The Handy Math Answer Book, Second Edition (2012)
FOUNDATIONS OF MATHEMATICS
FOUNDATIONS AND LOGIC
What are the foundations of mathematics?
The foundations of mathematics include how to formulate and analyze the language (you have to “speak” the right mathematical language to make meaningful mathematical statements), axioms (a statement accepted as true without proof), and developing logical methods in all mathematical studies. The most basic mathematical concepts in the foundations include numbers, shapes, sets, functions, algorithms, axioms, definitions, and proofs.
Why do so many philosophers study the foundations of mathematics?
There are three underlying reasons why philosophers often study the foundations of mathematics. First, these foundations have always been a part of scientific thought, with the abstract nature of mathematical objects offering unusual and often unique philosophical quandaries. Second, the subject offers a high level of technical sophistication, allowing philosophers to develop connections between models and patterns, laying the groundwork for many other sciences. And finally, the foundations of mathematics provides ways for philosophers (and mathematicians) to try out general philosophical doctrines in a specific scientific context.
What is logic?
Although it closely resembles mathematics (and is sometimes used as a basis for it), logic is a branch of knowledge or inquiry that is separate from mathematics and the sciences, but is still used by both fields in various ways. Simply put, logic is described as the systematic study of well-founded inference, in which there is a definite distinction between logical validity (also known as the formal properties of the inference process) and truth. This also means that a true result may come from an invalid argument (see below for definition of an argument). For example, “all cats are cute; Fluffer is a cat; therefore, Fluffer is cute,” is a valid inference; whereas, “all cats are cute; Fluffer is cute; therefore, Fluffer is a cat,” is an invalid inference, even if Fluffer really is a cat.
A bust of the Greek philosopher Aristotle, whose work Organon is the earliest known to introduce systematic logic, the foundation of mathematical logic.
What is the historical basis for mathematical logic?
Most mathematicians believe that systematic logic began with Aristotle’s collection of works titled Organon (Tool), in which he introduced his ideas on logic. In particular, Aristotle used general forms to describe logic, such as if all x are y; and all y are z; then all x are z. He presented three laws basic to all valid thought: the law of identity, or A is A (for example, an acorn will always yield a oak tree and nothing else); the law of contradiction, or A cannot be both A and not A (for example, an honest woman cannot be a thief); and the law of the excluded middle, or either or, in which A must either be A or not A (for example, a dog can be brown or not brown). In fact, author Ayn Rand divided her novel Atlas Shrugged into three parts after these three principles, as a tribute to Aristotle.
Was mathematics always based on a logical foundation?
No, not all of mathematics was always based on a logical foundation, but some ancient cultures did develop certain aspects of logic in their thought. The Greeks were probably one of the first cultures to understand logic’s role in mathematics and philosophy, and studied the subject extensively. For example, geometry, as presented by Greek mathematician Euclid (c. 325-c. 270 B.C.E.), had some foundations in logic. Greek scientist and philosopher Aristotle’s (384-322 B.C.E.) rules for syllogisms were also based on logic, and he wrote the first systematic treatise on logic. But his logic works were based on ordinary language—making them a matter of interpretation and subject to ambiguities.
It was not until the development of calculus that most of mathematics was put on a logical foundation. By the 17th century, people like German mathematician Gottfried Wilhelm Leibniz (1646–1716) began to demand a more regular and symbolic way to express logic. Logic truly became a part of mathematics around the mid-19th century, especially with the 1847 publication of English mathematician George Boole’s (1815–1864) The Mathematical Analysis of Logic and English mathematician Augustus De Morgan’s (1806–1871) Formal Logic. Thus, mathematics began to encompass symbolic logic, with precise rules to manipulate those symbols (see below for more about symbolic logic).
What is an argument?
In logic, an argument is not a “heated discussion,” although some mathematicians may argue over the validity of certain mathematical arguments. In this sense, an argument is a list of statements called premises, followed by a statement called the conclusion. Generally, an argument is valid if the conjunction of its premises implies its conclusion; stated differently, validity means that if all the premises are true, then so is the conclusion. But remember: The validity of an argument does not guarantee the truth of its premises, and thus it does not guarantee the truth of its conclusion. It only guarantees that if the premises are true, the conclusion will be true.
Of course, nothing is perfect, although mathematicians in the late 19th and early 20th centuries hoped it would be. They believed that all of mathematics could be described using symbolic logic and made purely formal. But in the 1930s, Austrian-American mathematician and logician Kurt Gödel (1906–1978) put a damper on such an idea by showing that not all truths could be derived by a formal logic system.
What were Aristotle’s contributions to logic?
Aristotle contributed syllogisms and propositions to the world of logic—in other words, the verbal versions of the formal deductive rules for logic. Aristotle believed that any logical argument could be explained in these standard forms. Today, we call these ideas Aristotelian logic, whereas Aristotle would have called it “analytics;” he would have referred to the term “logic” as “dialectics.” Although syllogisms are attributed to Aristotle, no doubt others came up with the same ideas with different ways of expressing them. But his teachings were well-known, even in his time. Thus, syllogistic logic would dominate Western cultural thought for more than 2,000 years.
Aristotle’s six books—Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics, and On Sophistical Refutations—were compiled around 1 B.C.E. Some historians don’t believe all the writings should be totally attributed to Aristotle, but were changed over time by teachers, lecturers, and students who studied his works. It is also thought that the books were used extensively in the school founded by Aristotle at the Lyceum (a form of educational school). The collection of books is often called the Organon, the name given by the followers of Aristotle (often referred to as the Peripatetics).
An example of a syllogism.
What are subjects and predicates found in Aristotelian logic?
Aristotelian logic is more like an English lesson: one subject and one predicate are used in a sentence (called a proposition). The subject is usually an individual entity (an object, house, city, man, animal); or it may be a class of entities (objects, houses, cities, men, animals). The predicate is the property or mode of existence that does or does not exist with a given subject. For example, a singular plant (subject) may or may not be blooming (predicate); all houses (subject) may or may not have two stories (predicate).
What do the terms propositions and syllogisms mean in Aristotelian logic?
There is a great deal to Aristotelian logic—too much to mention in this text. But overall, there are some general terms and properties. For example, propositions are sentences with two terms—a grammatical subject and predicate. A proposition has the properties of quality and quantity only. They can either be a negative or positive proposition in terms of quality; in terms of quantity, they are either universal or particular propositions. That means the four types of propositions are: universal affirmative (for example, “all men are mortal”); universal negative (for example, “no men are mortal”); particular affirmative (for example, “some men are strong”); and particular negative (for example, “some men are not mortal”). In most texts, these four proposition types are denoted by the letters A, E, I, and O, respectively.
What erroneous logic statement is often attributed to Aristotle?
It’s interesting to note that the classical syllogism is, “All men are mortal. Socrates is a man. Therefore Socrates is mortal.” But in reality, this is wrong, and is actually called a Peripatetic syllogism, a form not used by Aristotle. In fact, in Aristotelian logic, there are no singular terms. A more apt statement of Aristotelian logic would be “If all men are mortal, and all Greeks are men, then all Greeks are mortal.” To break part of this sentence down, in “All Greeks are men,” the “Greeks” is the subject, and the “men” is the predicate.
