
What, When, Where, How, Who?
Axiom
Introduction, Important Definitions and Related
Concepts:
In traditional
logic, an axiom or postulate is a
proposition that is not proved or demonstrated but
considered to be
selfevident. Therefore, its truth is taken for
granted, and serves as a starting point for deducing and
inferring other (theory dependent) truths. In
mathematics, the term axiom is used in two
related but distinguishable senses:
"logical axioms"
and "nonlogical
axioms". In both senses, an axiom is any
mathematical statement that serves as a starting point
from which other statements are logically derived.
Unlike
theorems, axioms (unless redundant) cannot be
derived by principles of deduction, nor are they
demonstrable by
mathematical proofs, simply because they are
starting points; there is nothing else they logically
follow from (otherwise they would be classified as
theorems). Logical axioms are usually statements
that are taken to be universally true (e.g., (A ∧
B) → A), while nonlogical axioms (e.g,
a + b = b + a) are
actually defining properties for the domain of a
specific mathematical theory (such as
arithmetic). When used in that sense, "axiom,"
"postulate", and "assumption" may be used
interchangeably. In general, a nonlogical axiom is not
a selfevident truth, but rather a formal logical
expression used in deduction to build a mathematical
theory. To axiomatize a system of knowledge is to show
that its claims can be derived from a small,
wellunderstood set of sentences (the axioms). There are
typically multiple ways to axiomatize a given
mathematical domain. Outside logic and mathematics, the
term "axiom" is used loosely for any established
principle of some field. The word "axiom" comes from the
Greek word ἀξίωμα
(axioma), a
verbal noun from the verb
ἀξιόειν (axioein),
meaning "to deem worthy", but also "to require", which
in turn comes from ἄξιος
(axios), meaning "being in balance", and hence
"having (the same) value (as)", "worthy", "proper".
Among the
ancient Greek
philosophers an axiom was a claim which could be
seen to be true without any need for proof. The logicodeductive
method whereby conclusions (new knowledge) follow from
premises (old knowledge) through the application of
sound arguments (syllogisms, rules of inference), was
developed by the ancient Greeks, and has become the core
principle of modern mathematics.
Tautologies excluded, nothing can be deduced if
nothing is assumed. Axioms and postulates are the basic
assumptions underlying a given body of deductive
knowledge. They are accepted without demonstration.
Logic (from
Classical Greek λόγος
logos; meaning word, thought, idea, argument,
account, reason, or principle) is the study of the
principles of valid
inference and
demonstration. As a
formal science, logic investigates and classifies
the structure of statements and arguments, both through
the study of
formal systems of
inference and through the study of arguments in
natural language. The field of logic ranges from core
topics such as the study of
fallacies and
paradoxes, to specialized analysis of reasoning
using
probability and to arguments involving
causality. Logic is also commonly used today in
argumentation theory.
^{
[1] }
Traditionally, logic was considered a branch of
philosophy, a part of the classical
trivium of
grammar, logic, and
rhetoric. Since the midnineteenth century formal
logic has been studied in the context of
foundations of mathematics, where it was often
called
symbolic logic. In 1903
Alfred North Whitehead and
Bertrand Russell attempted to establish logic
formally as the cornerstone of mathematics with the
publication of
Principia Mathematica.^{[2]}
However, except for the elementary part, the system of
Principia is no longer much used, having been largely
superseded by
set theory. As the study of formal logic expanded,
research no longer focused solely on foundational
issues, and the study of several resulting areas of
mathematics came to be called
mathematical logic. The development of formal logic
and its implementation in computing machinery is
fundamental to
computer science. Form is central to logic. It
complicates exposition that 'formal' in "formal logic"
is commonly used in an ambiguous manner. Symbolic
language is just one kind of formal logic, and is
distinguished from another kind of formal logic,
traditional
Aristotelian syllogistic logic, which deals solely
with
categorical propositions.

Informal logic is the study of
natural language
arguments. The study of
fallacies is an especially important branch of
informal logic. The dialogues of
Plato ^{
[3]} are a
good example of informal logic. Formal logic
is the study of
inference with purely formal content, where that
content is made explicit. (An inference possesses a
purely formal content if it can be expressed
as a particular application of a wholly abstract
rule, that is, a rule that is not about any
particular thing or property. The first rules of
formal logic that have come down to us were written
by
Aristotle.
^{
[4]} In
many definitions of logic, logical inference and
inference with purely formal content are the same.
This does not render the notion of informal logic
vacuous, because no formal language captures all of
the nuance of natural language.
