Mathematical constructivism[edit]

In the semantics of classical logic, propositional formulae are

assigned truth values from the two-element set
{
⊤
,
⊥
}
{\displaystyle \{\top ,\bot \}}

("true" and "false" respectively), regardless of

whether we have direct evidence for either case. This is referred to as
the 'law of excluded middle', because it excludes the possibility of any
truth value besides 'true' or 'false'. In contrast, propositional formulae
in intuitionistic logic are not assigned a definite truth value and are
only considered "true" when we have direct evidence, hence proof.
(We can also say, instead of the propositional formula being "true"
due to direct evidence, that it is inhabited by a proof in the Curry–
Howard sense.) Operations in intuitionistic logic therefore preserve
justification, with respect to evidence and provability, rather than
truth-valuation .
Intuitionistic logic is one of the set of approaches of constructivism in
mathematics. The use of constructivist logics in general has been a
controversial topic among mathematicians and philosophers (see, for
example, the Brouwer–Hilbert controversy). A common objection to
their use is the above-cited lack of two central rules of classical logic,
the law of excluded middle and double negation elimination. These
are considered to be so important to the practice of mathematics that
David Hilbert wrote of them: "Taking the principle of excluded middle
from the mathematician would be the same, say, as proscribing the
telescope to the astronomer or to the boxer the use of his fists. To
prohibit existence statements and the principle of excluded middle is
tantamount to relinquishing the science of mathematics altogether." [3]
Despite the serious challenges presented by the inability to utilize the
valuable rules of excluded middle and double negation elimination,
intuitionistic logic has practical use. One reason for this is that its
restrictions produce proofs that have the existence property, making it
also suitable for other forms of mathematical constructivism.
Informally, this means that if there is a constructive proof that an
object exists, that constructive proof may be used as an algorithm for
generating an example of that object, a principle known as the Curry–
Howard correspondence between proofs and algorithms. One reason
that this particular aspect of intuitionistic logic is so valuable is that it
enables practitioners to utilize a wide range of computerized tools,
known as proof assistants. These tools assist their users in the
verification (and generation) of large-scale proofs, whose size
usually precludes the usual human-based checking that goes into
publishing and reviewing a mathematical proof. As such, the use of
proof assistants (such as Agda or Coq) is enabling modern
mathematicians and logicians to develop and prove extremely
complex systems, beyond those which are feasible to create and
check solely by hand. One example of a proof which was impossible
to formally verify before the advent of these tools is the famous proof
of the four color theorem. This theorem stumped mathematicians for
more than a hundred years, until a proof was developed which ruled
out large classes of possible counterexamples, yet still left open
enough possibilities that a computer program was needed to finish
the proof. That proof was controversial for some time, but it was
finally verified using Coq.
Syntax[edit]
 The Rieger–Nishimura lattice. Its nodes are the propositional formulas in one
variable up to intuitionistic logical equivalence, ordered by intuitionistic logical
implication.
The syntax of formulas of intuitionistic logic is similar to propositional
logic or first-order logic. However, intuitionistic connectives are not
definable in terms of each other in the same way as in classical logic,
hence their choice matters. In intuitionistic propositional logic (IPL) it
is customary to use →, ∧ , ∨ , ⊥ as the basic connectives, treating ¬A
as an abbreviation for (A → ⊥). In intuitionistic first-order logic both
quantifiers ∃, ∀ are needed.