Syllogisms are composed of two premises and a conclusion; the conclusion comes from the two premises in a certain way. Apparently, Aristotle liked to name his logical offerings, so each word in the syllogism has a label. He said that each premise must have a term in common, which he called the middle term. The other terms he called extreme terms, divided into the major term, or the predicate of the conclusion and minor term, or subject of the conclusion. And logically, of course, the premise with the minor term is called the minor premise, and the one with the major term is called the major premise. Aristotle further divided the syllogisms into perfect syllogisms (“…which needs nothing other than what has been stated to make plain what necessarily follows…”) and imperfect syllogisms (“…needs either one or more propositions, which are indeed the necessary consequence of the terms set down, but have not been expressly stated as premises…”).
Who invented a way of analyzing syllogisms?
In 1880, English logician John Venn (1834–1923) presented a method to analyze syllogisms, now known as Venn diagrams. Venn initially criticized such diagrams in works by his contemporaries, especially those of English mathematicians George Boole (1815–1864) and Augustus De Morgan (1806–1871). But in 1880, Venn introduced his own, now famous, version of the diagrams in his paper On the Diagrammatic and Mechanical Representation of Prepositions and Reasonings. By 1881, along with correcting Boole’s work, Venn further elaborated on the diagrams in his book, Symbolic Logic. Today we are most familiar with Venn diagrams in connection with understanding sets.
Although Venn is credited with the diagrams, he was not the first person to use such geometric methods to represent syllogistic logic. German mathematician Gottfried Wilhelm Leibniz (1646–1716) used such graphic representations in his work. And even Swiss mathematician Leonhard Euler (1707–1783) is known to have presented diagrams that had a definite “Venn-ish” look a century before John Venn.
What are some examples of Venn diagrams?
Venn diagrams are schematic illustrations used in logic theory to show collections of sets and the relationship between them. Overlapping circles represent the sets (or the subjects and predicates in syllogistic logic); the standard way of presenting such diagrams include the intersection of two (order-two diagram) to three (order-three diagram) circles. Based on what circles intersect and the areas shaded, a conclusion about the sets may then be read directly from the diagram. Such illustrations can include the union of two sets, the intersection of two sets, the complement of a set, and the complement of the union of two sets. (For more information about sets, see “Set Theory,” below.)
In these examples of Venn diagrams, the top illustration represents an order-two diagram, and the bottom is an order-three diagram.
MATHEMATICAL AND FORMAL LOGIC
What is mathematical logic?
Mathematical logic is not the logic of mathematics, but is really the mathematics of logic composed of those parts of logic that can be modeled mathematically. Overall, it was invented to understand and present the work of Austrian-American mathematician and logician Kurt Gödel (1906–1978) and his interpretation of the foundations of mathematics in the early 20th century. Although mathematicians use mathematical logic to have rational and reasonable discussions of the many issues in the foundations of mathematics, not everything is agreed upon.
What is intuitionism?
There are some people within philosophy and mathematics who reject the formalism of mathematics and believe in intuitionism, which says that words and formulas have significance only as a reflection of the mind’s activity. Intuitionists believe that a theorem is meaningful only if it represents a mental construction of a mathematical or logical entity. This is different from the classical approach that states that the existence of an entity can be proven by refuting its non-existence. For example, if you said “A or B” to an intuitionist, he or she believes that either A or B can be proved; but if you said, “A or not A,” this is not allowed, since you cannot assume that it is always possible to either prove or disprove statement A.
What contributions did David Hilbert make to mathematics?
German mathematician David Hilbert (1862–1943) contributed a great deal to mathematical logic, as well as mathematics in general. In 1890 his proof of the theorem of invariants replaced earlier work on the subject and paved the way for modern algebraic geometry; by 1897, his algebraic number theory lead to many developments in that field. His contributions also included discoveries in number theory, mathematical logic, differential equations, multivariable calculus, Euclidean geometry, and even mathematical (theoretical) physics.
Hilbert is most well-known for presenting “Hilbert’s problems,” which originally were a set of 23 unsolved mathematical problems that he hoped would eventually lead to many more disciplines within the field of mathematics. His idea worked: As mathematicians attempted to solve the problems, their efforts led to mathematical discoveries in the 20th century, although a number of the problems have yet to be solved. (For more information about Hilbert, see “History of Mathematics.”)
What is a proposition in mathematical logic?
A proposition in mathematical logic is a statement that can be proven to be either true or false. For example, if you say, “the bear is black,” that is a proposition; but the statement “the bear is x,” cannot be true or false until a particular value for x is chosen; therefore, it is not a proposition.
What is symbolic logic?
Symbolic logic (also called formal logic) is mainly concerned with the structure of reasoning. It determines the meaning and relationship of statements used to represent specific mathematical concepts and provides a means to compose proofs of statements. Symbolic logic draws most notably on set theory. It uses variables combined by operations such as not or and, and assigns symbols to such (“~” and “&”, respectively).
What is propositional calculus?
Propositional calculus is not the calculus most of us hear about, but is considered by many to be the foundation of symbolic logic. (Actually, the term “calculus” is a generic name for any area of mathematics that deals with calculating; thus, arithmetic could be called the “calculus of numbers.”) Also known as truth-functional analysis, sentential calculus, or the calculus of propositions (or as seen above, any declarative sentence that is either true or false), propositional calculus deals with statements that can be assigned truth values. In general, it uses symbols to denote logical operators (such as and and or), and parentheses for grouping formulas.
What are truth values and truth functions?
As seen above, when discussing propositional calculus, a proposition is any declarative sentence that is either true (T) or false (F). Mathematicians refer to T or F as the truth value of the statement.
The combinations of such statements are known as truth functions, with their true values determined from the overall true values of their contents. Truth-functional analysis includes the following logical operators:
Negation—The negation of a statement is false if the original statement is true, and true if the original statement is false; it refers to “it is not the case that” or simply “not” in natural language.
Conjunction—The conjunction of two statements is true only if both are true and false in all other instances; it refers to “and” in natural language.
Alteration—Alteration (or disjunction) of two statements is false only if both are false and true in all other instances; it refers to “or” (and “either … or”) in natural language.
Conditional—Conditional (or implication) is false only if the antecedent is true and the consequent is false, and is true in all other instances; it refers to “if … then” or “implies” in natural language.
Biconditional—Biconditional (double implication or bi-implication) is true only if the two statements have the same value, either true or false; it refers to “if and only if…” in natural language.
What is a truth table?
A truth table is a two-dimensional array of truth values derived by determining the validity of arguments through assigning all possible combinations of truth values to the statements. This simple form of logic depends on a combination of certain statements, using terms such as “not” or “and,” along with the input values.
The first columns correspond to the possible input values and the last columns to the operations being performed; the rows list all possible combinations of true (T) or false (F) inputs, together with the corresponding outputs. The following is a truth table for the three most common binary operations of logic (“if…then,” “or,” “and”), using s and t as the statements:
Why are truth tables important to computers?
