## Monday, February 26, 2018

### George Boole and the Calculus of Thought

**by Richard Passov**

On the night of December 8th, 1864, George Boole, 49 years of age, in the grips of pneumonia, expired. He left a wife, Mary, and five daughters. Unfortunately, Mary had always carried two of his beliefs: the health benefits of long walks and the healing powers of homeopathic remedies. Late in a winter evening, after finishing his work at the University, under a cold rain, George walked the two miles to his home. Over the next several days, guided by her belief that the cure lay in the symptoms, Mary repeatedly doused George in cold water.

* * *

George was born in a small rail hub one hundred and fifty miles north of London, in the den of a cobbler who, said his wife, was *‘…good at everything except his own business of managing shop*.' Unable to afford an education, Boole's father nevertheless encouraged his son to study. Dutiful, George taught himself six languages, read the classics, read philosophy and studied his figures.

By twelve he had a reputation owing to his father's submission to the local newspaper of a poem translated from ancient Greek. Interested readers were surprised to discover that the translator was an unschooled boy. The Town Councilor, fancying himself an amateur mathematician, provided George access to a library filled with mathematics.

George calculated that math would be more remunerative than his first love, theology. Some things don't change. After enough self-study, hoping for tips from grateful parents he secured a position as an usher tutoring students in theater-sized classrooms. By nineteen, he opened his own school to which he gave the very unoriginal name: "The Classical, Commercial and Mathematical Academy."

Teaching during the day, studying at night, living with and the sole support for his extended family, George mastered the complete cannon of contemporary math. Mastered to such an extent that in 1837 he responded to an advertisement from a newly established mathematical journal with an astonishingly original piece of work.

David Gregory, a recent Cambridge graduate and editor of its math journal, read Boole's paper on Invariants - a field in mathematics that seventy five years later Einstein would rely upon to solve the riddle of General Relativity - and promptly invited the unknown author to Cambridge. Impressed by a quite self-effacing phenomenon working alone, Gregory proved generous. He mentored Boole through the process of journal writing then devoted three of the first six volumes of his journal to Boole's works. Stop at this point and you have a good story.

Emboldened by his publishing success, in 1839 Boole asked Gregory to support his candidacy for admission to Cambridge as an undergraduate math student. Much as today's young entrepreneur – think Zuckerberg or Gates – might be advised to avoid the high cost and limited value of pursuing a formal education, so Gregory advised Boole.

Of course Boole failed to find the wisdom in Gregory's advice. He funneled his disappointment into forty-one pages of math that stands as the basis for most practical solutions to differential equations. While Gregory was duly impressed by the work, he was also aware of a mysterious ailment, likely an undiagnosed cancer that would soon take his life. With his health in mind, he sent Boole's paper, accompanied by his highest recommendation, to the Royal Academy of Sciences.

While today's mid-level engineering student knows the basics of Boole's paper, the algebra of finite differences - a numerical method for solving differential equations - proved a considerable challenge for the Academy. After months of debate, the paper was given to two referees for a final judgment. While one recommended against publication, the second judge was more persuasive. Not only was Boole's work published, it became the first work of mathematics to be awarded the Royal Academy's coveted Gold Prize. Unfortunately, Gregory would not live to see the award ceremony.

* * *

During the years surrounding the turn of the 19th century, some seventy-five years after his death, London's mathematicians struggled under the legacy of Sir Isaac Newton. Unquestionably the most influential scientist of his time, when it came to managing his fame, Sir Isaac displayed a great capacity for shrewdness.

Twenty years after using *‘the Method of Fluxions and Fluents'* to unveil the foundations of classical physics, Newton saw the German polymath Gottfried Leibnitz's *‘Calculus'* in print and made the charge of plagiarism. For the next several years, Newton hounded Leibnitz, finally succeeding in presiding over a plenary meeting by the Royal Academy in which Newton, while chairing the review process, ghostwrote an indictment of Leibnitz. After publication, London's scientific establishment, believing they could remain at the vanguard of mathematics, turned their backs to the Continent.

Unfortunately, while Newton's contributions to physics endure, his method for developing his calculus proved an absolute dead-end. By the turn of the 19th century, in the field of higher math, France and Germany were far ahead.

Early in the 19th century a group of like-minded English academics formed the Analytical Movement to advocate for the ‘ *‘d'ism of the Continent versus the dot-age of the University' *– for Leibnitz over Newton. By the close of the 1830's, though much progress had been made, English Algebraists were fighting over the place of negative numbers. You *‘…may put a mark before one, which it will obey…but to attempt to take it away from a number less than itself is ridiculous.*' 1 (Frend, 18--)

As the arguments rolled into the 1840's, a new debate took hold. Through a series of colorful, public letters filled with Victorian insults, Augusts De Morgan, Chair of Mathematics at London University and Sir William Hamilton, University of Edinburgh's distinguished Professor of Logic and Metaphysics, began a feud over whether there was a place in logic for mathematicians.

