The Unknowable Thing-in-Itself

“Men are probably nearer the central truth in their superstitions than in their science.” ~Henry David Thoreau

Hume: The Destroyer of Reality
Certain factions of Eighteenth Century Enlightenment were committed to the idea the world was wholly phenomenal in nature, and we can have certain knowledge of it without appealing to innate mental concepts or a transcendental God. At the time, many philosophers believed the mind was a blank slate upon which experience wrote: matter (the external world) shaped mind. Bishop Berkeley drove the first nail into the coffin of this enterprise by arguing the mind, as a demonstrated ontological fact, could have no certain knowledge of anything beyond its own thoughts. David Hume drove the last nail in when, doing Berkeley one better, he extended the argument to say what the mind knew was hardly certain itself.

Hume’s great insight was this: though phenomenon A and phenomenon B are assumed to exist by observation, the necessary connection between them in a cause and effect relationship cannot be shown to be a phenomenon by observation. In other words, unlike object A and object B, necessary connection is not itself an object C for inspection: something we can touch with our fingers. When object A strikes object B and object B rolling away is judged to be the resulting effect of object A and object B being in close proximity, we are only assuming this as a mental exercise. We see only two objects on the table—nothing more. That rolling away should follow striking is only an idea in our brains: a habitual assumption we insert between the objects that is not a property of the objects themselves.

Hume had taken empiricism to its logical limits, and using only logic, he had left nothing standing. The curtain came crashing down on the Enlightenment.

Immanuel Kant, awakened from his “dogmatic slumber,” and wanting to rescue the foundations of knowledge for science, took up the challenge posed by Hume’s acid test empiricism; yet what he proposed not so much countered Hume as clarified the boundaries between what can be known and what cannot be known about our causal world.

Immanuel Kant

How Do We Know What We Know?
For Hume, there were only two types of knowledge: a priori and a posteriori. A priori knowledge means “prior to experience,” which translates as residing in the mind. A posteriori knowledge means “after experience,” A priori knowledge is comprised of self-evident axioms that do not require experience to form a judgment on their truthfulness, as with mathematical axioms: 2 + 3 = 5, for example. A posteriori truths, by comparison, require experience to form a judgment: Some birds fly south for the winter. Whereas the first type of knowledge is true by powers of deduction alone, the second type of knowledge, being rooted to mere sensation in experience by Hume’s estimation, may lead one to a serviceable assumption, but there is nothing certain or necessary in this assumption: Just because some birds fly south this winter is no predictor about any bird flying south next winter.

For the sake of being unbendingly consistent, Hume is less than thorough in his rush to a conclusion, which Kant demonstrates by adding two more categories to our knowledge: namely, analytic and synthetic knowledge.

With analytic knowledge, the predicate (answer) is always in the subject (question), as with the proposition: All bachelors are unmarried (bachelor means unmarried). Analytic propositions are categorically true as self-naming definitions, which for Kant included words that renamed concepts but not arithmetic, as numbers and their functions are distinct ideas: “5” is not renamed in “2 + 3 ”, as “2”, “+”, “3”, and “=” differ as concepts.

With synthetic knowledge, the predicate is not in the subject: All bachelors are unhappy (bachelor does not means unhappy). If the predicate is not in the subject, there are only two places it can be sought: either in the mind or in experience.

As analytic determinations are self-naming and require no experiential component, Kant regarded analytic a posteriori knowledge to be self-contradictory and undeserving of further investigation. Beyond this, analytic a priori knowledge (knowledge that is true by definition) changes nothing about the nature of self-evident axioms, and synthetic a posteriori knowledge (knowledge that is true by experience) changes nothing about the nature of firsthand matters-of-fact. However, positing the existence of synthetic a priori knowledge (knowledge that exists in the mind prior to and independent of experience) does raise issues about the nature of understanding itself. Before Kant, it was presumed the form of our understanding was in the substance of our understanding. Yet how is this demonstrated by mere sensation since, as Hume contends, there can never be anything other than impression generated by mere sensation in our minds?

