Unknowable Reality

Science, Mathematics and Mystery

Aleksandar I. Zecevic

Dept. of Electrical Engineering

Santa Clara University

The Unknowable and the

Counterintuitive

A fundamental question in the debate between science

and religion has to do with the fact that all religious

traditions emphasize that certain aspects of reality are

beyond our grasp. They speak of a “cosmic mystery”,

and what they have to say about it is often thoroughly

counterintuitive. Can scientifically minded individuals

accept such claims, given that they are at odds with our

everyday experience and cannot be verified by

experiment or simulation? If they do so, are they in

danger of being “intellectually dishonest”?

Are Certain Aspects of

Reality Counterintuitive?

Does the fact that certain theological claims are counterintuitive suggest that religion and science are incompatible? The answer to this question depends on what we think reality is really like.

We can’t know that for sure, of course, but what we do know suggests that physical reality is certainly not intuitive, and that some of the laws that govern it are very different from the “tidy” laws of classical physics.

We will now look at several results from physics and mathematics that challenge our “usual” concepts, and seem to be at odds with our everyday experience.

Example 1: The State of Superposition

The “state of superposition” in quantum particle is highly counterintuitive. Perhaps the best way to describe what this means is to consider an analogy with a coin toss. In the case of a coin toss, there are only two outcomes that our instruments can record – heads-up or tails-up. But what “state” is the coin in while it is still in the air?

We could perhaps describe this state by saying that the coin “exists” as a collection of unrealized possibilities which are mutually exclusive. Such a description would be entirely consistent with quantum mechanics, where seemingly incompatible states such as “spin up” and “spin down” can coexist when the particle is in a state of superposition.

Example 2: General Relativity

Einstein’s theory of gravity (which is known as general relativity) assumes that space is an active entity, rather than a mere “container” into which material objects are placed. Einstein’s conjecture was that spacetime and matter actually interact, and that spacetime becomes curved in the presence of matter (or energy). Its geometric form is determined by the way in which the masses are distributed. “Spacetime grips mass, telling it how to move; And mass grips spacetime, telling it how to curve.” John Wheeler

Example 3: Infinity, Self-Similarity

and Timeless Knowledge

To see a world in a grain of sand,

and a heaven in a wild flower,

hold infinity in the palm of your hand

and eternity in an hour.

William Blake

Blake’s poem is actually “scientifically correct” in several

ways. Fractals, for example, allow you to see a replica of

the “whole” in the tiniest part, and can fit in the palm of

your hand despite their infinite length. The phrase

“eternity in an hour”, on the other hand, makes sense in

the context of special relativity.

Special Relativity

Special relativity suggests that we must forego our natural tendency to treat space and time as separate entities, and should instead view them as components of a unified four dimensional spacetime. It also tells us that if we could move at the speed of light, the distinction between the past, present and future would disappear (this phenomenon is known as time dilation). For such an observer, there would only be an “eternal present moment”.

Fractals

A typical geometric figure has a “characteristic scale”, which corresponds to the size of its smallest feature. As a result, choosing a ruler length that is smaller than the characteristic scale ensures that we will accurately account for all the relevant details. In the case of fractals, we will fail to capture all the essential characteristics of the object no matter what resolution we choose, since new details are bound to emerge as we zoom in further. Such objects possess unlimited complexity.

The Koch Curve

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L = 1

STEP SEGMENT

LENGTH

NUMBER OF

SEGMENTS

n = 0 ε = 1 N(ε) = 1

n = 1 ε = 1/3 N(ε) = 4

n = 2 ε = (1/3)2 N(ε) = 42

L = 4/3

L = (4/3)2

Infinity in the Palm of Your Hand

The Koch curve actually has an "invisible" microscopic

structure whose cumulative length is infinite. Imagine

that we have a measurement device which can

distinguish segments that are as small as 10⁻¹⁵ meters

(this is roughly the diameter of an atomic nucleus). If our

initial line happened to be 1 cm long, such a device

would allow us to precisely record how the length of the

object evolves during the first 32 steps of the

construction process. The maximal length that we would

be able to measure in this way is (4/3)³², which is slightly

less than 100 meters. However, the length of the Koch

curve grows as L(n)=(4/3)ⁿ and therefore tends to infinity

as n increases.