Intuitionistic logic
From Wikipedia, the free encyclopedia
(Redirected from Constructive logic)
Jump to: navigation, search
Intuitionistic logic, sometimes more generally called constructive
logic, refers to systems of symbolic logic that differ from the systems
used for classical logic by more closely mirroring the notion of
constructive proof. In particular, systems of intuitionistic logic do not
include the law of the excluded middle and double negation
elimination, which are fundamental inference rules in classical logic.
Formalized intuitionistic logic was originally developed by Arend
Heyting to provide a formal basis for Brouwer's programme of
intuitionism. From a proof-theoretic perspective, Heyting’s calculus is
a restriction of classical logic in which the law of excluded middle and
double negation elimination have been removed. Excluded middle
and double negation elimination can still be proved for some
propositions on a case by case basis, however, but do not hold
universally as they do with classical logic.
Several systems of semantics for intuitionistic logic have been
studied. One of these semantics mirrors classical Boolean-valued
semantics but uses Heyting algebras in place of Boolean algebras.
Another semantics uses Kripke models. These, however, are
technical means for studying Heyting’s deductive system rather than
formalizations of Brouwer’s original informal semantic intuitions.
Semantical systems with better claims to capture such intuitions, due
to offering meaningful concepts of “constructive truth” (rather than
merely validity or provability), are Gödel’s dialectica interpretation,
Kleene’s realizability, Medvedev’s logic of finite problems,[1] or
Japaridze’s computability logic. Yet such semantics persistently
induce logics properly stronger than Heyting’s logic. Some authors
have argued that this might be an indication of inadequacy of
Heyting’s calculus itself, deeming the latter incomplete as a
constructive logic.[2]

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Mathematical constructivism[edit]
In the semantics of classical logic, propositional formulae are
assigned truth values from the two-element set
{
⊤
,
⊥
}
{\displaystyle \{\top ,\bot \}}

("true" and "false" respectively),

regardless of whether we have direct evidence for either case.
This is referred to as the 'law of excluded middle', because it
excludes the possibility of any truth value besides 'true' or
'false'. In contrast, propositional formulae in intuitionistic logic
are not assigned a definite truth value and are only considered
"true" when we have direct evidence, hence proof. (We can also
say, instead of the propositional formula being "true" due to
direct evidence, that it is inhabited by a proof in the Curry–
Howard sense.) Operations in intuitionistic logic therefore
preserve justification, with respect to evidence and provability,
rather than truth-valuation .
Measure theory[edit]
Classical measure theory is fundamentally non-constructive, since the classical definition
of Lebesgue measure does not describe any way to compute the measure of a set or the
integral of a function. In fact, if one thinks of a function just as a rule that "inputs a real
number and outputs a real number" then there cannot be any algorithm to compute the
integral of a function, since any algorithm would only be able to call finitely many values of
the function at a time, and finitely many values are not enough to compute the integral to any
nontrivial accuracy. The solution to this conundrum, carried out first in Bishop's 1967 book, is
to consider only functions that are written as the pointwise limit of continuous functions (with
known modulus of continuity), with information about the rate of convergence. An advantage
of constructivizing measure theory is that if one can prove that a set is constructively of full
measure, then there is an algorithm for finding a point in that set (again see Bishop's book).
For example, this approach can be used to construct a real number that is normal to every
base.

The place of constructivism in mathematics[edit]

Traditionally, some mathematicians have been suspicious, if not antagonistic, towards
mathematical constructivism, largely because of limitations they believed it to pose for
constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when
he wrote in Grundlagen der Mathematik, "Taking the principle of excluded middle from the
mathematician would be the same, say, as proscribing the telescope to the astronomer or to
the boxer the use of his fists".[3]
Errett Bishop, in his 1967 work Foundations of Constructive Analysis, worked to dispel these
fears by developing a great deal of traditional analysis in a constructive framework.
Even though most mathematicians do not accept the constructivist's thesis, that only
mathematics done based on constructive methods is sound, constructive methods are
increasingly of interest on non-ideological grounds. For example, constructive proofs in
analysis may ensure witness extraction, in such a way that working within the constraints of
the constructive methods may make finding witnesses to theories easier than using classical
methods. Applications for constructive mathematics have also been found in typed lambda
calculi, topos theory and categorical logic, which are notable subjects in foundational
mathematics and computer science. In algebra, for such entities as toposes and Hopf
algebras, the structure supports an internal language that is a constructive theory; working
within the constraints of that language is often more intuitive and flexible than working
externally by such means as reasoning about the set of possible concrete algebras and
their homomorphisms.
Physicist Lee Smolin writes in Three Roads to Quantum Gravity that topos theory is "the
right form of logic for cosmology" (page 30) and "In its first forms it was called 'intuitionistic
logic'" (page 31). "In this kind of logic, the statements an observer can make about the
universe are divided into at least three groups: those that we can judge to be true, those that
we can judge to be false and those whose truth we cannot decide upon at the present time"
(page 28).

