The Limits of Science to know Reality without a Thing-in-Itself

Mathematics and Ultimate Reality

By The Numbers
“There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.” ~Israel Gelfand

The far-ranging implications of mathematics being a form of synthetic intuition is important, since mathematics can both “understand” the workings of much in the phenomenal world by its rigid measure, and also, paradoxically, fashion unseen worlds by its malleable creativity. This, in a larger picture, highlights the dilemma of miraculous intuition: It is a tool that invents* as much as it describes. Mathematics nevertheless drops a toe in the noumenal pond, yet by demonstration of its infinite inventiveness it captures more spirit than substance. The truth is out there, as they say, only, noumenally speaking, it is not a there there.

Yet an agreeable corollary exists between mathematics, physics and chemistry. In biology, Mandelbrot fractals introduced a new geometry for describing the irregular features found in Nature, as well as a model for how a simple branching structure is the most efficient way to build complexity into natural structures. However, far from the hope of many, the artful efficiency of fractal mathematics only supplies a logic for the design of natural machines, not the ghost that animates them. That order exists in the Natural World is clear enough, but the question is whether numbers cause this order or only document it.

Consequently, theoretical algorithms are advanced to link biology to the numerical rationale of physics and chemistry. An algorithmic world is one where there are no free variables—no free choices. Even the slightest deviation in a pattern sets off another pattern that, in itself, is mathematically determined. No matter where or when the deviation occurs, it must always be water seeking its own level. Because of this determinacy nothing much above the subatomic level can be called truly random.

And yet, where are these complex algorithms to be discovered? What would be their instigating prime mover? And why should they favor hierarchical complexity over chaos?

We assume algorithms in Nature are organic and adaptable, yet what is to prevent a world of random sequences from becoming a hopeless Gordian knot residing motionless at the bottom of an abyss? Somehow, someway, it is reasoned, the fates conspire to keep threads from crossing in apocalyptic ways. Since we not only will our insights and actions into being but also order and rank them, we insert on the back end what cannot be shown to logically or mathematically proceed from the front end. As we discussed in the previous section, choices are made—not by mathematical determinacy but by a choosing will. This is easily demonstrated if we think about the complex algorithms we actually employ in technology: They constantly require modification as more information is gathered and facts on the ground change. Algorithms execute themselves where they are directed—they cannot wholesale change these tasks where a brand new thinking is required.

More curiously to the human animal, a machine may be able to make decisions based on the criteria of an input, but where a decision is needed simply to push things along—even if it is completely arbitrary—, how would the machine understand this irrational sequence and when to implement it? One could perhaps create a mathematical algorithms to occasionally insert a random command to mix things up, but such discontinuity for continuity’s sake is not truly random where, in the real world, decision makers intuit when to sweat the details and when to skip ahead. Grant it, a computer skipping ahead at random intervals may consistently achieve a similar result as the computer programmer skipping ahead, but it will not come from the same place: only the computer programmer can have a “gut instinct”. For technophiles who see this as a distinction without a difference, they miss the point of the very thing they hope to accomplish: through artificial algorithms, they achieve the critical mass of true consciousness in all its aspects that is not in any way artificial or contrived.

(*According to studies, when the brain is subjected to mild electrical current and the left hemisphere functions are suppressed, the right brain is briefly freed from the restrictions of logical and habitual thinking. This suppression not only allows the individual to see the world in a new light, but it stimulates creativity and improves mathematical ability. Contrary to analytical philosophers’ assertions, mathematics [at least at the level of intuition and not rote arithmetic] is not derived from logic [i.e., the left brain] but shares a kinship with the creative arts in the right hemisphere of the brain.)

The Aesthetics of Mathematics
The attraction to mathematics and algorithms do not arise from their difficulties, but from their perceived (or presumed) symmetry as ideas. Their hypnotic ability to generate patterns is based in part on the notion of “pleasing”. In other words, predictive capability and usefulness aside, the appeal in numbers, like everything else, is aesthetic.

