each hypostase (that is "notion of person") defines some "canonical" Kripke "multiverses". Perhaps search on "Kripke" in the archive, but I guess we will go back to this at some point. > > > The logical case for mathematical Platonism is based on the idea > that mathematical statements are true, and make existence claims. > That they are true is not disputed by the anti-Platonist, who > must therefore claim that mathematical existence claims are somehow > weaker than other existence claims -- perhaps merely metaphorical. > That the the word "exists" means different things in different contexts > is easily established. > > > > (Incidentally, this approach answers a question about mathematical and > empirical > truth. The anti-Platonists want sthe two kinds of truth to be > different, but > also needs them to be related so as to avoid the charge that one class > of > statement is not true at all. This can be achieved because empirical > statements rest on non-contradiction in order to achive correspondence. > If an empricial observation fails co correspond to a statemet, there > is a contradiction between them. Thus non-contradiciton is a necessary > but insufficient justification for truth in empircal statements, but > a sufficient one for mathematical statements). Even for math, non contradiction is not a sufficient criteria. This follows immediately from the second incompleteness theorem. PA cannot prove its own consistency (PA does not prove ~Bf). This means you will not get a contradiction by adding to PA the formula stating that PA is inconsistent (Bf). Sp PA + Bf, although quite insane in some sense, is actually consistent, but mathematically unreasonable (but useful in self-reference theory for getting a simple example of arithmetically unsound but consistent machine). > > > >>> Where is it shown the UD exists ? >> >> >> If you agree that the number 0, 1, 2, 3, 4, ... exist (or again, if >> you >> prefer, that the truth of the propositions: >> >> Ex(x = 0), >> Ex(x = s(0)), >> Ex(x = s(s(0))), >> ... >> >> is independent of me), then it can proved that the UD exists. It can >> be >> proved also that Peano Arithmetic (PA) can both define the UD and >> prove >> that it exists. > > But again this is just "mathematical existence". You need some > reason to assert that mathematical existence is not a mere > metaphor implying no real existence, as anti-Platonist > mathematicians claim. I do not think that is given by computationalism. When I say that there is an infinity of prime number, it is not a metaphor. I am not saying that prime numbers exists like electrons, only that the "physical existence of electron" emerge in the stable dreams of the lobian machines, and those dreams are reducible to relative and local finite computations which, relatively to universal numbers (which exist by CT), exist then, in the same sense than the prime number, that is the interpretation of formula like "ExP(x,y)" in the standard model of arithmetic (the one we learn at school). > Tell me also this, if you don't mind: are you able to doubt about the existence of "primary matter"? I know it is your main fundamental postulate. Could you imagine that you could be wrong? >>> >>> It is possible
Re: Arithmetical Realism
Bruno Marchal wrote: > Le 29-août-06, à 20:45, 1Z a écrit : > > > > > The version of AR that is supported by comp > > only makes a commitment about mind-independent *truth*. The idea > > that the mind-independent truth of mathematical propositions > > entails the mind-independent *existence* of mathematical objects is > > a very contentious and substantive claim. > > > You have not yet answered my question: what difference are you making > between "there exist a prime number in platonia" and "the truth of the > proposition asserting the *existence* of a prime number is independent > of me, you, and all contingencies" ? "P is true" is not different to "P". That is not the difference I making. I'm making a difference between what "exists" means in mathematical sentences and what it means in empiricial sentences (and what it means in fictional contexts...) The logical case for mathematical Platonism is based on the idea that mathematical statements are true, and make existence claims. That they are true is not disputed by the anti-Platonist, who must therefore claim that mathematical existence claims are somehow weaker than other existence claims -- perhaps merely metaphorical. That the the word "exists" means different things in different contexts is easily established. For one thing, this is already conceded by Platonists! Platonists think Platonic existence is eternal, immaterial non-spatial, and so on, unlike the Earthly existence of material bodies. For another, we are already used to contextualising the meaning of "exists". We agree with both: "helicopters exist"; and "helicopters don't exist in Middle Earth". (People who base their entire anti-Platonic philosophy are called fictionalists. However, mathematics is not a fiction because it is not a free creation. Mathematicians are constrained by consistency and non-contradiction in a way that authors are not. Dr Watson's fictional existence is intact despite the fact that he is sometimes called John and sometimes James in Conan Doyle's stories). The epistemic case for mathematical Platonism is be 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. (There is also disagreement about some axioms, such as the Axiom of Choice, and different mathematicians with different attitudes about the AoC will tend to get different answers -- a phenomenon which is easily explained by the formalist view I am taking here). The semantic case for mathematical Platonism is based on the idea that the terms in a mathematical sentence must mean something, and therefore must refer to objects. It can be argued on general linguistic grounds that not all meaning is reference to some kind of object outside the head. Some meaning is sense, some is reference. That establishes the possibility that mathematical terms do not have references. What establishes it is as likely and not merely possible is the obeservation that nothing like empirical investigation is needed to establish the truth of mathematical statements. Mathematical truth is arrived at by a purely conceptual process, which is what would be expected if mathematical meaning were restricted to the Sense, the "in the head" component of meaning. A possible counter argument by the Platonist is that the downgrading of mathematical existence to a mere metaphor is arbitrary. The anti-Platonist must show that a consistent standard is being applied. This it is possible to do; the standard is to take the meaning of existence in the context of a particular proposition to relate to the means of justification of the proposition. Since ordinary statements are confirmed empirically, "exists" means "can be perceived" in that context. Since sufficient grounds for asserting the existence of mathematical objects are that it is does not contradict anything else in mathematics, mathematical existence just amounts to concpetual non-contradictoriness. (Incidentally, this approach answers a question about mathematical and empirical truth. The anti-Platonists want sthe two kinds of truth to be different, but also needs them to be related so as to avoid the charge that one class of statement is not true at all. This can be achieved because empirical statements rest on non-contradiction in order to achive correspondence. If an empricial observation fails co correspond to a statemet, there is a contradiction between them. Thus non-contradiciton is a necessary but insufficient justification for truth in empircal statements, but a sufficient one for mathematical statements).
