I have a severe cold,. I can not even calculate what is 27/2 (no kidding). So due to lack of worthy arguments and to avoid to spread the virus in this group, I will stay away for a week at least ;.)

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On 3 mar, 00:32, "Stephen P. King" <stephe...@charter.net> wrote: > On 2/28/2012 8:20 PM, Alberto G.Corona wrote: > > > Dear Stephen, > > > A thing that I often ask myself concerning MMH is the question about > > what is mathematical and what is not?. The set of real numbers is a > > mathematical structure, but also the set of real numbers plus the > > point (1,1) in the plane is. The set of randomly chosen numbers { 1,4 > > 3,4,.34, 3} is because it can be described with the same descriptive > > language of math. But the first of these structures have properties > > and the others do not. The first can be infinite but can be described > > with a single equation while the last must be described > > extensively. . At least some random universes (the finite ones) can be > > described extensively, with the tools of mathematics but they donï¿½t > > count in the intuitive sense as mathematical. > > Dear Alberto, > > I distinguish between the existential and the essential aspects > such that this question is not problematic. Let me elaborate. By > Existence I mean the necessary possibility of the entity. By Essence I > mean the collection of properties that are its identity. Existence is > only contingent on whether or not said existence is self-consistent, in > other words, if an entity's essence is such that it contradicts the > possibility of its existence, then it cannot exist; otherwise entities > exist, but nothing beyond the tautological laws of identity - "A is A" > and Unicity <http://www.thefreedictionary.com/Unicity> - can be said to > follow from that bare existence and we only consider those "laws" only > after we reach the stage of epistemology. > Essence, in the sense of properties seems to require a spectrum of > stratification wherein properties can be associated and categories, > modalities and aspects defined for such. It is this latter case of > Essence that you seem to be considering in your discussion of the > difference between the set of Real numbers and some set of random chosen > numbers, since the former is defined as a complete whole by the set (or > Category) theoretical definition of the Reals while the latter is > contigent on a discription that must capture some particular collection, > hence it is Unicity that matters, i.e. the "wholeness" of the set. > I would venture to guess that the latter case of your examples > always involves particular members of an example of the former case, > e.g. the set of randomly chosen numbers that you mentioned is a subset > of the set of Real numbers. Do there exist set (or Categories) that are > "whole" that require the specification of every one of its members > separately such that no finite description can capture its essence? I am > not sure, thus I am only guessing here. One thing that we need to recall > is that we are, by appearances, finite and can only apprehend finite > details and properties. Is this limitation the result of necessity or > contingency? > Whatever the case it is, we should be careful not to draw > conclusions about the inherent aspects of mathematical objects that > follow from our individual ability to conceive of them. For example, I > have a form of dyslexia that makes the mental manipulation of symbolic > reasoning extremely difficult, I make up for this by reasoning in terms > of more visual and proprioceptive senses and thus can understand > mathematical entities very well. Given this disability, I might make > claims that since I cannot understand the particular symbolic > representations that I am a bit dubious of their existence or > meaningfulness. Of course this is a rather absurd example, but I have > often found that many claims by even eminent mathematicians boils down > to a similar situation. Many of the claims against the existence of > infinities can fall under this situation. > > > What is usually considered genuinely mathematical is any structure, > > that can be described briefly. Also it must have good properties , > > operations, symmetries or isomorphisms with other structures so the > > structure can be navigated and related with other structures and the > > knowledge can be reused. These structures have a low kolmogorov > > complexity, so they can be "navigated" with low computing resources. > > So you are saying that finite describability is a prerequisite for > an entity to be mathematical? What is the lowest upper bound on this > limit and what would necessitate it? Does this imply that mathematics is > constrained to some set of objects that only sapient entities can > manipulate in a way that such manipulations are describable exactly in > terms of a finite list or algorithm? Does this not seem a bit > anthropocentric? But my question is more about the general direction and > implication of your reasoning and not meant to imply anything in > particular. I have often wondered about many of the debates that go on > between mathematicians and wonder if we are all missing something deeper > in our