In many ways, truth tables are directly connected to digital logic circuits; in the case of computers, the terms would be AND, OR, NAND, NOR, NOT, XOR, or the “gates” that open and close in response to such terms. In such a circuit, values at each point can take on values of only true (1) or false (0); this is also known as the computer binary system. In general, there is also a three-valued logic, in which possible values are true, false, and “undecided.” A further generalization called fuzzy logic examines the “truth” as a continuous quantity ranging from 0 to 1. (For more about fuzzy logic and computers, see “Math in Computing.”)
What are logical operators in truth tables?
Logical operators in truth tables include such words as “and” or “or,” which are all represented by certain symbols (for more about logical operators in predicate calculus, see below). For example, “and” (also called the conjunction operator) is also referred to as a binary operator. It is one of the most useful logical operators, as in “p AND q”, represented by the symbols ∧ or &. The “or” (also called the disjunction operator) is also a binary operator, as in “p OR q”, and represented by the symbols 
and |. The “not” (also called the negation or inversion) operator is known as a unary operator, and is represented by the symbols ˜ or 
(in computer programming, NOT is often represented by the !). The “implies” (or implication operator) is also a binary operator; its symbols include ∴, ⊃, and Ã.
But note: Not all logical operators seem to represent words the way we are accustomed to using them, and many times they seem to contradict their proper definitions. But in a truth table, the logical operator means what it means—without the usual nuances of the English language.
What is a formula?
In mathematics, a formula is generally a rule, principle, or fact that is displayed in terms of mathematical symbols. (Although the Latin plural form of formula is “formulae,” “formulas” is the accepted common use in mathematics.) These equations express a definite, fixed relationship between certain quantities (usually expressed by letters), with their relationship indicated by algebraic symbols. For example, scientist Albert Einstein’s famous E = mc2 is a formula representing energy (E) equal to mass (m) times the speed of light (c) squared. The word “formula” is also used in logic; it is written as a propositional or sentential formula, or “a formula in propositional calculus is one that uses ‘and,’ ‘or,’ and so on.”
Who was responsible for expanding ideas of predicate calculus?
The German philosopher and mathematician Friedrich Ludwig Gottlob Frege (1848–1925) presented a way to rearrange sentences to make their logic clearer and to show how the sentences relate in various ways in his 1879 treatise Begriffsschrift (German for “Concept Script”). Before Frege began his work, formal logic (in the form of propositional or sentential calculus; see above) used such words as “and”, “or”, and so on, but the method could not break the sentences down into smaller parts. For example, formal logic could not show how the sentence, “Cats are animals,” actually entails, “parts of cats are parts of animals.”
Frege added words such as “all,” “some,” and “none,” using variables and quantifiers to rearrange the sentences, therefore making them more precise in their meaning. He also developed two of the major qualifier symbols for predicate calculus, the upside-down A 
and the backward E 
.
Frege’s work began modern logical theory, even though his work was considered defective in several respects and was considered awkward to use. By the 1910s and 1920s, Frege’s system was modified and streamlined into today’s predicate calculus.
What is predicate calculus?
Predicate calculus (also called first-order logic, functional calculus, or quantification theory) is a theory in symbolic logic that uses statements such as “there exists an object such that…” or “for all objects, it is the case that…”. It is a much more solid theory than propositional calculus, taking the interrelationship between sentences much further, but it is weaker than certain branches of mathematics, such as arithmetic and set theory.
What is a quantifier in predicate calculus?
A sentence or many sentences containing a variable (such as x) can be made into true or false propositions simply by using a quantifier. The quantifier actually assigns a truth value to the sentence, depending on the set of values allowed for that variable. There are two major quantifiers: the existential and universal quantifiers, which are represented by the logical operator symbols of 
and 
, respectively, although there are more exotic types of logic that use different quantifiers.
How is predicate calculus interpreted?
Predicate calculus may be a general system of logic, but it accurately expresses a large variety of assertions and provides many types of reasoning. It is definitely more flexible than Aristotle’s syllogisms and more useful (in many cases) than propositional calculus.
Predicate calculus makes heavy use of symbolic notation, using lower-case letters a, b, c, …,x, y, z,… to denote the subject (in predicate calculus, often referred to as “individuals”); upper-case letters M, N, P, Q, R,… to denote predicates. The simplest of assertions are formed by moving the predicate with the subject.
For example, using the “all” quantifier means that when you have an arbitrary variable, you must prove something true about that variable, and then prove that it does not matter what variable you chose, it will always be true. Thus, from propositional calculus, the sentence, “All humans are mortal,” becomes in predicate calculus, “All things x are such that, if x is a human, then x is a mortal.” This sentence may also be written symbolically under predicate calculus. (To compare, the sentence “x is a human” is not a statement in propositional calculus [see above] because it involves an unknown entity x; therefore, a truth value cannot be assigned without knowing what x represents.)
What are logical operators in predicate calculus?
Predicate calculus commonly uses seven special symbols—called logical operators— to express a formula (in this case, a formula is a meaningful expression built up from atomic formulas by repeated application of the logical operators). The following table lists the symbols and their meanings. (Note: Many of these symbols are also used as logical operators in truth tables; see below.)
What is the atomic formula of predicate calculus?
The atomic formula of predicate calculus is when a predicate and special case of a subject (individual) are written together. For example, if M is the predicate “to be a human,” and b is the subject (individual) “Socrates,” then Mbmeans the assertion “Socrates is a human.” This atomic formula is phrased “b is the argument of M.” Thus, M, as the predicate, may be applied to any subject, and that subject is then an argument of M. But if c is the subject “Vermont,” then Mc is a false assertion, because Vermont is not a human. Some predicates require more than one argument, thus, you can have formulas such as Mxy.
What is an algorithm?
The word “algorithm” is a distortion of Muhammad ibn Musa al-Khuwarizmi’s name (783-c. 850; also seen as al-Khowarizmi and al-Khwarizmi), the Persian mathematician who wrote about algebraic methods (for more about al-Khuwarizmi, see “History of Mathematics”). In general, an algorithm is a specific set of instructions that, if followed correctly, will lead to a recognizable end result. Simply put, a recipe is an example of an algorithm. For example, if there are two different recipes for making apple pie—one calling for cutting fresh apples for the filling, the other calling for apples from a can—the end results will be the same: an apple pie.
In mathematics, most algorithms include a finite sequence of steps that repeat, or require decisions using logic and comparisons until the final result is found (often called a computation). The best example is the long-division algorithm, in which the remainders of partial divisions are carried to the next digit or digits. For example, in the division of 1,347 by 8, a remainder of 5 in the division of 13 by 8 is placed in front of the 4, and 8 is then divided into “54,” and so on. More advanced use of algorithms are found in a type of logic called metamathematics (see below).
What is metamathematics?
Metamathematics is the study of mathematical reasoning in a general and abstract way, usually by trying to understand how theorems are derived from axioms. Thus, it is often called proof theory (for more information about axioms, see below). It does not study the objects of a particular mathematical theory, but examines the mathematical theories themselves with respect to their logical structure. Metamathematics is also used in logic to study the combination and application of mathematical symbols; this is often referred to as metalogic.
What is Gödel’s incompleteness theorem?