To make an inference from experience about a future event, we must first admit the answer is not in the question, as the answer does not presently exist: i.e., it can only be observed in the future. Since this entails time, and space, to establish a cause, we are assuming not only causation in making our inference but also space and time. If we extend Hume’s argument, these sensibilities must be fictions, too. However, though one can entertain the notion there is no necessity in causation, this feat is more difficult to pull off with space and time, since we are constantly making calculations and judgments about them that prefigure any active act of assuming their existence. No. There must be something real about space and time, and causation as well—if not as things-in-themselves that have the reality of objects, then as preconditions to every experience where objects are involved.

Know It When You See It
We therefore bring synthetic a priori categories for thinking to everything we encounter, and these blueprints* necessarily prefigure the operations of the mind in order to find dogmatic root in the mind. In other words, these ideas are transcendental, for originating in the mind, they give sensate reality the appearance of objectivity without having the quality of object-ness themselves. We apprehend space and time as having reality in absence of having physical proof for their existence. In Kantian terms they are only ideas. Thus, everything existing in experience requires something beyond experience to possess the character of being real, and from this unity in experience all knowledge about experience flows.

With synthetic a priori knowledge, Kant linked space to geometry, time (succession) to arithmetic, and causation to the physical sciences. In each case he argued the given area of knowledge required a form of synthetic intuition that must prefigure all-possible experience and all-possible knowledge. Hence, concepts like necessary connection were not habitual assumptions but in fact wired into the mind. More than that, not only was causation wired in the mind, but as space and time were as necessary as causation, they too were wired in the mind. Space is our outer intuition and time is our inner intuition, and both are required to make causation, and the world causation reveals, intelligible. Inescapably, Kant concluded that, outside our mind’s structured sensibility, space, time, and causation can have no reality-in-themselves. Though space and time can be said to have reality within phenomenal experience, this is only granted as a precondition of the mind. This precondition is therefore transcendental and idea and not real in the physical sense.

This is demonstrated by the nature of infinity, which, along with finitude, we ascribe as characteristics to space and time. Yet infinity can only prefigure space and time in sensibility—it cannot be shown to exist in space and time through sensibility. In other words: How can space and time go on without end? How can space and time have an end? How can reality have a first cause? These questions are nonsensical because they highlight the divide between what can exist in the mind as ideas (space, time, causation, and infinity) yet can never be show to exist in experience as phenomenal objects. Phenomenon C, to return to our original Humean illustration, is indeed not an object for inspection: it exists only in the mind.

“The difficulty, as Kant saw it, is in assuming infinities have actual reality in our received world. Infinities have actuality in mathematics, as with sets and whole numbers, but they do not have sensate reality. One could potentially count forever, or infinitely half distances, as with Zeno’s Paradox, but the tasks could never be completed in actuality. If something has no beginning, middle, or end, it has no boundaries; and if it has no boundaries, it can have no identity in the world of things. In short, a box cannot both have actual reality and have no sides. Kant concluded that what we suppose to be given to us in sense perception (space, time, and causation) exist only in the mind as modes of sensibility.

We cannot apply these concepts, or understand the world within these concepts, without first having a presuppositionless context in which to make them intelligible: namely, the ideation of infinity. This presuppositionless context is transcendentally idea in origin and not physical, since it cannot be deduced as a thing-into-itself from either logic (non-paradoxical as a concept) or from direct experience (non-relational as a thing).” ~from Omar's letter, Chapter Six of An Aversion to Ladders

(*Among those who agree with Kant is Noam Chomsky, who argues, through his concept of universal grammar, the mechanics of language are built into the mind.)

The Unknowable Thing-in-Itself
On first pass, it would appear Kant is siding with the rationalists by saying mind reveals the world of matter, and though ideas in the mind must prefigure matter in experience, ideas cannot be true simply because of their location. They must, as we have said, correspond to something real outside the mind. This is where the noumenal realm comes in to validate our empirical experience of phenomena in the world. Mind is only the imposition of a state upon something that is, in itself, identical with phenomenon, yet as a tradeoff cannot be known by itself as mind: That is, mind is the eye that sees everything but itself, and in absence of this eye’s point of view, mind becomes the everything it supposes to be something other than itself, which, of course, cannot be pictured.