Indeed, even more than explaining something, the appeal of an algorithmic world resides in its value as an end and not merely as a means, for its value and authority must be an extension of its beautiful design, which must be in some sense tautological to the intellect. Even though algorithms in their simplest presentations create patterns of aesthetic order, the beauty in patterns and their corresponding equations, as well as the value we attach to this symmetry, can only be a description of something else which infuses patterns and equations with notions of beauty and value.* These qualities cannot be a product of pure analytical deduction. Any judgment about value in equations as things-in-themselves would not be formulated or addressed in the equation itself. This non-functional, value-added judgment requires an admirer (observer).

(*It has been noted how artists and mathematicians approach the world similarly, and differ only in language. One is reminded of symbiotic relationship between Roger Penrose and M.C. Escher, where each inspired the other in the creation of impossible objects. To underscore the aesthetic dimension, this also explains why beauty, as a quality, would be a priority for both mathematicians and artists.)

Irrational Numbers and The Appearance of Symmetry
“Herein lies Shakespeare’s rub: The fact that a circle, the most symmetrical of things, should have the ratio of its circumference divided by its diameter come out to equal a real and irrational number* (pi = 3.14) means that no circle can be rounded off in perfect symmetry. The .14, with its infinity of non-recursive decimals places, is more than a clumsy fraction—it is the fabled eight hundred pound gorilla hiding in the untidy margin of our ruled books. ” ~from Omar's letter, Chapter Four of Icarus Transfigured

Among the many interesting real and irrational numbers is the Golden Ratio (1.6180339887...), which is found in biological structures and mathematics. It is also employed reverentially in the arts, architecture, and mystic religion. In like manner, the number “2” shares the appearance of symmetry with the circle and the Golden Ratio, yet it also leaves a messy fraction when we consider its square root.

It is ironic that we try to use the logic to round off experience with whole numbers, yet the closure we suppose to exist in experience is only undone by mathematics. We can conclude that the appearance of symmetry, like space and time, exist more in the mind as a transcendental idea that what it purports to describe out there in phenomenal experience.

The Mystery of Prime Numbers
“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate.” ~Leonhard Euler

Prime numbers are like the hard places the blind man feels in a dark room. There is curious regularity to their features, though the toucher is clueless to understand in any ontological sense what he cannot see, or how it connects to him.

Prime numbers, which are divisible only by one and themselves, have many irrational yet pattern-generating properties. In one scene of my book, I use reciprocals of the prime 7 (0.142857) written on a napkin. This fraction, when multiplied by numbers between 1 and 6, produces the same sequence of numbers through displacement:

1 X 142857 = 142857

2 X 142857 = 285714

3 X 142857 = 428571

4 X 142857 = 571428

5 X 142857 = 714285

6 X 142857 = 857142

When you reach the multiple 7:

7 X 142857 = 999999

The prime 17 (and other primes) reproduce this feat on a larger scale.

With mathematics we are often presented with beauty through pattern where there is no known meaning, and consequently it is easier to understand through these abstract instances how beauty is meaning, even if one cannot make the intellectual leap from one to the other, or from both to the Noumenon.

(*In geometry, beauty and symmetry are synonymous, and where particle physics is linked to geometrical symmetry, this is offered as perturbative proof to some unseen scientifically reductive design. We find this with Garrett Lisi’s use of E8 Lie Group algebra to find a convergence between quantum particles and Einstein’s gravity by way of geometry. However, this variant model on Superstring Theory predicts particles that have yet to materialize, including the graviton. Remember: The key to any Unified Field Theory must be beautiful symmetry, for when all is said and done, the artist will always trump the scientist, even within science.)

The Alpha Number
“If you square the charge of the electron and then divide it by the speed of light times Planck’s constant, all the dimensions [mass, time and distance] cancel out, yielding a so-called ‘pure number’—alpha, which is just slightly over 1/137. But why is it not precisely 1/137 or some other value entirely?” ~from Oglethrope.edu)

“If alpha [the fine structure constant] were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy.” ~Max Born, Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung

“It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the ‘hand of God’ wrote that number, and ‘we don't know how He pushed his pencil.’” ~Richard Feynman, The Strange Theory of Light and Matter)

I have already mentioned the fine structure constant in relation to Jung, Pauli, and synchronicity, but this prime number also has numeric value in Hebraic phrases such as “The Truth of God”, “Surrounding Brightness”, and “Crucifix”. There is a strange convergence between mathematics and conincidence, where Fate hides itself.