Re: Arithmetical Realism
Le 29-août-06, à 20:45, 1Z a écrit : > The version of AR that is supported by comp > only makes a commitment about mind-independent *truth*. The idea > that the mind-independent truth of mathematical propositions > entails the mind-independent *existence* of mathematical objects is > a very contentious and substantive claim. You have not yet answered my question: what difference are you making between "there exist a prime number in platonia" and "the truth of the proposition asserting the *existence* of a prime number is independent of me, you, and all contingencies" ? > Where is it shown the UD exists ? If you agree that the number 0, 1, 2, 3, 4, ... exist (or again, if you prefer, that the truth of the propositions: Ex(x = 0), Ex(x = s(0)), Ex(x = s(s(0))), ... is independent of me), then it can proved that the UD exists. It can be proved also that Peano Arithmetic (PA) can both define the UD and prove that it exists. > >> Tell me also this, if you don't mind: are you able to doubt about the >> existence of "primary matter"? I know it is your main fundamental >> postulate. Could you imagine that you could be wrong? > > It is possible that I am wrong. It is possible that I am right. > But you are -- or were -- telling me matter is impossible. Only when I use Occam. Without Occam I say only that the notion of primary matter is necessarily useless i.e. without explanatory purposes (even concerning just the belief in the physical proposition only) . This is a non trivial consequence of the comp hyp. (cf UDA). > But the negative integers exist (or "exist"), so it has > an existing predecessor. Yes. But the axiom Q1 "Ax ~(0 = s(x)" is not made wrong just because you define the negative integer in Robinson Arithmetic. The "x" are still for "natural number". The integer are new objects defined from the natural number. All right? To take another example, you can define in RA all partial recursive functions, but obviously they does not obey to the Q axioms, they are just constructs, definable in RA. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
"1Z" <[EMAIL PROTECTED]> wrote on August 29 > The version of AR that is supported by comp > only makes a commitment about mind-independent *truth*. The idea > that the mind-independent truth of mathematical propositions > entails the mind-independent *existence* of mathematical objects is > a very contentious and substantive claim. I'm very late in reading this thread. I assume AR is "Arithmetical Realism" and that *truth* in this thread implies alethic qualification of some sort. To me, a statement like "only use batteries with the same rated voltage" would seem only to be qualifiable as true or otherwise if related to factual content. Such a statement would not be meaningless and would contain information which could be worth preserving or using. I am wondering how much semantic loading Bruno's ideas of quantification are obliged to carry here. Quantifiers always worry me as they often seem to come up at a very early stage and they do always seem to carry with them a similar pattern to "only use batteries with the same rated voltage" and their meaning if any is never absolutely clear or clarifiable. Perhaps they cannot entail the aforementioned "mind-independent *existence* of mathematical objects". Or, at least, not without further qualification, rendering his theory possibly incomplete as theories tend to be. This is not the same as people saying "in spite of all we know about electricity, we do not know what electricity is", because of course we do know what electricity is, in context if not in metaphysics. [Bruno's defintiion of Arithmetic Realism I understand to be " Arithmetical Realism. All proposition pertaining on natural numbers with the form Qx Qy Qz Qt Qr ... Qu P(x,y,z,t,r, ...,u) are true independently of me. Q represents a universal or existential quantifier, and P represents a decidable (recursive) predicate. That is, proposition like the Fermat-Wiles theorem, or Goldbach conjecture, or Euclide's infinity of primes theorem are either true or false, and this independently of the proposition "Bruno Marchal exists". It amounts to accept, for the sake of my argument, that classical logic is correct in the realm of positive integers. Nothing more."] --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
