Science is the determination of the mathematical relationships inherent in the measureable properties of material reality. Oftentimes, associated with the mathematical relationships of science is a visual narrative pleasing to the human imagination. Such a narrative is essentially extrinsic to the science.
The Relationship of the Intellect and the Imagination
From the time of Aristotle it has been recognized that human intellectual knowledge is immaterial, an abstraction from material reality. However, Aristotle noted that the action of the intellect is dependent upon a phantasm, a composite of the sense knowledge obtained through the animal senses of man. Thus the human intellect, though immaterial in its nature and activity, is extrinsically dependent upon the sensual phantasm in order to function and to be aware. Without such a sensual phantasm, the human intellect is inactive.
The scope of the senses, including the sensual phantasm, is limited. Material reality, however, has properties beyond the scope of the senses, which properties can be measured instrumentally. Material properties smaller than this scope have been designated, micro, and larger than this scope, macro. Although the human intellect is unlimited, the human imagination is limited to the mid-range scope, which is that of human sensation. The dilemma arises when the sensual imagination attempts to constrain intellectual thought to its level of sensation.
In the Context of Science
A simple scientific relationship is that among the measured values of pressure, volume and temperature of a gas. Gas, confined to a tank, will increase in pressure when its temperature is increased. The science is the mathematical relationship among the instrumentally measured values of pressure, volume and temperature. Associated with the mathematics is a popular visual narrative. The gas is composed of molecules, depicted as tiny ‘micro-sized’ balls in motion. Pressure is depicted as the balls striking the internal walls of the tank. When the temperature is increased the balls move faster, so they strike the walls with greater force, thereby increasing the pressure. This is a harmless visual narrative at the level of human sensation and pleasing to the human imagination. In contrast, the mathematical relationship among the measured values of pressure, volume and temperature is the science.
However, there are scientific relationships, which are not accompanied by visual narratives satisfying to the human imagination. Some measured properties of light are related by equations, which are accompanied by a visual narrative of particles of light. Other properties of light are related by equations which are accompanied by a visual narrative of waves of light. Yet, the human imagination finds the particle and wave narratives visually incompatible in spite of the validity of the mathematical relationships among the measured values in the two different results.
The Classic Dilemma
Collimated light is unidirectional. When it is passed through a slit it produces a pattern of intensity, which may be viewed as a pattern of particles or quanta or photons of light distributed about a norm. The distribution may also be viewed as a continuous curve symmetrical about a single peak (See figure 5 at the end of reference 1).
When collimated light is passed through two adjacent slits the result is a pattern of intensity, which may be viewed as a pattern of particles or quanta of light distributed about several norms. The distribution may be viewed as a continuous curve of a series of individually symmetrical peaks (See figures 3 and 4 at the end of reference 1). This multi-peak pattern is best described mathematically, i.e. scientifically, as due to the interference of two waves of light emanating from point sources as they emerge from the slits.
The purpose of the experimental setup in reference 1 is to track light, quantum by quantum. In our human imagination a single isolated particle, i.e. a discrete particle, cannot act as a continuous wave.
This same purpose was that of the experiment described in reference 2. The rationale of this experiment is that interference requires two waves, whereas a single particle or quantum of light cannot behave as two waves. (This rationale suspends the validity of the quantum/wave mathematics applied at the micro-level and imposes a dictum of the sensory-level human imagination: Particles in the human imagination are discrete particles, not continuous waves.) The experiment proposes that collimated light as an individual quantum is in transit along one of two paths when a de-collimating device or beam-splitter is introduced or not introduced farther along where the two paths exit collimation and cross. The human imagination demands that the beam-splitter cannot change the particle into two waves once the particle is in transit along one or the other path. Nevertheless, the presence of the beam-splitter introduces wave interference. The human imagination reaches the conclusion, ‘Then one decides the photon shall have travelled by one route or by both routes after it has already done the travel.’ This is an imaginative impossibility.
The experiment in reference 1 is clearer than that of reference 2. When the collimation of the light is maintained even though the light is tracked photon by photon, the intensity pattern is of a single peak (figure 5) by detectors D3 and D4 identified in figure 2. When the collimated light is released from collimation by mixing the paths, it displays wave interference in the cumulative pattern of the individually tracked photons in figures 3 and 4 from detectors D1 and D2 of figure 2, respectively.
Another experiment uses two telescopes to maintain the collimation of light. Without the telescopes the light from the two sources interferes as waves. If the light, maintained as collimated by the two telescopes, is permitted to mix after exiting the telescopes it interferes as waves. (See reference 3).
The dilemma of the imagination is, ‘How can a single quantum be ‘mixed’ with nothing by a bean-splitter and thereby act as a wave?’
Attempts to Placate the Imagination within the Context of Physics
Some physicists pacify their imaginations by viewing the fundamental equation as that of the wave and view the detection of a quantum as 1 and the non-detection as 0, the two possible outcomes of a probability event. The wave function is viewed as an expression of probability, the outcome of which is yes, one or no, zero. An outcome is said to be the collapse of the wave function. As we shake a pair of dice in our cupped hands, the probability of the sum, seven, is one-sixth, but when the dice have landed the probability, which was one-sixth has collapsed to 1 (seven) or 0 (non-seven). Unlike the eleven discrete probabilities, which are the sums of two dice, a wave as a probability function, may be viewed as indefinite in probability between 0 and 1. In this view, the imagination states that the quantum does not exist except in the potency (probability) of a wave. The wave collapses due to an observation. As a result of observation, the quantum comes into existence or doesn’t. Upon observation, the probability function no longer exists. Thus a quantum exists, only if observed.
