The Things We Believe

We believe the most outrageous things. I don’t know if ‘believe’ even applies. According to many philosophers, belief begins with doubt and the things I’m talking about admit no doubt. These are things we just assume, things we simply take for granted, things that ‘must be exactly as they are’. 

For example, did you hear the one about the Universe?

• It came to be without cause, and it will cease to be without effect. (I like to lead with this one; it’s a real knee slapper.)

• It is 14 billion years old (10^10), 100 billion light years across (10^11), with a ‘normal life expectancy’ of 100 billion years (10^11). It consists of a trillion galaxies (10^12), averaging 100 billion stars each (10^11).  

• It is expanding everywhere, but not uniformly: 

  • Locally, a one meter length (10^0) expands by the width of one atom each year (10^-11 meters). 

  • But at the cosmic horizon, the Universe is racing out of view at the speed of light (10^15 meters/year) - a spread of 26 orders of magnitude.

  • And beyond the cosmic horizon, the unseen Universe is expanding at a rate of speed even greater than light.

  • Yet the rate of acceleration is incredibly gradual: 10^-11 meters per year per year.  

    
    

• Everything in the Universe is ‘moving’ at a uniform rate of speed, ‘c’, the speed of light, which means that everything is also ‘at rest’. 

  • Massless particles, like photons, move through space only; they are everywhere all at once all the time. 

  • Massive particles at rest move only through time. Like me during football season, they remain in one place indefinitely…but age nonetheless. 

  • Massive particles in motion move through space and time according to a ratio established by Heisenberg. That is why bodies in motion (you) age slowly relative to bodies at rest (me).

  • The three spatial dimensions are isomorphic, but one second of time = 300,000,000 meters.  Plus time is ‘rotated’ by √-1 relative to space. 

  • One year = 10^16 meters, the equivalent of 500 trips to Pluto and back. Busy, busy...but just think of the ‘Miles’ you’re accumulating.

    
    

How about Life?

• Earth is one unremarkable planet orbiting one unremarkable star in one unremarkable galaxy. 

• Earth formed about 4.5 billion years ago and conditions became compatible for life as we know it about 4.2 billion years ago. Living organisms first emerged about 4.0 billion years ago.

• Today, there is hardly a place on or near the surface of the planet that is not teeming with living organisms: from the top of Mt. Everest (5 miles > sea level) to the bottom of the Mariana Trench (7 miles < sea level), in glacial ice, on desert sand, and at the rims of active volcanoes. 

  
  

Or the one about Cause & Effect?

• The causal effects of an event diffuse through spacetime according to a smooth function (like the inverse square law). Everything is everywhere all at once, but the intensity of presence varies, often enormously, from region to region. Every event has a unique 4-d heat map. 

• A butterfly may flap its wings in Borneo and a tornado may touch down in Chicago…or not. 

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How about Mathematics?

• Every quantity (size, distance, count) can be represented by a point on a continuum known as the Real Number Line (the Set of Real Numbers).

• The order of events in some operations is irrelevant: x + y = y + x, so adding y units to x is the same as adding x units to y. Likewise, the order of certain operations is irrelevant: e.g. (x + y) + z = x + (y + z).  

  
  

Or Topology (Geometry)? 

• The world may be modeled as a hierarchy of sets that can embed one another, overlap one another, touch one another…or not. Every set has subsets, and every set may be a subset of another set, but no set may be a subset of itself. If B is a subset of A, A cannot be a subset of B. 

• If B and C are subsets of A, then B must be ≤ A and C must be ≤ A and B+C must be ≤ AI.  

• Every flat surface (plane) is two sided (obverse/reverse) and splits the Universe (3d) into two non-contiguous regions. Since there are an infinite number of planes in a 3d Universe, no two regions can be contiguous. 

• Every curved surface (spheroid) divides the Universe into two non-intersecting regions (inside/outside). Since there are an infinite number of spherical spaces in a 3d Universe, no two regions of space can be contiguous. 