Symbolic logic is the study of symbolic
abstractions that capture the formal features of
logical inference.^{[2]}^{[5]}
Symbolic logic is often divided into two branches,
propositional logic and
predicate logic.
Mathematical logic is an extension of
symbolic logic into other areas, in particular to
the study of
model theory,
proof theory,
set theory, and
recursion theory.
In
epistemology (theory of knowledge), a
selfevident proposition is one that is known to
be true by understanding its meaning without
proof. Some epistemologists deny that any
proposition can be selfevident. For most others,
the belief that oneself is
conscious is offered as an example of
selfevidence. However, one's belief that someone
else is conscious is not epistemically selfevident.
The following
metaphysical propositions are often said to be
selfevident:
 A finite whole is greater than any of its
parts It is impossible for the something
to be and not be at the same time in the same
manner.
Certain forms of argument from selfevidence are
considered fallacious or abusive in debate. For
example, if a proposition is claimed to be
selfevident, it is an argumentative
fallacy to assert that disagreement with the
proposition indicates misunderstanding of it. It is
sometimes said that a selfevident proposition is
one whose denial is selfcontradictory. It is also
sometimes said that an
analytic proposition is one whose denial is
selfcontradictory. But these two uses of the term
selfcontradictory mean entirely different
things. A selfevident proposition cannot be denied
without knowing that one contradicts oneself
(provided one actually understands the proposition).
An analytic proposition cannot be denied without a
contradiction, but one may fail to know that
there is a contradiction because it may be a
contradiction that can be found only by a long and
abstruse line of logical or mathematical reasoning.
Most analytic propositions are very far from
selfevident. Similarly, a selfevident proposition
need not be analytic: my knowledge that I am
conscious is selfevident but not analytic. An
analytic proposition, however long a chain of
reasoning it takes to establish it, ultimately
contains a
tautology, and is thus only a verbal truth: a
truth established through the verbal equivalence of
a single meaning. For those who admit the existence
of abstract concepts, the class of nonanalytic
selfevident truths can be regarded as truths of the
understandingtruths revealing connections between
the meanings of ideas. Claims of selfevidence
also exist outside of epistemology. Mathematics
(colloquially, maths or math) is the
body of knowledge centered on such concepts as
quantity,
structure,
space, and
change, and also the academic discipline that
studies them.
Benjamin Peirce called it "the science that
draws necessary conclusions".^{[2]}
Other practitioners of mathematics maintain that
mathematics is the science of pattern, and that
mathematicians seek out patterns whether found
in numbers, space, science, computers, imaginary
abstractions, or elsewhere.^{[3]}^{[4]}
Mathematicians explore such concepts, aiming to
formulate new
conjectures and establish their truth by
rigorous
deduction from appropriately chosen
axioms and
definitions.^{[5]
}Through the use of
abstraction and
logical
reasoning, mathematics evolved from
counting,
calculation,
measurement, and the systematic study of the
shapes and
motions of physical objects. Knowledge and use
of basic mathematics have always been an inherent
and integral part of individual and group life.
Refinements of the basic ideas are visible in
mathematical texts originating in the
ancient Egyptian,
Mesopotamian,
Indian,
Chinese,
Greek and
Islamic worlds.
Rigorous arguments first appeared in
Greek mathematics, most notably in
Euclid's
Elements. The development continued in
fitful bursts until the
Renaissance period of the
16th century, when mathematical innovations
interacted with new
scientific discoveries, leading to an
acceleration in research that continues to the
present day.^{[6]
}Today, mathematics is used throughout the
world in many fields, including
natural science,
engineering,
medicine, and the
social sciences such as
economics.
Applied mathematics, the application of
mathematics to such fields, inspires and makes use
of new mathematical discoveries and sometimes leads
to the development of entirely new disciplines.
Mathematicians also engage in
pure mathematics, or mathematics for its own
sake, without having any application in mind,
although applications for what began as pure
mathematics are often discovered later.^{[7]
}
The word "mathematics" (Greek: μαθηματικά or
mathēmatiká) comes from the
Greek μάθημα (máthēma), which means
learning, study, science, and
additionally came to have the narrower and more
technical meaning "mathematical study", even in
Classical times. Its adjective is μαθηματικός (mathēmatikós),
related to learning, or studious,
which likewise further came to mean mathematical.
In particular,
μαθηματικὴ τέχνη (mathēmatikḗ tékhnē),
in
Latin ars mathematica, meant the
mathematical art. The apparent plural form in
English, like the
French plural form les mathématiques (and
the less commonly used singular derivative la
mathématique), goes back to the Latin neuter
plural mathematica (Cicero),
based on the Greek plural τα μαθηματικά (ta
mathēmatiká), used by
Aristotle, and meaning roughly "all things
mathematical".^{[8]}
In English, however, the noun mathematics
takes singular verb forms.
according to or agreeing with the
principles of logic:
a logical inference. 