What Can We Conclude

From All This?

Modern science clearly suggests that many aspects

of physical reality are highly counterintuitive, and

have little to do with the “tidy” Newtonian view of

nature. Mathematics tells us something very similar,

particularly when it comes to dealing with objects

such as fractals or infinite sets, whose properties

have no counterpart in our everyday experience. In

view of that, the theological claim that certain aspects

of “reality” transcend our usual concepts and

categories doesn’t sound unreasonable at all.

Does Science Leave Room for

a “Cosmic Mystery”?

Since all religious traditions emphasize the

importance of “mystery”, it would be interesting

to examine whether science leaves any room for

this sort of speculation. The answer to this

question hinges on what we can and cannot

know about the nature of reality. We must, in

other words, draw a clear distinction between

the unknown and the unknowable.

How Do We Acquire

Scientific Knowledge?

What does it mean to “know” or “explain” something in

science and mathematics?

• In physics, we usually formulate predictive models,

and verify them by experiments and/or simulation.

Alternatively, we make empirical observations, and

then construct models that explain them.

• In math, we prove new theorems by systematically

tracing them back to existing theorems and/or

axioms.

We will now consider several examples where these

techniques fail us, and where we actually know what

we cannot know.

Example 1: Chaos Theory

Chaotic systems are infinitely sensitive to

initial conditions and changes in parameters (a

phenomenon that is sometimes referred to as

the “butterfly effect”). As a result, after a

sufficiently long time the outcomes of two

seemingly identical experiments will necessarily

be very different. This means that we can

empirically confirm only the short-term

behavior of such systems.

The Chaotic Behavior of Long Term

Weather Patterns

0 10 20 30 40 50 60 70-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Time

Sta

te x

1

Is Chaos Just Another

Form of Randomness?

What is the difference between throwing dice

and chaos? In the former case, there is

uncertainty from the outset, and we must use

probabilistic descriptions. In the case of chaos,

our models are completely deterministic. There

is short-term predictability, but long-term

unpredictability eventually prevails. Here we

have a mix of order and disorder, which is one

of the “trademarks” of complex dynamic

behavior.

Laws that can be “Broken”

0 50 100 150 200 250 300 350 400 450 5000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Numerical Simulation

If repeated experiments don’t work for chaotic systems, can

we use computer simulations instead? The difficulty here is

that we would need an enormous amount of input information

in order to maintain a given precision over longer simulation

intervals.

Simulation

Interval

Input Bits

(normal system)

Input Bits

(chaotic system)

[0 1] 10 10

[0 2] 20 102

[0 3] 30 103

: : :

[0 100] 1,000 10100

Example 2: Quantum Mechanics

Quantum mechanics is fundamentally indeterministic, since it cannot predict the outcome of a measurement. It only allows us to compute the probabilities of different outcomes when the experiment is repeated many times. Previous quantum measurements tell us nothing whatsoever about what will happen next (much like previous coin tosses have no bearing on future ones).

Do All Quantum Events Have a Cause?

The quantum state of superposition allows for the coexistence of many mutually exclusive possibilities. When we perform a measurement, we will see exactly one of them, but we can never know why that particular configuration materialized. It is entirely possible that another observer would see a totally different outcome under identical conditions. There is no “explanation” for this discrepancy – the concept of cause and effect does not apply in this case.

“Modern man has used cause and effect as ancient man used the gods to give order to the universe. This is not because it was the truest system, but because it was the most convenient.” Henri Poincaré

Example 3: Global Phenomena

Our understanding of the universe is inherently limited since all our observations are necessarily views from within. As a result, there may well be subtle regularities and global processes that we will never be able to identify.

Do we have any reason to believe that global processes and phenomena actually exist in the universe? Quantum mechanics suggests that we do.

Einstein’s Doubts about

Quantum Mechanics

Einstein never accepted the view that quantum

mechanics is inherently indeterministic. He voiced his

dislike for this interpretation with the famous phrase:

“God does not play dice with the universe.”