Because of this state of affairs, Kant concluded that the world of appearances, as delivered to us by way of synthetic a priori knowledge, cannot be the same thing as reality as it exists in itself, since our mind cannot think outside its spatial/temporal/causal box of understanding to comprehend total reality. He called this in-essence state the noumenon, or unknowable thing-in-itself.

Mind/Matter Duality in the Kantian/Schopenhauerean System: Phenomenal reality, as a construct, is perceived as something extending outside the mind, while simultaneously possessing duration inside mind. Time, providing inner cohesion to this construction, as well as the very notion of the mind that perceives, is the linchpin, and with its removal, the perceived duality between mind and matter evaporates. With duality removed, our whole concept of phenomenal reality ceases to exist, leaving us with reality as it exists in itself, which cannot be envisioned as an idea.

Kant’s Legacy of World-Strangeness
The Enlightenment strove to remove the transcendental dimension from the debate, and Kant, by paradoxical means of giving knowledge certainty by denying it objectivity, had shown that without a transcendental format in the mind that prefigures experience, experience, and any debate arising within it, could not occur.

In wanting to blend a priori rationalism and a posteriori empiricism, and consequently resolving the contradictions posed by each, the German philosopher from Königsberg barely comprehended what he created. He had slyly settled the debate between Newton and Leibnitz. Where the former had argued space and time were absolute things-in-themselves and the latter had argued they had no reality at all, Kant split the difference by granting absoluteness to space and time only as realities of the mind, not of matter. Yet he feared the interpretation of his philosophy would align him more with the radical subjectivism of Bishop Berkeley (reality is all mind) than with the soberer account of reality offered up by John Locke (reality is all matter). As a disciple of logic, he had little faith in where his logic had led him, and so retreated from the counter-intuitive implications of his noumenal aspect to reality.

Arthur Schopenhauer

Schopenhauer never forgave him for this, and it was left to him to save Kant from Kant, and Kant from his critics. In not shrinking from the bold vision, he finished the work his predecessor had started; and in so doing linked Kant to Eastern thought, provided a more nuanced view of synthetic intuition that fleshed out the arts and humanities, and strengthened Kant’s precepts through his own program that made their combined insights relevant in subsequent discoveries in mathematics and science. (6/19/10)

Giving Hume His Due
However, before totally discarding Hume, he is given a last word:

Kant had striven to save truth for science by erecting a wall between the phenomenal and noumenal realms, yet this wall, by the simple prospect of its existence, did not give synthetic knowledge any greater degree of certainty. The value in Kant’s insight lay in his determination that understanding exists first in the mind and then in experience. This is not a stamp of authority. It is a demonstration of the elastic nature inherent in intuition for constructing its own understanding.

Hume’s skepticism cannot be completely eradicated as a result, only faulted for failing to give a full account of the intuition that led to it. (6/19/10)

A Whole Greater Than The Sum Of Its Parts
“Analysis is the synthesis of the whole which it divides, and synthesis the analysis of the whole which it constructs ” ~F. H. Bradley

For Kant, the thing-in-itself was unknowable by our synthetic understanding in both a priori and a posteriori knowledge, though it is fair to say the inscrutability of infinity, as something that can be constructed in thought but not demonstrated to exist in experience, is a clue to the true ontological nature of reality. Infinity, as a metaphorical device, is not simply an expression of incalculable limitlessness, but it is symbolic of the nature of all irreconcilable paradox in logic and experience.

The logical empiricists, from Frege to Russell, rejected Kant’s concept of synthetic a priori knowledge. They were positivistic, believing rational assertions can have the weight of empirical evidence because they can be proven true by either mathematical or scientific proof. This belief notwithstanding, their analysis gave rise to paradox that—far from refuting Kant—added credence to his view.

