Darwin Among the Machines (10 page)

Boole also recognized that error and unpredictability, however foreign to the laws of Newtonian physics and formal logic, may be essential to our ability to think. “The slightest attention to the processes of the intellectual world,” concluded Boole, “reveals to us another state of things. The mathematical laws of reasoning are, properly speaking, the laws of right reasoning only, and their actual transgression is a perpetually recurring phenomenon. Error, which has no place in the material system, occupies a large one here.”
³³

An unbridged chasm separates our understanding of the logic of mental processes from our understanding of how these processes are executed in the brain. “One finds there only a confused mass in which nothing unusual appears but which nevertheless conceals some kind of filaments of a fineness much greater than that of a spider's web,” wrote Leibniz in 1702. “For the subtlety of the spirits contained in these passages is equal to that of light rays themselves.”
³⁴

Among the first to attempt to close the gap between neurology and mind was the English physician Alfred Smee (1818–1877), a prolific investigator whose contributions spanned numerous disciplines, from
The Potato Plant, Its uses and properties, together with the cause of the present malady
(1846) to a pioneering and widely reprinted sixpence broadsheet,
Accidents and Emergencies; A Guide for their Treatment before the arrival of Medical Aid
. The son of William Smee, accountant general to the Bank of England, Alfred grew up within the walled compound of the bank, spending long hours in an improvised laboratory on the ground floor of his father's residence, where he invented a new system for splinting fractures (1839), Smee's battery (1840), and other innovations that gained him fame if not reward. In 1841, at the age of twenty-two, he was appointed surgeon to the Bank of England, “a post which had been especially created for him by the directors . . . who thought that the bank could turn his scientific genius to good account.”
³⁵
Smee, who had a passion for all things electrical, invented the electrotype plate and by applying the technique to the printing of counterfeit-resistant English banknotes proved the directors' instincts right. Smee's two great interests were combined in a wide-ranging work of electrophysiology.
Elements of Electro-Biology; or, the Voltaic mechanism of Man
(1849), abridged and illustrated for a popular audience under the title
Instinct and Reason
in 1850. Smee introduced the use of electricity in diagnostic medicine and published a pamphlet entitled
The Detection of Needles, and other Steel Instruments,
impacted in the Human Frame
(1844)—an occurrence all too common in the industrial workplace of his day.

Smee worked both in theory and in the laboratory to explain the electrochemical basis of vision, sensation, memory, logic, and the origination and recombination of ideas. He believed that the mental powers of animals, human beings, and mechanisms were different not in kind but in degree. His definition of consciousness has seen scant improvement in 150 years. “When an image is produced by an action upon the external senses, the actions on the organs of sense concur with the actions in the brain; and the image is then a
Reality
. When an image occurs to the mind without a corresponding simultaneous action of the body, it is called a
Thought
. The power to distinguish between a thought and a reality is called
Consciousness
,” he wrote in his
Principles of the Human Mind deduced from Physical Laws
, published in 1849.
³⁶
As Leibniz envisioned the principles of digital computation, so Alfred Smee envisioned the crude beginnings of a theory of neural nets. “On attending the Physiological Lectures of Professor Mayo, I was remarkably struck with the unsatisfactory account of the functions of the brain, and I was surprised that so little appeared to have been done in connecting mental operations with that organ to which they were due,” he wrote in the introduction to his
Process of Thought Adapted to Words and Language, together with a description of the Relational and Differential Machines
.
³⁷

After considering what little was known concerning neural function at the time, Smee concluded that “every idea, or action on the brain, is ultimately resolvable into an action on a certain combination of nervous fibres, which is definite and determinable, and, regarding the sum total of the nervous fibres, is a positive result over a certain portion only, which has a distinct and clearly defined limit.”
³⁸
He was on the right track, but only half right, since he neglected the concept of neural inhibition that is central to the computational and representational powers of neural nets. His system was based on loosely defined analogies between the branching, combinatorial nature of the nervous system and the branching, combinatorial structure of language, logic, and ideas.

