I initially resisted paradoxes in my beloved math. Some try to patch up the traditions
like Russell's invention of theory of types. And Intuitionists have made some
efforts in the right direction but they have not completely challenged the 'simple and
believable', just shored it up a bit - they did not have today's examples of exactly
defined intelligence to guide them. But now, it is clear that the math paradoxes
are validly showing with traditional techniques that the traditional techniques are
invalid - what is a new basis for mathematics and logic besides the unknown,
fickle human brain. See Super-logic(25k)
[the following paragraph is a new insert - Nov. 97]
Some may question 'how can anything make something better than itself'? This may be
motivated by fear or pride in one's self or one's choice of myth. But my answer goes to
basics of the explicit words: 'what is better?', 'what is self?'. I start with some facts:
1. The hand cannot hold a block as tightly as a vise, yet the vise was made by
hands. 2. Genetic mutations in DNA may be caused by arbitrary gamma rays from
space. Some lucky chance genetic changes in a life form may make it better adapted
to its material environment. Man was created by an inferior ape in the jungle by stupid
chance. 3. Computers consistently and quickly evaluate more complex propositional
logic formulas than any human, yet they were made by humans. And idealized human nerve
connections can be modeled by a computer depending on experience and chance.
4. The BIG BANG cosmological start of our universe is simply observable as
'redder galaxies are dimmer'. It is modelable as photons being reddened by coming from
receding sources and its distance spreading them out.. All virtual particles come into
existence as self annihilationable pairs. So the BIG BANG to come from empty space, even
the nonexistent space and time implied by its relativity, may consist of temporarily
unannihilalated pairs of anti-particles(virtual photons). Both externally and
internally our universe as a whole is chaotic nothingness - thus it has no
consistent cause, it needs no consistent cause; the universe is self caused nothingness!
Yet the temporary local existence of some order is required to complete this over
all causeless, nothingness chaos. Here is the potential of humanity and its new species
outgrowth - computers. Well chosen actions complete the using up of temporary local order
necessary for complete nothingness - necessary for the self-causation of our
universe. This 'well chosen action' will usually require almost standard social
values of honest, logical cooperation to maximize its results.
This is the optimal objective of math and science development.
A continuing belief during all this was 'a correctly working computer cannot generate a paradox' - probably based on my long computer programming experience. There are clear, even if traditional, math logic proofs that propositional logic is consistent(preserves tautologies) and decide able (by truth table). This is equivalent to a digital transform - values of the propositional variables are '1' or '0' and the value of a propositional formula is '1' or '0'. The computer transform represents the evaluation of several PL formulas at once. Finite memory, deterministic (properly working and independent) computers implement only finite propositional logic and are thus consistent and decideable. In contrast to the human brain, computers are exactly defined by easily observable, reliable, discrete logic elements. However, any given computer can be completely observed and exactly predicted by a sufficiently larger and faster computer; the larger one only needs to use enough of the same kind of logic elements so it can exactly model the smaller one, sooner.
I had good experience in designing and constructing an IC array processor for
rhythmical 2D art. A coffee group philosophy discussion with CU Prof. Saalbach inspired me
to answer the problem of defining 'not' by just wiring up an IC and saying this is
'not'. And the closest a computer can get to a contradiction like 'p implies not
p' -is an IC inverter's output wired to its input - it just oscillates - the output
is not true and false at the same time - no computer contradiction. This kind of thing is
clarified by 'temporal logic', somewhat. See math in nor- formulas
Evolution omitted a window in the skull for detailed observation of the sequential
mechanism of thought. Besides being of no use in a dimly lit cave, it would be
embarrassing to clearly observe the unreliability and stupidity. So its understandable why
the ancients left time out of their logic. For a demo of how the correct use
of time removes paradox, see Interactive
Russell's Paradox. Also, one can understand how ideas of infinity arise from ignoring
time - 'one can always add 1 to any number and get a larger number' -
if it takes me longer than a lifetime to get the number and if I remember it can I still
do it? - in Paradise?