Austrian-American mathematician and logician Kurt Gödel (1906–1978) is best known for his studies in mathematical logic—in particular for his “incompleteness theorem” presented in 1931. This theorem shows that an infinite number of propositions that cannot be derived from axioms of a system may be proved by metamathematical means external to mathematics. In other words, mathematics abounds with questions that have a “yes or no” nature; the incompleteness theory suggests that such questions will always exist. (For more about Gödel, see “History of Mathematics.”)
Austrian-American mathematician and logician Kurt Gödel is best known for his studies in mathematical logic.
What does the philosophy of mathematics mean?
The philosophy of mathematics, or mathematical philosophy, may seem like the same thing. But to mathematicians, mathematical philosophy means many areas of study, such as ethics, logic, or metaphysics. The philosophy of mathematics is actually a branch of philosophy that studies the philosophical assumptions, implications, and foundations of the mathematics world. This branch of philosophy gives us an idea of just how mathematics fits into our modern world.
These philosophies tell us a great deal about the state of modern mathematics and logic. Like many abstract and complex studies, philosophies come and go; some good, some seemingly on the mathematical fringe. Currently, there is no single philosophy that truly defines our mathematical and logic foundations, especially when it comes to combining both mathematical knowledge and the application of mathematics to physical reality.
What are some philosophies of mathematical logic?
Over the centuries, there have been many mathematicians who have tried to figure out the complexity of mathematics. Fast-forward to modern times, and one finds that mathematical knowledge and logic have dramatically changed. In particular, in the late 20th and early 21st centuries, the development of predicate calculus and the digital computer has had a great impact on these studies. And out of these ideas—not to mention centuries of mathematics and logic groundwork—are some of the more interesting philosophical doctrines of mathematical thought:
· Formalism is the idea that mathematics is truly formal; therefore, it is only concerned with the algorithmic manipulation of symbols. In formalism, predicate calculus does not denote predicates—or anything else—meaning mathematical objects do not exist at all. This definitely fits into today’s world of computers, especially in the field of artificial intelligence. But this philosophy does not take into account human mathematical understanding, not to mention mathematical applications in physics and engineering.
· Set-theoretical Platonism sounds as if mathematicians are regressing back to Plato’s time. In reality, this philosophy is based on a variant of the Platonic doctrine of recollection, in which we are born possessing all knowledge and our realization of that knowledge is contingent on our discovery of it. In the set-theoretical Platonism, infinite sets exist in a non-material, purely mathematical realm. By extending our intuitive understanding of this realm, we can cope with the problems such as those encountered by the Gödel incompleteness theorem. But this philosophy, like the others, has a seemingly infinite number of gaps, especially the question of how can a theory of infinite sets be applied to a finite world.
· Constructivism was a “fringe” movement at the turn of the 21st century. Constructionists believe that mathematical knowledge is obtained by a series of purely mental constructions, with all mathematical objects existing only in the mind of the mathematician. But constructivism does not take into account the external world, and when taken to extremes, it can mean that there is no possibility of communication from one mind to another. This philosophy also runs the chance of rejecting the basic laws of logic. For example, if you have a mathematical problem with a yes or no nature, with the answer unknown, neither “yes” nor “no” is in the mind of the mathematician. This means that a disjunction is not a legitimate mathematical assumption—and thus, ideas such Aristotle’s law of the excluded middle (“either or”) are cast aside. And not many modern mathematicians want to throw out centuries of logic.
· Structuralism holds that mathematical theories describe structures—and that mathematical objects are defined by their place in such structures, but have no intrinsic properties. For example, if one knows that the number 1 is the first whole number after 0, then that is all that needs to be known. Even though it sounds simple, structuralism only relates to “… what kind of entity a mathematical object is, not to what kind of existence mathematical objects or structures have…” Thus, it has some interesting interpretations in philosophy.
· Fictionalism sounds like something you read that’s the opposite of non-fiction, but it’s actually a type of mathematical philosophy proposed around 1980. But in a way, the mathematician who brought out the proposal of fictionalism, Hartry Field, was talking about fiction after all: He believed that mathematics was dispensable, and should be considered “…as a body of falsehoods not talking about anything real.” This type of logic has not been readily accepted, as there are some logic and statement problems inherent in his philosophy.
· Social realism, or social constructivism, proposes that mathematics is mainly a construct of culture that is subject to change or even correction. It’s often said that this idea is the opposite of how mathematicians treat their field—that mathematics is more pure and objective. Social realism theorists believe this isn’t true, that mathematics is grounded by much uncertainty, and grows, like things in nature, through a sort of mathematical evolution.
Does everyone agree with axioms?
No, not everyone agrees with all axioms, the self-evident truth upon which knowledge must rest and other knowledge is built. For example, not all epistemologists (philosophers who deal with the nature, origin, and scope of knowledge) agree that any true axioms exist. However, in mathematics axiomatic reasoning is widely accepted, where it means an assumption on which proofs are based.
The word axiom (or postulate) comes from the Greek word axioma, and means “that which is deemed worthy or fit,” or “considered self-evident.” Ancient Greek philosophers used the term axiom as a claim that was true without any need for proof. In modern mathematics, an axiom is not a proposition that is self-evident, but simply means a starting point in a logical system. For example, in some rings (for more about rings, see “Algebra”), the operation of multiplication is commutative (said to satisfy the “axiom of commutativity of multiplication”), and in some it is not.
AXIOMATIC SYSTEM
What are axioms and postulates?
These two words are often treated as the same: in fact, some mathematicians consider the word axiom a slightly archaic synonym for postulate. Although both are considered to be a proposition (statement) that is true without proof, there are subtle differences.
An axiom in mathematics refers to a general statement that is true without proof, and is often related to equality, such as “two things equal to the same thing are equal to each other,” and those related to operations. They should also be consistent—or it should not be possible to deduce any contradictory statements from the axiom.
A postulate is also a proposition (statement) that is true without proof, but it deals with specific subject matter, such as the properties of geometric figures. Thus, it is not as general as an axiom. For example, Euclidean geometry is based on the five postulates known, of course, as Euclid’s postulates. (See below; for more about Euclid, see “History of Mathematics” and “Geometry and Trigonometry”).
What are some simple examples of axioms?
There are some very basic assertions when it comes to axioms. The following lists only a few classical axioms:
· Things which are equal to the same thing are also equal to each other.
· If equals be added to equals, the wholes are equal.
· If equals be subtracted from equals, the remainders are equal.
· Things that coincide with one another are equal to one another.
· The whole is greater than the part.
What is an axiomatic system?
An axiomatic system is a logical system that has a definite set of axioms; from these axioms, theorems can be derived. In each system, propositions (statements) are proved on the basis of a limited number of axioms or postulates—all with a few undefined terms. The other terms are defined on the basis of the undefined terms. One of the first axiomatic systems was Euclidean geometry.
Overall, an axiomatic system has several basic components: the undefined terms of the system (primitives); well-formed formulas, or how symbols are put into the system based on certain allowed rules, sometimes called defined terms; axioms, or what is also known as “self-evident truths” of the system; theorems, or statements that are proved based on axioms or other proven theorems; and finally, the rules of inference, or those that allow moves from certain formulas to other formulas.