Take, for example, Russell’s classification of paradoxes on how to define a container in relation to its contents: A set is something that contains numbers, yet as a defining rule to determine its contents, the set cannot be counted as part of its own contents. If it were indistinguishable from its contents, then there would be no boundary to define anything within a boundary. This quandary arises because of the limits of analytical logic. Sets as distinct ideas, or numbers as distinct ideas within a set, are transcendental. This means they are synthetic in our intellect and not matters of predicate-in-subject tautology.

By insisting arithmetic relied on analytic and not synthetic proof, the logical empiricists were committed to a project that eventually ran aground with the arrival of Gödel’s Incompleteness Theorem, where it was shown that a mathematical system of any complexity generates proofs that can neither be proved nor disproved within the system. This theorem went some ways towards vindicating Kant’s original view that arithmetic sprang from synthetic intuition.

The blindside prevalent in rational mathematics and logic can be found in the empirical sciences as well, though here Kant is less rejected and more ignored. (My following collection of entries will examine the empirical sciences in detail.)

Addressing our larger metaphysical dilemma, a whole is greater than the sum of its parts in a way that is impossible to quantify, which makes it directly unknowable as a quantity. Yet this is not the end of understanding but the beginning of an understanding that has little to do with quantity and everything to do with quality. As one set of questions leave off in the mind another commences in what can best be described as our differently-abled faculties.

Kant himself believed in God, yet saw our mind’s epistemological framework for understanding reality through transcendental ideation as lending no evidence to any argument for God’s Existence. It could not be argued God had infinite spatial or temporal dimensionality since space and time could not apply to anything other than our received world of experience. Kant, anticipating Kierkegaard, believed God could have no foundation in reason since reason applied only to matters of empirical inquiry. God required faith.

This is not to say one cannot make reasonable arguments for God’s Existence by novel applications of reason, where what reason cannot uncover is reason enough to allow for the possibility of God. This is the start of my philosophical inquiry. (6/19/10)

Origins of My Philosophy
The German Idealists (namely, Hegel, Fichte, and Schelling) rejected Kant’s idea of a noumenal realm, and with it its capacity to underwrite an empirical view of reality. They preserved Kant’s idea our ideas are innate and shape experience. They also followed his example of poor writing, as Kant was given to logic chopping and excessive jargon. The path they chose was therefore rationalistic. From Fichte’s idea we freely create the world through our ideas to Hegel’s concept ideas shape the world through us, ideas are, by the judgment of idealists, the fundamental substance of reality.

Schopenhauer was a lone wolf in going the other way, as he sought to ground Kant’s brand of empiricism in a world shortly to be revolutionized by science. What Kant and Schopenhauer did not anticipate was the essential empirical ideas of their day (those of Newton) were not so much to be built upon in the future as displaced. Popper and others in the philosophy of science maintained the transcendental distinction introduced by Kant (knowledge originates in the mind), yet rejected his noumenon for different reasons than had the Idealists: i.e., no idea, no matter how serviceable in empirical science, can be declared absolute by virtue of revelation.

Bryan Magee is the beginning of my journey with Kant and Schopenhauer, and he pointed out the impossibility of explaining reality via realism (with or without a transcendental distinction), and how a noumenon, far from being pure invention, accounts for unanswerable questions science and the philosophy of science have no interest in.

And as far as what science is interested in: John Horgan, in his excellent book, The End of Science, has taken the somewhat skeptical view that theoretical science, having reached a period of diminishing empirical returns, is likely nearing its end.

