Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Part I — GeneralPart II — Technical
Linguistic Phenomena in Mathematics
Mohan Ganesalingam
June 6, 2008
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
A Language for Mathematics
I Joint work with Thomas Barnet-Lamb.
I Long-term project.
I Only discuss linguistic aspects here.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics as a Linguistic Domain
I Clean, total semantics.
I Every mathematical term is formally defined.
I Can extract all syntactic and semantic information fromdefinitions.
I Adaptivity — language starts with a small core, expands viadefinitions.
I Benefits of both closed domains and open domains.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Example: Real Mathematics
Sylow’s Theorems
Let G be a finite group whose order is divisible by the prime p.Suppose pm is the highest power of p which is a factor of |G | andset
k =|G |pm
.
Then
1. the group G contains at least one subgroup of order pm,
2. any two subgroups of G of order pm are conjugate, and
3. the number of subgroups of G of order pm is congruent to 1modulo p and is a factor of k .
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Example: MIZAR (truncated)
theorem : : GROUP 10 :12f o r G being f i n i t e Group ,p being pr ime ( n a t u r a l number )holds ex P being Subgroup of G s t
P i s S y l o w p−s u b g r o u p o f p r i m e p ;
theorem : : GROUP 10 :14f o r G being f i n i t e Group ,p being pr ime ( n a t u r a l number ) holds
( f o r H being Subgroup of G s tH i s p−g r o u p o f p r i m e p holds
ex P being Subgroup of G s tP i s S y l o w p−s u b g r o u p o f p r i m e p. . .
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Comparison
Sylow’s Theorems
Let G be a finite group whose order isdivisible by the prime p. Suppose pm isthe highest power of p which is a factorof |G | and set
k =|G |pm
.
Then
1. the group G contains at least onesubgroup of order pm,
2. any two subgroups of G of orderpm are conjugate, and
3. the number of subgroups of G oforder pm is congruent to 1modulo p and is a factor of k.
theorem : : GROUP 10 :12f o r G be ing f i n i t e Group ,p be ing pr ime ( n a t u r a l number )ho ld s ex P be ing Subgroup of G s t
P i s S y l o w p−s u b g r o u p o f p r i m e p ;
theorem : : GROUP 10 :14f o r G be ing f i n i t e Group ,p be ing pr ime ( n a t u r a l number ) ho ld s
( f o r H be ing Subgroup of G s tH i s p−g r o u p o f p r i m e p ho ld s
ex P be ing Subgroup of G s tP i s S y l o w p−s u b g r o u p o f p r i m e p& H i s Subgroup of P) &
( f o r P1 , P2 be ing Subgroup of Gs t P1 i s S y l o w p−s u b g r o u p o f p r i m e p& P2 i s S y l o w p−s u b g r o u p o f p r i m e p
ho ld s P1 , P2 a r e c o n j u g a t e d ) ;
theorem : : GROUP 10 :15f o r G be ing f i n i t e Group ,
p be ing pr ime ( n a t u r a l number ) ho ld sc a r d t h e s y l o w p−s u b g r o u p s o f p r i m e ( p , G)
mod p = 1 &c a r d t h e s y l o w p−s u b g r o u p s o f p r i m e ( p , G)
d i v i d e s ord G ;
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Example: A New Language
Theorem 72 (“Sylow’s Theorems”)
Let G be a finite group whose order is divisible by a prime p. Let mbe the integer s.t. pˆm is the highest power of p which divides|G| and set
k = |G|/pˆm.
Then
1. the group G contains a subgroup of order pˆm,
2. any two subgroups of G of order pˆm are conjugate, and
3. the number of subgroups of G of order pˆm is congruent to 1modulo p and is a factor of k.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Comparison
Sylow’s Theorems
Let G be a finite group whose order isdivisible by the prime p. Suppose pm isthe highest power of p which is a factorof |G | and set
k =|G |pm
.
Theorem 72 (“Sylow’s Theorems”)
Let G be a finite group whose order isdivisible by a prime p. Let m be the in-teger s.t. pˆm is the highest power of pwhich divides |G| and set
k = |G|/pˆm.
Then
1. the group G contains at least onesubgroup of order pm,
2. any two subgroups of G of orderpm are conjugate, and
3. the number of subgroups of G oforder pm is congruent to 1modulo p and is a factor of k.
Then
1. the group G contains a subgroup oforder pˆm,
2. any two subgroups of G of order pˆmare conjugate, and
3. the number of subgroups of G of orderpˆm is congruent to 1 modulo p and isa factor of k.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Natural Formal Languages
I Fuse concepts from formal languages and natural languages.
I Requires solving substantive some problems, but...
I Find formal language features counteract weaknesses ofnatural language and vice versa.
I E.g. type (FL concept) defangs problems caused by ambiguity(NL concept).
I Major examples take too much space... .
I Below: Discuss a minor issue of particular interest to linguists.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H for some U in T, by definition of TH , andU ∩ H = i−1(U), so g−1(V ) = g−1(i−1(U)) = (i ◦ g)−1(U).
Sutherland, W. A., Introduction to Metric and TopologicalSpaces, OUP 1975, p. 52.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H for some U in T, by definition of TH , andU ∩ H = i−1(U), so g−1(V ) = g−1(i−1(U)) = (i ◦ g)−1(U).
