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 ;.)
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 don�t
> > 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 it�s
> > 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 »
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