This is where I pick up the thread. I believe in the noumenon as postulated by Kant and improved by Schopenhauer. With Schopenhauer, I reject Kant’s assertion the noumenal realm is entirely unknowable. Yet against Schopenhauer—and as demonstrated by his divided mind on the subject—I believe reality has purpose and value, and that purpose and value, originating with the noumenon, is synonymous with God or something very much like God. With the Idealists, I believe our ideas do not simply inform the nature of physical world but also reveal something ontological about their own nature. Whether through scientific inquiry or artistic creation, these ideas endeavor to complete the Noumenal Ideal as a project in terrestrial memory, and as indestructible matter/energy can only be conversed as it changes states, so too no idea born to eternity is lost to eternity. (6/25/10)

 

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

Space, Time, Infinity, and The Cosmos

Definitions
scientific reductivism: (aka, material reductivism, scientism) the belief science reveals total reality. The belief total reality is phenomenological in nature and consequently discoverable by scientific methodology and describable by scientific law.

transcendental idealism: the belief that we apprehend reality through ideation, as with concepts like space, time, and causation. Reality, however, as it exists in itself, transcends ideation, as well as the very notion of relational objects that arises from ideation.

Space, Time, and The Looking Glass
When we look up into the night sky, we are looking back into time. And yet, if the Universe was smaller at its beginning, then everything in that early Universe should be, by logic’s reckoning, close together. Instead, we see the most distant objects in the Universe everywhere we gaze in the heavens. They are not converging on a Big Bang event in a funnel, but appear far flung to the outermost frontier of our expanding Universe.

The solution, as always, comes down to the inscrutable observer and their point of view.

First, things closest to us in space are also closer to us is time. For example, a comet breaking up in Jupiter’s atmosphere happens closer to our time frame because Jupiter is not very far away. The same idea can be extended outward, moving from near-time events within our solar system to far-time events occurring on the other side of the Milky Way, as with supernovae. As we remain in place in our present moment and see further into space, our horizon line widens with our penetration deeper into time. Consequently, we see more things, and by the time we get to the furthest events in the Universe’s past, the horizon becomes a panoramic view of the entire history of the observable Universe. The funnel effect is the reverse of what logic would suppose: I, as a stationary observer, am at the narrow end of the funnel looking out.

In glimpsing deep into the realm of quasars, and further back to the dawn of the Big Bang, we are observing an almost fourteen billion year-old history. If we project our present moment into this record, events speed up as we move forward in space; and what awaits us at the end of our journey is an older Universe materially different from the early one we first spied in our telescope from the other side of Creation. It is like film in a projector: you cannot jump from one point in the story to another without passing through the intervening strip. For time is embedded in space, and the two cannot be decoupled in experience.

The Bait-and-Switch of Space/Time
“Relativity supports Kant’s view that space and time have subjectivity contingent on an observer, though with nothing like the Newtonian uniformity he supposed. Still, the assumption of space and time’s physical existence rests entirely on effects and geometry as a scientific concern. There is no true fabric to space/time. Space/time is an ingenious way of describing gravity, but it neither explains the true ontological nature of gravity nor the true ontological nature of space and time. Infinity, again, is something else. Science and mathematics have found ways to paper-over the enigma with limits, but such serviceable fixes should not be confused with demystification. Simply put: causation breaks down at the level of quanta, and no proposed number of preceding Big Bangs would ever explain a first Big Bang.” ~from Omar's letter, Chapter Six of An Aversion to Ladders

In classical relativity, space and time appear uniform when velocities are calculated for moving objects. When velocities approach the speed of light, a new physics is required, one where, owing to the vantage point of the observer, space and time give way to respect the speed of light. Whereas Einstein’s Theory of Special Relativity makes space and time relative to the observer, his Theory of General Relativity combines space and time into a single idea, space/time, and it in turn is made interchangeable with the concept of gravity. Here space/time is represented as two-dimensional geometry in the form of fabric, or grid, which serves to illustrate the warping effect on space/time by large masses such as stars.

Real space, however, is three-dimensional, not two-dimensional, so the idea of a flat, bendable substrate does not translate easily to a three-dimensional visual aid. What would the empty space over our heads look like if we were to curve it? It has no visual bottom, top, or thickness like the fabric depicted in the two-dimensional space/time model. More to the point, what would its boundary look like? In the two-dimensional model, the geometry simply stops like a table edge.

