The Mechanics of Reason
Aristotle fathered the syllogism, or at least was first to investigate it rigorously. He defined it as a formal argument in which the conclusion follows logically from the premises. There are four common statements of this type:
| All S (for subject) | is P (for predicate) |
| No S (for subject) | is P |
| Some S (for subject) | is P |
| Some S (for subject) | is not P |
Thus, Aristotle might say “All men are mortal” or “No men are immortal” as his subject. Adding an M (middle term), “Aristotle is a man,” as a minor premise, he could logically go on and conclude “Aristotle, being a man, is thus mortal.” Of course the syllogism unwisely used, as it often is, can lead to some ridiculously silly answers. “All tables have four legs. Two men have four legs. Thus, two men equal a table.”
Despite the weaknesses of the syllogism, nevertheless it led eventually to the science of symbolic logic. The pathway was circuitous, even devious at times, but slowly the idea of putting thought down as letters or numbers to be logically manipulated to reach proper conclusions gained force and credence. While the Greeks did not have the final say, they did have words for the subject as they did for nearly everything else.
Let us leave the subject of pure logic for a moment and talk of another kind of computing machine, that of the mechanical doer of work. In the Iliad, Homer has Hephaestus, the god of natural fire and metalworking, construct twenty three-wheeled chariots which propel themselves to and fro bringing back messages and instructions from the councils of the gods. These early automatons boasted pure gold wheels, and handles of “curious cunning.”
Man has apparently been a lazy cuss from the start and began straightway to dream of mechanical servants to do his chores. In an age of magic and fear of the supernatural his dreams were fraught with such machines that turned into evil monsters. The Hebrew “golem” was made in the shape of man, but without a soul, and often got out of hand. Literature has perpetuated the idea of machines running amok, as the broom in “The Sorcerer’s Apprentice,” but there have been benevolent machines too. Tik-Tok, a latter-day windup man in The Road to Oz, could think and talk and do many other things men could do. He was not alive, of course, but he had the saving grace of always doing just what he “was wound up to do.”
Having touched on the subject of mechanical men, let us now return to mechanical logic. Since the Greeks, many men have traveled the road of reason, but some stand out more brightly, more colorfully, than others. Such a standout was the Spanish monk Ramón Lull. Lull was born in 1232. A court page, he rose in influence, married young, and had two children, but did not settle down to married domesticity. A wildly reckless romantic, he was given to such stunts as galloping his horse into church in pursuit of some lady who caught his eye. One such escapade led to a remorseful re-examination of himself, and dramatic conversion to Christianity.
He began to write books in conventional praise of Christ, but early in his writings a preoccupation with numbers appears. His Book of Contemplation, for example, actually contains five books for the five wounds of the Saviour, and forty subdivisions for the days He spent in the wilderness. There are 365 chapters for daily reading, plus one for reading only in leap years! Each chapter has ten paragraphs, symbolizing the ten commandments, and three parts to each chapter. These multiplied give thirty, for the pieces of silver. Beside religious and mystical connotations, geometric terms are also used, and one interesting device is the symbolizing of words and even phrases by letters. This ties in neatly with syllogism. A sample follows:
… diversity is shown in the demonstration that the D makes of the E and the F and the G with the I and the K, therefore the H has certain scientific knowledge of Thy holy and glorious Trinity.
This was only prologue to the Ars Magna, the “Great Art” of Ramón Lull. In 1274, the devout pilgrim climbed Mount Palma in search of divine help in his writings. The result was the first recorded attempt to use diagrams to discover and to prove non-mathematical truths. Specifically, Lull determined that he could construct mechanical devices that would perform logic to prove the validity of God’s word. Where force, in the shape of the Crusades, had failed, Lull was convinced that logical argument would win over the infidels, and he devoted his life to the task.
Renouncing his estate, including his wife and children, Lull devoted himself thenceforth solely to his Great Art. As a result of dreams he had on Mount Palma, the basis for this work was the assumption of simple premises or principles that are unquestionable. Lull arranged these premises on rotating concentric circles. The first of these wheels of logic was called A, standing for God. Arranged about the circumference of the wheel were sixteen other letters symbolizing attributes of God. The outer wheel also contained these letters. Rotating them produced 240 two-term combinations telling many things about God and His good. Other wheels prepared sermons, advised physicians and scientists, and even tackled such stumpers as “Where does the flame go when the candle is put out?”
Unfortunately for Lull, even divine help did not guarantee him success. He was stoned to death by infidels in Bugia, Africa, at the age of eighty-three. All his wheelspinning logic was to no avail in advancing the cause of Christianity there, and most mathematicians since have scoffed at his naïve devices as having no real merit. Far from accepting the Ars Magna, most scholars have been “Lulled into a secure sense of falsity,” finding it as specious as indiscriminate syllogism.
Yet Lull did leave his mark, and many copies of his wheels have been made and found useful. Where various permutations of numbers or other symbols are required, such a mechanical tool is often the fastest way of pairing them up. Even in the field of writing, a Lullian device was popular a few decades ago in the form of the “Plot Genii.” With this gadget the would-be author merely spun the wheels to match up various characters with interesting situations to arrive at story ideas. Other versions use cards to do the same job, and one called Plotto was used by its inventor William Wallace Cook to plot countless stories. Although these were perhaps not ideas for great literature, eager writers paid as much as $75 for the plot boiler.
Not all serious thinkers relegated Lull to the position of fanatic dreamer and gadgeteer. No less a mind that Gottfried Wilhelm von Leibnitz found much to laud in Lull’s works. The Ars Magna might well lead to a universal “algebra” of all knowledge, thought Leibnitz. “If controversies were to arise,” he then mused, “there would be no more reason for philosophers to dispute than there would for accountants!”
Leibnitz applied Lull’s work to formal logic, constructed tables of syllogisms from which he eliminated the false, and carried the work of the “gifted crank” at bit nearer to true symbolic logic. Leibnitz also extended the circle idea to that of overlapping them in early attempts at logical manipulation that foreshadowed the work that John Venn would do later. Leibnitz also saw in numbers a powerful argument for the existence of God. God, he saw as the numeral 1, and 0 was the nothingness from which He created the world. There are those, including Voltaire whose Candide satirized the notion, who question that it is the best of all possible worlds, but none can question that in the seventeenth century Leibnitz foresaw the coming power of the binary system. He also built arithmetical computers that could add and subtract, multiply and divide.
A few years earlier than Leibnitz, Blaise Pascal was also interested in computing machines. As a teen-ager working in his father’s tax office, Pascal wearied of adding the tedious figures so he built himself a gear-driven computer that would add eight columns of numbers. A tall figure in the scientific world, Pascal had fathered projective geometry at age sixteen and later established hydrodynamics as a science. To assist a gambler friend, he also developed the theory of probability which led to statistical science.
Another mathematical innovation of the century was that of placing logarithms on a stick by the Scot, John Napier. What he had done, of course, was to make an analog, or scale model of the arithmetical numbers. “Napier’s bones” quickly became what we now call slide rules, forerunners of a whole class of analog computers that solve problems by being actual models of size or quantity. Newton joined Leibnitz in contributing another valuable tool that would be used in the computer, that of the calculus.