Unknowable Reality. Science, Mathematics and Mystery. Aleksandar I. Zecevic Dept. of Electrical Engineering Santa Clara University

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1 Unknowable Reality Science, Mathematics and Mystery Aleksandar I. Zecevic Dept. of Electrical Engineering Santa Clara University

2 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?

3 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.

4 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.

5 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

6 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.

7 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.

8 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.

14 The Koch Curve

15 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.

16 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.

17 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.

18 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.

19 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.

20 The Chaotic Behavior of Long Term Weather Patterns

21 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.

22 Laws that can be Broken

23 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. Simulati Input Input on Bits Bits Interval (normal (chaotic system) system) [0 1] [0 2] [0 3] : : : [0 100] 1,

24 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).

25 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

26 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.

27 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.

28 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.

29 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.

30 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

31 Example 4: Unprovable Propositions in Geometry A

32 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.

33 An Alternative to Euclid s Fifth Axiom

34 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.

35 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.

36 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.

37 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.

38 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

Quantum Entanglement. Chapter Introduction. 8.2 Entangled Two-Particle States

Quantum Entanglement. Chapter Introduction. 8.2 Entangled Two-Particle States Chapter 8 Quantum Entanglement 8.1 Introduction In our final chapter on quantum mechanics we introduce the concept of entanglement. This is a feature of two-particle states (or multi-particle states) in