Saturday, 16 April 2016

Logic, information and the world mind

This is an excerpt from an upcoming book: Einstein's Enlightenment.

Logic is the branch of philosophy which studies valid reasoning. When Aristotle introduced the subject of logic to philosophy, one of his most significant contributions was the logical syllogism where the truth of some statements is obvious merely due to the form of the statements. For example:

1) All men are mortal
2) Socrates is a man
3) Therefore, Socrates is mortal

It is obvious, given the first two statements, that the third must be true.

Later three basic logical operators were identified: ‘and’, ‘or’ and ‘not’. These operators could be used to describe relationships between logical variables which may have the value of either ‘true’ or ‘false’. All possible relationships between logical variables may be described using these operators leading to the field of Boolean Algebra. In turn logical relationships cast in terms of Boolean Algebra serve as the foundations from which all mathematics may be derived. This astonishing understanding shows that all mathematics is but the implications held within binary true/false relationships.

A more concrete example of the power of logic is that the three logical operators may be cast in the form of electronic circuits and used to construct a computer. General purpose computers or universal Turing machines are capable of computing all mathematically computable functions.

Therefore, logic is a central discipline at the basis of mathematics, computation and much of science.  Charles Sanders Peirce, the great American philosopher, made a number of fundamental advances in these fields and was able to envision the central role played by logic within philosophy. He understood logic primarily as the rules by which the ‘world mind’ composing the universe evolves, an observation very much in the spirit of Einstein’s Enlightenment. A first step towards this conclusion is Peirce’s contention that logic forms the rules of thought or mind (1):

The term ”logic” … in its broader sense, it is the science of the necessary laws of thought

Second, Peirce viewed ‘thought’ and ‘mind’ as not necessarily associated with brains but rather as general properties of nature (1):

Thought is not necessarily connected with a brain. It appears in the work of bees, of crystals, and throughout the purely physical world; and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there. … Not only is thought in the organic world, but it develops there

Peirce was a first class logician, mathematician and scientist but he understood these within a context of mind or thought; logic, mathematics and science merely expressed the rules for clear thought. He portrayed mind or thought as a property of nature and thus ventured beyond the usual materialistic view of science.

To a surprising extent Peirce demonstrated many of these basic relationships between logic and mind a half century earlier than the pioneers of our current information and computer ‘revolutions’. Sadly, his work was ignored and he was left largely without credited.

Peirce was the first to prove that all logical propositions, including Boolean algebra and mathematics, could be constructed from relations between two primitive fundamental logical quantities: ‘not-and’ and ‘not-or’. He used this insight to understand that these two logical quantities could be implemented in electronic circuits and thus was the first to suggest that all logical statements could be cast in the form of electronic circuitry. He and his colleagues actually developed a wiring diagram of a ‘mechanical logic machine’ (2).

In this he very much foretold the basis of our current understanding of computation. Given his vision that the processes of logic operate within a universal property of thought or mind he also presaged the current view among physicist that the universe is primarily engaged in the computation of an evolving logic (3).

Peirce’s great understanding of the relationship between logic and thought or mind was ignored by researchers largely because of his meager status on the peripheries of academia but also because he was so far ahead of his time. It was not until 1943 that Warren McCulloch and Walter Pitts rediscovered the same fundamental relationships underlying logic and thought (4):

Many years ago one of us, by considerations impertinent to this argument, was led to conceive of the response of any neuron as factually equivalent to a proposition which proposed its adequate stimulus. He therefore attempted to record the behavior of complicated nets in the notation of the symbolic logic of propositions. The "all-or-none" law of nervous activity is sufficient to insure that the activity of any neuron may be represented as a proposition. Physiological relations existing among nervous activities correspond, of course, to relations among the propositions; and the utility of the representation depends upon the identity of these relations with those of the logic of propositions.

This seminal understanding of logical propositions is fundamental to both the fields of artificial intelligence and cognitive psychology.

The philosophy developed by Peirce puts logic at its core, it insists that the world acts logically. This logic is played out through semiotic, the relationship among object, sign, and an interpretant or receiver of the sign.

