Hume's principle
Encyclopedia
Hume's Principle or HP—the terms were coined by George Boolos
—says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection
) between the Fs and the Gs. HP can be stated formally in systems of second-order logic
. Hume's Principle is named for the Scottish philosopher David Hume
.
HP plays a central role in Gottlob Frege
's philosophy of mathematics. Frege shows that HP and suitable definitions of arithmetical notions entail all axioms of what we now call second-order arithmetic
. This result is known as Frege's theorem
, which is the foundation for a philosophy of mathematics known as neo-logicism.
's A Treatise of Human Nature
. Hume there sets out seven fundamental relations between ideas. Concerning one of these, proportion
in quantity
or number
, Hume argues that our reasoning about proportion in quantity, as represented by geometry
, can never achieve "perfect precision and exactness", since its principles are derived from sense-appearance. He contrasts this with reasoning about number or arithmetic
, in which such a precision can be attained:
Note Hume's use of the word number
in the ancient sense, to mean a set or collection of things rather than the common modern notion of "positive integer". The ancient Greek notion of number (arithmos) is of a finite plurality composed of units. See Aristotle
, Metaphysics
, 1020a14 and Euclid
, Elements, Book VII, Definition 1 and 2. The contrast between the old and modern conception of number is discussed in detail in Mayberry (2000). The credit Frege tries to give to Hume is therefore probably not deserved, and Hume certainly would have rejected at least some of the consequences Frege draws from HP, in particular, the consequence that there are infinite sets.
was to be characterized in terms of one-to-one correspondence had previously been used to great effect by Georg Cantor
, whose writings Frege
knew. The suggestion has therefore been made that Hume's Principle ought better be called "Cantor's Principle". But Frege criticized Cantor on the ground that Cantor defines cardinal number
s in terms of ordinal number
s, whereas Frege wanted to give a characterization of cardinals that was independent of the ordinals. Cantor's point of view, however, is the one embedded in contemporary theories of transfinite numbers, as developed in axiomatic set theory.
George Boolos
George Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.- Life :...
—says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
) between the Fs and the Gs. HP can be stated formally in systems of second-order logic
Second-order logic
In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
. Hume's Principle is named for the Scottish philosopher David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
.
HP plays a central role in Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
's philosophy of mathematics. Frege shows that HP and suitable definitions of arithmetical notions entail all axioms of what we now call second-order arithmetic
Second-order arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. It was introduced by David Hilbert and Paul Bernays in their...
. This result is known as Frege's theorem
Frege's theorem
Frege's theorem states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob Frege in his Die Grundlagen der Arithmetik , published in 1884, and proven more formally in his Grundgesetze der Arithmetik , published...
, which is the foundation for a philosophy of mathematics known as neo-logicism.
Origins
Hume's Principle appears in Frege's Foundations of Arithmetic, which quotes from Part III of Book I of David HumeDavid Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...
's A Treatise of Human Nature
A Treatise of Human Nature
A Treatise of Human Nature is a book by Scottish philosopher David Hume, first published in 1739–1740.The full title of the Treatise is 'A Treatise of Human Nature: Being an Attempt to introduce the experimental Method of Reasoning into Moral Subjects'. It contains the following sections:* Book 1:...
. Hume there sets out seven fundamental relations between ideas. Concerning one of these, proportion
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
in quantity
Quantity
Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...
or number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
, Hume argues that our reasoning about proportion in quantity, as represented by geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, can never achieve "perfect precision and exactness", since its principles are derived from sense-appearance. He contrasts this with reasoning about number or arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
, in which such a precision can be attained:
Algebra and arithmetic [are] the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty. We are possessed of a precise standard, by which we can judge of the equality and proportion of numbers; and according as they correspond or not to that standard, we determine their relations, without any possibility of error. When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal; and it is for want of such a standard of equality in [spatial] extension, that geometry can scarce be esteemed a perfect and infallible science. (I. III. I.)
Note Hume's use of the word number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
in the ancient sense, to mean a set or collection of things rather than the common modern notion of "positive integer". The ancient Greek notion of number (arithmos) is of a finite plurality composed of units. See Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, Metaphysics
Metaphysics (Aristotle)
Metaphysics is one of the principal works of Aristotle and the first major work of the branch of philosophy with the same name. The principal subject is "being qua being", or being understood as being. It examines what can be asserted about anything that exists just because of its existence and...
, 1020a14 and Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, Elements, Book VII, Definition 1 and 2. The contrast between the old and modern conception of number is discussed in detail in Mayberry (2000). The credit Frege tries to give to Hume is therefore probably not deserved, and Hume certainly would have rejected at least some of the consequences Frege draws from HP, in particular, the consequence that there are infinite sets.
Influence on set theory
The principle that cardinal numberCardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
was to be characterized in terms of one-to-one correspondence had previously been used to great effect by Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
, whose writings Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
knew. The suggestion has therefore been made that Hume's Principle ought better be called "Cantor's Principle". But Frege criticized Cantor on the ground that Cantor defines cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
s in terms of ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s, whereas Frege wanted to give a characterization of cardinals that was independent of the ordinals. Cantor's point of view, however, is the one embedded in contemporary theories of transfinite numbers, as developed in axiomatic set theory.
External links
- Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
: "Frege's Logic, Theorem, and Foundations for Arithmetic" -- by Edward Zalta. - "The Logical and Metaphysical Foundations of Classical Mathematics,"
- Arche: The Centre for Philosophy of Logic, Language, Mathematics and Mind at St. Andrew's University.