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2. Tomorrow is a long Time Ago

What is it and why should it matter that I view the world through a metaphor?  What possible effect does it have on the quality of life?  Belief and truth are cousins, not identical twins.  Beliefs, whether false or true, are important to us to the extent they affect our behavior and feelings.  Out of my own view of the world as finite and discrete, with its infinities being processes in time, the metaphor arises of a gigantic sphere as large or as larger than our entire universe.  Upon the surface of the sphere an immortal robot ant wanders randomly forever, eventually retracing its own steps in a path coinciding for a while with a previous path.  The stepping of the ant is reifying one by one the possibilities in the matrix of all things possible.  David Hume believed that when we die we become what we were before we were born.  But what were we before we were born?  Were we not a potentiality waiting to be realized?

Does existence take a discrete and finite but an infinitely repetitive form?  If so, then its apparent infinities must occur as an inconceivably vast plenitude of turnovers, a domain of immortal atheists and bad pennies.  Think of an imaginary stochastic stylus visiting contiguous World States as a random walk on an infinite surface such as that of a sphere.[i] In an eternal process, it would repeat segments of its journey.  The common revulsion toward the Eternal Return misses its vastness, its quality of forever being first and only, its rich variety, its myriad parallel lives of each individual, even its affinity for unsentimental physics.[ii]  However, if the great Newton can insist hypotheses non fingo, neither do I offer theories here but simply point out the speculative nature of human thought on final things. We are, however, simply fishing in a cistern of rain until and unless we can know the final nature of existence, an achievement perhaps beyond human possibility.

Nobody knows what space and time are, nor do we understand the necessity for existence as opposed to absolute nothingness instead.  Our public knowledge of physical things, those which are manifested by controlled independently replicated experiments, only works to various levels of probability, never to certainty.  Our only certain knowledge is never about the world but about ideal abstractions.  We know that 2 + 2 = 4 under a given set of rules.  We know that an English sonnet is written in the iambic pentameter.  We know that a unicorn has one horn.  We do not know with absolute certainty that the external world exists at all or that a candle cannot burn in a vacuum no matter where it may be located in the universe.  We assume so and rate such things as a round earth as being highly probable.  What is the probability the earth is round.  We may rank it as almost 1.0 but quite.  Always absolute certainty about the space-time world eludes us.

Among the numerous incoherent assumptions often present in what passes for deep thinking, look at the following six:  Space is infinitely divisible.  Space is openly infinite in extent (not to be confused with a closed infinity, such as the surface of a sphere).  Point singularities with infinite density and curvature exist.  Time has a unique past and future.  Time is infinitely divisible.  Infinity is an instantaneity rather than a process in time.

I do not wish to say these assumptions about nature are untrue, only that they are incoherent and cannot be clearly defined without self-contradiction.  Among other such things, they lead to such statements as the one often quoted from Pascal, “Nature is an infinite sphere in which the center is everywhere and the circumference is nowhere.”  We have elite mathematicians and scientists who claim that Pascal’s statement makes sense to them.  None of them give a lucid explanation of what it is they clearly understand.  It is simply an incoherent extrapolation, which, if true about nature, is not understood clearly by anyone, any more than extrapolations to infinite density or infinite curvature are lucid concepts.

If we treat incoherent assumptions as artifacts, useful perhaps as analytic conveniences but not physical characteristics, we come up with a different world view of the system.  Instead of being infinitely divisible, space is grainy and discrete with smallest units, such as suggested by the Planck scale.  Space is closed upon itself and has an ultimate outer size depending upon curvature.  No sphere can expand until its surface is perfectly flat.  Size has inward and outward limits if we are to avoid incoherence of thought.  We suppose as well a limit to density and curvature such that point singularities based on extrapolation into infinities do not exist as physical entities.  Time is round like a ball or some other single-surfaced geometry with past and future containing the same vast set of elements.  Time is discrete as suggested by the Planck scale, and infinity is a process in time, not a number.