These ideas were completed with my wrestling Turing's halting problem into JavaScript - all
finite automatons will ultimately cycle their results: Interactive:
Computer cycling: The key confusion of Turing's halting problem is infinite memory -
was it logically necessary as in Turing's machine - an infinite memory would allow a
computer to never repeat - it could just store more 0's before the next 1 in its memory
forever: 010010001... Liar's, Zen's, Russell's, Goedel's and Turing's paradoxes do not
carry over to real computers!
[start 22Apr 98 incert]
Fundamental idea:
If math or logic is to apply to the observed physical world in a given time period, their axioms, basis or implications should be consistent with the observed physical world in the given time period.
I add 'in a given time period' to preclude 'ideas' that assume or imply some 'physical' event at some arbitrarily large time, like 'we can always add 1 to any number and get a larger number'. Since one merely asks for an n that takes longer to generate than P.Maximum size of universe:
The maximum speed of physical objects is well established to be the speed of light in a vacuum. Hence, given a specific time period: P, anything further than light can travel during P from something cannot affect or be affected by that thing. Hence, for any given time period the effective universe is finite.
Maximum energy of the universe:
Again I consider the universe in a given finite time period. By the above the effective universe has a finite size. If it has infinite energy it has, by relativity, infinite mass and so infinite density. There are many problems with this. For one, everything is accelerated to the speed of light to the center of mass, everything gets unboundedly heavy from its increase in energy, its time stops. But we don't observe everything accelerating unboundedly or its radiation getting unboundedly red shifted , so our universe has finite energy for us inside it.
Minimum size of things:
Heisenberg's uncertainty principle is also well established. It states 'the energy of a particle multiplied by its lifetime must be greater than Planks constant to be detected'. Hence, for something to be detectable in a time period P, its energy can not be 'arbitrarily' small since it would take an arbitrarily long time to detect it. So the infinitesimals of traditional calculus are unrealistic. But, even I have found it easy to replace basic calculus ideas with ones of similar or greater usefulness with finite techniques - see derivatives without infinitesimals Thus Zeno's paradox of 'late starting but faster runner cannot catch up because he must catch up half way, first' is not realistic since the size of the catch up places is unboundedly small and will soon take longer to detect than the race to finish. Incidentally, other solutions use reduction in time required to 'catch up half way'. Shows time is a necessary part of logic.[temporal logic, forced on 'proving computers', is a modern admission of need for time as in integral part of logic],
So infinite sets are unrealistic:
Our universe cannot be divided into an infinite number of parts in any finite time period. Because it has finite effective size and its observable parts can not be infinitesimal in a finite time period.
Human brain has a finite number of states at any time, a maximum number of changes in any finite time period:
Biological experimentation demonstrates the 2 state possibilities of nerve stimulation(on or off) and the finite number of nerves in any given brain. And there is a maximum speed of neural transmission of the state changes, and there is a minimal nerve length in any given brain. Also the period of 'on-off' has a lower bound along with an upper bound of the individual's life time, so the total number of states in a lifetime is bounded.
Artificial brains can solve any physical problem:
Modern technology makes it clear that all significant physical processes can be adequately digitized. In particular, all significant physical problems in a given time period can be typed out in a finite number of letters. The same is true for the answers. This is a digital transform. See my simple proof that my JavaScripted -ANN can learn any digital transform, 1 synapse at a time: -ANN.
[end 22Apr 98 incert]
Some sarcastic comments in sci.logic prompted more thinking on 'why has traditional math and logic been so successful? How can a human "understand" a computer's super-logic?'. My answer came from realizing how math in nor-formulas and interactive -ANN can teach humans by experience: we can visually remember each sequence of non-human, mechanical steps, after we observe we can imitate, if we are very careful.`[I plan to show internalsteps in demos]. It's like writing a computer program: one tries to think like the computer, think in complete detail how it interprets its language. One might say 'you can never rise above trivial mechanical steps that way'. But, -ANN can activate synapse formulas by activating their name nerve, can generalize by activating several synapse formulas by one name or accept equivalent ones. Since it can learn, synapse by synapse, any consistent, finite digital transform, it can learn anything consistent, that we can. Any typed general solution to any typed problem can be put into finite-mechanical-electrical, non-human form. Similarly, humans can not be super-logical by imitating themselves as long as significant details of brain operations are unknown or not clearly and simply experienced visually.
