John Mikes wrote:
Brian,
nothing could be more remote for me than to argue 'math' (number's
application and theories) with you. I thinkyou mix up* 'counting'* for
the stuff that serves it. As I usually do, I looked up Google for the
Peano axioms and found nothing in them that pertains to the
origination of numbers. They USE them and EXPLAIN sich usage. Use what????
Indeed, counting and what I'm referring to as numbers are different.
Counting is a mental process while numbers have nothing to do with mind
though the mind may apprehend and understand numbers to some extent.
Counting is not the origin of numbers. Counting inspired the discovery
of numbers as elucidated by people like Peano. Numbers are idealized
models for the process of counting much like how a rectangle is an
idealized model for the blueprint of an architectural structure's
foundation. Rectangles are not found in nature and neither are numbers;
both are abstractions of things we see in nature.
Yet numbers and rectangles (and many other abstractions) have a
suspiciously good use for modeling things in nature.
I wonder if you have an example where application of numbers is
extractable from ANY quantity the numbers refer to?
<Three plus four> is not different from <blue plus loud>, <sound plus
speed>, /_whatever_/, meaningless words bound together. UNless - of
course - you as a human, with human logic and complexity, UNDERSTAND
the amount *three* added to a _comparable_ amount of *four *and RESULT
in /_*seven* pertaining to the same kind of amount._/
I only mean to reference the difference between numbers and the quantity
they point to. In an important way, <3+4> is different from your other
examples in that <3+4> can be translated into a language devoid of human
baggage and symbolically manipulated so as to show an equivalence
between the symbols <3+4> and <7>.
/_ _/
//
/_Axioms_/ however sounds to my vocabulary like inventions helping to
justify our theories. Sometimes quite weird.
And *Brent* was so right: /"...I don't think the existence of some
number of distinct things is the same as the "existence" of
numbers...."/ - Tegmark's quoted "accounted for..." is not "consists
of".
/_To 'explain' _/something by a conceptualization does not
substitute for the existence and justification of such conceptualization.
Axioms are statements. Do humans need to exist in order for the
statement "the galaxy is approximately a spiral shape" to exist? How
about "3+4=7", does that require humans to exist in order for the
statement to exist? What about the existence of the statement "the
president of the US is male"; if all the humans were to die out, that
statement would still exist. Statements are uttered by humans but do
not depend on humans for their existence. This is how axioms exist
independent of humans, because they are statements. The notation
differs and are invented but what is being referred to by the symbols is
independent of humans. Moreover, I'm not talking about the truth of
statements; I'm talking about the statements themselves not requiring
anyone to utter them in order to exist.
Numbers do not physically exist; so if physical existence is the only
form of existence you permit, then numbers do not exist... in the same
sense that math might as well be about Luke Skywalker, who does not
exist physically. However, math has a suspiciously good use in nature
like I said, unlike a novel about Luke Skywalker.
Does it make sense that 'numbers existed' when nobody was around to
*/_K N O W or U S E??_/*
Especially when they did not/_ *C O U N T*_/ anything? BTW: what are
those abstract symbols you refer to as numbers?
(and this question is understood for times way before humans and human
thinking).
Sorry I asked
John M
Does it make sense? Let me ask you a question. Way back when, in the
earliest stages of counting, let's assume there was a point at which a
hundred thousand was the furthest anyone had counted to. Now.. Did the
number 1,000,000 exist at this stage of counting? I think it did. A
million and all of its successors.
Bruno,
-------------
Hmm... Lawvere has tried to build an all encompassing universal
mathematical structure, but he failed. It was an interesting failure as
he discovered the notion of topos, (discovered also independently by
Groethendieck) which is more a mathematical mathematician than a
mathematical universe.
Also Tegmark is not aware that Digital Mechanism entails the non
locality, the indeterminacy and the non cloning of matter, and that DM
makes the physical into a person-modality due to the presence of the
mathematician in the arithmetical reality.
Quanta are special case of first person plural sharable qualia.
-------------
I'm not looking for a truly all-encompassing mathematical structure.
What I'm looking for is a mathematical structure in which all
mathematical structures can be embedded. By mathematical structure, I
mean there is a symbol set S consisting of constant symbols, relation
symbols, and function symbols, and the pairing of a set with a list of
rules that interpret the symbols. In Tegmark's papers on "ultimate
ensemble TOE" and "the mathematical universe," he refers to what I call
a mathematical structure as a "formal system" (and also mathematical
structure).
The structure I'm looking for wouldn't encompass anything that isn't a
mathematical structure, like a category with no objects/elements.
Tegmark argues that reality is a mathematical structure. What's cute
about his argument is that while invoking the concept of a TOE, his
argument is independent of what that TOE might be. He defines a TOE to
be a complete description of reality. Whether or not this can be
expressed in a finite string is an open problem as far as I know. (I
doubt it can.) He argues that a complete description of reality must be
expressible in a form that has no human baggage and I would add to that
is something that exists independent of humans in the sense that while
the symbols used to provide that complete description will depend on
humans, what is pointed to by the symbols is not.
Tegmark argues that reality is a mathematical structure and states that
an open problem is finding a mathematical structure which is isomorphic
to reality. This might or might not be clear: the mathematical
structure with the property that all mathematical structures can be
embedded within it is precisely the mathematical structure we are
looking for.
I am confident that I have found such a structure but only over a fixed
symbol set; I need such a structure to be inclusive of all symbol sets
so as to cast away the need to refer to a symbol set. The technique I
used was to use NFU, new foundations set theory with urelements--which
is known to be a consistent set theory, to first find the set of all
S-structures. Then I take what I believe is called the reduced product
of all S-structures. Then I show that all S-structures can be embedded
within the reduced product of all S-structures. Admittedly, there is
nothing at all deep about this; none of my arguments are deeper than
typical homework problems in a math logic course.
My next move is to find justification for the existence of a math
structure with the important property that all structures can be
embedded within it --independent of the symbol set-- and thus
eliminating the need to refer to it.
One thing I wonder is how to define all your notions such as
"mathematician," "n-brains," "n-minds," and "digital mechanism" in terms
of mathematical structures. I'm particularly interested in defining
something that models awareness and using it to find self-aware
structures such as "mathematicians."
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-l...@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.