Taking the same top-down approach to semantic analysis that would be followed by the artificial intelligence industry in another hundred years, Smee developed a method for parsing natural language by means of a geometric series of symbols (“cyphers”) that would render the meaning of any given sentence exact. “This mode of notation may, at first sight, appear more complicated than ordinary language,” he wrote, “but if carefully studied, it will be found to
afford us an artificial mode of reasoning, which, although immensely inferior to that which is in actual operation by the elaborate machine furnished us by nature, yet as far as it goes, may be conducted by fixed and immutable laws.”
³⁹
By analysis of “this most exact form of language,” Smee made the leap between mind and mechanism, concluding that “it is apparent that thought is amenable to fixed principles. By taking advantage of a knowledge of these principles it occurred to me that mechanical contrivances might be formed which should obey similar laws, and give those results which some may have considered only obtainable by the operation of the mind itself.”
⁴⁰

Unlike later proponents of neural and semantic nets, Smee made no grand promises of thinking machines, but merely suggested the development of small-scale logical automata for research. “When the vast extent of a machine sufficiently large to include all words and sequences is considered, we at once observe the absolute impossibility of forming one for practical purposes, inasmuch as it would cover an area exceeding probably all London,” he cautioned. “Nevertheless, those lesser machines containing but a few elements, exemplify the principles of their operation, and demonstrate those laws of induction, deduction and relation, the right use of which cannot fail to render our thoughts more accurate, and our language more precise.”
⁴¹

Smee understood the inescapable bureaucracy and rigidly enforced assumptions of formal systems. He suggested, with an unspoken nod to the thirteenth-century
Ars Magna
of Ramon Lull, that one of his differential machines “might be beneficially brought into use by those who use fixed and unchangeable creeds; for if they be arranged correctly then any deviation from them would be immediately registered. It must be apparent that such a machine would not estimate the quality of the creed, but only show whether any new creed, or portion of creed, coincided or not with the former creed. For whether the creed inferred a belief in the true God, in Mohammed, in ibises, crocodiles, or saints, in the power of the Virgin, or winking pictures of her, or the qualities of relics, or the virtues of images, or in the parties' own inspiration, the effect would be the same.”
⁴²

“By using the relational and differential machines together,” concluded Smee, “from any definite number of premises the correct answer may be obtained, by a process imitating as far as possible, the natural process of thought.”
⁴³
But Smee advised his public to “rely upon the abilities which it has pleased Providence to give to them, and not seek assistance from extraneous sources,” and made only passing reference to the potentials of electrical logic machines, keeping the prospect of so upstaging nature to himself. “In animal bodies we
really have electro-telegraphic communication in the nervous system,” he had written in
Instinct and Reason
, juxtaposing micrographic plates of brain tissue with electrotyped illustrations of how he imagined the electrical network to be configured in the brain. He built his own simple electric telegraph, a system “of a somewhat similar character, as it communicates intelligence from one spot to another,”
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and connected it to a thermometer in his greenhouse so as to transmit an alarm signal when extreme temperatures threatened his collections of exotic plants. In 1849 he suggested “in a remote and imperfect manner” how to construct an artificial ear that would translate sound into electrical signals and expressed “no doubt but that a perfect acoustic telegraph could be made, which shall be acted upon by sounds, and have the power of transmitting them to any distance.”
⁴⁵

Speculating how vision might be processed by eye and brain, Smee introduced concepts we now know as pixelization, bit mapping, and image compression. He suggested both digital facsimile and analog television at a time when photography was still in its formative years. “From my experiments I believe that it is sufficiently demonstrated that the light falling upon the [optic] nerve determines a voltaic current which passes through the nerves to the brain,” he wrote. “From this fact we might make an artificial eye, if we did but take the labour to aggregate a number of tubes communicating with photo-voltaic circuits. . . . Having one nervous element, it is but a repetition to make an eye; and . . . there is no reason why a view of St. Paul's in London should not be carried to Edinburgh through tubes like the nerves which carry the impression to the brain.”
⁴⁶