In the 1930s, Einstein and his two graduate students,

Podolski and Rosen, produced what they thought was

a decisive argument in favor of their view that quantum

mechanics is an incomplete theory.

The EPR Paradox

The EPR paradox is best described in terms of an

experiment in which an atom with spin zero disintegrates

into two smaller particles (say A and B) that fly off in

different directions. According to quantum mechanics, prior

to a measurement the state of each particle ought to be a

superposition of “up” and “down” spins.

If we happen to measure spin “up” for particle A, it is

guaranteed that we will subsequently measure spin “down”

for B, regardless of the distance between them (as a result

of a quantum property known as entanglement). This is

clearly a global phenomenon.

Spukhafte Fernwirkung?

Einstein referred to this phenomenon as “spukhafte

Fernwirkung” or “spooky action at a distance.” He

suggested that particles A and B were really subject to

“hidden” deterministic laws, and are both in a state of

definite spin which is decided at their “birth”.

Einstein was proved wrong, but not in his lifetime. The

decisive experiment was performed some 25 years after

his death by French physicist Alain Aspect and his

collaborators.

Quantum Mechanics is Non-Local

“As a result of Bell’s theorem and the experiments

it stimulated, a supposedly purely philosophical

question has now been answered in the

laboratory: There is a universal connectedness. …

Any objects that have ever interacted continue to

instantaneously influence each other. Events at

the edge of the galaxy influence what happens at

the edge of your garden.”

Bruce Rosenblum

Example 4: Unprovable

Propositions in Geometry

A

Implications of Unprovability

The existence of unprovable propositions presents us with

two logically equivalent options. We can either adopt

such propositions as axioms (i.e. truths), or we can choose

their opposites as axioms.

In making such decisions, it is completely irrelevant

whether one of the choices is less intuitive than the other.

The fact that non-Euclidean geometries seem to defy our

conventional understanding of spatial relationships doesn't

undermine their validity in the least.

An Alternative to Euclid’s Fifth Axiom

Example 5: Gödel’s Theorem

Mathematical knowledge is based on axioms and formal rules which allow us to derive theorems from them. Until the early1930s, it was almost universally believed that every formal mathematical statement can be classified as True or False (a property known as “completeness”).

Gödel showed that this is not the case, and that every sufficiently complex formal system necessarily contains unprovable propositions. This famous result is known as the Incompleteness Theorem.

The “Mystery” is Alive and Well …

These examples suggest that certain facts about

nature are unknowable. In that respect, science

is clearly open to the existence of a “cosmic

mystery”, and actually reinforces this idea. How

we interpret the mystery is a different matter –

we may embrace it and give it a “personal”

character, or we could adopt a “neutral” attitude

and simply recognize that we are not “in

complete control”.

Who is Being More “Rational” Here?

Would it be fair to say that one of these views is more

“rational” than the other one? Mathematics teaches us

that unprovable propositions present us with two

logically equivalent options. We can either adopt such

propositions as axioms, or we can choose their

opposites. In making such decisions, it is completely

irrelevant whether one of the choices is less intuitive

than the other.

If we apply this criterion to the nature of the “cosmic

mystery” (which certainly qualifies as an “unprovable

proposition”) the religious interpretation seems to be just

as acceptable as the secular one.

Religion and Knowledge

Whatever we may think of theological claims, it is important to remember that knowledge is not the enemy of religion - it is the enemy of superstition. Theology sees all knowledge as fundamentally good, but stresses that certain truths cannot be known by reason alone. The fact that science acknowledges that such unknowable truths exist can help remove certain perceived “barriers” to religious belief.

As we continue to learn more about the structure of the universe, we should never lose sight of the fact that a profound mystery lies at the heart of the cosmic order. This should instill a sense of awe and humility in all of us, regardless of whether our world view is religious or secular.

A Touch of Humility…

“Man, formerly too humble, begins to think of himself as almost God. … In all this, I feel a grave danger, the danger of what might be called cosmic impiety. The concept of “truth” as something outside human control has been one of the ways in which philosophy hitherto has inculcated the necessary element of humility. When this check upon pride is removed, a further step is taken on the road towards a certain kind of madness. … I am persuaded that this intoxication is the greatest danger of our time.”

Bertrand Russell