  
  

Now before I spoil the fun and give you the punch lines, let’s take a moment to celebrate what we’ve accomplished. After all, it’s taken us more than 10,000 years to get where we are today. Polymath Roger Penrose has dubbed this period, “The Road to Reality.”

What can we say about this cosmic quilt that we’ve managed to stitch together from various absurdly shaped patches of mismatched cloth? We can say that it provides an astoundingly functional model of reality, one that has enabled us to decode the human genome, to send satellites into space, and to glimpse even the oldest stars and galaxies as they first formed. 

We can also say that this highly functional model is almost entirely unrelated to reality. Almost. The world we know and sometimes love can be thought of as an n-1 dimensional slice of an n-dimensional block. Everything noted above is an abstraction from, or a special case of, the actual ‘real world’.

As much as this ‘standard model’ of reality has enabled phenomenal practical achievements, it has also led to some conclusions that are plainly inconsistent with the world we know.

For example, it fails to account properly either for ‘motion’ or for ‘recursion’. It cannot explain how an arrow released from a taut bow string can travel many meters to hit a distant target. It allows a lumbering tortoise to beat fleet footed Achilles in a road race. It cannot account for the fact that two events, separated in space and/or time, may nonetheless influence one another. Plus it fails to account for good ole Karma – when what goes around comes around - or for the equal and opposite ‘reaction’ that accompanies every ‘action’ (Newton). 

In other words, it accounts for almost nothing important…and yet it works beautifully for everything else. To bastardize a line from Wittgenstein, it surely is important nonsense. 

Of course, we have devised ‘work arounds’ (aka hacks) to get us past these shortcomings. One such hack, for example, is calculus. It was ‘invented’ by Newton and Leibniz to put Zeno in his place: it failed, but the collateral by-products are amazing. When you join your fellow ‘space colonists’ on Europa, be sure to thank Sir Isaac and Herr Gottfried. 

Calculus supports the illusion of continuity where there is none and it allows us to navigate across spacetime as though it were continuous, but it tells us squat about how the universe is actually structured.

But I promised you punch lines…so here we go! The phenomenal world appears to be continuous…but it’s not! The actual world is quantized into Planck size bubbles, known by Iron Chefs everywhere as ‘quantum foam’. The phenomenal world is an n-1 dimensional surface of n-dimensional reality.  

Spacetime is not substructural; it is an emergent property of the Cosmos. Spacetime is a lattice on which events are positioned according to how they relate to one another across various parameters. We model these relationships using manufactured concepts like ‘sequence’ and ‘proximity’.  

Because spacetime is epiphenomenal ‘scale’ is also a secondary category. The true deep structure of Cosmos is fractal. The Universe is self-similar - always, everywhere, and across all scales. 

According to Penrose, the structure of the cosmos is conformal, i.e. angles (relationships) are conserved; size, duration, distance, scale and mass (weight) are not: 

“At the foundation is relationship” (Martin Buber); ontogenesis occurs when potential entities co-actualize by “granting (each other) reck”. (Anaximander via Heidegger) “In the beginning was the Logos…” (John 1: 1)

“All things came to be through him (Logos) and without him nothing that came to be came to be.” (John 1: 3a)

You live in a glass house…no, not that kind of glass house! You live in a Hall of Mirrors – an array, hexagonal at a minimum, of mutually reflecting mirrors. One image repeats indefinitely in every direction. Pretty cool, huh?

“What came to be through him (Logos) was life.” (John 1: 3b - 4) Therefore, “the hills are alive” (Sound of Music), as is the Earth (Gaia), as is the Cosmos. (Teilhard de Chardin, The Coming of the Cosmic Christ)

Biologists and science fiction writers notwithstanding, as far as we know, life has evolved only once and only here on Earth. It is possible that something recognizable as ‘life’ may have emerged elsewhere in the Universe, but we have no evidence to support such a conjecture. Every living organism we know of is descended from one, single, 4 billion year old primordial organism. “Thank you,” doesn’t seem to cover it.