Einstein argued space and time could not be regarded as objective outside its subject-dependent experience. That is, there can be no objective present, past, or future in time, and no privileged location in space. In his mathematical formulation, space/time, though finite, is not absolute in the Newtonian sense. In considering the hole paradox generated by his General Relativity Theory, Einstein concluded that points in space and time were meaningless in absence of matter, saying, “People before me believed that if all the matter in the universe were removed, only space and time would exist. My theory proves that space and time would disappear along with matter.”

The question is: In absence of space, time, and matter, what exists?

It is proposed that the Universe has a point beyond which it does not extend, proceeding from the Big Bang and expanding some ninety-three billion light years at its diameter. Yet if the Universe has a point beyond which it does not extend (dare we call it an edge), how is this point demarcated? There is a difference between the observable Universe and the Universe itself, as the first requires an observer, and the second can only be guessed at.* This complicates our notions of space/time boundaries—as to whether or not they exist, and as to where they would be in relation to their contents if they did exist. We reach a point where Einstein cannot help us.

This “boundary” is not simply a question of space/time being counter-intuitive as an idea, but also being counter-intuitive to experience. As a space/time Universe involves math—and requires matter—, we have no way of contrasting it with nothingness as a concept. Nothingness is only the point (if not the place) where mathematics cease. We are led to believe more is being explained than is being explained by space/time, and we are told we must abandon the difficulties space/time presents to our reason, since these difficulties lead to only unproductive questions. Yet this amounts to something of a bait-and-switch for those looking to have larger questions about the nature of the Universe answered.

(*It is argued that if the Universe proper was smaller than the observable Universe, then we might see double. That is, we might see multiple copies of galaxies, as their light would encircle the Universe more than once. This could never be determined with certainty since each copy would represent a different cosmic time in the life of the galaxy.)

Space/Time versus Infinity
For example, space/time does not concern itself with infinity. This is not simply an issue for boundless space and time (which many physicists believe aptly describes our Universe) but for causation itself, as well as the very beginning of the Universe. Again, we are told nothing existed before the Big Bang. And yet, inexplicably, something the size of a proton existed the split second before space and time exploded into existence. Does this something-from-nothing argument satisfy the intellect in positing a first cause to reality? Can we be assured there were no Big Bangs prior to the one that began our Universe? If not, do these Big Bangs extend infinitely into the past? As Schopenhauer was fond of saying, if the past were infinite, then it would be impossible we should exist, for an infinite amount of time would need to have lapsed for us to get to a present moment. Clearly there is a gap between physics and cosmology, and space/time does not bridge this gap, even though many in physics are content to leave the impression it does.

Space/Time versus Gravity
Calling space/time constructed is at least true. But this begs the question of what prefigures this construction as raw materials?

As space and time are integrally linked with matter/energy, and as space and time are synonymous with the force of gravity, what are space and time? What is gravity, for that matter? If space/time/gravity are a form of energy, then this energy is clearly playing by a different set of rules than other physical forces. Since energy can neither be created nor destroyed, only conserved as it changes states, then where is the space/time energy coming from as the Universe flies apart? Such a relationship between object and ether has been likened to the creation of sound waves by sound-emitting sources. Yet a sound wave, which is a disruption of molecules of air, is a percussive force passing through gas—not a gas itself.

Moreover, as it is easier to imagine gravity being a quantum force* than imagine space and time playing this role, then why is this force many billions of times weaker than the other quantum forces? The gap that separates the physics of the very large (gravity) from the physics of the very small (electromagnetism, strong nuclear force, weak nuclear force) is the primary problem that concerns modern physics. A hypothetical graviton particle is advanced as a way of beefing up gravity and connecting it to the other quantum forces. Advantageously, gravitons, being the force carrier for gravity, would lend physical reality to space and time, since space and time are—again—identical to gravity. Yet where is this validating graviton to be found? Why in other proposed dimensions of space, of course.

This is a little like trying to prove the existence of a flashlight by illuminating it with a flashlight.

