by DK
Algebraic logic
Boole’s investigations into logic began in an unconventional way. In 1847, a friend, British logician Augustus De Morgan, became involved in a dispute with a philosopher about who deserved the credit for a particular idea. Boole was not directly involved, but the event spurred him to set down his ideas concerning how logic could be formalized with mathematics, in his 1847 essay Mathematical Analysis of Logic.
Boole wanted to discover a way to frame logical arguments so that they could be manipulated and solved mathematically. In order to achieve this, he developed a type of linguistic algebra, in which the operations of ordinary algebra, such as addition and multiplication, were replaced by the connectors that were used in logic. As in algebra, Boole’s use of symbols and connectives allowed for the simplification of logical expressions.
The three key operations of Boole’s algebra were AND, OR, and NOT; Boole believed these were the only operations necessary to perform comparisons of sets, as well as basic mathematical functions. For example, in logic, two statements may be connected by AND, as in “this animal is covered in hair” AND “this animal feeds its young with milk,” or by OR, as in “this animal can swim” OR “this animal has feathers.” The statement “A AND B” is true when A and B are both individually true, whereas the statement “A OR B” is true if one or both of A and B is true. In Boolean terms, such statements can be given as, for example: (A OR B) = (B OR A); NOT (NOT A) = A; or even NOT (A OR B) = (NOT A) AND (NOT B).
Boole’s binaries
In 1854, Boole published his most important work, An investigation into the laws of thought. Boole had studied the algebraic properties of numbers and realized that the set {0, 1}, together with operations such as addition and multiplication, could be used to form a consistent algebraic language. Boole proposed that logical propositions could have only two values—true or false—and could not be anything in between.
In Boole’s logical algebra, truth and falsity were reduced to binary values: 1 for true and 0 for false. Starting out with an initial statement that was either true or false, Boole could then construct further statements and use the AND, OR, and NOT operations in order to determine whether or not these further statements were true.
Boolean algebra makes it possible to prove logical statements by performing algebraic calculations.
Ian Stewart
British mathematician
One plus one is one
Despite the resemblance, Boole’s true and false binary of 1 and 0 is not the same as binary numbers. Boolean numbers are entirely different from the mathematics of real numbers. The “laws” of Boole’s algebra allow statements that would not be permitted by other forms of algebra. In Boole’s algebra, there are only two possible values for any quantity, either 1 or 0. There is also no such thing as subtraction in Boole’s algebra. For example, if statement A, “my dog is hairy,” is true, it has a value of 1, and if statement B, “my dog is brown” is true, it also has a value of 1. A and B can be combined to make the statement “my dog is hairy OR my dog is brown,” which is also true, and also has a value of 1. In Boolean algebra, OR behaves like + (aside from 1 + 1 = 1) and AND behaves like × (see Logic gates).
The furthest thing from my mind has been those efforts which try to establish an artificial similarity [between logic and algebra].
Gottlob Frege
Visualizing results
One way of visualizing Boole’s algebra is in the form of diagrams invented by British logician John Venn. In his work Symbolic Logic (1881), Venn developed Boole’s theories employing what became known as Venn diagrams. These depict relations of inclusion (AND) and exclusion (NOT) between sets. They consist of intersecting circles, each one representing a distinct set. A two-circle Venn diagram represents propositions such as: “All A are B,” while a three-circle diagram represents propositions involving three sets (such as x, Y, and Z).
The results of a statement in Boolean algebra can also be assessed using a truth table, in which all possible input combinations are tried and written out. These truth tables were first used by American logician Charles Saunders Peirce in 1893, nearly 30 years after Boole’s death. For example, the statement A AND B can only be considered true if both A and B are true. If one or both of A and B are false, then A AND B is false. Therefore, out of the four possible combinations of A and B, only one results in a true answer. On the other hand, for A OR B, there are three possible combinations in which that statement is true, as it will only be false if both A and B are false. More complex statements can also be assessed by drawing truth tables. For example, A AND (B OR NOT C) is true when A and B are both true and C is false, and is false when A is false and both B and C are true. Out of eight possible combinations of true and false, there are three in which the statement is true and five in which it is false.
These Venn diagrams represent three of the most basic functions in Boolean algebra: the functions for AND, OR, and NOT. The three-circle diagram represents a combination of two functions: (x AND Y) OR Z.
Limitations
One drawback in Boole’s system of algebra was that it contained no method of quantification: there was no simple way of expressing a statement such as “for all x,” for example. The first symbolic logic with quantification was produced in 1879 by German logician Gottlob Frege, who objected to Boole’s attempts to turn logic into algebra. Frege’s work was followed by Charles Sanders Peirce and another German logician, Ernst Schröder, who introduced quantification into Boole’s algebra and produced substantial works using Boole’s system.
This logic module is used for teaching how logic gates function in electronic circuits. The gates can be connected to lights or buzzers which go on and off depending on the output.