How are some parts of an axiomatic system further defined?
There are several terms that further define an axiomatic system. All of them are slightly intertwined, depending on the system.
The absence of contradiction—or the ability to prove a proposition (statement) and its negative are both true—is known as consistency. Independence is not necessary to an axiomatic system, but consistency is definitely necessary. The opposite of consistency in an axiomatic system is inconsistency.
An axiomatic system is called independent if no other axioms can be derived (or proved) from other axioms in the system; in other words, the entire axiomatic system will be termed independent if all of its underlying axioms are independent. The independence of a system is usually determined after the consistency. An axiomatic system that is dependent has some axioms that are redundant; this is also called redundancy.
A page from Euclid’s Elements, found near Oxyrhynchus, Egypt, c. 1896, is currently kept at the University of Pennsylvania.
An axiomatic system is complete if no additional axiom can be added to the system without making the new system dependent or inconsistent. In other words, the aim is to prove or disprove any statement about the objects in the system from the axioms alone. In complete systems, every true proposition about the defined and undefined terms can be proved from the axioms. Systems with the logic based on true or false propositions connected by “and,” “or,” and “not” are complete, as are those that include quantifiers. More complex systems, such as set theory, are not considered complete.
What is an undefined term?
In terms of axiomatic systems, undefined terms are also called primitives. And although it sounds like “double-speak,” these primitives are object names, but the objects they name are left undefined. (The axioms are statements within the system that make assertions about the primitives.) If a meaning is attached to a primitive, it is called an interpretation.
Undefined terms are also found in a mix of axiomatic systems and geometry, in which definitions are formed using known words or terms to describe a new word. There are three words in geometry that are not formally defined—point, line, and plane—because they cannot be described without using words that are themselves undefined. These terms are fundamentally important in the study of geometry, because they are needed to further describe even more complex objects such as circles and triangles. (For more about geometry, see “Geometry and Trigonometry.”)
What are theorems, corollaries, and lemmas?
In mathematics and logic, a theorem is a statement demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is usually based on some general principle that makes it part of a larger theory; it differs from an axiom in that a proof is required for its acceptance. Some of the more well-known theorems are named after their discoverers, such as the Pythagorean theorem (involving right triangles) and Fermat’s last theorem. It is interesting to note that American Richard Feynman (1918–1988), one of the most brilliant physicists of the 20th century, stated that any theorem, no matter how difficult to prove in the first place, is viewed as “trivial” by mathematicians once it has been proved. Thus, according to Feynman, there are only two types of mathematical objects: trivial ones and those that have not yet been proved.
A corollary is a theorem that has been proved in only a few steps from an established theorem, or one that follows as a direct consequence of another theorem or axiom. And finally, a lemma is a theorem proved as a preliminary or intermediate step in the proof of another, more basic theorem; or a brief theorem used to prove a larger theorem.
What is a proof?
A proof is simply the process of showing a theorem to be correct, although the process itself might not be simple. These mathematical arguments are often quite rigorous, and they are used to demonstrate the truth of a given proposition. The result of the proven statement is a theorem.
Are there different types of proofs?
There are several different types of proofs in mathematical logic. Direct proofs are based on rules that result in one true proposition from two propositions. They show that a given statement is true by simply combining existing theorems with or without some mathematical manipulations. For example, if you have two sides of a triangle with the same length, a definition and theorem show that a line bisecting their vertex produces two congruent triangles—a direct proof that the angles at the other two vertices have the same size.
In logic, indirect proofs are also called “proofs by contradiction,” and are known in Latin as reductio ad absurdum (“reduced to an absurdity”). This type of proof initially assumes that the opposite of what you are trying to prove is true; from this assumption, certain conclusions can be drawn. One then searches for a conclusion that is false because it contradicts given or known information. Sometimes, a given piece of information is contradicted, which shows that, since the assumption leads to a false conclusion, the assumption must be false. If the assumption is false (the opposite of the conclusion one is trying to prove), then it is known that the goal conclusion must be true. All of this has therefore been shown “indirectly.”
What are some well-known axiomatic systems?
One of the most well-known axiomatic systems was developed by the Greek mathematician Euclid (c. 325-c. 270 B.C.E.). He presented 13 books of geometry and other mathematics, titled Elements (or Stoicheion in Greek). Included in these books were theorems about geometry and numbers derived from five postulates about points, lines, circles, and angles, four axioms about equality, and one axiom stating “the whole is greater than the part.” A more modern axiomatic system is the axiomatic set theory, which is based on eight axioms and three undefined terms.
Finally, a disproof is a single instance that contradicts a proposition. For example, the disproof of “all primes are odd” is the true statement “the number 2 is a prime and not odd.” If a disproof exists for a proposition, then the statement is false.
How are computers involved in determining proofs?
Numerous mathematicians and computer scientists are adapting computer systems to automate proofs, a computer process called automated theorem proving (ATP) or automated deduction, currently the most well-developed subfield of automated reasoning. In the 1970s, the first attempts were made to prove a theorem by computer. The so called four-color theorem was attempted by an ATP computer, which involved the theorem’s mapping problem: Using no more than four colors, the computer had to color regions of a map so that no two adjacent regions had the same color. Its proof relied on meticulous computer testing of many separate cases, all of which cannot be verified by hand.
Yet another more recent automated theorem prover is the Vampire, developed in the Computer Science Department at the University of Manchester by Andrei Voronkov and Kryštof Hoder (and previously with Alexandre Riazanov). To date, this ATP computer has won the “world cup for theorem provers,” also called the CADE ATP System Competition (International Conference on Automated Deduction [CADE], associated with the Association for Automated Reasoning) eleven times.
Although becoming less and less in numbers, some mathematicians (mostly purists) do not believe these computer-assisted proofs are valid. Not only could numbers be wrong, but the logic entered into the computer may not be correctly interpreted by the computer. They believe that only humans can understand the nuances and have the intuition needed to develop a theorem’s proof. But this is also often a proof’s dilemma: Many proofs would take decades—and reams of paper—to finish and prove, if you could prove them at all.
Have any recent proofs been discovered?
It seems that as long as there are mathematicians, there will always be someone who tries to solve mathematical questions using proofs. For example, a researcher at the Georgia Institute of Technology recently came up with a new proof, using a game-type scenario, that real numbers are not countable. Another researcher (Grigori Yakovlevich Perelman; for more about him, see “History of Mathematics”) uncovered the proof of the Poincaré conjecture, or that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes, a conjecture first proposed by H. Poincaré in 1953. And what is the dream of some mathematicians? To eventually have formal proofs of all the central theorems in mathematics right at their fingertips, something some have compared to DNA sequencing—as in the “sequencing of the mathematical genome.”
What is deduction and induction and how are they used in mathematics?
Deduction in logic is when conclusions are drawn from premises and syllogisms (for more information on these terms, see above). In this instance, a deduction is a form of inference or reasoning such that the conclusion is true if the premises are true; or, based on general principles, particular facts and relationships are derived. Deductive logic also means the process of proving true statements (theorems) within an axiomatic system; if the system is valid, all of the derived theorems are considered valid. For example, if it is known that all dogs have four legs and Spot is a dog, we logically deduce that Spot has four legs; other examples of deductive reasoning include Aristotle’s syllogisms.
