On 10 Oct 2011, at 08:21, Stephen P. King wrote:

On 10/6/2011 12:04 PM, Bruno Marchal wrote:On 04 Oct 2011, at 21:59, benjayk wrote:Bruno Marchal wrote:On 03 Oct 2011, at 21:00, benjayk wrote:I don't see why. Concrete objects can be helpful to grasp elementary ideas aboutnumbers for *some* people, but they might be embarrassing forothers.Well, we don't need concrete *physical* objects, necessarily, butconcrete"mental" objects, for example measurement. What do numbers meanwithout anyconcrete object, or measurement? What does 1+1=2 mean if therenothing tomeasure or count about the object in question?It means that when you add the successor of zero with itself youget the successor of one, or the successor of the successor of zero.Bruno Marchal wrote:The diophantine equation x^2 = 2y^2 has no solution. That fact does not seem to me to depend on any concreteness, and I would say that concreteness is something relative. You seem to admit that naivematerialism might be false, so why would little "concrete" piecesonstuff, or time, helps in understanding that no matter what: thereareno natural numbers, different from 0, capable to satisfy the simple equation x^2 = 2y^2.This is just a consequence of using our definitions consistently.Not really. In this case, we can indeed derived this from ourdefinitions and axioms, but this is contingent to us. The very ideaof being realist about the additive and multiplicative structure ofnumbers, is that such a fact might be true independently of ourcognitive abilities.We don't know if there is an infinity of twin primes, but we canstill believe that "God" has a definite idea on that question.That the diophantine equation x^2 = 2y^2 has no solution, isconsidered to be a discovery about natural numbers. It is not aconvention, or the result of a vote, nor of a decision. For theearly Pythagoricians that was a secret, and it seems they killedthe one who dare to make that discovery public (at least in somelegend).Of coursewe can say 1+2=3 is 3 just because we defined numbers in the waythat thisis true, without resorting to any concreteness.Yes. Mathematical realism stems from the intuition that abstractentities can have theor own life (relations with other abstract orconcrete entities).My point is that we can't derive something about the fundamentalnature ofthings just by adhering to our own definitions of what numbersare, sincethese ultimately are just a bunch of definitions,You are right. We need some philosophical principles (like comp) tounderstand that eventually we don't need those philosophicalprinciple. In the case of comp, we can understand why some(relative) numbers will bet on it, and why some other numbers willnot. In fine, it is like with the south american, we can feel themenough close to us to listen to them.whereas the "actual" thingthey rely on (what numbers, or 0 and succesor actually are),remains totallyundefined.Not with comp. An apple becomes something very complex when definedin pure number theory. It will involve infinite sets of longcomputations, complex group of symmetries, etc. But it is definable(in principle) from numbers (some including LUM observers).So whatever we derive from it is just as mysterious asconsciousness, or matter, or whatever else, since the basis istotallyundefined.The problem does not consist in finding the ultimate definitions,but to agree on elementary propositions, and to explain the rest,of as much as possible from them.Bruno Marchal wrote:If it isn't, the whole idea of an abstract machine as an independent existing entity goes down the drain, and with it the consequences of COMP.Yes. But this too me seems senseless. It like saying that we cannotprove that 17 is really prime, we have just prove that thefiollowingline .................cannot be broken in equal non trivial parts (the trivial partsbeingthe tiny . and the big ................. itself). But we have no yet verify this for each of the following: ................. ................. ................. ................. ................. ................. ................. ................. ................. ................. ................. etc. On the contrary: to understand arithmetic, is quasi-equivalent withthe understanding that a statement like 17 is prime, isindependent ofall concrete situation, in which 17 might be represented.Lol, the funny thing is that in your explantion you used concretethings,namely ".".Is that a problem?Of course concrete is relative.I think so.It's concreteness is not really relevant,the point is that numbers just apply to countable or measurablethings.Yes. The natural numbers are somehow the type of the finitediscrete or discernible entities.Without being countable natural numbers don't even make sense.In order for COMP to be applicable to reality, reality had to becountable,Raaah.... Not really. The big 3-thing *can* be countable, becausefrom inside it will be non countable. The important reality is notthe big 3-thing seen from outside, because no one can go there. The"real" reality with comp is epistemological. It is the living ideasfrom *inside*.but it doesn't seem to me to be countable.Because you are inside. (assuming comp, ...).Abstract machines might exist, but just as ideas.The point of platonism is that ideas, despite being epistemologicaldoes exist, and are somehow more real than the big intellectualconstruction, which in fine is shown to not really matter, and canbe very simple.Show that they existbeyond that, and then the further reasoning can be taken moreseriously. Ifnumbers, and abstract machines exist just as ideas, everythingderived fromthem will be further ideas. You can't unambigously conclude fromsome ideasomething about reality.Reality is an idea itself.[SPK] Whose "idea" exactly?

`LUN's idea. (an idea occurring in the mind of the Löbian universal`

`numbers in virtue of their complex relations with infinities of other`

`numbers).`

`And that idea might be correct. "Idea" is not pejorative for an`

`objective rational immaterialist.`

If there is no one to whom Reality has a meaning does it have ameaning? No.

There are many one, and even more.

You seem to assume that meaningfulness exist in the absence of asubject to whom that meaning obtains. That is a contradiction.

