On Apr 15, 11:33=A0am, mimus <tinmimu...@[EMAIL PROTECTED]
> wrote:
> On Mon, 14 Apr 2008 10:01:18 -0700, The Starmaker wrote:
> > On Apr 12, 7:20=A0am, "Suresh M.Sc" <suresh.ra...@[EMAIL PROTECTED]
> wrote:
>
> >> Mathematicsisthe king of arts and the queen of all sciences.
> >> Isn't?WhatisMathematics actually? Give me a best definition for
> >> Mathematics. Thanks.
>
Is math an instinct?
The cerebral substrates of arithmetic:
How and why did human beings evolve the ability to do mathematics?
We all possess the ability to cope with mathematics--if only we
recognize what's required. We all possess, if not literally a gene,
then at least an inherent ability not just for arithmetic but for real
mathematics: algebra, calculus, and the rest. A recent Darwinian
explanation for the origin of this ability, is based on the idea that
being able to handle abstract ideas and relation****ps confers key
evolutionary advantages.
Human infants have a rudimentary number sense, this sense is as basic
as our perception of color, and that it is wired into the brain. The
invention of symbolic systems of numerals started us on the climb to
higher mathematics. We are now approaching the crossroads where
numbers and neurons intersect. The structure of the brain shapes our
mathematical abilities, and our mathematics opens up a window on the
human mind.
It seems we have a number sense, the human mind seems to have an
innate grasp of mathematics. Place value systems (such as the Arabic
numeral system we use) arose independently in four separate
civilizations--evidence of a universal sense of number. A rudimentary
number sense is wired into our brains at birth. Experiments show that
chimps, like us, use symbols to denote numbers.
In the same mathematical reasoning that inspired Plato with visions of
eternal ideals, we find evidence for a provocative theory of
evolutionary change. The evolution of language is the surest
indication of a new kind of strictly internal brain activity, one
neither stimulated by the environment nor tied to physical activity.
Out of this "off-line thinking" emerged not only the syntax necessary
for speech, but also the symbolic logic essential to mathematics.
Enhanced symbolic abilities let early hominids think in this "off-
line" manner, while asking and answering "what if" questions about
tools, predators, habitats or prey.
Mathematics is a great artistic triumph of the race, one made possible
by an innate human ability. Language evolved in two stages and its
main purpose was not communication. The ability to think
mathematically arose out of the same symbol-manipulating ability that
was so crucial to the very first emergence of true language. Combining
a number sense with symbolic abilities, we use abstractions to
manipulate quantities, leading to arithmetic and potentially to
calculus and number theory. Abstract models describe concrete things--
from rotating clock faces to rattles**** skins, use higher math
abilities. Mathematics is more than arithmetic. Real mathematics
involves making logical arguments about abstract objects.
Though its deepest structure shares an evolutionary origin shared with
language, math frequently calls upon a neurological number sense,
naturally strong in some, weak in others. Consequently, poets may
command powers of abstraction akin to those of mathematical geniuses,
yet still falter in doing simple algebra. But in any manipulation of
symbols, verbal or mathematical, we can easily see faculties that set
one of the earth's creatures apart from all others. Exploring the
mysterious beginnings of the mind's symbolic powers, takes us a long
way toward understanding what it means to be human.
If people are endowed with a "number instinct" similar to the
"language instinct"-as recent research suggests-then why can't
everyone do math? Why, then, can't we do math as well as we speak? The
answer is that we can and do-we just don't recognize when we're using
mathematical reasoning. Mathematics merely involves a relatively high
level of abstraction--but one we can all cope with, if we work at it.
Doing mathematics is very much like running a marathon. It does not
require any special talent, and 'fini****ng' is largely a matter of
wanting to succeed."
In a way similar Chomsky's theory that we are all born with "hard-
wired" linguistic ability, the mental process of making logical
connections between abstract objects and the mental process needed to
construct sentences have the identical structure. Thus, we can see
that the genetic heritage that gives us all the ability to communicate
by language also gives us the ability to do mathematics.
The Math Gene: How Mathematical Thinking Evolved & Why Numbers Are
Like Gossip
http://www.amazon.com/exec/obidos/ASIN/0465016197/
The Number Sense: How the Mind Creates Mathematics
http://www.amazon.com/exec/obidos/ASIN/0195132408/
> > Mathematics is the new Holy Bible! It's an invention. It was invented
> > for the same
> > reason the bible was invented, to explain the universe to others.