2. 
reasoning in accordance with the
principles of logic, as a person or the
mind:
logical thinking. 
3. 
reasonable; to be expected:
War was the logical consequence of such
threats. 
4. 
of
or pertaining to logic. Of, relating
to, in accordance with, or of the nature
of logic. Based on earlier or otherwise
known statements, events, or conditions;
reasonable: Rain
was a logical expectation, given the
time of year. Reasoning or capable
of reasoning in a clear and consistent
manner.
capable of or reflecting the
capability for correct and
valid reasoning; "a logical
mind" [ant:
illogical] 
2. 
based on known statements or
events or conditions; "rain
was a logical expectation,
given the time of year" [syn:
legitimate] 
3. 
marked by an orderly,
logical, and aesthetically
consistent relation of
parts; "a coherent
argument" [syn:
coherent] [ant:
incoherent] 
4. 
capable of thinking and
expressing yourself in a
clear and consistent manner;
"a lucid thinker"; "she was
more coherent than she had
been just after the
accident" [syn:
coherent] (thinking or
acting) according to the
rules of logic
Example:
It is logical to assume that
you will get a higher salary
if you are promoted; She is
always logical in her
thinking. Of or pertaining
to logic; used in logic; as,
logical subtilties. Bacon.
According to the rules of
logic; as, a logical
argument or inference; the
reasoning is logical.
Prior. Skilled in logic;
versed in the art of
thinking and reasoning; as,
he is a logical thinker.
Addison.
(From the technical term
"logical device", wherein a
physical device is referred
to by an arbitrary "logical"
name) Having the role of. If
a person (say, Les Earnest
at SAIL) who had long held a
certain post left and were
replaced, the replacement
would for a while be known
as the "logical" Les
Earnest.Nonlogical
axioms are formulas that
play the role of
theoryspecific assumptions.
Reasoning about two
different structures, for
example the natural numbers
and the integers, may
involve the same logical
axioms; the nonlogical
axioms aim to capture what
is special about a
particular structure (or set
of structures, such as
algebraic groups). Thus
nonlogical axioms, unlike
logical axioms, are not
tautologies. Another name
for a nonlogical axiom is
postulate.
Almost every modern
mathematical theory starts
from a given set of
nonlogical axioms, and it
was thought that in
principle every theory could
be axiomatized in this way
and fomalized down to the
bare language of logical
formulas. This turns out to
be impossible and proved to
be quite a story. This is
the role of nonlogical
axioms, they simply
constitute a starting point
in a logical system. Since
they are so fundamental in
the development of a theory,
it is often the case that
they are simply referred to
as axioms in the
mathematical discourse, but
again, not in the sense that
they are true propositions
nor as if they were
assumptions claimed to be
true. For example, in some
groups, the operation of
multiplication is
commutative; in others it is
not. Thus, an axiom
is an elementary basis for a
formal logic system that
together with the
rules of inference
define a deductive system.
Examples Arithmetic,
Euclidean geometry, linear
algebra, real analysis,
topology, group theory, set
theory, projective geometry,
symplectic geometry, von
Neumann algebras, ergodic
theory, probability, etc.
All these theories are based
on their respective set of
nonlogical axioms.
Arithmetic In all this
formalism, the
Peano axioms constitute
the most widely used
axiomatization of
arithmetic; these are a
set of nonlogical axioms
strong enough to prove
several relevant facts of
number theory and they
allowed G�del to establish
his
second incompleteness
theorem The language is
<math>\mathfrak{L}_{NT} =
\{0, S\}\,<math> where
<math>0\,<math> is a
constant symbol and
<math>S\,<math> is a unary
function. The postulates
are: <math>\forall x \lnot (Sx
= 0) <math> <math>\forall x
\forall y (Sx = Sy \to x =
y) <math> <math>((\phi(0)
\land \forall x\,(\phi(x)
\to \phi(Sx))) \to \forall
x\phi(x)<math> for any
<math>\mathfrak{L}_{NT}\,<math>
formula <math>\phi\,<math>
with one free variable.
There is a standard
structure is <math>\mathfrak{N}
= <\N, 0, S>\,<math> where
<math>\N\,<math> is the set
of natural numbers,
<math>S\,<math> is the
successor function and
<math>0\,<math> is naturally
interpreted as the number 0.