Mathematicians who have made major contributions to

constructivism[edit]
 Leopold Kronecker (old constructivism, semi-intuitionism)
 L. E. J. Brouwer (forefather of intuitionism)
 A. A. Markov (forefather of Russian school of constructivism)
 Arend Heyting (formalized intuitionistic logic and theories)
 Per Martin-Löf (founder of constructive type theories)
 Errett Bishop (promoted a version of constructivism claimed to be consistent with
classical mathematics)
 Paul Lorenzen (developed constructive analysis)

B virus infection is caused by a herpes virus. B virus is also

commonly referred to as herpes B, monkey B virus, herpesvirus
simiae, and herpesvirus B.
The virus is found among macaque monkeys, including rhesus
macaques, pig-tailed macaques, and cynomolgus monkeys (also
called crab-eating or long-tailed macaques). Macaque monkeys are
thought to be the natural host for the virus. Macaques infected with B
virus usually have no or only mild symptoms. Macaques housed in
primate facilities usually become B virus positive by the time they
reach adulthood. However, infection in macaques can only be
transmitted during active viral shedding through body fluids.
Infection with B virus is extremely rare in humans. When it does
occur, the infection can result in severe brain damage or death if the
patient is not treated soon after exposure (see Risks for Infection and
Treatment sections). Infection in humans is typically caused by
animal bites or scratches or by mucosal contact with body fluid or
tissue.

CAUSE AND INCIDENCE

B virus infection is caused by the
zoonotic agent Macacine herpesvirus
1...
RISK FOR INFECTION
Persons at greatest risk for B virus
 infection are veterinarians, laboratory
workers, and others...
SIGNS AND SYMPTOMS
Initial symptoms of B virus infection in
humans include fever, headache, and
vesicular skin lesions...
TRANSMISSION
B virus infection in humans occurs
only rarely. Possible routes of
transmission include...
FIRST AID AND
TREATMENT
When B virus infection occurs in
humans, it is often fatal unless treated
right away...
PREVENTION
Adherence to appropriate laboratory
and animal facility protocols will
greatly reduce the risk of B virus
transmission...