The author of reference 1, concludes, ‘Ho-hum another experimental proof of quantum mechanics’. Another expression rejecting the need to placate the human imagination has been expressed as ‘Shut up and calculate!’ (See reference 4).
The need to placate the human imagination has been carried to two extremes. One extreme is to claim that observation creates material reality, such as ‘The moon is demonstrably not there, unless someone is looking.’ (See reference 5). The other extreme is to claim that there are as many universes as there are possible material outcomes. A quantum we observe is valid within our universe, but not universally valid across multi-verses. What exists in our universe in our observation is characteristic of our universe. All other possibilities exist in a multitude of universes encompassed in what appears in our universe to be a probability function, the wave (reference 6).
The middle course is to recognize that the compatibility of the discrete and the continuous is a sticky wicket, not to the intellect, but to the sensual imagination. The photon vs. wave phenomena of light is just one example where the human imagination tends to deny the compatibility between the discrete and the continuous.
The Central Problem, Imagining the Concept of Continuity
The most fundamental concept in mathematics is the counting of discrete elements. Material objects are easily counted. Another concept in the basic development of mathematics is the fractionation of a discrete element. Because mathematics is an abstraction from material reality, it is intellectually possible not only to fractionate an element into two parts, it is possible to fractionate it into an unlimited number of parts. You can’t do that with an apple or any material thing even of an apparently uniform structure. Our sensual imagination cannot keep up with our intellect. Our intellect has conceived the abstract idea of continuity.
An independent variable is continuous over its domain if its value can be specified to any arbitrary precision.
Suppose I had a building with a frontage of thirty meters. I could delineate two contiguous parallel parking spaces of ten meters each demarcated with painted end stripes twenty centimeters wide and having one stripe at the midpoint. That would be a 20.6 meter, continuous segment, of the frontage. However, if I asked a painter to paint the three twenty centimeter stripes to a precision of 10^(-11) meters, he would think I was crazy. He would note that the diameter of the electron cloud of the hydrogen atom was 10^(-10) meters and I was asking for a painted line to the precision of a tenth of the diameter of a hydrogen atom. He would point out that the abstract concept of mathematical continuity to such a precision did not apply to the context of the level of human sensation and specifically to the painting of lines in a street. He wouldn’t deny the validity of the mathematical concept of continuity. The painter would merely note that the human imagination did not have the power to concretize the abstract concepts of mathematics in the manner and context, which I proposed.
One of my favorite apparent incompatibilities of a mathematical concept with material reality, as viewed by the human imagination, involves the probability of the sequence of a deck of playing cards, namely 1 in 8.06 x 10^67. Intellectually we are content with emulating a ‘random selection’ from a set of that size by shuffling the deck. Yet a set of that size can only be logical, far beyond the scope of the human imagination. It would be foolish to claim that the shuffled result was the collapse of the fundamental reality, the probability function, into a materially observable event, where the collapse of the probability function was essentially due to the observation. It would likewise be foolish to claim that the shuffled result was a material event in my universe, but that the entire set of probable mutations of a deck of cards must exist in some other material universe(s), rather than solely in human logic. In both views, shuffling brings one material event into observable existence out of ‘real’ probabilities or ‘real’ worlds equaling 8.06 x 10^67. These views require material reality to conform to human thought in its subjection to the human imagination. The proper view is to require human judgments about reality to conform to reality, while also recognizing the immateriality of human thought and logic, which frees the human intellect from subordination to the human imagination, i.e. subordination solely to sense knowledge.
Note that adding two jokers to the deck would increase the ‘real’ probabilities and the number of ‘real’ worlds by a factor greater than 1000 to 2.3 x 10^71. The mass of the earth is a mere 5.97 x 10^36 nanograms.
The mathematics of probability concerns the identification of logical sets solely according to their fractional composition of logical elements. When this mathematics is applied to material sets, the material properties of the elements are completely irrelevant. The IDs of the elements are purely nominal. The elements have no relevant material properties. Material elements may be used only in emulation of the mathematics, while ignoring their material properties. Thus, the mathematics may be used as a tool of ignorance of the material properties of the elements.
To the extent that a wave function is considered a probability function, it renders the mathematics a tool of ignorance, i.e. a tool to compensate for ignorance of the material properties underlying the phenomena being studied. It is a serious error to take probability as fundamental reality or as a characteristic of reality, rather than as a mathematic tool of abstract logic.
1. Excerpts from “A Delayed Choice Quantum Eraser” by Yoon-Ho et al Phys. Rev. Lett. 84:1-5 (2000) with commentary by Ross Rhodes. http://www.bottomlayer.com/bottom/kim-scully/kim-scully-web.htm
2. Wheeler’s delayed choice experiment by Alain Aspect
3. Wheeler’s Classic delayed choice experiment
4. Copenhagen interpretation
5. Video of the faith and science conference in 2011, at time 1:09:45
6. Chapter 11, Quantum Time, by Sean Carroll