Let us call “knowledge” every mental representation of reality that can be transferred from one mind to another. This perhaps is the difference between knowledge and thinking. A thought can occur inside a mind without a specific expression in a specific language, and it may very well never leave the mind in which it occurred. A piece of knowledge, in contrast, is supported and carried by a slice of reality, since it is reality that it must traverse in order to reach any other mind. Consequently, all knowledge is necessarily finite. A piece of knowledge has weight; it has a size that can be measured in numbers of symbols or packets of symbols. In particular, a thought that cannot be transferred (cannot leap from one mind to another) does not attain the status of knowledge. Any piece of knowledge carrier begins and ends, it occupies a space, be it a written text, a musical score, a painting, a sculpture, or a scientific theory.
I commented earlier that the central intuition that the SM we seek must fulfill is its usefulness for understanding reality. But what purpose does understanding reality serve? There is one utility that is plain to see in history—to survive. This is the crucial point of connection between two concepts of key importance: natural selection and cultural selection. This idea leads us to two others. In effect, an understanding of reality that anticipates uncertainty ought to enjoy two types of universality. One of these is internal independence: the understanding should be as independent as possible of the mind that devised it; in other words, as little preconceived ideology should be invested in the process as possible. The second is external independence: understanding should be as independent as possible of the particular slice of reality that we want to understand, and also of the time and place in which this understanding arises. In addition, the comprehension of reality always involves observation, and the manner in which observing is done may suffer from limitations that change with time and place. This means that if observation changes (improves), understanding is also liable to change (improvement). This is the third intuition: the ability of science to progress. So here we have the three first intuitions that the SM we seek must meet: universality, the capacity for anticipation, and the capacity for progress.
These three first intuitions require the mind to be capable of perceiving and recording reality and the changes it undergoes. This means that in addition to the concepts of reality and its understanding, a third fundamental concept must mediate between them: the perception or observation of reality. This, then, is the tripod of the conceptual schema on which the SM must rest:
Reality
There is a triple zero hypothesis prior to the establishment of scientific understanding: (1) reality exists; (2) a mind capable of understanding it also exists; and (3) a certain interaction between them is possible. In particular, the mind is capable of perceiving reality and of organizing successive perceptions in order to observe it. With these opening words, we have already named the three fundamental concepts of the schema we want to build: reality, observation, and comprehension. The necessary definitions are given below.
Slice of Reality R(Ω, τ)
Slice of reality R(Ω, τ) is a distribution of matter, energy, and information contained within a Ω region of space and a τ period of time.
Real Object RO
Real object RO is a slice of reality for a particular instant tЄτ being fixed, i.e., R(Ω, tЄτ).
Real Phenomenon RP
Real phenomenon RP is a slice of reality for r points of the space of a volume v
C
Ω being fixed, i.e., R(rЄv
C
Ω, τ).
The object of a piece of scientific knowledge always refers to a slice of reality. We will assume that the reality exists even when there is no observer in a position to perceive it. The perception of reality depends on the time and place. To the naked eye, perceivable reality is in fact very limited: the slice of reality may be imperceptible because it is too large or too small, too opaque or too transparent, too far away or too close, too quick or too slow, too complex, and so forth. Over the course of history the mind has managed to widen perceivable reality with the help of instruments that act as exosomatic extensions of the ability to see (telescopes, microscopes, high- and low-speed cameras, scanners that operate at different frequencies, etc.).
Observation (of Reality)
To perceive reality implies a kind of conversation between a mind and a slice of reality. The mind devises a representation of a slice of reality by using some kind of language. When the perceptions are programmed in accordance with preconceived criteria, the perception is called observation. When the observation arises from imposing certain particular conditions, the observation is called experimentation. These are the definitions linked to this concept that we require.
Language Lm
Language Lm is the collection of m words (letters, magnitudes, variables, notes, lines, symbols, etc.) that are combined to compose phrases (propositions, equations, images, sounds, etc.), and with them texts that are employed to represent a slice of reality.
A perception Lm R
A perception Lm R of a slice of reality R is a text of words and propositions in a language L
m
that represents a slice of reality R with a particular spatial resolution ΔΩ and a particular temporal resolution Δτ.
The Spatial Resolution ΔΩ
The spatial resolution ΔΩ of a piece of knowledge is the size of the region of space in which the representation is (or is considered to be) invariant. The temporal resolution
Δτ of a representation is the length of the period of time in which the representation is (or is considered to be) invariant. The size
|L
m
R|
of a perception
L
m
R is the number of symbols (letters, words, propositions, etc.) used in it. The universe of reference
U of a slice of reality R is a set of slices of reality R
j constructed or selected bearing in mind the differences with R: {R,R
j
} for j = 1,2,…n.
An Observation O
An observation O of a slice of reality R is the set of representations in accordance with L
m
of all the m slices of reality of a universe of reference included in the slice R to be observed:
$$ O = \left\{ {L_{m} R,L_{m} R^{j} } \right\} \, j = 1,2, \ldots m. $$
The Size |O|
The size |O| of an observation is the number of words in L
m
of the representation.