In Peirce’s philosophy, it is the shunting of signs between objects and interpretants - which themselves may act as signs - that causes the universe to learn and evolve. Our minds are described as subsumed within this greater world mind. According to his philosophy, information bearing signs are perhaps the most fundamental entity in the universe (5):

all this universe is perfused with signs, if it is not composed exclusively of signs


Figure 4: Peircean semiotics describes logic and information transfer as a triadic relationship among objects, signs which signify the objects, and interpretants which draw conclusions when receiving signs. In the case pictured, the fire is the object, the smoke is a sign signifying the fire and a human who sees the smoke is the interpretant who interprets the smoke as signifying a fire. Peirce consider semiotics as the basic rule governing the relationship of entities.

This view is remarkably similar to that of the current ‘information revolution’; the understanding, across many branches of science, that information is perhaps the most fundamental entity in our universe. How could Peirce have scooped current researchers by over fifty years? Why doesn’t our information age rhetoric acknowledge the contributions made by Peirce’s philosophy? The answer is that Peirce’s philosophy was ignored, in fact it was nearly lost.

Upon his death, Peirce’s widow sold his largely unpublished papers to Harvard’s library. There they moldered for many years and many still remain unpublished. Although currently gaining in recognition Peirce’s work is still largely unknown. Perhaps Peirce was simply too far ahead of his times to be understood.

Instead it was the mathematical description of information, developed by Claude Shannon (6) in 1948, which would form the foundation of information theory. The business of Bell Labs, where Shannon was a researcher, was to understand the nature of information and Shannon’s work is widely regarded as the foundation of our modern understanding of information and its context.

In his description, information is related to probability; information is a function of probability, if a high probability is assigned to a possible outcome of an event and if it happens to be the actual outcome then there is not much information gain but if a low probability is assigned to an actual outcome then a lot of information is received. A complete set of possible outcomes may be assigned probabilities and the resulting distribution constitutes a probabilistic model of the event. 
The information contained in evidence regarding the actual outcome implies a recalculation of all the probabilities making up the model. The model is updated in accord with Bayes’ theorem. In this understanding information does not exist in isolation, something must be informed and that something is a model. This view of information’s reliance on a model is analogous to Price’s view of a sign’s reliance on an interpretant.

Only now is an important implication of Shannon’s theory becoming apparent, that although information may be a fundamental component of the universe it is far from simple. Information does not exist outside of the complex context of a probabilistic model (7). Long before Shannon, Peircean semiotic captured that notion, that information or signs do not have an existence outside of a context of objects which generate the sign and interpretants which interpret them.

One of the many technical contributions which Peirce made to logic was his development of existential graphs. This system of logical notation served to greatly simplify complicated logical expressions and his work initiated a program within the study of logic towards notations which would reveal the underlying simplicity of logical expressions. After all, in the end, once all the values of variables are stated, every logical expression is simply either true or false.

The program for the simplification of logical notation which Peirce initiated was concluded in 1969 with the publication of George Spencer-Brown’s Laws of Form (8). Spencer-Brown developed an ultimate simplification of logical notation which uses only one symbol. As Louis Kauffmann explains (9):











Using this symbol two types of logical operations may be described, the first a type of repetition with two marks in succession, the second adding no new information. We may think of the mark as the crossing of a distinction; from that view the first operation merely states that to twice state a position in regard to a distinction is the same as only stating it once. The second logical operation is a type of negation with one mark inside another which cancel each other to nothing. From the view of distinction, we can interpret this to say that if we cross a distinction twice it is the same as not crossing it at all.

The mark may be conceived of as a type of negation and our two operations as two types of double negatives. The first is redundant as in ‘I cannot, I cannot go to sleep’, the second ‘cannot’ is redundant and provides no further meaning than does ‘I cannot go to sleep’. The second type of double negative is cancellation as in ‘I cannot not go to sleep’. Here the negatives cancel and the meaning is ‘I will go to sleep’.