(*This is a curiosity: As the photon is the force carrier for electromagnetism, and the W and Z bosons are the force carrier for the weak nuclear force, and the gluon is the force carrier for the strong nuclear force, what is the force carrier for gravity?)

Spooky Action at a Distance
Given the disconnection between quantum mechanics and the Theory of Relativity, one can readily grasp why the former gave Einstein fits. For example, with quantum entanglement, subatomic particles can become entangled and share properties, and regardless how much space may come to separate them, when one is collapsed and disentangled by an act of observation, its pair collapses too. This, in simultaneity (if not in a transference of information about the simultaneity [which would violate causality]), is a clear violation of the speed of light, which is the only absolute constant in Einstein’s scheme. If this is true, and not an error in calculations as Einstein believed, then it casts doubt on the phenomenal reality of space and time as anything that separate matter in a metaphysical sense.

Schopenhauer, building from Kant, argued this was in fact the case in the century before quantum entanglement was proposed and demonstrated. He reached the conclusion space and time have no reality-in-themselves and, therefore, constitute no barrier in the underlying ontology of all phenomena. (This idea is similarly illustrated in Heisenberg’s Uncertainty Principle, where electrons, when considered as particles and not waves, can change location within their shells instantaneously without passing through an intervening space.)(6/10/10)

Space versus Other Kinds of Constructed Space
“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 An Aversion to Ladders

Kant has been faulted for not anticipating Non-Euclidian geometry, where curved lines making up hyperbolic and elliptical shapes only appear to be straight and Euclidian when viewed as shorter line segments. Nothing in sense intuition anticipates this idea, and this can be appreciated if you think about an airplane flying from Russia to the United States, where flying over Greenland makes for a shorter trip than flying in a straight line. As a conceptual matter, space and time can be constructed outside the sensibility of the mind, but in real world space and time, boring a straight line through the mantle of the Earth would still get you to the United States faster by way of a straight line. (I am not trying to be funny here, but make a subtle point.)

Because space and time can be constructed as an exercise does not mean they have any greater degree of reality by virtue of having an expanded context. Mathematics does not need real space and time or even objects to be intelligible in itself: numbers and lines will symbolically do. Put succinctly, the ontological reality of space and time may not settled science, but ontology is not needed in mathematics.

Regardless, mathematics can drop a toe into the transcendental pond, and with constructions like hyper-cubes and hyper-spheres, four-dimensional models of ideas that cannot be shown to exist in real space can nevertheless have mathematical space. In the case of the hyper-sphere, a model for the Universe is advanced where an end-run is made around infinity by placing the contents of the Universe on the surface of a finite ball-like surface, where, in theory, if you travel in a straight line you will wind up where you started.Two problems arise with this: Because the Universe is expanding, one would never complete the trip all the way around the curved space since the starting point would continually get further away. Secondly, if space is finite, then what is this finite spherical space expanding into?

Adding one more dimension to this puzzle: It as been observed in data retrieved from the WMAP satellite, which has set out to map the Universe, that space/time is not nearly as curved as one may have supposed. In fact, it is mostly flat, as Euclid would have supposed. Of course, a lack of curvature in space/time at this scale does not refute Einstein’s calculable space/time, which has been demonstrated as an effect in bending light around stars. However, a flat Universe bolsters the idea space and time extend infinitely, which in no way addresses the enigma of infinity. This leaves something to the imagination for those who do not despair of voids.

Arguably, the more demonstrable the void is to the circle-closing mind, the more the heart can take comfort in its non-existence.

(*Among the many interesting real and irrational numbers is the golden ratio [1.6180339887...], found in biology and mathematics, and employed by the arts, architecture, and mystic religion.) (5/27/10)

The Center of The Universe After All
Our ability to gaze back to the earliest visible events in the Universe has been greatly improved by the launch of the aforementioned WMAP satellite, whose primary mission is making the most accurate map of the early Universe to date. The nature of the data retrieved so far is astonishing, but little of it is widely publicized.