Infinity and Mathematical Hoodwink
“Mathematics also bolsters us in our belief we penetrate to the heart of things; yet again, we are not on site but dealing with a mute sub-contractor, one whose answers come in the form of nods. Bertrand Russell once said, 'Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.' Einstein, putting it another way, remarked, 'As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.' ” ~from Omar's letter, Chapter Thirteen of Icarus Transfigured

According to advocates of Superstring Theory, this point signifies an actual physical infinity. In fact, it can signify all-possible physical infinities in a tenth dimension. Yet like infinite sets in mathematics, this is something of a trick, or as Kant would say, “a mere idea”.

An infinity in mathematics is like a black box where, because you can put things into it and take things out of it, you are lulled into believing you are dealing with things and not simple concepts. For example, if you take the number “17” out of a set containing infinite numbers, then it becomes an infinity distinct from one where 17 is present. However, this distinction hardly constitutes anything distinctly real as a phenomenon, as in the case where one differentiates between a basket containing sixteen apples and one containing seventeen apples. With infinite sets, you can never say what exactly the set is in non-abstract terms or demonstrate concretely what it contains beyond a symbolic term like “infinity”.* Arising from this confusion of concepts for things-in-themselves, current theoretical science makes little distinction between ideas that can be represented in equations and things that can be demonstrated to exist by empirical scientific criteria.

(*With a Cantor set, we infinitely exclude the middle third from a line segment, and in the manner of Zeno’s Paradox, we find we have an infinite number of line segments. With Mandelbrot sets, we discover we cannot measure the length of a fractal, but we can measure its roughness at any given point. Yet due to its infinite length, an infinite amount of roughness can be applied. Everywhere we look in Nature, we see fractal math. We can even say it is Nature’s will to infinity, which we are doomed to graph finitely.)

Mathematics versus God
Some say mathematics is God. But does this symbology represent literal ontology or only a substitution? To call mathematics God risks being a variation on the God-is-Nature argument, where the pantheism is only meant to glorify Nature, not God. God becomes only a value-added adjective where the reason for its inclusion is rarely articulated.

Again, because we can equate mathematics with beauty, we can equate mathematics with (V)alue. Yet Value in this in-Itself sense transcends uses, and even symbolic representation. Likewise, value-in-itself, found in the aesthetic appeal of numerical language and logic, is—again—not analytically reductive. The ultimate meaningful meaning of mathematics is not only nonverbal, but also non-numerical.

 

The Limits of Theoretical Science

The Lost Value in Paradox
Paradoxes, if taken seriously, would end any academic career vested in the proposition that reality in itself is finite and wholly rational. However, some in science openly embrace paradox, not as a check on what they can know with certainty in a finite world but as what they can apply to a finite world without check. Superstring guru, Brian Greene, speculates on the “mathematical” possibility of their being a near infinite number of Brian Greene clones on distant worlds in distant universes engaging in exactly the same behavior as him at precisely the same time. Truly, black holes are passé, and dark energy is so last year. So to overcome the collective yawn resulting from wow fatigue, Mr. Greene crunches celestial bottles of beer on the wall in the name of science, and with barely a syllable of skepticism from the science-adorning media. It is because we do not respect the paradoxes of space, time, and infinity, and because we are not humbled by the limitations they place upon our understanding, that we are routinely overwhelmed by fantastical schemes that are increasingly underwhelming to our soul.

Many Worlds and Many Questions: Physics Jumps the Shark
Wave function collapse, where indeterminate light waves become specific particles on observation, creates objective reality by means of a subjective experience. This bait-and-switch may well be empirical, but it is hardly satisfactory by any scientifically reductive definition of reality. Moreover, the experimenter affects the experiment, because where instruments constructed to observe atoms are also made of atoms, they too are drawn into the ghost world of indeterminacy. The price of nailing down a particle is to displace its unreality in an infinite chain of observers observing observers.