Sutherland, W. A., Introduction to Metric and TopologicalSpaces, OUP 1975, p. 52.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
I (α[U] for some U in T) and β[U].
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
I (α[U] for some U in T) and β[U].
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
I (α[U] for some U in T) and β[U].
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
I (α[U] for some U in T) and β[U].
I Compositional Analysis in First-order Logic:
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
I (α[U] for some U in T) and β[U].
I Compositional Analysis in First-order Logic:
I ∃U.(in(U,T) ∧ α̂[U]) ∧ β̂[U]
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
I (α[U] for some U in T) and β[U].
I Compositional Analysis in First-order Logic:
I ∃U.(in(U,T) ∧ α̂[U]) ∧ β̂[U]
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematics and Logic
I Then V = U ∩ H︸ ︷︷ ︸α[U]
for some U in T and U ∩ H = i−1(U)︸ ︷︷ ︸β[U]
.
I (α[U] for some U in T) and β[U].
I Compositional Analysis in First-order Logic:
I ∃U.(in(U,T) ∧ α̂[U]) ∧ β̂[UOO
unbound variable
]
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Varying the Quantifier
I α[U] for some U in T and β[U].
I *α[U] for every U in T and β[U].
I Asymmetric treatment of quantifiers required.
I All standard logics fail.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Discourse Representation Theory
x
John(x)
y
dog(y)owns(x , y)
⇒hungry(y)
All John’s dogs are hungry.(I.e. If y is a dog which John owns, then y is hungry.)
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Discourse Representation Theory
x
John(x)
y
dog(y)owns(x , y)
⇒hungry(y)
All John’s dogs are hungry.(I.e. If y is a dog which John owns, then y is hungry.)
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Discourse Representation Theory
x
John(x)
y
dog(y)owns(x , y)
⇒hungry(y)
All John’s dogs are hungry.(I.e. If y is a dog which John owns, then y is hungry.)
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Discourse Representation Theory
x
John(x)
y
dog(y)owns(x , y)
⇒hungry(y)
tired(y)
All John’s dogs are hungry.(I.e. If y is a dog which John owns, then y is hungry.)
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Donkey Sentences and DRT
x y
farmer(x)donkey(y)owns(x , y)
⇒beats(x , y)
Every farmer who owns a donkey beats it.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Donkey Sentences and DRT
x y
farmer(x)donkey(y)owns(x , y)
⇒beats(x , y)
Every farmer who owns a donkey beats it.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Donkey Sentences and DRT
x y
farmer(x)donkey(y)owns(x , y)
⇒beats(x , y)
Every farmer who owns a donkey beats it.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematical Variables as Referents
x
natural number(x)greater than(x , 1)
⇒y
is prime(y)divides(y , x)
Every natural number which is greater than 1 has a prime divisor.
n
natural number(n)greater than(n, 1)
⇒p
is prime(p)divides(p,n)
If n > 1 is a natural number, then there is a prime p such that p|n.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematical Variables as Referents
x
natural number(x)greater than(x , 1)
⇒y
is prime(y)divides(y , x)
Every natural number which is greater than 1 has a prime divisor.
n
natural number(n)greater than(n, 1)
⇒p
is prime(p)divides(p,n)
If n > 1 is a natural number, then there is a prime p such that p|n.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Mathematical Variables as Referents
x
natural number(x)greater than(x , 1)
⇒y
is prime(y)divides(y , x)
Every natural number which is greater than 1 has a prime divisor.
n
natural number(n)greater than(n, 1)
⇒p
is prime(p)divides(p,n)
If n > 1 is a natural number, then there is a prime p such that p|n.
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
DRT in Action
U
in(U,T)α̂[U]
]
β̂[U]
α[U] for some U in T
and
β[U]
U
in(U,T)⇒
α̂[U]
]
β̂[U]
α[U] for every U in T
and
β[U]
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
DRT in Action
U
in(U,T)α̂[U]
]β̂[U]
α[U] for some U in T and β[U]
U
in(U,T)⇒
α̂[U]
]β̂[U]
α[U] for every U in T and β[U]
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
DRT in Action
U
in(U,T)α̂[U]
β̂[U]
α[U] for some U in T and β[U]
U
in(U,T)⇒
α̂[U]
β̂[U]
*α[U] for every U in T and β[U]
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
DRT in Action
U
in(U,T)α̂[U]
β̂[U]
α[U] for some U in T and β[U]
U
in(U,T)⇒
α̂[U]
β̂[U]
*α[U] for every U in T and β[U]
Mohan Ganesalingam Linguistic Phenomena in Mathematics
Part I — GeneralPart II — Technical
Conclusion
I Mathematical language contains variables, normally associatedwith formal languages.
I But a semantic theory fitted to formal languages (PredicateCalculus) cannot describe their behaviour.
I Need to use a theory designed for natural languages (DRT).
I Such hybridised formal language/natural language conceptsrecur throughout mathematics.
I We need a ‘natural formal language’ to describe mathematics.
Mohan Ganesalingam Linguistic Phenomena in Mathematics