>
> Mathematics is the description of form, and both form and its
description
> at bottom and simplest are binary (try to go simpler), best and most
> simply described in the form (heh) of two laws, best and most simply
> symbolized as
>
> ()()=3D()
>
> and
>
> (())=3D
>
> and renderable in English as
>
> "Crossing from one binary state to the other more than once still leaves
> you in the second state"
>
> and
>
Informally, the Peano axioms may be stated as follows:
There is a natural number 0.
Every natural number a has a successor, denoted by S(a) or a'.
There is no natural number whose successor is 0.
Distinct natural numbers have distinct successors: a =3D b if and only
if S(a) =3D S(b).
If a property is possessed by 0 and also by the successor of every
natural number which possesses it, then it is possessed by all natural
numbers. (This axiom, also known as axiom of induction, ensures that
the proof technique of mathematical induction is valid.)
In mathematics, the Peano axioms (or Peano postulates) are a set of
second-order axioms [extension of propositional logic] proposed by
Giuseppe Peano which determine the theory of arithmetic. The axioms
are usually encountered in a first-order form, where the crucial
second-order induction axiom is replaced by an infinite first-order
induction schema, and Peano Arithmetic (PA) is by convention the name
of the widely used system of first-order arithmetic given using this
first-order form. However, Peano arithmetic is essentially weaker than
the second order axiom system, since there are nonstandard models of
Peano arithmetic, and the only model for the Peano axioms (considered
as second-order statements) is the usual system of natural numbers (up
to isomorphism).
http://en.wikipedia.org/wiki/Peano_axioms
http://en.wikipedia.org/wiki/Giuseppe_Peano
"Axiom", in classical terminology, referred to a self-evident
assumption common to many branches of science. A good example would be
the assertion that
When an equal amount is
taken from equals, an
equal amount results.
At the foundation of the various sciences lay certain basic hypotheses
that had to be accepted without proof. Such a hypothesis was termed a
postulate. The postulates of each science were different. Their
validity had to be established by means of real-world experience.
Indeed, Aristotle warns that the content of a science cannot be
successfully communicated, if the learner is in doubt about the truth
of the postulates.
The classical approach is well illustrated by Euclid's elements, where
we see a list of axioms (very basic, self-evident assertions) and
postulates (common-sensical geometric facts drawn from our
experience).
A1 Things which are equal to the same thing are also equal to one
another.
A2 If equals be added to equals, the wholes are equal.
A3 If equals be subtracted from equals, the remainders are equal.
A4 Things which coincide with one another are equal to one another.
A5 The whole is greater than the part.
P1 It is possible to draw a straight line from any point to any other
point.
P2 It is possible to produce a finite straight line continuously in a
straight line.
P3 It is possible to describe a circle with any centre and distance.
P4 It is true that all right angles are equal to one another.
P5 It is true that, if a straight line falling on two straight lines
make the interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles.
http://planetmath.org/encyclopedia/Axiom.html
http://www.mathgym.com.au/history/pythagoras/pythgeom.htm
----------------------------
Peano's Axioms
Sure, every school child knew the rules of counting, addition,
subtraction, multiplication, and division; but whence came these
"rules"? What reason do we have to believe that these rules are valid.
Euclid (c. 300 BC) had placed a firm foundation under plane geometry
with his set of five axioms, from which the whole of classical
geometry could be derived deductively. This was Peano's goal for
arithmetic. To reach this goal, he began, as Euclid did, with five
axioms:
Axiom 1. 0 is a number.
Axiom 2. The successor of any number is a number.
Axiom 3. If a and b are numbers and if their successors are equal,
then a and b are equal.
Axiom 4. 0 is not the successor of any number.
Axiom 5. If S is a set of numbers containing 0 and if the successor of
any number in S is also in S, then S contains all the numbers.
=2E..If there is some property P which we believe to be true of all the
whole numbers, mathematicians agree that we cannot simply show a few
examples where P is true and then conclude that it is, therefore, true
for all numbers. Since the set {0, 1, 2, 3,...} is infinite, we cannot
conclude from a finite number of examples, no matter how large, that
property P holds for all the numbers in {0, 1, 2, 3,...}. However,
Peano's fifth axiom, the principle of mathematical induction gives us
the means for determining whether property P is true for all numbers
in {0, 1, 2, 3...}. The induction axiom says that we need carry out
only two steps to determine the truth of P for the infinite set of
whole numbers.