Every observation of reality consists, then, of perceptions of reality. It is quite possible for a reality to be perceived but for it to be difficult or impossible to observe. For example, the elliptical trajectory of a planet around a star is perfectly observable if it is perceivable because we can break down the movement into spatial and temporal elements in order to ascertain the differences between similarities (a single planet around a single star, different planets around the same star, different planets around different stars, etc.). The behavior of a galaxy can be perceived in an instant, but its evolution over time is difficult to observe due to its slowness in relation to the time allotted to an observer. In this situation, however, it is always possible to construct a universe of reference using the perception of different galaxies of different ages in different conditions. In contrast, what we call a mystical experience can be perceived, but it is very difficult to observe. With these intuitions, we can already put forward two of the fundamental concepts of the schema we wish to build: reality and understanding (of reality).
The concept of observation can be summed up as a construction achieved by means of differences between similar realities.
Comprehension (of the Observation of Reality)
Arriving at an understanding of reality is the central concept of the SM. Curiously, it admits of a definition symmetrical with the earlier concept of observation. If the observation is a construction built up of differences between similarities, then understanding can be defined as a construction achieved by similarities between different realities. As in the case of observation, understanding requires a language.
A Comprehension C
A comprehension C (of an observation O of a slice of reality R in relation to a universe of reference {R,R
j
}) is a representation in accordance with L
m
of the possible intersections between the slice of reality R and the other slices of reality that make up the universe of reference R
j, that is to say, an understanding is a representation of similarities between differences.
That is, for a particular language L
m
, understanding can be represented as
$$ C = \{ R \cap R^{j} ,R \cap R^{i} \cap R^{j} ,R \cap R^{i} \cap R^{j} \cap R^{k} , \ldots \} \;for\quad i,j,k = 1,2, \ldots ,q $$
In other words, understanding is made up of everything shared in common by the various slices of reality of the prepared universe under observation.
Accurate observation of the motion of the planets, for example, leads to a differential equation that compresses all understanding of such a slice of reality. And from such an understanding, it is possible to anticipate and reconstruct the motion of any planet in any galaxy in the universe.
However, some things are not scientifically understandable. In the case of a mystical experience, scientific comprehension seems unattainable. The experience in itself is perceivable, yet planning an observation is practically impossible—how can one define a universe of observation with an unrepeatable slice of reality?
The degree of intelligibility of scientific understanding calls for a number of additional definitions: The size of an area of comprehension |C| is the number of words in L
m
of the comprehension C. The universality
U of an area of comprehension C is the reunion of all the slices of reality that share this comprehension.
$$ U = \cup_{\upsilon } \{ L_{m} R^{\upsilon } |C \subset L_{m} R^{\upsilon } \} \quad\upsilon = 1,2, \ldots_{{}} $$
An observation and an understanding are always finite. Yet the domain of validity of an area of comprehension, its universality U, may be infinitely large. This is not to say that two infinite universalities must necessarily be the same in size. For example, the laws of classical mechanics and those of relativistic mechanics are finite. However, both theories have infinite universalities even though classical mechanics will always be contained in relativistic mechanics when the reverse is not true. Cantor’s (1915) theory is clear in this respect. There are infinite natural numbers, infinite rational numbers, and infinite real numbers, yet the infinity of real numbers is greater than the infinity of rationals, and the infinity of rationals is larger than the infinity of naturals.
The Degree of Universality
It is possible to establish an order between different universalities, U1 and U2, simply by defining that the degree of universality of U1 is greater than that of U2 if the first contains the second. This criterion is fundamental if the SM is, as we propose, to demand the maximum universality possible.
The Degree of Intelligibility μ
The degree of intelligibility μ of an area of understanding depends on the relationship between the size of this understanding and the size of the observation that preceded it. There are two aspects to understanding: one derives from what is common to what is different, and the other derives from the simplest expression of the first. In this latter respect, the greater the compression, the greater the understanding, enabling us to formulate the degree of intelligibility in the following manner:
$$ \mu = 1 - \left| C \right|/\left| O \right| \, \mu \in \left[ {0,1} \right] $$
If μ = 1, intelligibility is at its maximum and it occurs when maximum comprehension corresponds to maximum compression: |C| < < |O|. At the other extreme, intelligibility is at its lowest when μ = 0, i.e., when the observation is in itself also the best understanding: |C| = |O|. Chaitin-Kolmogorov complexity theory illustrates this way of seeing things (see the “Selection of the Comprehension (OC) of a UO” section).
There are, then, degrees of intelligibility. The degree of intelligibility of planetary motion is high since the size of the understanding is finite (Newton’s three laws and the law of gravitation), whereas the size of the observation may be made infinitely large. Another slice of reality, such as the behavior of a family group of gorillas, will undoubtedly result in the size of the observation and the size of the understanding being much closer. Different ways of understanding a single reality can, therefore, be arranged in order according to their degree of intelligibility. Kepler’s laws, for example, are a good understanding of the motion of the planets around the sun, but Newton’s laws have a greater degree of universality and intelligibility. The degree of intelligibility will also lead to a good criterion if it is appropriate, as is the case, that the SM should demand the highest possible intelligibility.