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 about numbers for *some* people, but they might be embarrassing for others.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 there nothing to measure or count about the object in question?It means that when you add the successor of zero with itself you getthe 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 naive materialism might be false, so why would little "concrete" pieces on stuff, or time, helps in understanding that no matter what: there are no 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 idea ofbeing 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 can stillbelieve 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 the earlyPythagoricians that was a secret, and it seems they killed the one whodare to make that discovery public (at least in some legend).Of coursewe can say 1+2=3 is 3 just because we defined numbers in the way thatthisis 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 numbers are, since these 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 will not. In fine,it is like with the south american, we can feel them enough close tous to listen to them.whereas the "actual" thingthey rely on (what numbers, or 0 and succesor actually are), remainstotallyundefined.Not with comp. An apple becomes something very complex when defined inpure 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 as consciousness, or matter, or whatever else, since the basis is totally undefined.The problem does not consist in finding the ultimate definitions, butto agree on elementary propositions, and to explain the rest, of asmuch 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 cannot prove that 17 is really prime, we have just prove that the fiollowing line ................. cannot be broken in equal non trivial parts (the trivial parts being the 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 with the understanding that a statement like 17 is prime, is independent of all concrete situation, in which 17 might be represented.Lol, the funny thing is that in your explantion you used concrete things, 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 measurable things.Yes. The natural numbers are somehow the type of the finite discreteor 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, because frominside it will be non countable. The important reality is not the big3-thing seen from outside, because no one can go there. The "real"reality with comp is epistemological. It is the living ideas from*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 can bevery 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 from someideasomething about reality.Reality is an idea itself.

[SPK]

`Whose "idea" exactly? If there is no one to whom Reality has a`

`meaning does it have a meaning? No. You seem to assume that`

`meaningfulness exist in the absence of a subject to whom that meaning`

`obtains. That is a contradiction.`

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 simple conceptually than any something you can multiply them by.No. Otherwise we would understand 0, 1, 2, before we understood "one of something", 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. Butit ismore complex to understand, because we mentally have to add thesomething inorder for the numbers to have meaning beyond intellectual mastrubation.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 of`

`substitution exists. This existence cannot be then eliminated by some trick.`

Bruno Marchal wrote:But comp needs only that you belief that the elementary arithmetical truth does not depend on you or us (little ego). Are you thinking that if an asteroid rips of humanity from the cosmos, 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 the number 17 is prime, 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 intuit thatis 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 set`

`then there would be no Mandelbrot set. It is not more complicated than that.`

So, the number 17 is always prime because we defined numbers in theway. IfI define some other number system of natural numbers where I just declare that 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. Butyou will only propose something else universal, and I have alreadysaid that I am not sanguine about numbers in particular. I wouldprefer to use the combinators, or the lambda expression, but naturalnumbers are well known, and that is why I use them in this list. Thelaws of mind and matter are independent of the initial theory, oncethat theory verify the condition of being sigma_1 complete =sufficiently strong to represent the partial computable functions, andto 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 justify itself.`

You are justasserting the truth of you own axioms when you say that number 17 isprime,which is as good as saying my axiom is "everything goes" and I derivefromthat that you are in reality living inside the belly of an invisible pink unicorn.Yes, my proposal of declaring 17 to not be prime is ridiculous,because itdoesn't fit with our conceptions of what properties numbers ought tohave,or ought to be able to have. But these conceptions come from our senseperceptions, and imagination, were we can count and measure things.So whenyou want to apply numbers to the fundamental realtiy, which as such obviously is not countable, nor measurable, your natural numbers are as weird 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 interpretation andskip 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 a numbers.We assume DIGITAL mechanism, and my goal is just to show that thisleads to a reversal physics/machine psychology making the hypothesistestable. The question of using numbers or java programs is a questionof implementation and engineering, like using a mac or a PC.Bruno http://iridia.ulb.ac.be/~marchal/ <http://iridia.ulb.ac.be/%7Emarchal/>

[SPK]

`Try to state your result with no action or verb terms.`

`Implementation is not something that has any meaning absent some`

`physical process. OTOH, we do not need to assume that the physical is`

`primitive nor the abstract. To claim such is a straw man.`

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