Induction is a term usually used in regards to probability, in which the conclusion can be false even when the premises are true. In contrast to deduction, the premise gives grounds for the conclusion, but does not necessitate it. Inductive logic generates “correct” conclusions based on observation or data. (But note that not all inductive logic leads to correct generalizations, making the validity of many such arguments probabilistic or “iffy” in nature.)
One can see how both these processes work in the scientific world, especially in the scientific method, in which general principles are inferred from certain facts. For example, by observation of events (induction) and from principles already developed (deduction), new hypotheses are formulated. Hypotheses are then tested by applications; and as the results satisfy the conditions of the hypotheses, laws are developed by induction. Future laws are then often developed, many of them determined by deduction.
What are some examples of paradoxes throughout history?
The oldest paradoxes may be from the Greek Epimenides the Cretan (lived sometime during the 6th century B.C.E.), who stated, “All Cretans are liars.” If this statement is true—and any other culture you would care to put in the Cretans’ place—then the implication is that the statement is a lie. This is also called the “Liar’s paradox.”
The number of paradoxes continued almost ad infinitum since then. Some of the more popular ones include those listed as Zeno’s paradoxes. They were named after Greek philosopher Zeno of Elea (c. 490 B.C.E.), a disciple of the philosopher Parmenides, who believed that reality was an absolute, unchanging whole—and, thus, many things we take for granted, such as motion, were simply illusions. In order to defend his master’s highly debated philosophy, Zeno developed his paradoxes. Most of Zeno’s paradoxes are still highly debated by modern mathematicians and philosophers, thus proving another paradox: Nothing truly changes throughout history—or does it?
What is a conclusion in logic?
A conclusion is a statement (proposition) found by applying a set of logical rules (syllogisms) to a set of premises. In addition, the final statement of a proof is called the proof’s conclusion. For example, in a statement that includes “if … then,” the result following the “then” in the statement is called the conclusion.
What is modus ponens?
The Latin term modus ponens means “mode that affirms,” or in the case of logic, stands for the rule of detachment. This rule (also known as a rule of inference) pertains to the “if…then” statement and forms the basis of most proofs: “If p then q,” or if p is true, then the conclusion q is true. Or simply, it is often seen as the following:
If p, then q.
p. Therefore, q.
To see this another way:
p ⇒ q: ‘If it is raining, then there are clouds in the sky.”
p: “It is raining.”
q: “There are clouds in the sky.”
There are several ways to break down the modus ponens. The argument form has two premises: The “if-then” (or conditional claim), or namely that pimplies q; and that p (called the antecedent of the conditional claim) is true. From these two premises it can be logically concluded that q (called the consequent of the conditional claim) must be true as well; in other words, if the antecedent of a conditional is true, then the consequent must be true.
What is a fallacy?
A fallacy is an incorrect result—in this case, one arrived at through misleading reasoning when examining a logical argument. One of the more common fallacies in logic is thinking incorrectly that if “p implies q” is true, “then q implies p” is also true. The idea of such invalid arguments was well-known in the past: With Greek mathematician Aristotle’s syllogisms, an argument was valid if it adhered to all the laws; to be false, it only needed to break one law. Euclid, another Greek mathematician, was known to have written an entire book on fallacies in geometry, but the book has since been lost.
English logician, philosopher, and mathematician Bertrand Russell came up with a puzzling paradox involving sets.
What is a paradox?
In logic, paradoxes are statements that seem to be self-contradictory or contrary to one’s expectations. These arguments imply both a proposition and its opposite. One of the most famous paradoxes was stated by English logician Bertrand Russell (1872–1970) in 1901 dealing with sets: “If sets that are not members of themselves are normal, is the set of normal sets itself normal?” (For more information about Russell’s paradoxes in set theory, see below.)
What are some paradoxes that deal with space and time?
There are numerous paradoxes that deal with the counterintuitive aspects of continuous space and time. One of the most well-known is the dichotomy (or racetrack) paradox. Before an object can travel a distance d, it must keep traveling “in halves.” In terms of the racetrack, in order to reach the end of the course, a person would have to first reach the halfway mark, then the halfway mark of the remaining half, then the halfway mark of the final fourth, then of the final eighth, and so on ad infinitum (to infinity). Therefore, the distance can never truly be traveled to reach the end of the racetrack.
When is a paradox not a paradox?
There are many proposed paradoxes that are not really one—even though they sound like it. One just-for-fun, simple example of a non-paradox can be called a “friends” paradox. Brought out in a study on friendships by sociologists Scott L. Feld, it demonstrates the seeming paradox of why your friends seem to have more friends than you do. Follow us here: There are 12 people who have a friend who has 12 friends, but there is only one person who has a friend who has only one friend. There is also no one who has a friend who doesn’t have any friends. There is only one friend who has 12 friends. Thus, the number 12 gets counted only once when you compute the average number of friends that people have, but it gets counted 12 times when you calculate the average number of friends that their friends have. Thus, it seems like a paradox that your friends have more friends than you do, when in reality, it’s merely that you’re more likely to be friends with someone who has more friends than with someone who has fewer friends. If you’re totally confused, don’t worry. That’s the nature of logic and the paradox.
The Achilles and the tortoise paradox is a version of the tortoise and the hare, but with a very different resolution than the well-known fable. In this paradox, Achilles gives the slower tortoise a head start; Achilles starts when the tortoise reaches point a. But by the time Achilles reaches a, the tortoise has already moved beyond that point, to point b; when Achilles reaches b, the tortoise is at point c, and so on ad infinitum. Since this process goes on forever, Achilles can never catch up with the tortoise.
Another paradox is the arrow paradox. In this case, an arrow in flight has a certain position at a given instant in time, but that is indistinguishable from a motionless arrow in the same position. So how is the arrow’s motion perceived?
Finally, one of the most interesting and insightful paradoxes is attributed to Socrates—thus, the Socrates’ paradox. It is based on Socrates’ statement, “One thing I know is that I know nothing.”
SET THEORY
What is set theory?
Set theory is the mathematical theory of sets and is associated with logic; it is also considered the study of sets (collections of objects or entities that can be real or concepts) and their properties. (For more about sets, see below.) Under formal set theory, three primitives (undefined terms) are used: S (the set), I (the identity), and E (the element). Thus, the formulas Sx, Ixy, Exy mean “x is a set,” “x is identical to y,” and “x is an element of y,” respectively.
Overall, set theory fits in with the aims of logic research: to find a single formula theory that will unify and become the basis for all of mathematics. And as it turns out, sets lead directly to a vast amount of data encompassing all of modern mathematics. There are also a number of different set theories, each having its own rules and axioms. No matter what version, set theory is not only important to mathematics and logic, but to other fields as well, such as computer technology, and atomic and nuclear physics.
What are naïve and axiomatic set theory?
The naive set theory is not one that takes everything for granted. It is actually a branch of mathematics that attempts to formalize the nature of the set using the fewest number of independent axioms possible. But it is not the answer to formalizing sets, as it quickly leads to a number of paradoxes. Because of this, mathematicians use a more formal theory called the axiomatic set theory, which is a version that uses axioms taken as uninterpreted rather than as formalization of pre-existing truths (for more about axiomatic systems, see elsewhere in this chapter).