ymptoms, Diagnosis, & Treatment

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Symptoms
• Most people infected with chikungunya virus will develop some
symptoms.
• Symptoms usually begin 3–7 days after being bitten by an infected
mosquito.
• The most common symptoms are fever and joint pain.
• Other symptoms may include headache, muscle pain, joint swelling,
or rash.
• Chikungunya disease does not often result in death, but the
 symptoms can be severe and disabling.
• Most patients feel better within a week. In some people, the joint
pain may persist for months.
• People at risk for more severe disease include newborns infected
around the time of birth, older adults (≥65 years), and people
with medical conditions such as high blood pressure, diabetes,
or heart disease.
• Once a person has been infected, he or she is likely to be protected
from future infections.
Diagnosis
• The symptoms of chikungunya are similar to those of dengue and
Zika, diseases spread by the same mosquitoes that transmit
chikungunya.
• See your healthcare provider if you develop the symptoms
described above and have visited an area where chikungunya
is found.
• If you have recently traveled, tell your healthcare provider when and
where you traveled.
• Your healthcare provider may order blood tests to look for
chikungunya or other similar viruses like dengue and Zika.
Treatment
• There is no vaccine to prevent or medicine to treat chikungunya
virus.
• Treat the symptoms:
Get plenty of rest.
Drink fluids to prevent dehydration.
Take medicine such as acetaminophen (Tylenol®) or
paracetamol to reduce fever and pain.
Do not take aspirin and other non-steroidal anti-inflammatory
drugs (NSAIDS until dengue can be ruled out to reduce
the risk of bleeding).
If you are taking medicine for another medical condition, talk to
your healthcare provider before taking additional medication.
Cardinality[edit]
To take the algorithmic interpretation above would seem at odds with classical notions
of cardinality. By enumerating algorithms, we can show classically that the computable
numbers are countable. And yet Cantor's diagonal argument shows that real numbers have
higher cardinality. Furthermore, the diagonal argument seems perfectly constructive. To
identify the real numbers with the computable numbers would then be a contradiction.
And in fact, Cantor's diagonal argument is constructive, in the sense that given
a bijection between the real numbers and natural numbers, one constructs a real number
that doesn't fit, and thereby proves a contradiction. We can indeed enumerate algorithms to
construct a function T, about which we initially assume that it is a function from the natural
numbers onto the reals. But, to each algorithm, there may or may not correspond a real
number, as the algorithm may fail to satisfy the constraints, or even be non-terminating (T is
a partial function), so this fails to produce the required bijection. In short, one who takes the
view that real numbers are (individually) effectively computable interprets Cantor's result as
showing that the real numbers (collectively) are not recursively enumerable.
Still, one might expect that since T is a partial function from the natural numbers onto the real
numbers, that therefore the real numbers are no more than countable. And, since every
natural number can be trivially represented as a real number, therefore the real numbers
are no less thancountable. They are, therefore exactly countable. However this reasoning is
not constructive, as it still does not construct the required bijection. The classical theorem
proving the existence of a bijection in such circumstances, namely the Cantor–Bernstein–
Schroeder theorem, is non-constructive and no constructive proof of it is known.
Axiom of choice[edit]
The status of the axiom of choice in constructive mathematics is complicated by the different
approaches of different constructivist programs. One trivial meaning of "constructive", used
informally by mathematicians, is "provable in ZF set theory without the axiom of choice."
However, proponents of more limited forms of constructive mathematics would assert that ZF
itself is not a constructive system.
In intuitionistic theories of type theory (especially higher-type arithmetic), many forms of the
axiom of choice are permitted. For example, the axiom AC11 can be paraphrased to say that
for any relation R on the set of real numbers, if you have proved that for each real
number x there is a real number y such that R(x,y) holds, then there is actually a
function F such that R(x,F(x)) holds for all real numbers. Similar choice principles are
accepted for all finite types. The motivation for accepting these seemingly nonconstructive
principles is the intuitionistic understanding of the proof that "for each real number x there is
a real number y such that R(x,y) holds". According to the BHK interpretation, this proof itself
is essentially the function F that is desired. The choice principles that intuitionists accept do
not imply the law of the excluded middle.
However, in certain axiom systems for constructive set theory, the axiom of choice does
imply the law of the excluded middle (in the presence of other axioms), as shown by
the Diaconescu-Goodman-Myhill theorem. Some constructive set theories include weaker
forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory.
Measure theory[edit]
Classical measure theory is fundamentally non-constructive, since the classical definition
of Lebesgue measure does not describe any way to compute the measure of a set or the
integral of a function. In fact, if one thinks of a function just as a rule that "inputs a real
number and outputs a real number" then there cannot be any algorithm to compute the
integral of a function, since any algorithm would only be able to call finitely many values of
the function at a time, and finitely many values are not enough to compute the integral to any
nontrivial accuracy. The solution to this conundrum, carried out first in Bishop's 1967 book, is
to consider only functions that are written as the pointwise limit of continuous functions (with
known modulus of continuity), with information about the rate of convergence. An advantage
of constructivizing measure theory is that if one can prove that a set is constructively of full
measure, then there is an algorithm for finding a point in that set (again see Bishop's book).
For example, this approach can be used to construct a real number that is normal to every
base.