quest. > For example, why is it that there are multiple and different set > theories that have as axioms concepts that are so radically different. > Witness the way that a set theory be such that it assumes the continuum > hypothesis is true while other set theories assume that the continuum > hypothesis is false. This arbitrariness would seem to indicate that > mathematics is more like a game that minds play where all that matters > is that all the "moves" are consistent with the "rules". But what if > this is just a periphery symptom, an indication of something else where > all we are thinking of is the bounding surface of the concepts? > > > So the demand of computation in each living being forces to admit > > that universes too random or too simple, wiith no lineal or > > discontinuous macroscopic laws have no complex spatio-temporal > > volutes (that may be the aspect of life as looked from outside of our > > four-dimensional universe). The macroscopic laws are the macroscopic > > effects of the underlying mathematical structures with which our > > universe is isomorphic (or identical). > > But why must what we do be reducible to some definable set of > procedures? Is there not a kind of prejudice in that idea, that all that > we can know and experience must follow some definable set of rules? > Could it be that what is describable and delimited to follow a set of > rules in the content of our knowledge, where as the processes of the > world are inscrutable on their own. It is only after we sapient and > intercommunicating beings have evolved concepts and explanations that > there is something that we can identify as being, for example, "random" > or "simple" or "complex" or spatio-temporal" or ... or some finite > combination thereof. > > > > > And our very notion of what is intuitively considered mathematical: > > "something general simple and powerful enough" has the hallmark of > > scarcity of computation resources. (And absence of contradictions fits > > in the notion of simplicity, because exception to rules have to be > > memorized and dealt with extensively, one by one) > > I like this attention that you are focusing on "scarcity of > resources". Are you considering that it is a situation that occurred due > to per-existing conditions or is it more of the result of an > optimization process? For example, a tiger has tripes and large teeth > and other features because those features just happen to be the one's > that "won the competition" for ensuring the survival of more tigers than > a set of features that might have been expressed by just some random > occurrence? I have pointed out a article by Stephen Wolfram that > discusses how most systems in Nature happen to express behaviors and > complexities that are such that the best possible computational > simulation of those system by a computational system given physically > possible resource availability is the actual evolution of those systems > themselves. Could it be that a physical system in a real way is "the > best possible computational simulation" of that particular system in > that particular world? This would act as a natural mapping between the > category of possible physical systems and the category of computations, > in the sense that any computation is ultimately a transformation of > information such that the generation of a simulation of some kind of > process occurs. > > > > > Perhaps not only is that way but even may be that the absence of > > contradictions ( the main rule of simplicity) or -in computationa > > terms- the rule of low kolmogorov complexity _creates_ itself the > > mathematics. That is, for example, may be that the boolean logic for > > example, is what it is not because it is consistent simpleand itï¿½s > > beatiful, but because it is the shortest logic in terms of the > > lenght of the description of its operations, and this is the reason > > because we perceive it as simple and beatiful and consistent. > > . > > I believe that the absence of contradictions is an imposed rule of > a sort since it is only necessary to have logical non-contradiction to > reproduce (copy) a given structure. I argue that this is the case > because there is not a priori logical reason why a logical system based > on a particular set of axioms should be ontologically prefered. The set > of {0,1} maybe be a small set of possible variations that can be > associated but why not { i, 1) or {Real Numbers} or {Complex Numbers}? > We must be careful that we do not conflate the particular means by which > we actually do think with the Nature of Reality itself. One thing we > have been taught by Nature in the most forceful way is that Nature does > not respect any preference of framing, coordinate system, or basis. Why > would it necessarily prefer a particular logical system? > To communicate about a structure would fall under this > no-contradiction rule because to communicate coherently and effectively > one must have, at some point in the communicative scheme, a means to > generate a copy of the referent of the message. The so-called very weak > anthropic principle states that observers can only observe themselves in ... > > leer más » -- 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.