Hamilton, the logician, seemingly unaware that math and logic had once been the same, wrote of math that it is *‘… in fact not a useful discipline at all except in what it has with some of the most despised pursuits and that if prosecuted far, it positively unfits the mind for the useful employment of its faculties on any other subject.'*

In 1847, with the debate at its peak, Boole sent De Morgan a work entitled: ** "The Mathematical Analysis of Logic."** In the introduction, Boole wrote that he was inspired by the debate between Hamilton and De Morgan to

*‘…resume an almost forgotten train of thought...that in addition to whatever logic may be concerned with, it also has a deeper system of relations that has everything to do with…the constitution of the mind.'*

Boole was the first to clearly see that *‘…It is not of the essence of mathematics to be conversant with the ideas of number and quantity.'* Mathematics, specifically Symbolic Algebra, he argued, could be applied to human reasoning. While the effort was incomplete, De Morgan recognized its genius and stepped in where Gregory had left off.

Through correspondence, De Morgan learned of Boole's predicament of working full time as a headmaster while pursuing math in the late evenings. He encouraged Boole to apply for the Professorship of Mathematics at the newly established Queens College in Cork, Ireland. Without Boole's knowledge, he secured recommendations. The effort was a success. In 1849, Boole won a full Professorship and at the age of 34, separated from his parents and moved to Ireland.

After seven years of encouragement from De Morgan, in 1854 Boole produced his masterpiece: "** An Investigation of the Laws of Thought**." In the opening sentence, he described his work as a

*‘treatise… to investigate the fundamental laws of … the mind by which reasoning is performed'*and to apply the

*‘process of math'*, through which he believed he could

*‘…discover the rules of thought.'*

* * *

Self-taught, working alone, Boole believed he had found an exact agreement between how the mind operates during reasoning and the how it operates when using the language of Algebra. Exact, except in one aspect; what he called his special *‘Law of Thought.'*

Across more than 300 pages, Boole developed his *‘Calculus of Thought.'* He showed that, like all math, it can be manipulated through functions. If intermediate steps are uninterruptable, so long as the rules are followed, the end result will always be an interpretable *‘truth.' *

The square root of negative one – an imaginary number - has no stand-alone -interpretation, yet it is a stepping-stone in the mathematics of trigonometry. So *‘middle terms'* in the *Calculus of Thought* always lead to final terms that can be interpreted once translated back into the ‘…*medium of thought – into language*…' Language mapped into symbols, manipulated strictly through the process of math, always arrives at an interpretable statement.

The choice of symbols is arbitrary. The key is to fix meaning to symbols and then only admit rules that ‘… *founded exclusively upon the … fixed sense or meaning of the symbols employed…'* return to the symbols. If numbers are the symbols then the only allowable operations that can be performed on the numbers are those that return back to the set of numbers. Applying the rules to the symbols, ‘*theorems'* are discovered, one after another, as if jumping from one stone to another in a ‘*deductive'* process. If you follow this ‘*process of math'*, then according to Boole you will always land on *‘truths' *and you will, if time permits, discover *‘…all of the truths…'* that can be known.

In his masterpiece, Boole played the game of math on reasoning. He assigned meaning to arbitrary symbols; say an ‘*x*' is taken to represent *‘men.'* And suppose ‘*y*' represents *‘tall.'* He chose the multiplication symbol to represent the set that contains the relationship between the things - ‘*tall men*.' Next, he showed the ‘*properties*' of ‘*relationships*': If ‘*x*' is men and ‘*y*' is tall then ‘*xy*' equals ‘*yx*'; tall men are the same as men who are tall. And this holds true, not just for a ‘thing' and its ‘relationships' but for any series of things and their relationships. In other words, the communicative rule of Algebra holds for Boole's *Calculus of Thought*.

Relying on his definition of *‘reasoning' *– operations of thought that result in interpretable language - Boole developed rules for reasoning that in all cases, save one, prove out a one-to-one correspondence with the rules of Algebra on Numbers. The exception is Boole's special *‘Law of Thought.'*

If ‘*X*' is the universe of men, then what is ‘*XX*'? ‘A*ll men who are part of the set of all men…*' remain ‘…*all men*.' In the *‘Calculus of Thought'*, *XX = X*.

According to Boole, all that separates Algebra on Numbers from the ‘*Calculus of Thought'* is the need to confine the numerical value of symbols to those natural numbers that satisfy the equation: *X ^{2} = X.*

Boole manipulates ** X^{2} = X** into its equivalent,

**, then shows that the only numbers satisfying this simple algebra are**

*X(1-X) = 0*

__‘1' or ‘0'.__According to Boole, the only values necessary to map the laws of thought are the quantities *‘1' and '0'*. From here, discovering the *‘rules of thought'* is done in the same manner that one would discover the rules of Algebra on Numbers: through *‘…a system of processes carried on by the aid of symbols having a definite interpretation, and subject to laws founded upon that interpretation alone.'*

After deriving his Law of Thought from within his Calculus, Boole showed that his law is actually Aristotle's first Axiom of Logic:

*"It is impossible that the same quality should both belong and not belong to the same thing … This is the most certain of all principles … For it is by nature the source of all the other axioms."* *Aristotle*

If X is or all of something, then, according to Boole's calculus, (1 – X) is everything but the given thing. And what is common to both X and 1 – X, or (X)(1-X), is 0 as there cannot be something that both belongs and does not belong to the same set. Once again, the only possible values for X are '1' and ‘0.'