To get around this problem of unreality, some physicists set Occam’s razor on its head by keeping assumptions about the Natural World in place and righting the logic: i.e., making atoms of ghosts instead of ghosts of atoms. The Many World Interpretation proposes that every quantum particle exists in all-possible states at all-possible times, and when it is forced into a single state by an act of observation, those other possible states split off into alternative worlds in different dimensions of space. For example, Schrodinger’s famed cat, with its fate tied to the fate of an altered subatomic particle, never dies in an act of observation, it simply continues on, infinitely and immortally, in another universe where the subatomic particle is still undetermined.

As we have discussed, where infinity is concerned, the concept of “moments” is mathematically meaningless since each moment is an infinity in itself. Yet Many Worlds assumes, for the sake of multiplying universes, that infinity can be pared down to a linear construction like a road, where it only moves in one direction with one idea at a time. Or it is analogized to a branching tree, so as to placate our needs for logical order and forward-moving narrative. But what exactly is being copied?

Moreover, in what sense can there be “choices” in moments anymore than there are moments? Or “non-choices” in moments, for that matter? A lack of seriousness is underscored in the idea among some Many Worlders that when we choose between two choices, the choice not taken by us is taken by our doppelganger in an alternate universe. First, there is the assumption we choose between distinct options in distinct events at distinct times. At momentous occasions in our life, we may indeed be able to characterize our dilemma as choosing between two things, but it is perhaps the gift (or curse) of autism that I routinely game dozens of calculations in such moments—not only of options, but options within options, and options within options within options. Is there a doppelganger created with each twist and turn of my restless mind? And what about the plethora of microorganisms living in my body, whose combined weight (I am told) outweighs the remainder of me? What of the choices they make within me, as to whether to turn up or down, or toward my liver or toward my small intestine? Do microorganisms rise to the level of having fates, making choices, or meriting doppelgangers? And how do you define and differentiate between their distinct choices since microorganisms are constantly moving? For that matter, how do you do the same for me? Does raising my elbow five centimeters as opposed to five and a half centimeters constitute me choosing between two calibrations? Or choosing one calibration from a virtual infinity of calibrations? Is there a distinct elbow in a distinct universe for each of these calibrations across the gradation of microseconds that presumably comprised the minutia of my willful life? (The only thing that can end these fanciful scenarios about other worlds is exhaustion, for infinity, light years beyond our imagination in theory, is never exhausted in practice.)

In truth, we have two vague terms modifying each other, for the notion of “choosing” is no better defined as a distinct thing-in-itself than “moment”. This whole scheme falls apart in the realm of language, for Many Worlders allow the words “choice” and “moment” to dictate the narrow confinement of their thinking. Furthermore, since science attaches little value to the concept of being, being, as an indefinable quantity that prefigures any choice, is subsumed into the game pieces shuffled around on fantastical game boards; little existential gravitas is involved. Many Worlds thus explains nothing about the underlying ontological nature of reality beyond its facile need to build suburbs onto suburbs.

Many Worlds and Many Wills
Will, which underlies all choosing, is, like being, simply assumed extemporaneously in Many Worlds.

Even at the level of bacteria, we have a concept of will—even free will where genetically identical E coil display independent movements in space and time. Should we insist that more than three dimensions of space exist, and these spaces come into being as a byproduct of a willer willing, then we have not only mushrooming spaces proliferating to infinity but also mushrooming wills. Will, given this extraordinary power to create new realities at will, overturns the concept of biological determinism at the global level, since for every action there are an infinity of reactions. The Many Worlder might counter with: as all action is taken in sum, no action is free to not be taken. However, given the destructive power of infinity to make nonsense of any claim, we can answer by saying we never reach that sum in actuality, regardless if it is theoretically determined or not as a question of mathematics.

Moreover (and perhaps unintended) Will becomes a metaphysical necessity since it is the one carry-over ingredient to connect all these worlds. According to Many Worlds, parallel universes cannot physically connect, so what, then, nudges the new copy of the universe into existence if not something from the old copy in the old universe? The Many Worlds Interpretation stresses that minds and observers are not required as they are in the Copenhagen Interpretation, because there is no indeterminacy. Yet how can this be the case since an ontological will is needed to keep choosing and pushing things along? For that matter, where is the first willed action to set off the causal cascade of splitting branches? Is there an original action (first cause) in among the copies?

