Typographical Number Theory
Encyclopedia
Typographical Number Theory (TNT) is a formal axiom
atic system describing the natural numbers that appears in Douglas Hofstadter
's book Gödel, Escher, Bach
. It is an implementation of Peano arithmetic that Hofstadter uses to help explain Gödel's incompleteness theorems
.
Like any system implementing the Peano axioms, TNT is capable of referring to itself (it is self-referential).
. Instead it makes use of a simple, uniform way of giving a compound symbol to each natural number:
The symbol S can be interpreted as "the successor of", or "the number after". Since this is, however, a number theory, such interpretations are useful, but not strict. We cannot say that because four is the successor of three that four is SSSS0, but rather that since three is the successor of two, which is the successor of one, which is the successor of zero, which we have described as 0, four can be "proved" to be SSSS0. TNT is designed such that everything must be proven before it can be said to be true. This is its true power, and to undermine it would be to undermine its very usefulness.
. These are
More variables can be constructed by adding the prime symbol
after them; for example,
are all variables.
In the more rigid version of TNT, known as "austere" TNT, only
are used.
and "a times d" is written as
The parentheses are required. Any laxness would violate TNT's formation system (although it is trivially proved this formalism is unnecessary for operations which are both commutative and associative). Also only two terms can be operated on at once. Therefore to write "a plus b plus c", we must write either
+ c)
or
)
is a true statement in TNT, with the interpretation "3 plus 3 equals 6".
, i.e. the turning of a statement to its opposite, is denoted by the "~" or negation operator. For instance,
is a true statement in TNT, interpreted as "3 plus 3 is not equal to 7".
By negation, this means negation in Boolean logic
(logical negation), rather than simply being the opposite. For example, if I were to say "I am eating a grapefruit", the opposite is "I am not eating a grapefruit", rather than "I am eating something other than a grapefruit". Similarly "The Television is on" is negated to "The Television is not on", rather than "The Television is off". This is a subtle difference, but an important one.
and ∃
.
Note that unlike most other logical systems where qualifiers over sets require a mention of the element's existence in the set, this is not required in TNT because all numbers and terms are strictly natural numbers or logical boolean statements. It is therefore equivalent to say ∀a:(a ∈ N):∀b:(b ∈ N): (a + b) = (b + a) and ∀a:∀b:(a + b) = (b + a)
For example:
("For every number a and every number b, a plus b equals b plus a", or more figuratively, "Addition is commutative.")
("There does not exist a number c such that c plus one equals zero", or more figuratively, "Zero is not the successor of any (natural) number.")
apart from the Atom symbols are used in Typographical Number Theory, and they retain their interpretations.
Atoms are here defined as strings which amount to statements of equality, such as
1 is not equal to 2:
2 plus 3 equals five: = SSSSS0
2 plus 2 is not equal to 3:
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
atic system describing the natural numbers that appears in Douglas Hofstadter
Douglas Hofstadter
Douglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics...
's book Gödel, Escher, Bach
Gödel, Escher, Bach
Gödel, Escher, Bach: An Eternal Golden Braid is a book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll"....
. It is an implementation of Peano arithmetic that Hofstadter uses to help explain Gödel's incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
.
Like any system implementing the Peano axioms, TNT is capable of referring to itself (it is self-referential).
Numerals
TNT does not use a distinct symbol for each natural numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
. Instead it makes use of a simple, uniform way of giving a compound symbol to each natural number:
| zero | 0 |
| one | S0 |
| two | SS0 |
| three | SSS0 |
| four | SSSS0 |
| five | SSSSS0 |
The symbol S can be interpreted as "the successor of", or "the number after". Since this is, however, a number theory, such interpretations are useful, but not strict. We cannot say that because four is the successor of three that four is SSSS0, but rather that since three is the successor of two, which is the successor of one, which is the successor of zero, which we have described as 0, four can be "proved" to be SSSS0. TNT is designed such that everything must be proven before it can be said to be true. This is its true power, and to undermine it would be to undermine its very usefulness.
Variables
In order to refer to unspecified terms, TNT makes use of five variablesVariable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
. These are
- a, b, c, d, e.
More variables can be constructed by adding the prime symbol
Prime (symbol)
The prime symbol , double prime symbol , and triple prime symbol , etc., are used to designate several different units, and for various other purposes in mathematics, the sciences and linguistics...
after them; for example,
- a
' , b' , c' , a, a
are all variables.
In the more rigid version of TNT, known as "austere" TNT, only
- a
' , a, a etc.
are used.
Addition and multiplication of numerals
In Typographical Number Theory, the usual symbols of "+" for additions, and "·" for multiplications are used. Thus to write "b plus c", we write- (b + c)
and "a times d" is written as
The parentheses are required. Any laxness would violate TNT's formation system (although it is trivially proved this formalism is unnecessary for operations which are both commutative and associative). Also only two terms can be operated on at once. Therefore to write "a plus b plus c", we must write either
+ c)
or
)
Equivalency
The "Equals" operator is used to denote equivalence. It is defined by the symbol "=", and takes roughly the same meaning as it usually does in mathematics. For instance,- (SSS0 + SSS0) = SSSSSS0
is a true statement in TNT, with the interpretation "3 plus 3 equals 6".
Negation
In Typographical Number Theory, negationNegation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
, i.e. the turning of a statement to its opposite, is denoted by the "~" or negation operator. For instance,
- ~(SSS0 + SSS0) = SSSSSSS0
is a true statement in TNT, interpreted as "3 plus 3 is not equal to 7".
By negation, this means negation in Boolean logic
Boolean logic
Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
(logical negation), rather than simply being the opposite. For example, if I were to say "I am eating a grapefruit", the opposite is "I am not eating a grapefruit", rather than "I am eating something other than a grapefruit". Similarly "The Television is on" is negated to "The Television is not on", rather than "The Television is off". This is a subtle difference, but an important one.
Quantifiers
There are two quantifiers used: ∀Universal quantification
In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
and ∃
Existential quantification
In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. It is denoted by the logical operator symbol ∃ , which is called the existential quantifier...
.
Note that unlike most other logical systems where qualifiers over sets require a mention of the element's existence in the set, this is not required in TNT because all numbers and terms are strictly natural numbers or logical boolean statements. It is therefore equivalent to say ∀a:(a ∈ N):∀b:(b ∈ N): (a + b) = (b + a) and ∀a:∀b:(a + b) = (b + a)
- ∃ means "There exists"
- ∀ means "For every" or "For all"
- The symbol : is used to separate a quantifier from other quantifiers or from the rest of the formula. It is commonly read "such that"
For example:
- ∀a:∀b:[ (a + b) = (b + a) ]
("For every number a and every number b, a plus b equals b plus a", or more figuratively, "Addition is commutative.")
- ~∃c:Sc = 0
("There does not exist a number c such that c plus one equals zero", or more figuratively, "Zero is not the successor of any (natural) number.")
Atoms and propositional statements
All the symbols of propositional calculusPropositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
apart from the Atom symbols are used in Typographical Number Theory, and they retain their interpretations.
Atoms are here defined as strings which amount to statements of equality, such as
1 is not equal to 2:
- ~ S0=SS0
2 plus 3 equals five: = SSSSS0
2 plus 2 is not equal to 3:
- ~[ (SS0 + SS0) = SSS0 ]