Boole’s legacy
It was not until some 70 years after Boole’s death that the potential of his ideas was fully grasped. American engineer Claude Shannon used Boole’s Mathematical Analysis of Logic to establish the basis of modern digital computer circuits. While working on the electrical circuitry for one of the world’s first computers, Shannon realized that Boole’s two-value binary system could be the basis of logic gates (physical devices that move based on Boolean functions) in the circuitry. Aged just 21, Shannon published the ideas that would form the basis of future computer design in A Symbolic Analysis of Relay and Switching Circuits, published in 1937.
The building blocks of codes now used to program computer software are based on the logic formulated by Boole. Boolean logic is also at the heart of how internet search engines work. In the early days of the internet, the AND, OR, and NOT commands were commonly used to filter results to find the specific thing being searched for, but advances in technology allow people today to search using more natural language. The Boolean commands have simply become silent: a search for “George Boole,” for example, has an implied AND between the two words, so that only web pages containing both names will appear in the results.
Logic gates, which are physical electronic devices implementing Boolean functions, form an important part of computer circuitry. This table shows the various symbols for each type of logic gate. Truth tables show the possible outcomes of various inputs into the gate.
GEORGE BOOLE
Born in Lincoln in 1815, George Boole was the son of a shoemaker who passed his love of science and mathematics on to him. When his father’s business collapsed, the 16-year-old George took up a post as an assistant schoolmaster to support his family. He began to study mathematics seriously, starting by reading a book on calculus. He later published work in the Cambridge Mathematical Journal, but still could not afford to study for a degree.
In 1849, as a result of his correspondence with Augustus De Morgan, Boole was appointed professor of mathematics at the new Queen’s College in Cork, Ireland, where he remained until his premature death at the age of 49.
Key works
1847 Mathematical Analysis of Logic
1854 An investigation into the laws of thought<
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1859 Treatise on differential equations
1860 Treatise on the calculus of finite differences
See also: Syllogistic logic • Binary numbers • The algebraic resolution of equations • Venn diagrams • The Turing machine • Information theory • Fuzzy logic
IN CONTEXT
KEY FIGURE
August Möbius (1790–1868)
FIELD
Applied geometry
BEFORE
3rd century CE A Roman mosaic of Aion, Greek god of eternal time, features a zodiac shaped like a Möbius strip.
1847 Johann Listing publishes Vorstudien zur Topologie (Introductory Studies in Topology).
AFTER
1882 Felix Klein describes the Kleinsche Flasche (Klein bottle), a shape composed of two Möbius strips.
1957 In the US, the B. F. Goodrich Company produces a patent for a conveyor belt based on the Möbius strip.
2015 Möbius strips are used in laser beam research, with potential application in nanotechnology.
A Möbius strip can be made from a simple length of paper. It can be colored in with a crayon in one continuous movement without taking the crayon away from the paper. The shape has a single surface; this can be tested by following the surface of the shape with the eye.
Named after 19th-century German mathematician August Möbius, a Möbius strip can be created in seconds by twisting a strip of paper through 180°, then joining its two ends together. The shape that results has some unexpected properties, which have advanced our understanding of complex geometrical figures—a branch of study called topology.
The 19th century was a creative period for mathematics, and the exciting new field of topology spawned many new geometrical shapes. Much of this impetus came from German mathematicians, including Möbius and Johann Listing. In 1858, the two men independently investigated the twisted strip, which Listing is said to have discovered first.
Once formed, the Möbius strip has only one surface—an ant crawling along that surface would be able to cover both sides of the paper in one continuous movement without crossing the edge of the paper. In geometry, it is considered a classic example of a “nonorientable” surface. This means that when you trace your finger around the complete strip, the left and right sides of the paper are reversed. The Möbius strip is the simplest nonorientable, two-dimensional surface that can be created in three-dimensional space.
Experimenting with the Möbius strip produces other unexpected results. For instance, if you draw a line around the center of the strip and then cut along it, the shape does not divide in half. Rather, it produces a longer, continuous twisted loop. Alternatively, draw a line about a third of the way across the width of the strip, then turn the scissors 90° and cut along its length: the result is one twisted loop linked to a second, thinner twisted loop that is twice as long.
A Roman mosaic dating from c. 200 CE includes what may be the earliest representation of a Möbius strip, which is thought to represent the eternal nature of time.
Space, industry, and art
The Möbius strip shape sometimes occurs naturally, such as in the movement of magnetically charged particles within the Van Allen radiation belts that surround Earth and in the molecular structure of some proteins. Its properties have been put to use in everyday applications, too. In the early 20th century, the Möbius strip shape was used in continuous-loop recording tapes to provide double the playback time. There are also Möbius strip roller-coasters, such as the Grand National at Blackpool Pleasure Beach in northern England.
