focus (plural, foci) Each CONIC SECTION has associated with it one or two special points each called a
focus of the conic.
See also ELLIPSE; HYPERBOLA; PARABOLA.
formal logic (symbolic logic) In mathematics, the
systematic study of reasoning is called formal logic. It
analyzes the structure of ARGUMENTs, as well as the
methods and validity of mathematical deduction and
proof.
The principles of logic can be attributed to ARISTOTLE (384–322 B.C.E.), who wrote the first systematic
treatise on the subject. He sought to identify modes of
inference that are valid by virtue of their structure, not
their content. For example, “Green and blue are colors;
therefore green is a color” and “Cows and pigs are reptiles; therefore cows are reptiles” have the same structure
(“A and B, therefore A”), and any argument made via
this structure is logically valid. (In particular, the second
example is logically sound.) This mode of thought
allowed EUCLID (ca. 300–260 B.C.E.) to formalize geometry, using deductive proofs to infer geometric truths
from a small collection of AXIOMs (self-evident truths).
No significant advance was made in the study of
logic for the millennium that followed. This period was
mostly a time of consolidation and transmission of the
material from antiquity. The Renaissance, however,
brought renewed interest in the topic. Mathematical
scholars of the time, Pierre Hérigone and Johann Rahn
in particular, developed means for representing logical
arguments with abbreviations and symbols, rather than
words and sentences. GOTTFRIED WILHELM LEIBNIZ
(1646–1716) came to regard logic as “universal mathematics.” He advocated the development of a “universal
language” or a “universal calculus” to quantify the
entire process of mathematical reasoning. He hoped to
devise new mechanical symbolism that would reduce
errors in thinking to the equivalent of arithmetical
errors. (He later abandoned work on this project,
assessing it too daunting a task for a single man.)
In the mid-1800s GEORGE BOOLE succeeded in creating a purely symbolic approach to propositional
logic, that part which deals with inferences involving
simple declarative sentences (statements) joined by the
connectives:
not, and, or, if … then…, iff
(These are called the NEGATION, CONJUNCTION, DISJUNCTION, CONDITIONAL, and the BICONDITIONAL, respectively.) He successfully applied it to mathematics, thereby
making a significant step to achieving Leibniz’s goal.
In 1879 the German mathematician and philosopher Gottlob Frege constructed a symbolic system for
predicate logic. This generalizes propositional logic by
including QUANTIFIERs: statements using words such as
some, all, and, no. (For example, “All men are mortal”
as opposed to “This man is mortal.”) At the turn of the
century DAVID HILBERT sought to devise a complete,
consistent formulation of all of mathematics using a
small collection of symbols with well-defined meanings.
English mathematician and philosopher BERTRAND
RUSSELL, in collaboration with his colleague ALFRED
NORTH WHITEHEAD, took up Hilbert’s challenge. In
1925 they published a monumental work. Beginning
with an impressively minimal set of premises (“self-evident” logical principles), they attempted to establish
the logical foundations of all of mathematics. This was
an impressive accomplishment. (After hundreds of
pages of symbolic manipulations, they established the
validity of “1 + 1 = 2,” for example.) Although they
did not completely reach their goal, Russell and Whitehead’s work has been important for the development of
logic and mathematics.
Six years after the publication of their efforts, however, KURT GÖDEL stunned the mathematical community by proving Hilbert’s (and Leibniz’s) goal to be
futile. He demonstrated once and for all that any formal system of logic sufficiently sophisticated to incorporate basic principles of arithmetic cannot attain all
the statements it hopes to prove. His results are today
called GÖDEL’S INCOMPLETENESS THEOREMS. The vision
to reduce all truths of reason to incontestable arithmetic was thereby shattered.
Understanding the philosophical foundations of
mathematics is still an area of intense scholarly research.
See also ARGUMENT; DEDUCTIVE/INDUCTIVE REASONING; LAWS OF THOUGHT.
formula Any identity, general rule, or general expression in mathematics that can be applied to different
values of one or more quantities is called a formula.