Re our butterfly, yes, it may (or may not) flap its wings and, yes, a tornado may (or may not) touch down. If the butterfly flaps and a tornado touches, we may say that the one caused the other. Likewise, if there is no flap and no tornado, we may say that the butterfly’s restraint saved Chicago’s Loop. 

But it is just as likely that a flap prevented a tornado or that a lack of flapping made the ‘Windy City’ windier.  Post hoc ergo propter hoc: something happened (or not), then something else happened (or not). That’s all there is to the tyrannical principle of cause and effect. Everything is the cause of everything else and therefore nothing is the cause of anything. Causality itself is an imaginary category of explanation. 

Order is a function of correlation, not causation. The process of correlation is inertial; there is an innate tendency of patterns to repeat. Ergo, fractals.

The Set of Real Numbers is infinitely large…yet it excludes far more than it includes. For example, it has no room for infinite, or infinitesimal, quantities. But we need those quantities to make math (e.g. calculus) work. We need to make room in our model for ‘unreal numbers’ (e.g. hyperreal and p-adic numbers). 

Fun fact, there are an infinite number of hyperreal infinitesimals between the smallest possible positive real number and zero. P-adic numbers represent quantities on a logarithmic scale but the larger the quantity represented, the closer the numeral is to zero: ¼ > ½.

Re alleged ‘operational symmetry’, we know that reality is not agnostic when it comes to sequence. Everyone (over the age of 2) knows that adding more to less is different from, and usually better than, adding less to more. And as far as the order of operations is concerned, do you wish to be cremated before or after death?

The ‘standard model’ of reality assumes that quantity is always conserved. Again, what we assume to be incontrovertible need not be so. Quantitative relationships (measurements) need not recapitulate topological relations (sets). In our A, B, C example (above), if B and C are both subsets of A, then as long as they do not intersect, B and/or C may be greater than A, and/or B+C may be greater than A. In fact, both B and C may have positive ‘real number’ volumes while the volume of A may be infinitesimal.

We imagine (above) that we live in a 360° world; we imagine it even though we know it’s not true! Consider an electron, or a proton, or any other massive particle; they live in a world where symmetry requires at least 720° of space. Photons do live in a 360° world, but gravitons need only 180° to exhibit symmetry.  

We imagine we live in a world whose topology (geometry) is ‘orientable’. Process is vectored. Things have beginnings and ends; and so they do… sometimes. But, as above, our ‘orientable universe’ is an n-1 dimensional surface of a ‘non-orientable’ n-dimensional actual world.

The Universe is an n-dimensional correlate of a Mobius Strip. Locally, the strip consists of segments that are straight, flat and two-sided; globally, the strip is circular, twisted, and one-sided. As with calculus, we can treat any smoothly curving surface as if it were a sequence of linear tangents; but that doesn’t make it so! 

So yes, we do take a lot of things for granted. And that’s just fine for getting us from point A to point B. ‘Reality’, as we know it, is essentially a hack. It works! But it tells us next to nothing about the world that underlies it. 

Our phenomenal world is like a well-defined crouton floating on top of a bowl of French Onion soup. When we look beneath the ‘solid surface’ we find a much more loosely ordered Cosmos (broth) capable of supporting many alternatively structured  phenomenal worlds. 

Bottom line: many of the things we think of as ‘obvious’ are simply matters of habit, testimonies to intellectual sloth and/or to our lack of imagination. Determining the absolutely rock bottom conditions necessary to support a real Universe is the eternal task of metaphysicians. May the Force be with you!

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Josef Albers, German, 1888–1976. Homage to the Square series, 1950–1976. Oil on Masonite. Various dimensions. Albers’ Homage to the Square mirrors the essay’s exploration of perception and reality by using precise, repetitive forms to reveal how context transforms experience and meaning.