Spencer-Brown used two relationships between the marked symbols to indicate these two types of negation, the first a repetition and the second a cancellation:











Using this deceptively simple notation it is possible to almost immediately see the solution to some very complicated logical puzzles. For example, Spencer-Brown demonstrates the solution to a famous puzzle posed by Lewis Carrol (10):



This amazing ability of Spencer-Brown’s system to simplify complicated logical problems is just one aspect of its power. It has been demonstrated that it can also be used to derive the axioms of Boolean Algebra and hence all of mathematics. It is an amazing fact that all of mathematics may be derived merely from the nature of a binary distinction (10).

The great mathematician and physicist Louis Kauffman has made extensive investigations into Spencer-Brown’s research and has concluded (8):

It remained for Spencer-Brown (some fifty years after Peirce and Nicod) to see the relevance of an arithmetic of forms underlying his notation and thus putting the final touch on a development that, from a broad perspective, looks like the world mind doing its best to remember the significant patterns that join logic, speech and mathematics.

Indeed, the image of a ‘world mind’ at the basis of the universe appears almost inescapable in the philosophical systems of both Peirce and Spencer-Brown. This may be disturbing to some as it denies a long-held scientific assumption that materialism is the basic property of the universe but this would seem unavoidable once one concedes that information is also a basic constituent. After all mind may only be a euphemism for information processing which, it now seems clear, is a basic natural operation.

We should remember that Spencer-Brown’s powerful system is based upon only two simple observations concerning the nature of distinctions. We should also remember that the bit, the fundamental unit of information, is merely the unit of information which records a binary distinction.


Figure 5: George Spencer-Brown

Some researchers have marveled at ‘the unreasonable effectiveness of mathematics in the natural sciences’ (11); why are so much of the natural sciences so well described by the patterns of mathematics? We may perceive the outlines of an answer:

1)    Information is perhaps the most fundamental component of reality.
2) Information is composed of yes/no binary distinctions.
3)     Any effective description of reality, such as mathematics, must be derivable from the nature of binary distinctions.  

Inherent to Spencer-Browns conception of ‘distinction’ is both a ‘mark’ to mark the distinction and an ‘observer’ to interpret it. The concluding sentence of Laws of Form, is:

We see now that the first distinction, the mark and the observer are not only interchangeable, but, in the form, identical.

It has been noted that Spencer-Brown’s paradigm meshes well with Peircean semiotics (8) if we equate distinction with object, mark with sign and observer with interpretant. The choice of the word ‘observer’ should be understood here in the Einsteinian manner as a generalized entity able to react to measurements or data or some other instantiation of a sign. I suggest the word ‘model’ be used to guard against anthropocentric interpretations which may be attached to ‘observer’.

Spencer-Brown’s logic should be understood as an extension of Peirce’s philosophical program, one which presages Einstein’s Enlightenment. Although the model of logic and semiotics developed by Peirce and Spencer-Brown may be suggestive of a world mind capable of processing and building knowledge it lacks details. Clearly it offers some insight into the relationship between models (interpretants) and that which is modeled (objects), a relationship which might be considered to be knowledge, however details of the mechanisms which relates object to sign and sign to model are lacking.

If semiotics is to serve as a useful paradigm for knowledge it should be expected to encompass science. Indeed, science, understood as a process of Bayesian inference, may be cast in a triad form compatible with semiotics: phenomena, data, and model. The inclusion of Bayesian inference brings a particular strength to the semiotic paradigm. Bayesian experimental design provides a mechanism for extracting data (signs) from the phenomena (object) so as to maximize the expected knowledge gain of the model (interpretant). It also provides the mechanism for using data to most effectively update models in the form of the Bayesian update.

Probability distributions employed as models within Bayesian inference have the property of information entropy; the amount of information which separates the model from certainty. If the log of the probabilities used in the inference is to base 2 then this information is measured in bits. Bits are the basic unit of distinction. The model’s entropy is the number of binary distinctions which separate it from certainty. While models treated within Bayesian inference may be separated from certainty by any finite number of distinctions, the special case of one bit separations from certainty is the case where Bayesian inference becomes equivalent to classical logic. In Spencer-Brown’s terms single bit models are ‘observers’ of the first distinction.