For example, the formula for the area A of a circle is:
A = πr
2
formula 199
focus of the conic.
See also ELLIPSE; HYPERBOLA; PARABOLA.
formal logic (symbolic logic) In mathematics, the
systematic study of reasoning is called formal logic. It
analyzes the structure of ARGUMENTs, as well as the
methods and validity of mathematical deduction and
proof.
The principles of logic can be attributed to ARISTOTLE (384–322 B.C.E.), who wrote the first systematic
treatise on the subject. He sought to identify modes of
inference that are valid by virtue of their structure, not
their content. For example, “Green and blue are colors;
therefore green is a color” and “Cows and pigs are reptiles; therefore cows are reptiles” have the same structure
(“A and B, therefore A”), and any argument made via
this structure is logically valid. (In particular, the second
example is logically sound.) This mode of thought
allowed EUCLID (ca. 300–260 B.C.E.) to formalize geometry, using deductive proofs to infer geometric truths
from a small collection of AXIOMs (self-evident truths).
No significant advance was made in the study of
logic for the millennium that followed. This period was
mostly a time of consolidation and transmission of the
material from antiquity. The Renaissance, however,
brought renewed interest in the topic. Mathematical
scholars of the time, Pierre Hérigone and Johann Rahn
in particular, developed means for representing logical
arguments with abbreviations and symbols, rather than
words and sentences. GOTTFRIED WILHELM LEIBNIZ
(1646–1716) came to regard logic as “universal mathematics.” He advocated the development of a “universal
language” or a “universal calculus” to quantify the
entire process of mathematical reasoning. He hoped to
devise new mechanical symbolism that would reduce
errors in thinking to the equivalent of arithmetical
errors. (He later abandoned work on this project,
assessing it too daunting a task for a single man.)
In the mid-1800s GEORGE BOOLE succeeded in creating a purely symbolic approach to propositional
logic, that part which deals with inferences involving
simple declarative sentences (statements) joined by the
connectives:
not, and, or, if … then…, iff
(These are called the NEGATION, CONJUNCTION, DISJUNCTION, CONDITIONAL, and the BICONDITIONAL, respectively.) He successfully applied it to mathematics, thereby
making a significant step to achieving Leibniz’s goal.
In 1879 the German mathematician and philosopher Gottlob Frege constructed a symbolic system for
predicate logic. This generalizes propositional logic by
including QUANTIFIERs: statements using words such as
some, all, and, no. (For example, “All men are mortal”
as opposed to “This man is mortal.”) At the turn of the
century DAVID HILBERT sought to devise a complete,
consistent formulation of all of mathematics using a
small collection of symbols with well-defined meanings.
English mathematician and philosopher BERTRAND
RUSSELL, in collaboration with his colleague ALFRED
NORTH WHITEHEAD, took up Hilbert’s challenge. In
1925 they published a monumental work. Beginning
with an impressively minimal set of premises (“self-evident” logical principles), they attempted to establish
the logical foundations of all of mathematics. This was
an impressive accomplishment. (After hundreds of
pages of symbolic manipulations, they established the
validity of “1 + 1 = 2,” for example.) Although they
did not completely reach their goal, Russell and Whitehead’s work has been important for the development of
logic and mathematics.
Six years after the publication of their efforts, however, KURT GÖDEL stunned the mathematical community by proving Hilbert’s (and Leibniz’s) goal to be
futile. He demonstrated once and for all that any formal system of logic sufficiently sophisticated to incorporate basic principles of arithmetic cannot attain all
the statements it hopes to prove. His results are today
called GÖDEL’S INCOMPLETENESS THEOREMS. The vision
to reduce all truths of reason to incontestable arithmetic was thereby shattered.
Understanding the philosophical foundations of
mathematics is still an area of intense scholarly research.
See also ARGUMENT; DEDUCTIVE/INDUCTIVE REASONING; LAWS OF THOUGHT.
formula Any identity, general rule, or general expression in mathematics that can be applied to different
values of one or more quantities is called a formula.
For example, the formula for the area A of a circle is:
A = πr
2
formula 199