The place of constructivism in mathematics[edit]

Traditionally, some mathematicians have been suspicious, if not antagonistic, towards
mathematical constructivism, largely because of limitations they believed it to pose for
constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when
he wrote in Grundlagen der Mathematik, "Taking the principle of excluded middle from the
mathematician would be the same, say, as proscribing the telescope to the astronomer or to
the boxer the use of his fists".[3]
Errett Bishop, in his 1967 work Foundations of Constructive Analysis, worked to dispel these
fears by developing a great deal of traditional analysis in a constructive framework.
Even though most mathematicians do not accept the constructivist's thesis, that only
mathematics done based on constructive methods is sound, constructive methods are
increasingly of interest on non-ideological grounds. For example, constructive proofs in
analysis may ensure witness extraction, in such a way that working within the constraints of
the constructive methods may make finding witnesses to theories easier than using classical
methods. Applications for constructive mathematics have also been found in typed lambda
calculi, topos theory and categorical logic, which are notable subjects in foundational
mathematics and computer science. In algebra, for such entities as toposes and Hopf
algebras, the structure supports an internal language that is a constructive theory; working
within the constraints of that language is often more intuitive and flexible than working
externally by such means as reasoning about the set of possible concrete algebras and
their homomorphisms.
Physicist Lee Smolin writes in Three Roads to Quantum Gravity that topos theory is "the
right form of logic for cosmology" (page 30) and "In its first forms it was called 'intuitionistic
logic'" (page 31). "In this kind of logic, the statements an observer can make about the
universe are divided into at least three groups: those that we can judge to be true, those that
we can judge to be false and those whose truth we cannot decide upon at the present time"
(page 28).

Recommend on Facebook
Tweet

Symptoms
• Most people infected with chikungunya virus will develop some
symptoms.
• Symptoms usually begin 3–7 days after being bitten by an infected
mosquito.
• The most common symptoms are fever and joint pain.
• Other symptoms may include headache, muscle pain, joint swelling,
or rash.
• Chikungunya disease does not often result in death, but the
 symptoms can be severe and disabling.
• Most patients feel better within a week. In some people, the joint
pain may persist for months.
• People at risk for more severe disease include newborns infected
around the time of birth, older adults (≥65 years), and people
with medical conditions such as high blood pressure, diabetes,
or heart disease.
• Once a person has been infected, he or she is likely to be protected
from future infections.
Diagnosis
• The symptoms of chikungunya are similar to those of dengue and
Zika, diseases spread by the same mosquitoes that transmit
chikungunya.
• See your healthcare provider if you develop the symptoms
described above and have visited an area where chikungunya
is found.
• If you have recently traveled, tell your healthcare provider when and
where you traveled.
• Your healthcare provider may order blood tests to look for
chikungunya or other similar viruses like dengue and Zika.
Treatment
• There is no vaccine to prevent or medicine to treat chikungunya
virus.
• Treat the symptoms:
Get plenty of rest.
Drink fluids to prevent dehydration.
Take medicine such as acetaminophen (Tylenol®) or
paracetamol to reduce fever and pain.
Do not take aspirin and other non-steroidal anti-inflammatory
drugs (NSAIDS until dengue can be ruled out to reduce
the risk of bleeding).
If you are taking medicine for another medical condition, talk to your
healthcare provider before taking additional medication.
Classical measure theory is fundamentally non-constructive, since the classical definition
of Lebesgue measure does not describe any way to compute the measure of a set or the
integral of a function. In fact, if one thinks of a function just as a rule that "inputs a real
number and outputs a real number" then there cannot be any algorithm to compute the
integral of a function, since any algorithm would only be able to call finitely many values of
the function at a time, and finitely many values are not enough to compute the integral to any
nontrivial accuracy. The solution to this conundrum, carried out first in Bishop's 1967 book, is
to consider only functions that are written as the pointwise limit of continuous functions (with
known modulus of continuity), with information about the rate of convergence. An advantage
of constructivizing measure theory is that if one can prove that a set is constructively of full
measure, then there is an algorithm for finding a point in that set (again see Bishop's book).
For example, this approach can be used to construct a real number that is normal to every
base.

