- ... true.
^{1} - In mathematics one first searches for the most powerful and
general consistent assumptions and then tries to simplify without
weakening them. There has long been and still is a controversy
about mathematical truth discussed in the section on mathematics.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... explicit.
^{2} - Physicists make their assumptions explicit by formulating them
mathematically. One current exception is the measurement problem in
quantum mechanics. This is the need for a philosophical
interpretation to explain the actualization of probability
densities into experimental observations. Physicists are notorious
for being less rigorous than mathematicians because their first
priority is explaining the experimental record. This translates to
less concern with making assumptions explicit.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... approaches.
^{3} - The foundations of mathematics is facing a similar challenge to
the Platonic philosophy that justifies the axioms of set theory.
This is discussed in the section on mathematics.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... NOT.
^{4} - Computer design also involves clocking that controls when logic
states change and memory that preserves states over time.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... empirical
^{5} - Some may object that internal experience beyond communication
cannot count as empirical evidence. The claim that it must, that it
is the most certain of empirical facts, is at the core of realistic
monism.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... memory
^{6} - By memory in this context I mean memory one can consciously
recall. The brain is constantly building neural connections at
different levels of its organization. Many of these are a form of
memory. For example when we learn to ride a bike we build some
conscious memories, but much of the work is in building memories of
how to maintain balance which are outside of consciousness.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... fields
^{7} - Fields can be discrete or continuous. In the former case field
values are only defined at discrete points, the only locations that
exist.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... networks
^{8} - Artificial neural networks are electronic circuits that emulate
to some degree the structure of biological neurons. They are
trained to perform tasks by strengthening connections that lead to
better results and weakening those that do not. They are
particularly useful in pattern recognition applications where it
can be extremely difficult to develop an analytical solution.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... replaced.
^{9} - A change of consciousness will only be detectable if one can
compare memories with current sensation. A change could affect not
only current sensation, but also how we experience memory. The
empirical evidence will always have this caveat.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... numbers
^{10} - The ordinal numbers are a class of sets that generalize
induction on the integers in an open ended way. Higher levels of
induction in set theory are defined by defining more complex
ordinal numbers.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... reflection.
^{11} - All ordinals beyond the integers are defined as infinite sets.
However a fragment of them, called recursive ordinals, have a
structure that can be output by a recursive process or ideal
computer (see note 12)
program. It is such ordinals that can characterize the level of
abstraction and self reflection of a finite physical entity like
the human mind.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... computer
^{12} - An ideal
computer can run forever error free and has access to unlimited
storage.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Hypothesis.
^{13} - Cantor proved that one could not pair up every real number with
a unique integer. He claimed that this implied there must be more
reals than integers. This is an obviously correct argument for
finite sets, but can be questioned for infinite ones. The Continuum
Hypothesis is the conjecture that the reals are the smallest set
larger than the integers. It has been shown that both the Continuum
Hypothesis and its negation are consistent with the standard axioms
of set theory[4].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... way.
^{14} - `[Feferman's note] CH [Continuum Hypothesis] is just the most
prominent example of many set-theoretical statements that I
consider to be inherently vague. Of course, one may reason
confidently
*within*set theory (e. g., in ZFC [Zermelo Frankel axiomization of set theory plus the Axiom of Choice]) about such statements*as if*they had a definite meaning.'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Feferman99.
^{15} - I have argued for a philosophy of mathematical truth that
limits objectively meaningful mathematical questions to those
relevant to ultimate destiny in a finite but potentially infinite
universe. Such questions are logically determined by a list of
events that an ideal computer program could enumerate[2].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... problem
^{16} - The problem is to determine if an ideal computer will run
forever or eventually halt. The determination must be made by an
algorithm that produces an answer for every possible computer
program in a finite number of steps. The problem is solvable for
some examples, but there is no general solution. The question of
whether a finite formal mathematical system is consistent is
equivalent to the halting problem for some computer program easily
constructed from the axioms of the formal system.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... universe
^{17} - Roger Penrose has argued that quantum effects in the brain
allow mathematicians to transcend the limitation of Gödel's
proof[21]. This is not
necessary to explain the mathematically capable human mind. All
that requires is the enormous diversity of biological evolution as
discussed below. There is no significant evidence to support
Penrose's idea.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... path.
^{18} - The proof is trivial. If unlimited resources are available and
there is no need to select which approach is correct, all
possibilities can be explored, each with ever expanding resources.
As a practical matter, the possibilities will be restricted, but
all alternatives that cannot definitely be excluded must be
allowed.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... infinite.
^{19} - Previous boundaries on the size of the universe have all been
greatly expanded. Cosmology is of necessity a speculative science
that is continually changing. For example, it has recently been
discovered that `dark energy' accelerates the expansion of the
universe.[24]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... emerging.
^{20} - Although Buddhism has a rich intellectual tradition its primary
focus for personal transformation is `the refinement and
utilization of first-person, introspective methods in Buddhist
contemplation'[5, 142]. We
may have much to learn from Buddhist introspective empiricism.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... systems.
^{21} - Singer says `The ethical point of view does, as we have seen,
require us to go beyond a personal point of view to the standpoint
of an impartial spectator. Thus looking at things ethically is a
way of transcending our inward-looking concerns and identifying
ourselves with the most objective point of view possible - with, as
Sidgwick put it, ``the point of view of the universe''[25, 334].' This suggests that, from an
ethical point of view, one should care about the future evolution
of consciousness, but that is not a focus of his work.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... panpsychist
^{22} - `Every element [of matter] contains, at least to an
infinitesimal degree, some germ of inwardness and spontaneity, that
is to say of consciousness.'[8,
225]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... hominization
^{23} - `[De Chardin's note] A maximum--to be followed or not by a
redescent? This question can only be decided by reference to a
subsequent paragraph devoted to the ``activation'' of the human
evolutionary force.' De Chardin sees hominization as a process
where `it [humanity] is provided with special linking organs which
not only assure rapid communication between the elements but little
by little transform their aggregate into a sort of organism which
it would be wrong to consider as simply metaphorical'[8, 59].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... panpsychist
^{24} - `So we could say that the universe --``all that is''-- is
indeed personal, is ``conscious'' in some way that we cannot fully
comprehend. This is no more unreasonable an assumption or belief
than believing that another person is conscious. Personally, I do
feel this to be the case. But this does not require me to go beyond
the ``mere'' ``material'' world and its transcendent patterns. The
world that is, is profound enough.'[18, 215]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... traditions
^{25} - Kurzweil's previous book on artificial intelligence was
*The Age of Spiritual Machines*[17].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... capacity
^{26} - I use `mental capacity' because I think `intelligence' is too
narrow in this context.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... technology
^{27} - Biological neural circuits are much slower than electronic
equivalents.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... unaided
^{28} - Kurzweil foresees a merging of human and machine capabilities
and consciousness by using nano technology to create massive direct
electronic neural interfaces to the human brain.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... planet.
^{29} - Eventually we may develop the technology to send robotic space
ships to distant solar system with all our knowledge and the
technology to reproduce our civilization.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...\samepage
^{30} - The ideas in this paper are more fully developed in
*What is and what will be: Integrating spirituality and science*[1]. There is a related video, `Mathematical Infinity and Human Destiny', on Google Video.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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