For Will to function across the board, it must either reside outside any and all universes, or prefigure any and all action. In other words, Will must be noumenal. Wouldn’t it be simpler, then, to combine the concept of infinity with the concept of Will outside the realm of proliferating universes where paradox begets paradox? And while we are at it, can't we just leave the superfluous universes out of it?

Einstein and More Duct Tape
The chief difficulty (or virtue, depending on your viewpoint) of space/time is how the malleability of Einstein’s idea lends itself to a nearly inexhaustible generation of “speculative phenomena” by means of mathematical abstraction.* From black holes in universes to universes in black holes, it is only a matter of time before every conceivable combination of matter, space, and time are presented in an unfalsifiable theory.

With the advent of black holes, a paradox involving The Conservation of Information was created. It is argued all information about a physical phenomena is conserved in its effect, no matter how fragmented it becomes in time. A theoretical black hole, where information goes in but does not come out, violates this fundamental rule. The fix, so devised by theoretical physicist Leonard Susskind, was to say that in addition to being a devouring well, a black hole is also holographic, and all the information about what passes over its event horizon is preserved as a two-dimensional record at its surface. Consequently, nothing that appears in a third dimension of space actually occurs in the place where it is alleged to be, according to Susskind. All the information of the Universe is stored along its “edge” and projected. Of course, issues are raised about the nature of the third dimension of space, and a whole new roll of mathematical duct tape is needed to explain this feature.

Yet another fantastical speculation posed by space/time is the possibility of time travel, where prior to another fantastical speculation—parallel universes—, The Grandfather Paradox prevented a time traveler from going into the past and killing his grandfather, thus preventing the grandson’s eventual birth and, necessarily, his act of time travel murder. A parallel universe, it is contended, would allow another copy of the universe to come into existence and cancel out the paradox of a grandson who was never born. Yet at what point would the new universe be created? At the moment of the murder? Or at the moment of the time travel? (The multiverser would doubtless say both.)

In any case, would two universes differ in only this one regard: a father was never born in one even as a son walks around in another? Since there is a near infinity of causal connections contingent on phenomena being as they are in specific space and time, would not altered universes going forward from the time of the murderous act be subject to incalculable contradictions? Contradictions that could potentially tear the universe to bits and make it impossible it should ever exist? Again, the resourceful multiverser would argue that a new universe would be created at each impasse to get around the contradiction. Yet here we have bizarre logic being employed because a bizarre theory requires it, and only because the bizarreness of ghostly atoms is judged to be more offensive than the bizarreness of multiverses.

More to the heart of this nonsense, if time travel were possible by a technologically advanced civilization, as by commuting through wormholes, then it is clear we are living in one version of our universe where they have decided (to date) not to reveal themselves. This is either a demonstration of great cosmic wisdom in not intervening, or it is merely an example of self-reinforcing delusion when applied to a belief in parallel universes.

Long before we reach these headache-inducing scenarios, the construction of a time machine (again, based on Einstein’s space/time) is needed to generate them. Yet this contrivance is even more improbable than any theoretical scenario to which it gives rise. When one examines the engineering schematics for concepts like time travel or teleportation, they either do not go far beyond crude examples of quantum entanglement, which cannot be enacted on anything much larger than a subatomic particle, or they require an infinite amount of energy that no proposed advanced civilization could harness. Of course, this is all gaping-hole science fiction plot. By the time anyone realizes the improbability of any of this being true by virtue of all that is required to make it remotely probable in real world terms, Einstein’s space/time will have to be abandoned in order to start over with a clean slate and a clear head.

(*With the explosion of exotica arising from Einstein’s space/time, it has been necessary to not only explain fine-tuning in a general sense, but contrive necessity where inhabitants of our universe must be protected from potentially destructive forces such as black holes. Roger Penrose, rising to the challenge, devised a cosmic censorship hypothesis to hide black holes behind an event horizon. Beyond such fixes, it is also necessary to protect the very notions of space and time themselves, and consequently the deterministic notion of causation for science to have any predictive value. Here we see the ultimate futility of Einstein’s logic, where the whole bedrock of science threatens to unravel by its own mischievous design.)