# Ones and Zeros

Mathematician and logician George Boole died 150 years ago today, on 8th December, 1864. Today also marks the start of the year-long schedule of events UCC are running to commemorate Boole, culminating in the bicentenary of his birth on 2nd November 2015 (see GeorgeBoole.com for more).

George Boole was born in Lincoln, the eldest son in a family of modest means. For details of his life as a self-taught mathematician to first professor in UCC (then Queens College Cork) in 1849, where he lived until his death see the detailed biography here.

Boole had a large impact on mathematics, providing the basis for invariant theory, and working on differential and difference equations, and probability. Developments of his work such as set theory and boolean algebra are taught to school children today.

However, of most interest philosophically are *The Mathematical Analysis of Logic*, and its successor *An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities* published in 1854. These proposed that ideas expressed in language can be expressed in algebraic form. This combination of philosophical logic and algebra, as DeMorgan said “would not have been believed until it was proved.”

### Boole’s Investigation of the Laws of Thought

Boole starts his *Investigation of the Laws of Thought* with the point (clear to a linguist) that words are signs. However words are not the only form of signs. “Arbitrary marks” or sounds can be used as long as they have a fixed interpretation.

Boole suggested that we use single letters for classification of what we are thinking about. So we might say *x* is the class of all white things and *y* the class of all sheep. Therefore *xy* is the class of all white sheep. We can add new letters for new classes (eg. *z* is the class of horned things, then *xyz* is the class of horned white sheep), to arrive at the level of detail we require.

We can use our operations on these classes, some of which are similar to their use in normal sums, and some are not. For example *xy* looks like a multiplication sum. We can see in boolean algebra *xy* = *yx* (the class of all sheep that are white is the same as the class of all white sheep), just as 5X2 = 2X5. Similarly *x(yz)*=*x(zy)*

Other operations that look like normal sums act differently however. Logically we can see *xx* (all sheep that are sheep) must be equal to *x*, or in notation *x ^{2}*=

*x*.

Boole suggests using + to combine classes. However these classes for Boole must be mutually exclusive – “cows” and “sheep” can be combined but not “sheep” and “white objects”. (This is one of the differences between Boole and modern Boolean algebra.) Combining cows (*x*) and sheep (*y*) gets us a big herd of both, whatever way we do it: *x*+*y*=*y*+*x* (just as 5+2=2+5). And combining white cows and white sheep gets us a white herd of both, no matter whether we start with a combined herd or separate out the white (*z*) ones first: *z*(*x*+*y*)=*zx*+*zy*. Similarly Boole uses – to show separation, so taking the cows from our herd of cows and sheep is denoted as *xy*–*x*=*y*.

In chapter III, Boole introduces 0 to mean “nothing” and 1 to represent the Universe. Using these and the laws above, he derives *x*(1 − *x*) = 0 which is the Principle of Contradiction, the second of the classic laws of thought. (This equation is also apparently the root of Boole’s inspiration to combine algebra and logic, stemming from a spiritual experience he had at seventeen).

### Boole’s Legacy

I suspect the reader is now wondering how this could be possibly be useful. (If the reader actually wants to explore further, see here). Boole supplied an answer: at the end of the *Investigation* Boole applies his notation to philosophical texts by Baruch Spinoza (1632-77) and others, to test them logically. This was the first example of how Boole’s methods could strengthen the role of logic, a task taken further by others.

Bertrand Russell, who made efforts to have Boole’s papers printed, said that *An Investigation of the Laws of Thought* was (though perhaps not absolutely literally) “the work in which pure mathematics was discovered” (McHale, 1985). Boole’s work tended towards making mathematics more abstract, an aspect that probably appealed to Russell who read him and who attempted with Frege to reduce mathematics to symbolic logic. While ultimately that project failed, it led to further development of logic, including the development of precise notation for logic in philosophy. Ultimately this resulted in the Vienna Circle movement and the development of analytic philosophy.

However Boolean logic itself is best known for its association with computing.

Babbage had already worked on his Difference Engine, a programmable mechanical computer. He met Boole in 1862 and they discussed how the Difference Engine might be altered: it is intriguing to speculate what might have happened if they had worked together. Around the same time William Stanley Jevons (1835-82) who worked on Boole’s logical system created the Logic Piano (1870), a working mechanical computer.

Boolean algebra was further developed by Jevons, Schröder, Huntington and others, removing many of its perceived shortcomings.

In 1937 Claude Shannon (1916-2001) realised that boolean algebra could be employed to simplify the arrangement of electromechanical relays used for telephone call routing. More importantly, he also realised that those arrangements of relays could be used to solve problems. This principle of using switches to perform logic enabled the creation of digital computers. In 1943 Claude Shannon and Alan Turing met, and agreed that their work was compatible; indeed the computer Tommy Flowers was building at the time in Bletchley Park was based on Boolean logic. The work of Boole, once seen as marginal, is now central to the modern world.

### Sources

Des McHale (1985) George Boole: His life and work, Boole Press (now republished).

James Gasser (2004) “George Boole” in *Dictionary of Irish Philosophers*, pp. 26-9.

Janet Heine Barnett (2013) Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce (pdf) on MMA.org

GeorgeBoole.com – the UCC website celebrating the 200th anniversary of George Boole’s birth.

New Apps: The Spinoza-Clarke dispute in Boole (the origins of analytic philosophy, reconsidered)

NYU.EDU The Role of Formalization: Boole’s influence on Frege

### Further Reading

Irish Times: How George Boole’s zeroes and ones changed the world

Yovisto: George Boole – The Founder of Modern Logics

The Riverside (UCC): George Boole’s untimely death (including a mention of Boole meeting Babbage, plus extracts from letters)

Boole’s books on Project Gutenburg: An Investigation of the Laws of Thought and The Mathematical Analysis of Logic

Also on Project Gutenburg, Philosophy and Fun of Algebra by Mary Everest Boole (also in audio), mathematician, educationalist and wife of George Boole.

Mary Everest Boole’s exchange of letters with Charles Darwin

Those Amazing Boole Girls: about George and Mary’s five remarkable daughters including Alicia, a mathematician, Lucy, a professor of chemistry, and Ethel (Voynich) author The Gadfly.

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Shameless self-promotion: there is, of course, my own book (unfortunately overpriced at amazon at the moment):

http://www.amazon.com/exec/obidos/ISBN=0780334264

Ones and Zeros: Understanding Boolean Algebra, Digital Circuits, and the Logic of Sets

Meant to be understood by the lay reader.

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