`Arithmetical reality is full life and full of subjects. (even without`

`comp, i.e. without us being there).`

Bruno Marchal wrote:1, 2, 3,... make only sense in terms of one of something, two of something,... OK, we could say it makes sense to have one of nothing, two of nothing, etc, but in this case numbers are superfluous, and all numbers, and all computations are equivalent.I think that 0, 1, 2, and many others are far more simpleconceptuallythan any something you can multiply them by.No. Otherwise we would understand 0, 1, 2, before we understood"one ofsomething", which clearly is not true.This does not follows.We understand 1 through "one apple", etc...It is only simpler in terms of being simpler to write down,because wesimply eliminate the mention of the "something" that is counted.But it ismore complex to understand, because we mentally have to add thesomething inorder for the numbers to have meaning beyond intellectualmastrubation.What do you propose as an alternative theory?My point is just that if we say "yes" to the doctor, then we haveliterally no choice on this matter.[SPK]To assume Yes Doctor is to assume that the physical reality ofsubstitution exists.

Yes.

This existence cannot be then eliminated by some trick.

`The existence is not eliminated, it is reduce to a modal existence,`

`whose definition is extracted from the UD argument. The math shows it`

`to be of the type persistent stable partially sharable deep "dream".`

Bruno Marchal wrote:But comp needs only that you belief that the elementaryarithmeticaltruth does not depend on you or us (little ego).Are you thinking that if an asteroid rips of humanity from thecosmos,the number 17 would get a non trivial divisor? That does not make sense, I think.Of course an asteroid won't influence that the number 17 has nonon-trivialdivisor, because we defined the numbers in a way so that thenumber 17 isprime, which is true regardless what happens.All right then. That was my point.The point is that a definition doesn't say anything beyond it'sdefinition.This is deeply false. Look at the Mandelbrot set, you can intuitthat is much more than its definition. That is the base of Gödel'sdiscovery: the arithmetical reality is FAR beyond ANY attempt todefine it.[SPK]If it where not possible to define the rules of a Mandelbrot setthen there would be no Mandelbrot set. It is not more complicatedthan that.

`I agree. But many definable sets are not recursively enumerable. for`

`example the set of codes computing the factorial functions, or the set`

`of codes of total computable functions. Only sigma_1 are computable,`

`above you have many set having relevant information (about numbers)`

`but not at all computable, and, for machine, the arithmetical reality`

`is *very* rich.`

So, the number 17 is always prime because we defined numbers inthe way. IfI define some other number system of natural numbers where I justdeclarethat number 17 shall not be prime, then it is not prime.No. You are just deciding to talk about something else.Who says that your conception of natural numbers is right, and mine is wrong?Then you have to tell me what axioms you want me to make a change.But you will only propose something else universal, and I havealready said that I am not sanguine about numbers in particular. Iwould prefer to use the combinators, or the lambda expression, butnatural numbers are well known, and that is why I use them in thislist. The laws of mind and matter are independent of the initialtheory, once that theory verify the condition of being sigma_1complete = sufficiently strong to represent the partial computablefunctions, and to emulate the UD.[SPK]The fact that we can have this discussion tells us something,but it is not that we can believe that a theory alone can justifyitself.

Comp does not justifies itself (unlike the Löbian machine, actually).

`On the contrary comp justifies that you need an act of faith, you have`

`to bet on something impossible to justify (like a substitution level,`

`notably).`

You are justasserting the truth of you own axioms when you say that number 17is prime,which is as good as saying my axiom is "everything goes" and Iderive fromthat that you are in reality living inside the belly of aninvisible pinkunicorn.Yes, my proposal of declaring 17 to not be prime is ridiculous,because itdoesn't fit with our conceptions of what properties numbers oughtto have,or ought to be able to have. But these conceptions come from oursenseperceptions, and imagination, were we can count and measurethings. So whenyou want to apply numbers to the fundamental realtiy, which as suchobviously is not countable, nor measurable, your natural numbersare asweird as mine, because they both miss the point that reality is not countable.Of course we can do a lot of interpretation to rescue our theory,forexample by interpreting something beyond numbers into numbers viaGödel, butthen we could as well just use our capability of interpretationand skip thenumber magic.The numbers are just more pedagogical. When you say "yes" to thedoctor he can put a java program on a disk, or a combinators, butusually people will see only 0 and 1, and still call that anumbers. We assume DIGITAL mechanism, and my goal is just to showthat this leads to a reversal physics/machine psychology making thehypothesis testable. The question of using numbers or java programsis a question of implementation and engineering, like using a macor a PC.Bruno http://iridia.ulb.ac.be/~marchal/[SPK]Try to state your result with no action or verb terms.Implementation is not something that has any meaning absent somephysical process.

`Implementation is definable in Peano Arithmetic. The universal number`

`u implements the function phi_i when for all number x we have`

`phi_u(<i, x>) = phi_i(x). And this definition can be written in`

`arithmetic (that is using only s, 0, + and * (and logical symbols).`

OTOH, we do not need to assume that the physical is primitive northe abstract. To claim such is a straw man.

`Sure. But once we say yes to the doctor, we have to take the`

`consequences into consideration.`

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 everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.