The place of constructivism in mathematics[edit]

Traditionally, some mathematicians have been suspicious, if not antagonistic, towards
mathematical constructivism, largely because of limitations they believed it to pose for
constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when
he wrote in Grundlagen der Mathematik, "Taking the principle of excluded middle from the
mathematician would be the same, say, as proscribing the telescope to the astronomer or to
the boxer the use of his fists".[3]
Errett Bishop, in his 1967 work Foundations of Constructive Analysis, worked to dispel these
fears by developing a great deal of traditional analysis in a constructive framework.
Even though most mathematicians do not accept the constructivist's thesis, that only
mathematics done based on constructive methods is sound, constructive methods are
increasingly of interest on non-ideological grounds. For example, constructive proofs in
analysis may ensure witness extraction, in such a way that working within the constraints of
the constructive methods may make finding witnesses to theories easier than using classical
methods. Applications for constructive mathematics have also been found in typed lambda
calculi, topos theory and categorical logic, which are notable subjects in foundational
mathematics and computer science. In algebra, for such entities as toposes and Hopf
algebras, the structure supports an internal language that is a constructive theory; working
within the constraints of that language is often more intuitive and flexible than working
externally by such means as reasoning about the set of possible concrete algebras and
their homomorphisms.
Physicist Lee Smolin writes in Three Roads to Quantum Gravity that topos theory is "the
right form of logic for cosmology" (page 30) and "In its first forms it was called 'intuitionistic
logic'" (page 31). "In this kind of logic, the statements an observer can make about the
universe are divided into at least three groups: those that we can judge to be true, those that
we can judge to be false and those whose truth we cannot decide upon at the present time"
(page 28).

Bib
Biography[edit]
Early in his career, Brouwer proved a number of theorems that were in the emerging field of
topology. The main results were his fixed point theorem, the topological invariance of degree,
and the topological invariance of dimension. The most popular of the three among
mathematicians is the first one called the Brouwer Fixed Point Theorem. It is a simple
corollary to the second, about the topological invariance of degree, and this one is the most
popular among algebraic topologists. The third is perhaps the hardest.
Brouwer also proved the simplicial approximation theorem in the foundations of algebraic
topology, which justifies the reduction to combinatorial terms, after sufficient subdivision
of simplicial complexes, of the treatment of general continuous mappings. In 1912, at age 31,
he was elected a member of the Royal Netherlands Academy of Arts and Sciences.[6] He was
an Invited Speaker of the ICM in 1908 at Rome[7] and in 1912 at Cambridge, UK.[8]
Brouwer in effect founded the mathematical philosophy of intuitionism as an opponent to the
then-prevailing formalism of David Hilbert and his collaborators Paul Bernays, Wilhelm
Ackermann, John von Neumann and others (cf. Kleene (1952), p. 46–59). As a variety
of constructive mathematics, intuitionism is essentially a philosophy of the foundations of
mathematics.[9] It is sometimes and rather simplistically characterized by saying that its
adherents refuse to use the law of excluded middle in mathematical reasoning.
Brouwer was a member of the Significs Group. It formed part of the early history
of semiotics—the study of symbols—around Victoria, Lady Welby in particular. The original
meaning of his intuitionism probably can not be completely disentangled from the intellectual
milieu of that group.
In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life, Art
and Mysticism described by Davis as "drenched in romantic pessimism" (Davis (2002),
p. 94). Arthur Schopenhauer had a formative influence on Brouwer, not least because he
insisted that all concepts be fundamentally based on sense intuitions.[10][11][12] Brouwer then
"embarked on a self-righteous campaign to reconstruct mathematical practice from the
ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to