Bruno Marchal wrote: > Le 28-août-06, à 16:47, 1Z a écrit : > > > > > > > Bruno Marchal wrote: > > > >> AR eventually provides the whole comp ontology, although it has > >> nothing > >> to do with any commitment with a substantial reality. > > > > If it makes no commitments about existence,. it can prove nothing about > > ontology. > > > > > Absolutely so. But I said that comp makes no commitment about primary > physical stuff. It makes no other ontological commitment. >As I said more than 10 times to you is that comp, > through AR makes a commitment about the existence of (non substantial) > numbers. The version of AR that is supported by comp only makes a commitment about mind-independent *truth*. The idea that the mind-independent truth of mathematical propositions entails the mind-independent *existence* of mathematical objects is a very contentious and substantive claim. > You tend to beg the question through your assumption that only primary > physical matter exists. AFAICS, I am only asuming that *I* exist. (I could also you tend to beg the guqestio that ruth is existence...) > But then comp is false or the UDA reasoning is false, but then just > show where, please. Where is it shown the UD exists ? > Tell me also this, if you don't mind: are you able to doubt about the > existence of "primary matter"? I know it is your main fundamental > postulate. Could you imagine that you could be wrong? It is possible that I am wrong. It is possible that I am right. But you are -- or were -- telling me matter is impossible. > > Bruno Marchal wrote: > > > >> In both comp and the quantum, a case can be made that the > >> irreversibility of memory (coming from usual thermodynamics, or big > >> number law) can explain, through physical or comp-physical > >> interactions, the first person feeling of irreversibility. > >> But with comp we do start from a basic "irreversibility": 0 has a > >> successor but no predecessors. > > > > ...among the natural numbers. Does COMP really prove > > that negative numebrs don't exist ? > > > Who said that? You can already define the negative integer in Robinson > Arithmetic, and prove the existence of each negative integer. The > common algebraical construction of the integer as couple of natural > number togeteher with the genuine equivalence relation can be done in > RA. RA or PA proves only that 0 has no predecessor among the natural > numbers. But the negative integers exist (or "exist"), so it has an existing predecessor. All you are aying is that in Platoia there are structures with the same one-way quality as time, well, of course there are. Every structure exists in Platonia, if Paltonia exists. That doesn't explain why we see only one particular structure (which is still only B-series). > Actually, as I have said, RA can already define all partial recursive > functions, i.e. all function which are programmable in your favorite > programming language. (No need of CT here, unless your favorite > programming language belongs to the future). > Despite this RA is very weak and has almost no ability to generalize. > Peano Arithmetic PA, which is just RA + the induction axioms, is much > clever, and most usual mathematics (including Ramanujan's work) can be > done by PA. > > Bruno > > > > > http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
Le 28-août-06, à 16:47, 1Z a écrit : > > > Bruno Marchal wrote: > >> AR eventually provides the whole comp ontology, although it has >> nothing >> to do with any commitment with a substantial reality. > > If it makes no commitments about existence,. it can prove nothing about > ontology. Absolutely so. But I said that comp makes no commitment about primary physical stuff. As I said more than 10 times to you is that comp, through AR makes a commitment about the existence of (non substantial) numbers. You tend to beg the question through your assumption that only primary physical matter exists. But then comp is false or the UDA reasoning is false, but then just show where, please. Tell me also this, if you don't mind: are you able to doubt about the existence of "primary matter"? I know it is your main fundamental postulate. Could you imagine that you could be wrong? > Bruno Marchal wrote: > >> In both comp and the quantum, a case can be made that the >> irreversibility of memory (coming from usual thermodynamics, or big >> number law) can explain, through physical or comp-physical >> interactions, the first person feeling of irreversibility. >> But with comp we do start from a basic "irreversibility": 0 has a >> successor but no predecessors. > > ...among the natural numbers. Does COMP really prove > that negative numebrs don't exist ? Who said that? You can already define the negative integer in Robinson Arithmetic, and prove the existence of each negative integer. The common algebraical construction of the integer as couple of natural number togeteher with the genuine equivalence relation can be done in RA. RA or PA proves only that 0 has no predecessor among the natural numbers. Actually, as I have said, RA can already define all partial recursive functions, i.e. all function which are programmable in your favorite programming language. (No need of CT here, unless your favorite programming language belongs to the future). Despite this RA is very weak and has almost no ability to generalize. Peano Arithmetic PA, which is just RA + the induction axioms, is much clever, and most usual mathematics (including Ramanujan's work) can be done by PA. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Arithmetical Realism
Bruno Marchal wrote: > AR eventually provides the whole comp ontology, although it has nothing > to do with any commitment with a substantial reality. If it makes no commitments about existence,. it can prove nothing about ontology. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
