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... 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.
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... 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.
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... 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.
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... NOT.4
Computer design also involves clocking that controls when logic states change and memory that preserves states over time.
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... empirical5
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.
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... memory6
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.
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... fields7
Fields can be discrete or continuous. In the former case field values are only defined at discrete points, the only locations that exist.
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... networks8
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.
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... 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.
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... numbers10
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.
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... 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.
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... computer12
An ideal computer can run forever error free and has access to unlimited storage.
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... 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].
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... 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.'
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...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].
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... problem16
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.
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... universe17
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.
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... 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.
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... 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]
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... 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.
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... 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.
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... panpsychist22
`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]
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... hominization23
`[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].
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... panpsychist24
`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]
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... traditions25
Kurzweil's previous book on artificial intelligence was The Age of Spiritual Machines[17].
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... capacity26
I use `mental capacity' because I think `intelligence' is too narrow in this context.
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... technology27
Biological neural circuits are much slower than electronic equivalents.
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... unaided28
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.
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... 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.
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...\samepage30
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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Mountain Math Software
home consulting videos book QM FAQ contact
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