But Smee's loyalty was to the vegetable kingdom, not the kingdom of machines. “There is nothing to prevent man from forming an elaborate engine, which should work by change of matter [i.e., electricity] . . . but . . . he must, with the Psalmist, exclaim, ‘Such knowledge is too wonderful and excellent for me.'”
⁴⁷
Smee devoted the rest of his life to horticulture and ecology, publishing a monumental volume,
My Garden; its Plan and Culture together with a general description of its geology, botany, and natural history
(1872), illustrated with thirteen hundred plates. “Its author has endeavoured, so to speak, to catch Nature, animate and inanimate, in a trap of some seven acres and a half, and to chronicle all its everyday features with a sort of Boswellian fidelity,” wrote the
Saturday Review
.
⁴⁸
Babbage died alone, obsessed by the unfulfilled promise of his machines; Smee died at peace, surrounded by a garden full of grandchildren and plants. “Had Smee lived a few years later,” wrote D'arcy Power, “he would have made himself a great reputation as an electrical engineer.”
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Hobbes's ratiocination, Leibniz's
calculus ratiocinates
, Babbage's mechanical notation, Boole's laws of thought, and Smee's conceptual cyphers all attempted to formalize the correspondence among a mechanical system of things, a mathematical system of symbols, and our mental system of thought and ideas. All approaches to formalization from the time of Hobbes until today have been haunted by similar questions: Is the formalization consistent? Is it complete? Does it correspond, in whole or part, to the real world? To the way we think? These questions hinge on the definition of consistency and completeness, available in two different strengths. A formal system is syntactically, or internally, consistent if and only if the system never proves both a statement and its negation and syntactically complete if one or the other is always proved. The system is semantically consistent, under a particular external interpretation, if and only if it proves only true statements and semantically complete if all true statements can be proved.

In 1931, Austrian logician Kurt Gödel (1906–1978) expanded the horizons of mathematics by proving, for both definitions, that no formal system encompassing elementary arithmetic can be at the same time both consistent and complete. Within any sufficiently powerful and noncontradictory system of language, logic, or arithmetic, it is possible to construct true statements that cannot be proved within the boundaries of the system itself.

Gödel achieved this conclusion by a technique now known as Gödel numbering, whereby all expressions within the language of a given formal system are assigned unique identity numbers and thereby forced to obey the manipulations of a strictly arithmetic bureaucracy from which it is impossible to escape. (“Gödel, having grown up in the Austrian
Kaiserreich
, famous for its bureaucracy, must have been familiar with the process,” says my mother, no stranger to bureaucracy, being Swiss.) The Gödel numbering, like the characteristic numbering of Leibniz, is based on an alphabet of primes. But Gödel, unlike Leibniz, provided an explicit coding mechanism so that translation between compound expressions and their Gödel numbers remains a two-way street.

“Metamathematical concepts (assertions) thereby become concepts (assertions) about natural numbers or sequences of such, and therefore (at least partially) expressible in the symbolism of the system . . . itself,” wrote Gödel in the introduction to his proof.
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By some ingenious twists of logic and number theory. Gödel constructed a formula, the Gödel sentence, “which asserts its own unprovability” even though it can be perceived by reasoning outside the system as
being true. The Gödel sentence is loosely equivalent to a self-referential statement that says, “This statement cannot be proved.” But saying this with words and saying it with mathematics are two different things. The Gödel numbering enables the formalization of this self-reference by means of a sentence (G) saying, in effect, “The sentence with Gödel number
g
cannot be proved,” where the details of the system are manipulated so that the Gödel number of G is
g. G
cannot be proved within the specified system and so it is true. Since, assuming consistency, its negation cannot be proved, the Gödel sentence is therefore formally undecidable, rendering the system incomplete. Where Leibniz and his followers had dreamed of a universal coding that would allow the calculation of all truths, Gödel showed that even a system as simple as ordinary arithmetic could never be made complete. Thus Gödel brought Leibniz's dream of a universal, all-encompassing formalization to an end.