For Aristotle as well as Boole, this is the sapphire that makes the ocean blue; the law from which all others are derived. And it is the birth of our digital age, as computer science was born from this *‘general truth'*.

* * *

Boole died 60 years before the first digital computers were developed. Where is the connection? One obituary notes that Boole *‘... endeavored to write for a very limited audience and to every degree, succeeded.' *True, but not entirely. A few contemporaries recognized the beginnings of a new science of math in Boole's work. Pierce, an Algebraist, showed how to reduce Boole's Calculus to truth tables while John Venn, another contemporary, created Venn Diagrams to graphically depict Boole's concepts of joining or intersecting sets.

Nevertheless, Boole's work was almost forgotten. Set theory developed around the turn of the 20^{th} Century and with each advance credit was taken from Boole until he seemed almost no longer associated with a branch of mathematics for which he is rightfully one of the founders. He would be completely forgotten if not for chance which, in 1936, caused a young math student at the University of Michigan to wander into a course on Logic.

After studying the *Calculus of Thought*, Claude Shannon realized it was the solution the world had been looking for and wrote a master's thesis entitled ‘*A Symbolic Analysis of Relay and Switching Circuits*' that showed how Boole's Calculus turned circuit design, and everything else to do with computers, from art to science. Shannon's thesis has been described as the single most important master's thesis of the twentieth century.

* * *

In addition to his other pursuits, Boole dabbled in verse. His prose was Victorian and generally observational in nature. That he took to poetry is not a surprise. For much of his formative years, he lived outside of his surroundings, watching his world through a singular intelligence, sharing his math with strangers.

In his second year at Queens College, Boole met Mary Everest then visiting an uncle, John Ryall, head of the Classics department. Though just nineteen, Mary was accustomed to traveling alone while her father rested in declining health and her second uncle surveyed the great Himalayas. George and Mary shared something in that first meeting and George agreed to visit her from time to time at her home in London and to continue feeding her growing interest in all things mathematical.

Two years into their infrequent tutorship, Mary informed Boole of the death of her father and the offer from her uncle to join his family in India. Twice her age, yet not willing to let what may have been his only true friend go so far, George proposed.

Mary devoted the years after her husband's untimely death to promoting his works but kept his poetry private. In her later years, she wrote:

*‘Soon after our marriage, I found a sheet of paper covered with blank verse…'What is this?' I asked. ‘Some poetry of mine.' I read half through the paper…Then I walked over to the fire and dropped the paper into the flames, asking him never to write verses, but, if he had any poetry in his composition, to let it out in talk to me. He promised compliance but with a curious smile...'*

* * *

Many believe that Boole lies at the heart of a ‘binary,' true/false view of life. He does not. Boole is about truth only in the sense of truth being a logically consistent outcome. In his masterpiece, Boole applies his calculus to Spinoza's treatise on the existence of God and shows that despite the excessive length, *‘… there are no … conclusions expressive of … any of the elementary propositions…'*

Boole wrote, *‘Thought as the limit of an indefinite process of abstraction … is not enough to comprehend the mind for …* *the empire of Truth is, in a certain sense, larger than that of Imagination.' *

He went further, warning that *‘… to supersede the employment of common reasoning or to subject it to the rigor of technical forms would be the last desire of one who knows the value of that intellectual toil and warfare which imparts to the mind … vigor and teaches it to … rely upon itself ...'*

Though he did not foresee computers, Boole thought deeply about the implications of a path through human reasoning made by math. In our current times of math once again promising to discover everything, we can still read the conclusion of his masterpiece. There, the self-taught man who believed he had used the *‘perfect science'* to discover the *‘Right Path through human reasoning'* left us this:

*‘…If the mind, in its capacity of formal reasoning, obeys, whether consciously or unconsciously, mathematical laws, it claims through its other capacities of sentiment and action, through its perceptions of beauty and of moral fitness, through its deep springs of emotion and affection, to hold relation to a different order of things. There is, moreover, a breadth of intellectual vision, a power of sympathy with truth in all its forms and manifestations, which is not measured by the force and subtlety of the dialectic faculty.'*

* * *

P.S. He also wrote this:

*"We have as an elemental part of our nature the ability to observe imperfection and from that imperfection deduce the laws of Nature – deduce perfection. This ability is our capacity to ‘abstract' and it may well be the case that a different capacity to abstract, even in the face of a universal order of things, would lead to a different perception of that order."*

*---------------------------------------------------*

Having traveled far and wide in his youth, Richard Passov has spent the past twenty years on the banks of the Hudson River, New York. After more than 25 years working in technology and life sciences companies, Richard helped found Iota Biosciences, focused on closed loop modulation of neural pathways. A member of the Finance Committee of OneAcreFund and OffAssigment, he is a Lecturer at the Fung Institute of Engineering Leadership at University of California, Berkeley, currently researching the impact of the declining cost of computation on problem definition. He has graduate and undergraduate degrees from the University of California at Los Angeles along with a graduate degree from the University of California, Berkeley.

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