Geometry
Probably the most famous
very early set of axioms is
the 4 + 1
postulates of Euclid.
This turns out to be
incomplete, and many more
postulates are necessary to
completely characterize his
geometry (Hilbert
used 23). "4 + 1" because
for nearly two millennia the
fifth (parallel) postulate
(through a point outside a
line there is exactly one
parallel) was suspected of
being derivable from the
first four. Ultimately, the
fifth postulate was found to
be independent of the first
four. Indeed, one can assume
that no parallels through a
point outside a line exist,
that exactly one exists, or
that infinitely many exist.
These choices give us
alternative forms of
geometry in which the
interior angles of a
triangle add up to less
than, exactly or more than a
straight line respectively
and are known as
elliptic,
Euclidean and
hyperbolic geometries.
In
mathematical logic, a
theorem is a type of
abstract object, one
token of which is a
formula of a
formal language which
can be derived from the
rules of the
formal system that is
applied to the formal
language; another token of
which is a statement in
natural language, that
can be
proved on the basis of
explicitly stated or
previously agreed
assumptions. In all
settings, an essential
property of theorems is that
they are derivable using a
fixed set of
inference rules and
axioms without any
additional assumptions. This
is not a matter of the
semantics of the
language: the expression
that results from a
derivation is a
syntactic consequence of
all the expressions that
precede it. In
mathematics, the
derivation of a theorem is
often interpreted as a proof
of the truth of the
resulting expression, but
different
deductive systems can
yield other interpretations,
depending on the meanings of
the derivation rules. The
proofs of theorems have two
components, called the
hypotheses and the
conclusions. The
proof of a mathematical
theorem is a logical
argument demonstrating that
the conclusions are a
necessary consequence of
the hypotheses, in the sense
that if the hypotheses are
true then the conclusions
must also be true, without
any further assumptions. The
concept of a theorem is
therefore fundamentally
deductive, in
contrast to the notion of a
scientific
theory, which is
empirical. Although
they can be written in a
completely symbolic form,
theorems are often expressed
in a natural language such
as English. The same is true
of proofs, which are often
expressed as logically
organised and clearly worded
informal arguments intended
to demonstrate that a formal
symbolic proof can be
constructed. Such arguments
are typically easier to
check than purely symbolic
ones — indeed, many
mathematicians would express
a preference for a proof
that not only demonstrates
the validity of a theorem,
but also explains in some
way why it is
obviously true. In some
cases, a picture alone may
be sufficient to prove a
theorem. Because theorems
lie at the core of
mathematics, they are also
central to its aesthetics.
Theorems are often described
as being "trivial", or
"difficult", or "deep", or
even "beautiful". These
subjective judgements vary
not only from person to
person, but also with time:
for example, as a proof is
simplified or better
understood, a theorem that
was once difficult may
become trivial. On the other
hand, a deep theorem may be
simply stated, but its proof
may involve surprising and
subtle connections between
disparate areas of
mathematics.
Fermat's last theorem is
a particularly wellknown
example of such a theorem.
Logically most theorems
are of the form of an
indicative conditional:
if A, then B. Such a
theorem does not state that
B is always true,
only that B must be
true if A is true. In
mathematics, a proof
is a convincing
demonstration that some
mathematical statement
is necessarily true, within
the accepted standards of
the field. A proof is a
logical argument, not an
empirical one. That is,
the proof must demonstrate
that a proposition is true
in all cases to which it
applies, without a single
exception. An unproven
proposition believed or
strongly suspected to be
true is known as a
conjecture. Proofs
employ
logic but usually
include some amount of
natural language which
usually admits some
ambiguity. In fact, the vast
majority of proofs in
written mathematics can be
considered as applications
of
informal logic. Purely
formal proofs are considered
in
proof theory. The
distinction between
formal and informal proofs
has led to much examination
of current and historical
mathematical practice,
quasiempiricism in
mathematics, and
socalled
folk mathematics (in
both senses of that term).
The
philosophy of mathematics
is concerned with the role
of language and logic in
proofs, and
mathematics as a language.
Regardless of one's
attitude to formalism, the
result that is proved to be
true is a
theorem; in a completely
formal proof it would be the
final line, and the complete
proof shows how it follows
from the
axioms alone by the
application of the rules of
inference. Once a theorem is
proved, it can be used as
the basis to prove further
statements. A theorem may
also be referred to as a
lemma if it is used as a
stepping stone in the proof
of a theorem. The axioms are
those statements one cannot,
or need not, prove. These
were once the primary study
of philosophers of
mathematics. Today focus is
more on
practice, i.e.
acceptable techniques.