Models within Bayesian inference are moved towards certainty by incorporating the implications of data (sign) into the model (interpretant); through the process of Bayesian updating. Mathematically this is the unique method of moving a model towards certainty (12); that is of increasing knowledge. 

Bayesians have frequently committed the error of assuming that knowledge is a human property and that Bayesian inference is solely descriptive of human activities. Spencer-Brown did not indulge in this misconception; he maintained a focus on the world mind (10):

Thus we cannot escape the fact that the world we know is constructed in order (and thus in such a way as to be able) to see itself.

This is indeed amazing.

Not so much in view of what it sees, although this may appear fantastic enough, but in respect of the fact that it can see at all.

But in order to do so, evidently it must first cut itself up into at least one state which sees, and at least one state which is seen. In this severed and mutilated condition, whatever it sees is only partially itself.

Like Peirce, Spencer-Brown did not have a large impact on the scientific community. His work is provocative and its reception was varied but muted. Some extreme claims have been made for its mathematical importance. The book jacket of some editions of Laws of Form contains an endorsement from Bertram Russell “Not since Euclid’s Elements have we seen anything like it’. The book jacket of my edition contains excerpts from reviews in Nature and the British Journal of the Philosophy of Science; both describe it as ‘a work of genius’. Others however have dismissed it as a work of mysticism with little mathematical content. That view has not been mitigated by Spencer-Brown’s subsequent claim to be the reincarnation of the Buddha and not just any old Buddha but rather the Buddha who comes only once every 2500 years. Perhaps as a result his work has been largely shunned within the mathematical community.

A ‘world mind’ may at first seem more akin to a supernatural rather than a scientific concept but it is clearly resolved within a scientific context if we accept that:

1)      Information is a fundamental component of the universe.
2)     The universe evolves through numerous nested Darwinian processes (universal Darwinism).
3)     Darwinian processes are isomorphic to Bayesian inference. The process of evolution accumulates information into knowledge through an evidence-based processes.

This scientific context describes a continual universal process of evolution which processes information into knowledge. This is a process, aptly described as that of a ‘world mind’, in which the world comes to better know itself.

This view, initiated by Peirce, places logic or the ‘necessary laws of thought’ guiding the world mind at the foundation of metaphysics or the study of the nature of reality.

Bibliography

1. Peirce, Charles S. Collected Papers of Charles Sanders Peirce. 1906. p. 101. Vol. 6.
2. Wikipedia. Charles Sanders Peirce bibliography. Wikipedia. [Online] [Cited: 1 10, 2016.] https://en.wikipedia.org/wiki/Charles_Sanders_Peirce_bibliography.
3. Lloyd, Seth. Programming the Universe. s.l. : Vintage; Reprint edition, 2007.
4. A Logical calculus of the ideas immanent in nervous activity. McCulloch, Warren and Pitts, Walter. 1943, Journal of Mathematical Biophysics, Vol. 5.
5. Peirce, C.S. Collected papers of Charles Sanders Peirce: The basis of pragmatism. 1906. Vol. Voume 5.
6. A mathematical theory of communications. Shannon, Claude. 1948, Bell System Technical Journal.
7. Campbell, John O. Darwin does physics. s.l. : CreateSpace, 2015.
8. The Mathematics of Charles Sanders Peirce. Kauffman, Louis H. 2001, Cybernetics & Human Knowing, Vol.8, no.1–2, pp. 79–110.
9. Kauffman, Louis. Laws of Form - An exploration in mathematics and foundations. s.l. : Unpublished rough draft.
10. Spencer-Brown, G. The Laws of Form. New York : E.P. Dutton, 1979.
11. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Wigner, Eugene. 1960, Communications on Pure and Applied Mathematics 13, pp. 1–14.
12. Jaynes, Edwin T. Probability Theory: the logic of science. Cambridge : Cambridge University Press, 2003.