What is Russell’s Paradox?
Russell’s Paradox is one of the most famous of the set theory paradoxes. It first appears when studying the naive set theory: In this case, R is the set of all sets that are not members of themselves; from there, R is neither a member of itself nor a member of itself. The paradox sets in when you try to reason how a set appears to be a member of itself if and only if it is not a member of itself. Symbolically, it is expressed as follows:
Discovered by Welsh mathematician and logician Bertrand Arthur William Russell (1872–1970) in 1901, the paradox sparked a great deal of work (and controversy) in logic, set theory, and especially in philosophy and foundations of mathematics. The reason why it became so important was its affect on mathematics: It created problems for those who based mathematics on logic, and it also indicated that something was wrong with Georg Cantor’s intuitive set theory. (For more about Russell and the paradox, see “History of Mathematics.”)
What is a set?
Simply put, a set is a collection of objects or entities; these are called the elements of the set. The number of elements in a set can be large, small, finite, or infinite. The informal notation used for sets is sometimes seen as x = {y, z, …}, with brackets used to contain the elements within the set. It is stated as, “x is a set consisting of the elements y, z, and so on.” But more commonly, sets are seen as capital letters and elements as lowercase letters, such as a is an element of set A.
Who developed set theory?
German mathematician George (Georg) Ferdinand Ludwig Philipp Cantor (1845–1918) is most well-known for his development of set theory (for more information on Cantor, see “History of Mathematics”). His Mathematische Annalen was a basic introduction to set theory, in which he built a hierarchy of infinite sets according to their cardinal number. In particular, using one-to-one pairing, he showed that the set of real numbers has a higher cardinal number than does the set of rational fractions.
Unlike most subjects in mathematics, Cantor’s set theory was his creation alone. But like many brilliant, revolutionary thinkers throughout history, his ideas were highly criticized by his contemporaries. This strong opposition contributed to the multiple nervous breakdowns he suffered throughout the last 33 years of his life, which ended tragically in a mental institution.
How do you interpret sets?
There are several ways to look at sets. Two sets (or more) are considered identical if, and only if, they have the same collection of objects or entities. This is a principle known as extensionality. For example, the set {a, b, c} is considered to be the same as set {a, b, c}, of course, because the elements are the same; the set {a, b, c} and the set {c, b, a} are also the same, even though they are written in a different order.
It becomes more complex when sets are elements of other sets, so it is important to note the position of the brackets. For example, the set {{a, b}, c} is distinct from the set {a, b, c} (note that the brackets differ); in turn, the set {a, b} is an element of the set {{a, b}, c}. (It is a set included between the outside brackets.)
Another example that shows how sets are interpreted includes the following: If B is the set of real numbers that are solutions of the equation x2 = 9, then the set can be written as B = {x: x2 = 9}, or B is the set of all x such that x2= 9. Thus B is {3, -3}.
What are open and closed sets?
Formally, an open set is one in which every element in the set has a neighbor in the set or does not include its boundary. A closed set is one that does include a boundary of a set, or one where some elements have neighbors not in the set.
What is a null or empty set?
A null or empty set contains no elements; an empty set is considered to be a subset of every other set. The opposite of an empty set is, logically, a nonempty set, or one that is not empty. The notations for empty set are { } and ∅, but not ( ), as it is sometimes written in texts. Interestingly enough, an empty set is considered to be both open and closed for any set X.
What is the set theory approach to arithmetic?
The set theory approach to arithmetic is defined in terms of the non-negative whole numbers 0, 1, 2, 3, …. These numbers are identified with specific sets based on the placement and number of brackets. For example, mathematicians identify 0 with the empty set { }; 1 is identified with {{ }}; 2 is identified with {{}, {{}}}; 3 is identified with {{ }, {{ }}, {{ }, {{ }}}}, and so on (each bracket an interpretation of the empty set and 1, or {{}}).
What are the basic symbols used to operate on sets?
When set theory founder Georg Cantor developed the symbols for sets, he used a single horizontal overbar to denote a set with no structure besides order; thus, it represented the order type of the set. A double bar meant that there was no order from the set, also called the cardinal number of the set (see below).
The more common symbols accepted in today’s set theory include the following:
Symbols Used to Operate Sets
How are some symbols used in operations on sets?
There are many ways to operate on sets. The following lists some of the more simple operations on sets, in which E, F, and G are sets:
What are the basic set operations?
There are several basic set operations, the most common being the intersection of sets, union of sets, and the complement of sets. The following lists these operations (note: the first two operations obey the associative and communtative laws, and together they obey the distributive law):
Intersection—The intersection of two sets is the set of elements common to the two sets. For example, the intersection of sets A and B is the set of elements common to both A ∩ B. This is usually written as A D B. Thus, if A = {1, 2, 3, 4} and B = {3, 4, 5}, then the intersection of A and Bwould be {3, 4}.
Union—The union of sets is the combining of members of the sets. For example, the union of two sets A and B is the set obtained by combining members of sets A and B. This is usually written as A ∪ B. Thus, if A = {1, 2, 3, 4} and B = {3, 4, 5}, then the union of A and B would be {1, 2, 3, 4, 5}.
Complement or complementation—When the set of all elements under consideration must be specified, it is called the universal set. And if the universal set is U = {1, 2, 3, 4, 5} and A = {1, 2, 3}, then the complement of A (or A’) is the set of all elements in the universal set that are not A, or {4, 5}. The intersection between a set and its complement is the empty or null set (∅); the union of a set and its complement is the universal set.
What are cardinal and ordinal numbers and finite sets in set theory?
Cardinal and ordinal numbers are used in reference to numbers: Ordinal numbers are used to describe the position of objects or entities arranged in a certain sequence, such as first, second, third, and so on; cardinal numbers are natural numbers, or 0, 1, 2, 3, and so on. (For more information about cardinal and ordinal numbers, see “Math Basics.”)
But cardinal numbers used in set theory describe the number of members in a set. Both ordinal and cardinal numbers are further used to describe infinite sets, and are prefaced with “first ordinal” or “first cardinal” infinities. The first ordinal infinity applies to the smallest number greater than any finite ordered set of natural numbers. The first cardinal infinity applies to the number of all the natural numbers. (For more about infinite sets, see below.)
A finite set is one that is not infinite; it can be numbered from 1 to n, for some positive integer n. This number n is also called the set’s cardinal number; thus, for a certain set A, the cardinality is denoted by card(A). There are a number of rules to cardinal numbers and finite sets. For example, if two sets bisect, then they are said to have the same cardinality (or power). The empty set is considered to be a finite set, with its set’s cardinal number being 0.
What is a universal set?
A universal set actually applies to sets that are not universal, but are chosen from a specific type of entity, such as sets of numbers or letters. Thus, the set of all the elements in a set theory problem are collectively called the universal set. In reality, the “set of all things” does not exist because there is no largest or all-inclusive set, and so the true universal set is not recognized in standard set theory.
What is a subset and proper subset in set theory?