In Aristotle, Archimedes, Gauss, Euler, and numerous other perceptive and deep minds, infinity is not a muddled and difficult concept.  It is simply a closed process in time.  Blake’s metaphor of holding infinity in the palm of your hand works either as logic or poetry.  As a common example, the system of natural numbers is infinite.  These are, in the decimal system, a wheel with ten spokes revolving without stopping, not a mysterious number that can be larger or smaller, requiring lesser or greater time than other closed repetitive processes.   We find infinity lucidly and simply in sampling with replacement or in even the simplest computer loops.  Infinity is not a mystery until it is presented as a spatial or temporal flat surface or line that never ends and never returns, e.g., the infinite flat plane or an eternal time line treated as forever and distinct in two opposite direction, the past and the future.

The continuum in its pure form occurs in the mathematical line or surface described as a loci of dimensionless points with an infinite number of points between any two that you care to choose.  For this reason, in a continuous probability distribution, such as the normal curve, it is impossible to be at a point on the axis.  It is impossible in any continuum, for example the real number axis.  Easily intuited is the statement that we can never be exactly at pi because it has unending decimal extension without terminating in a string of zeros or without becoming fixed in repeated sequences.   All irrational numbers are like pi in that we cannot be exactly at them.  Harder to see is that rational numbers, which have decimal equivalents that terminate in a tail of zeros or become periodic in their decimal extension, cannot be precisely located on the real number axis in terms of pure continuums.  The rational number 2 cannot be precisely located on the axis because its decimal equivalent is 2.0 … .  In order to locate the number 2 precisely we must be able to name the next of previous number on the real number axis.  To do this would require that we convert the axis to discrete rather than continuous by truncating the tail of zeros, ending at say the millionth decimal point, for example.  Then we can state the adjacent numbers to a million places.  Or if we truncate at 2.0, then the adjacent numbers are 1.9 and 2.1.  However, on a continuum an infinite number of real numbers exist between 2.0 and either of the next and previous.  The conclusion is simply that the real number axis cannot be treated as a pure continuum if we wish to locate precisely a real number.  Yet no agreement concerning the real axis rule for defining rigorously the location of numbers exists.  The great mathematician Hermann Weyl has stated in his book on the continuum, “The notion that an infinite set is a ‘gathering’ brought together by many arbitrary individual acts of selection, assembled and then surveyed as a whole by consciousness is nonsensical; ‘inexhaustibility’ is essential to the infinite.”[iii]  In the forward to the same book, John Archibald Wheeler characterized Weyl’s position as “… we can adopt the continuum and give up absolute logical rigor, or adopt rigor and give up the continuum.”[iv]

Physical nature exists only by change.  Nothing in space-time exists except by constant discrete movements or jumps.  A timeless world cannot exist at all, for it would be frozen without motion or change of any kind.  Even an observation or any thought involves motion in the brain.  In an episode of Star Trek a character is allowed to exist outside of time.  In such a case, he could move and think and speak, all of which are timeful events involving changes.  The treatment of time as beginning only with the big bang, associates time not simply with change but with the arrow of time reflecting the second law of thermodynamics, but not with reversible time, such as photographed vibrations that can be run backward on a screen without our noticing the difference, as in a movie of a pendulum swinging.  When we watch a motion picture in a theater, the big screen is blank much of the time, for it takes time for one still picture to jump to the next.  In that interval, the movie does not exist, yet we see it as continuous.  Its “motion” is a series of discrete jumps from one configuration to the next.  So it is with the existence of our world.       

It would be unseeming for a mathematical layman to make Draconian pronouncements on the transfinite world of Cantor.  However, one may rely on Leopold Kroneker, Henri Poincare, Hermann Weyl, Ludwig Wittgenstein and other master minds to do it for me.  Their objections were never to its counter intuitive conclusions but on the obvious questionable step of treating infinity as a number instead of a process in time.  Their criticisms have never been answered but simply dealt with by throwing the baby out with the bath or else by simply ignoring serious criticisms of a prize so rich as to be called a paradise and defended as such.  Poincaré called transfinite numbers a mathematical disease.  Wittgenstein denounced set theory as nonsensical and wrong.  It is the case at times that mathematics advances into new territory by glossing over objections to the advance, for example those raised by Bishop Berkeley concerning infinitesimals.  The calculus set the infinitesimal to zero in solving Zeno’s paradoxes, such as the Stadium paradox of how one can ever travel from A to B.  The problem of never getting to a destination because first one must go half way and half way again, ad infinitum, becomes no problem at all if space is discrete like Pete Rose’s hits on the way to overtaking Ty Cobb’s record of career hits in the major leagues.  The logical flaw in calculus could be ignored since it got the right answers.  The subsequent work on limits and non-standard analysis simply redefined the problem in order to justify the success of calculus at getting the correct answers, but Berkeley remains relevant to physical questions in his opinions concerning the “ghosts of departed numbers.”

