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愒
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愀
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
worlds that are non-contradictory with their existence and we can argue
many implications of this principle, but note that it places an observer
and a world into a mutually defining role such that they as concepts
stand or fall together. Is there a principle to applies to individual
entities, so that we can consider what similar principle applies to any
object, say an electron or a massive black hole?
It could be that Boolean logic appears to use as the most simple,
beautiful and consistent type of logic because the act of observation
itself is contained to met the kind of optimization terms that you are
pointing toward? I would also point out that the topological dual of
Boolean algebras has the same appearance as what we consider when we
think of the idea of "atoms in a void". Is this a mere coincidence?
Could the logical way we observe the world define in a very real way
exactly the kind of things that we observe? We usually observe, via our
eyes and touch, objects composed of collections of points merely because
we are, predominately, observing the world with a position basis
<http://en.wikipedia.org/wiki/Quantum_state#Basis_states_of_one-particle_systems>?
Onward!
Stephen
Dear Albert,
One brief comment. In your Google paper you wrote, among other
interesting things, "But life and natural selection demands a
mathematical universe
<https://docs.google.com/Doc?docid=0AW-x2MmiuA32ZGQ1cm03cXFfMTk4YzR4cn...>somehow".
Could it be that this is just another implication of the MMH idea? If
the physical implementation of computation acts as a selective pressure
on the multiverse, then it makes sense that we would find ourselves in a
universe that is representable in terms of Boolean algebras with their
nice and well behaved laws of bivalence (a or not-A), etc.
Very interesting ideas.
Onward!
Stephen
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