Yet, we or computers can never exactly model a complete universe because the model must include ourselves modeling it! But we can still beat out other things that make worse models in the race to use up temporary order sooner.
[Jan99 insert]
More on human or machine logic: If we want our thoughts in our brains or electrons in computers to predict physical events, we first need to discover, in some way, how physical events are predictable. [Most religious 'theory' seems to proudly claim "gods are not slaves of human reason"]
[end Jan99 insert]
[end Jan99 insert]
Infinity is of no use unless it is defined in -ANN lessons. Now, having majored in
real analysis in grad school, I was very familiar with traditional ideas of infinity. So I
started examining interesting cases where infinity was used and easily found it was
either not needed or the results were very suspect(many paradoxes) - see calculus without limits . In particular it's easy to binarily
'differentiate' polynomials without Infinitesimals: instead of making a delta
x less than 1, scale x up: derivatives
without infinitesimals Finally, arguments in comp.ai, sci.logic, etc. prompted efforts
to make a simpler paradox, Russell like, than Goedel's but as this was difficult(hard to
use ideas one does not like or are really impossible to understand to fight themselves
- a mad dog may bite its own tail, but it's dangerous to help it). I now simply
point out 'one paradox should be enough' - human math is sick but too many have had faith
in it for too long to be easily, generally rejected. To ignore the math contradictions
is to use "blind faith" in the worst way! But much traditional human math
may be translated into automaton math. We can still have a bit of Paradise on earth.
Consider, 'one can always get a larger number by adding 1 to any given number'. This has been adored for centuries. But what is doing the adding? Who can 'take' the number anyone may think they are giving? Does 'any given number' include those that would take a computer 100 years to print out? What can always verify that the result is larger? Nothing, but maybe a god, certainly not a computer since it has reality limits. It can not correctly add 1 to the largest number it can remember. Why would one try to use something in the real world that is of use only by God in Paradise? Peano arithmetic is only for the gods!
If one could really prove something requiring 'infinity', using a finite automaton,
then that aspect of 'infinity' has just been reduced in this way to the finite!
The following obvious theorem makes this rigorous. And it rigorously proves all
responses of human thought is equivalent to one simple finite formula of only
'or', 'not', 'true' and 'false' elements. Also, a good packing and unpacking program
applied to these inputs and outputs as a whole will find an efficient reduced form of the
information that requires fewer nor-gates.
The axiom-proof-theorem system in math can be viewed as a knowledge packing system. But in
keeping with super rigor, an axiom-proof-theorem system must be defined for a completely
observable, finite intelligence. Also axiom-proof-theorem must be interpreted in a digital
transform way: The 'theorem' may be a transform(nor-formula) constructed by 'proof' from
simpler 'axiom' transforms. A simple unpacking problem would be to start with axioms and
theorem and generate proofs. But the most ambitious math unpacking system would be an
automatic proof system that would fill a given memory with those theorems from the axioms
that let the intelligence maximize entropy sooner in a given environment.
Any higher logic or infinity process that could do better is equivalent to another
nor-formula and could be just a nor-gate improvement to the pack process.
The following theorem shows that nothing more complex than the 'nors'
(equivalent to a 'read only memory') is required in a device(machine or brain) for any
consistent set of digital responses to finite digital inputs. And the eye nerve inputs
and muscle outputs of a human brain are equivalent to digital numbers.
This means any transformation of digital input to digital output, any 'black box' or
fixed(pre-learned) response system that cannot be duplicated with a finite nor-formula is
inconsistent(non-repetitive)! Digital transform=nor-formula (10k)
E-mail author: R Massey on 'blind faith math'
Mar98.
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