Simply put, a subset is a portion of a set. If set B is a subset of set A, then all elements of set B are also elements in set A. If A and B are equal, then both sets are subsets of themselves; the empty set is also considered a subset of every other set. A proper subset is a subset other than the set itself.
When is a set a superset?
A superset is one that contains all the elements of a smaller set. For example, if B is a subset of A, then A is a superset of B; in other words, A is a superset of set B if every element in B is in A. Like a proper subset, there are also proper supersets (or a superset that is not the entire set).
What does it mean if a set is countable?
If a set is countable (or denumerable), it means that it is finite. This also means that the set’s members can be matched in a one-to-one correspondence, in which each element in one set is matched exactly with one element in the second, and vice versa, with all the natural numbers, or with a subset of the natural numbers. Mathematicians often say, “A and B are in one-to-one correspondence,” or “A and B are bijective.” (For more about one-to-one correspondence, see “Math Basics.”) In set theory, all finite sets are considered to be countable, as are all subsets of the natural numbers and integers. But sets such as real numbers, points on a line, and complex numbers are not countable.
What is Zermelo’s axiom of choice?
Although it sounds like something on a Greek restaurant menu, Zermelo’s axiom of choice is actually a fundamental axiom in set theory. It states that given any set of mutually exclusive nonempty sets, there is at least one set that contains exactly one element common with each of the nonempty sets.
This was one of David Hilbert’s problems that needed to be solved by mathematicians of his day (for more about David Hilbert, see elsewhere in this chapter, and in “History of Mathematics”). German mathematician Ernst Friedrich Zermelo (1871–1953) took on the task, and in 1904 he developed what is called the well-ordering theorem, which says every set can be well ordered based on the axiom of choice.
This brought fame to Zermelo, but it was not accepted by all mathematicians who balked at the lack of axiomatization of set theory (for more about axiomatic set theory, see above). Although he finally did axiomatize set theory and improve on his theorem, there were still gaps in his logic, especially since he failed to prove the consistency in his axiomatic system. By 1923, German mathematician Adolf Abraham Halevi Fraenkel (1891–1965) and Norwegian mathematician Albert Thoralf Skolem (1887–1963) independently improved Zermelo’s axiomatic system, resulting in the system now called Zermelo-Fraenkel axioms (Skolem’s name was not included, although another theorem is named after him). This is now the most commonly used system for axiomatic set theory.
What is combinatorics?
Combinatorics is a branch of mathematics—overall called combinatorial mathematics—that studies the enumeration, combination, and permutation of sets and the mathematical relations that involve these properties, defined as:
Enumeration—Sets can be identified by the enumeration of their elements; in other words, determining (or counting) the set of all solutions to a given problem.
Combination—Combination is how to count the many different ways elements from a given set can be combined. For example, the 2-combinations of the 4-set {A,B,C,D} are {A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}.
Permutation—Permutation is the rearrangement of elements of a set into a particular order, often a one-to-one correspondence. The number of permutations of a particularly sized set with n members is written as the factorial n!. For example, a set with 4 members would have 4 in first place to 1 in the last place. This would equal 4 × 3 × 2 × 1 = 4!, or 24, permutations of 4 members. (For more information about factorials, see “Algebra.”)
What is an ordered pair?
An ordered pair is two quantities—usually written as (a, b)—that have a significant order; thus, (a, b) does not equal (b, a). Ordered pairs are used in set theory to define members in a function.
Ordered pairs are also valuable in linear equations and graphing, in which the x coordinate is the first number and the y coordinate is the second number, or (x, y). They are used on a grid to locate a point. (For more information about ordered pairs and graphs, see “Geometry and Trigonometry.”)
How do functions pertain to sets?
A function in sets pertains to a correspondence between two sets, called the domain and range; each member of the domain has exactly one member of the range. It is often called a many-to-one (or sometimes one-to-one) relation. For example, f = {(1,2), (3,6), (4, -2), (8,0), (9,6)} is a function, with each set of numbers being ordered pairs. This is because it assigns each member of the set {1, 3, 4, 8, 9} exactly one value in the set {2, 6, -2, 0, 6}. It never has two ordered pairs with the same x and different y values. In this case, the domain is {1, 3, 4, 8, 9} and the range is {2, 6, -2, 0, 6}.
To show an example that is not a function, f = {(1,8), (4,2), (3,5), (1,3), (6,11)} is not a function because it does not assign each member of the set exactly one value: It assigns x = 1 to both y = 8 and y = 3, or it has two ordered pairs that have the same x values to two different y values, (1, 8) and (1, 3). (For more information about functions, see “Algebra.”)
The expanses of the universe seem infinite to us, but in mathematics the concept of infinity reaches even beyond the edges of the universe toward numbers that are inconceivably large.
How else is the term function used?
Unfortunately (as with many mathematical terms), there is often more than one function for the word “function.” For example, contrary to the definition above, function can also mean the relationships that map single points in the domain to multiple points in the range—called multivalued functions—mainly used in the theory of complex functions. To further confuse matters, there are also functions called non-multivalued functions.
What are some strange characteristics of infinity?
There are some strange ideas about infinity—the most interesting one being that it’s not just a bunch of numbers that go on forever. There are certain properties, too. For example, infinity doesn’t always equal infinity; and infinity minus infinity does not equal zero; or even infinity over infinity (infinity/infinity) does not equal one. Why don’t the “normal” rules of mathematics apply here? The biggest problem is infinity itself—or is it “themselves”? After all, one infinity in the questioned equation may be larger than the other—and you wouldn’t know.
There is also the paradox of infinity. For example, take the following two sequences:
ab = 1, 2, 3, 4, 5, 6 …
cd = 2, 4, 6, 8, 10…
ab contains all natural numbers, while cd contains all even natural numbers. Although you would think cd would have half the number of terms, and thus would be a “smaller” infinity, they both have an infinite number of terms— thus the paradox of infinity. That is because most of us think of infinity as a number rather than a concept.
Where did the symbol for infinity originate?
Infinity is represented by the symbol ∞, a sign introduced by John Wallis in 1655 in his treatise, “De sectionibus conicus.” Historians believe that Wallis, a classical scholar, adopted the sign from the late Roman symbol for “1,000.” Whether it was from there or another source, the result was (and remains) the same: “a figure-eight on its side,” as many people call the infinity symbol.
Are there different types of infinity in mathematics?
At the end of the nineteenth century, German mathematician George (Georg) Ferdinand Ludwig Philipp Cantor (1845–1918) showed that different orders of infinity existed and that the infinity of points on a line was of a greater order than that of prime numbers. Since that time, mathematicians have managed to divide the topic of infinity into even more precise terms.
To most of us, the universe represents infinity, but in mathematics, infinity is the unbounded quantity that is greater than every real number. In fact, it is called potential infinity in mathematics, in which the potential for infinity exists with natural numbers because you can always mention a number greater than any given number. Directed infinity applies to an infinity in direction z and is an infinite numerical quantity that is a positive real multiple of the complex number z; a directed infinity with an unknown direction is known as a complex infinity. Still another “type” of infinity in mathematics is completed infinity, which refers to the size of an infinite set (such as all the points on a line).