As I have said, WMAP’s findings have strengthened the flat universe model of the Universe. In fact, the Universe is so flat its critical density is described as having the “special value” of N=1, and this “fine-tuned”* status has been maintained since shortly after the Universe’s violet inception. A Theory of Inflation, where there is a sudden expansion in the size of the early Universe, accounts for this unusual equilibrium and flatness. (This inflation is noteworthy, since a new physics would be required to explain how this rapid expansion could violate the speed of light.)

Another perplexing problem posed by the Big Bang Theory emerges on our event horizon, and cosmically dates from the time when the background radiation began to cool and become homogeneous. Among the oldest celestial features at this distance, we can see galaxies that are, paradoxically, too far apart to see each other. In other words, the Universe is too young at this stage in its development for emitted light from these remote sources not to connect causally. Inflation would explain this anomaly, too, although some in the scientific community are not sold on this multi-tasked fix.

In fact, there are cosmologists who have never been comfortable with the Big Bang model of the Universe, and its theological overtones can be no small part of their misgivings. More to this point, the WMAP data has emboldened advocates of Intelligent Design and others employing the just-right Anthropic Principle to explain the “creation” of our “special” Universe. Indeed, the most extraordinary discovery of WMAP has come from looking at the cosmic microwave background itself, for if the “boundary” of the observable universe is equal to the entire universe (and there is no reason so far to suppose it would be otherwise), then this has profound ramifications for our place in the Universe.

“Results from Wilkinson Microwave Anisotropy Probe (WMAP) appear to run counter to Copernican expectations. The motion of the solar system, and the orientation of the plane of the ecliptic are aligned with features of the microwave sky, which on conventional thinking are caused by structure at the edge of the observable universe. [6][7]

Lawrence Krauss is quoted as follows in the referenced Edge.org article:[8]   ‘But when you look at CMB map, you also see that the structure that is observed, is in fact, in a weird way, correlated with the plane of the earth around the sun. Is this Copernicus coming back to haunt us? That's crazy. We're looking out at the whole universe. There's no way there should be a correlation of structure with our motion of the earth around the sun — the plane of the earth around the sun — the ecliptic. That would say we are truly the center of the universe.’” ~Quoted from The Copernican Principle in Wikipedia 

To reestablish the validity of The Copernican Principle, scientists have proposed another fix, namely, dark energy:

“When the universe was just 400,000 years old, matter and radiation decoupled and left a remnant radiation that still pervades the entire universe today. By measuring the tiny temperature fluctuations of this CMB radiation, scientists can learn things about the universe such as its shape, size, and rate of expansion. In the latter case, the observations show that the universe is expanding at an ever-accelerating rate, leading scientists to speculate about the existence of dark energy, new laws of gravity, and other possible – and often exotic – theories.

But what if the universe’s accelerating expansion is just an illusion? As Caldwell and Stebbins explained, this scenario is entirely plausible if the Copernican principle is loosened a bit. If, instead of the universe being homogenous and isotropic as the Cosmological Principle states, there is rather ‘a peculiar distribution of matter centered upon our location,’ then the universe would be centered on a low-density, matter-dominated void. Such a universe would be non-accelerating, and there would be no need for dark energy or other similar theories.” ~A Test of the Copernican Principle From Physorg.com

“Although dark energy may seem a bit contrived to some, the Oxford theorists are proposing an even more outrageous alternative. They point out that it's possible that we simply live in a very special place in the universe - specifically, we're in a huge void where the density of matter is particularly low. The suggestion flies in the face of the Copernican Principle, which is one of the most useful and widely held tenants in physics…” ~Dark Energy v. The Void: What if Copernicus was Wrong? From Physorg.com

(*Fine-tuning in science is a problem, where tinkering is needed to make a given theory sync with what is observed. There is also the just-so problem, as in the example where if the strong nuclear force was two-percent stronger, then stars would burn through their fuel in minutes and not millions of years, and resultantly life could not emerge around them. Why should this improbable balance exist?)