The Möbius strip’s form has inspired artists and architects. Dutch artist M.C. Escher created a notable woodcut of ants endlessly patrolling the shape. Impressive Möbius strip buildings are being constructed to minimize the impact of the sun’s rays. The shape is used in the universal symbol for recycling and also suggested in the mathematical symbol for infinity (∞), echoing the eternity image in the ancient Roman mosaic.
Our lives are Möbius strips, misery and wonder simultaneously. Our destinies are infinite, and infinitely recurring.
Joyce Carol Oates
American novelist
AUGUST MÖBIUS
Born near Naumberg in Saxony, Germany, in 1790, August Ferdinand Möbius was the son of a dance teacher. At the age of 18, he entered the University of Leipzig to study mathematics, physics, and astronomy, and later studied in Göttingen under the great German mathematician Carl Friedrich Gauss. In 1816, Möbius was appointed professor of astronomy at Leipzig and stayed there for the rest of his life, writing treatises on Halley’s Comet and other aspects of astronomy.
Möbius is associated with a number of mathematical concepts, including Möbius transformations, the Möbius function, the Möbius plane, and the Möbius inversion formula. He also conjectured a geometrical projection known as a Möbius net. Möbius died in Leipzig in 1868.
Key works
1827 The Calculus of Centers of Gravity
1837 Textbook of Statics
1843 The Elements of Celestial Mechanics
See also: Graph theory • Topology • Minkowski space • Fractals
IN CONTEXT
KEY FIGURE
Bernhard Riemann (1826–66)
FIELD
Number theory
BEFORE
1748 Leonhard Euler defines the Euler product, linking a version of what will become the zeta function to the sequence of prime numbers.
1848 Russian mathematician Pafnuty Chebyshev presents the first significant study of the prime counting function π(n).
AFTER
1901 Swedish mathematician Helge von Koch proves that the best possible version of the prime counting function relies on the Riemann hypothesis.
2004 Distributed computing is used to prove that the first 10 trillion “nontrivial zeros” lie on the critical line.
In 1900, David Hilbert listed 23 outstanding mathematical problems. One of them was the Riemann hypothesis, which is still agreed to be one of the most important unsolved problems in mathematics. It concerns the prime numbers—numbers that are only divisible by themselves or 1. Proving the Riemann hypothesis would solve many other theorems.
The most noticeable thing about prime numbers is that the larger they are, the more widely spread out they get. Of the numbers between 1 and 100, 25 are prime (1 in 4); between 1 and 100,000, 9,592 are prime (about 1 in 10). These values are expressed through the prime counting function, π(n), but π here is not related to the mathematical constant pi. Inputting n into π gives the number of primes between 1 and n. For example, the number of primes up to 100 gives π(100) = 25.
Finding the pattern
For centuries, mathematicians’ fascination with primes has led them to seek a formula that would predict the values of this function. Aged just 14, Carl Gauss found a rough answer, and he was soon able to find an improved version of the prime counting function that could predict the number of primes between 1 and 1,000,000 as 78,628, which is accurate to 0.2 percent.
The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers.
Enrico Bombieri
Italian mathematician
A new formula
In 1859, Bernhard Riemann constructed a new formula for π(n), which would give the most accurate estimates possible. One of the inputs needed for this formula is a series of complex numbers defined by what is now called the Riemann zeta function, ζ(s).
The numbers that are needed to confirm Riemann’s formula for π(n) are those complex numbers (s) for which ζ(s) = 0. Some of these—the “trivial zeros”—are easy to find; they are all the negative even integers (-2, -4, -6, and so on). Finding the others (the “nontrivial zeros”— all other values for which ζ(s) = 0) is more difficult. Riemann only calculated three. He believed that nontrivial zeros have one thing in common: when they are plotted on the complex plane, they all lie on “the critical line,” where the real part of the number is 0.5. This belief is called the Riemann hypothesis.
The uranium atom is one example of a heavy atom whose nucleus follows the same statistical behavior as prime numbers, making it extremely difficult to predict.
A solution
In 2018, British mathematician Michael Atiyah, then aged 89, said he had found a simple proof for the Riemann hypothesis. He died a few months later, the proof unverified.
Although proving the Riemann hypothesis would validate the zeta function's status as the best predictor of the distribution of primes, it still would not allow prime numbers to be fully predicted. Their distribution is to some extent chaotic. But the hypothesis does pin down the blend of predictability and randomness the primes obey. This blend is exactly that exhibited by the energy levels of the nuclei of heavy atoms, according to quantum theory. This profound connection means the hypothesis may one day be proved not by a mathematician, but by a physicist.
BERNHARD RIEMANN
The son of a pastor, Bernhard Riemann was born in Germany in 1826. Initially fascinated by theology, he was persuaded to change his degree to mathematics by Carl Gauss, under whom he then studied at the University of Göttingen. The result was a series of breakthroughs that remain influential today.