Zeno was from Elea and was defending the Eleatic school of philosophy that viewed the world as static, as being only one complete and constant thing, the appearance of change being illusory.  This is close to Gödel’s belief, shared apparently by Einstein, in a block universe with elements existing all at once so that time is an illusion.[v]  The Eleatic outlook is similar to my own definition of reality as being the humungous but finite set of all possible variations of the world.  How we realize this moment (the “present”) out of the matrix of possibilities in space-time nobody understands since nobody knows what space and time are. 

Where did our universe come from?  Was it a fluctuation to extreme density in a super space to which it is in the process of expanding back into an average density of that space?  Are universes simply fluctuations of density in a primary space so that the system of existence is an infinite process of forming worlds without end?  Or is our universe the one and only, cycling through a process of expanse and collapse?  What do our beliefs matter regarding these questions?

If the beliefs we adopt toward such things do not matter to our behavior, then they are at best the enjoyment of intellectual games.  Of some behavioral importance, however, questions of purpose arise.  The interpretations of reality given here seem pointless, just a system that has no awareness of itself or any particular goals.  It just is.  We do not seem to find in purely natural processes in space-time any Holy Grail.  What about the question of God?  Doubtless many versions of Deity can be grafted on to stochastic or mechanistic interpretations of the world.  As carefully rationalist and scientific a person as the brilliant Martin Gardner believes in a personal God and a life of prayer.  Many persons of high intelligent have Deistic notions that affect their behavior but I have never made sense of the oblong blurs that pass for Deity.  Perhaps the term “God” is without intelligible meaning as the Vienna positivist claimed.  However, it a word that refers to feelings of the oceanic kind of mysticism and to consequences that range from the angelic to the diabolic.

As for this Ten Volume Suicide Note that is 120 essays long, this one essay simply admits that my metaphor on final things may be no comfort at all, but would suggest that the robot immortal ant will walk in these tracks again, first and forever, perhaps a quadrillion universes from now, which for the dead is an eye-blink long.  If it sounds strange that we live a myriad of lives forever, bear in mind that whatever the truth is about final things it will seem very, very strange and unlikely to us.  My world view is that of a mysterian, by which I mean that the remark of Haldane has it right:  The world is not only stranger than we think it is, but stranger than we can think.  Let us tend our garden.

          



[i] Or a Wiener process, i.e., a continuous-time random walk with a jump at every instant, wandering through a humongous sphere of world-states, not to be confused with a random walk on an infinite flat lattice on which the probability of return is a Polya random walk constant, 0.3405373296… .   The constant refers only to a first return.  Multiple returns are soon out of the question.  Also the Polya constants will be smaller for increased dimensions.  The Polya constant is unity for one or two dimensions. 

 

[ii] Martin Gardner made what I consider a doubtful remark concerning the so-called “block universe” in which everything exists at once and time is an illusion (as found in Kant and Gödel).  Gardner says, “In such a cosmos, free will obviously is also an illusion, and it is impossible for anyone to alter either the future or the past.” [Gardner, Martin, Are Universes Thicker than Blackberries? W. W. Norton & Co., New York, 2003, p 69.]  This would be true only of certain block models with only one possibility, viz., a single determinate reification process.  If we think of existence as a vast sphere of discrete World States and the mysterious shift from one World State to the next as stochastic and containing personal time-lines for many life variations, then free will flourishes through choice of direction.  The individual has many time lines.  The eternal return in a block universe can be modeled in far richer ways than its critics consider.  A more acceptable outcome, at least in the little English village of Ockham in Surrey County, would be to allow a random walk through the world states rather than to invent a single Laplacian stroll in which being born has a rigidly determined butterfly effect that has led inevitably, at just this time and place, to my typing the following period.

 

[iii] Weyl, Hermann, The Continuum, p. 23, Dover, New York, 1994, translation from German to English by Pollard & Bowl.

 

[iv] Ibid. p. xii.

[v] An elaboration of this statement about Gödel and Einstein is treated as the leit motif of Palle Yourgrau’s World Without Time, Basic Books, New York, 2005