First, we must show that 0 has property P.
Second, we must show that, for any number n, if n has property P, then
this implies that the succesor of n, namely n+1, also has property P.
Imagine a line of an infinite number of dominoes standing upright. Two
things need to happen for all the dominoes to fall with a single push:
(1) the first domino must fall, and
(2) each domino in the line must be close enough to the next domino so
that when it falls it causes the next domino to fall.
Note than both conditions must hold in order for all the dominoes to
fall. If the first domino does not fall, then the chain reaction never
begins. On the other hand, if the first domino does fall but somewhere
along the line the separation between a domino and its successor is so
great that they never make contact, then the chain reaction stops
there. This is the essence of Peano's fifth axiom.
One of the great beauties of the Peano axioms is that they make
possible the generation of an infinite set of numbers from a finite
number of symbols. Essentially, Peano hands us two items, the number 0
and the concept "successor", and an "instruction manual," i.e, his
five axioms, and promises us that with just these we can build an
entire system of arithmetic.
As a first step in building this arithmetic system, we may generate
all the natural numbers(positive integers) as follows. Let us agree
that if n is any number, rather than writing out "the succesor of n,"
we will write "s(n)." Now Peano has given us 0. Axiom 1 says that 0 is
a number and Axiom 2 says that the successor of any number is a
number. Therefore, s(0) is a number, s(s(0)) is a number, s(s(s(0)))
is a number, and so on. Clearly this process, if continued forever, is
sufficient to generate all the natural numbers. As each new natural
number is "born" by our process, we may want to give it a name. For
instance we may wish to call s(0) "Sam" or "Samantha" or, preferably,
"1." To our newly generated s(s(0)), which clearly is s(1), we will
give the name "2." We shall call s(s(s(0)))=3Ds(s(1))=3Ds(2) by the name
"3" and so forth. Thus the set {1, 2, 3,...} is generated.
Having generated the natural numbers, we can next begin to define
operations on those numbers. For example, Peano defines addition as
follows: For any natural numbers n and k: i. n+0=3Dn and ii. n+s(k)=3Ds(n
+k), where s(k) still means "the successor of k." So the addition "2 +
1" is interpreted as 2 + 1 =3D 2 + s(0) =3D s(2+0) =3D s(2) =3D 3.
Multiplication can be defined in a similar way and then subtraction is
defined in terms of the addition of inverse elements and division is
defined in terms of the multiplication of inverse elements. In order
to do this, the negative integers are defined as the additive inverses
(or opposites) of the natural numbers and the rational numbers of the
form 1/n, where n is not 0, are defined as multiplicative inverses of
the natural numbers and their opposites. In this way, we expand the
number system to include all the integers and all the rational numbers
of the form 1/k, where k is not 0. Then we must account for division
of the form p/q where p/q turns out to be neither an integer nor the
multiplicative inverse of an integer. Thus we bring in the rest of the
rational numbers to allow for this. In this way, we can continue to
build up a richer number system. We will need irrational numbers to
account for the solutions of equations such as x2 =3D 2 and complex
numbers to account for solutions of equations such as x2 + 1 =3D 0. The
im****tant thing is that all these expansions of the number system can
be accomplished by definitions, without adding any more axioms or
primitive terms to Peano's original system. That is not to say that
this expansion is trivial; for example, it took some true
inventiveness on the part of mathematicians such as Dedekind
(1831-1916), Cantor (1845-1918) to come up with an acceptable
definition of a real number. However, once this was accomplished,
mathematicians could feel more comfortable about the foundations of
the real number system, which has been the setting for the development
of arithmetic, algebra, geometry, and modern mathematical analysis
including the calculus. Thus, starting with 0, the idea of a successor
for each number, and his five axioms, Peano provided a simple but
solid foundation upon which to construct the edifice of modern
mathematics.
http://www.bookrags.com/sciences/mathematics/peano-axioms-wom.html
> "Crossing from one binary state to the other and back again leaves you
in
> the original state."
>
> Note that these are _not_ arbitrary "axioms"; they are verifiable
> observations, in a plethora of concrete examples; they are natural laws.
>
> The arithmetic and algebra arising from the above two laws comprise the
F2=
> group of laws; the F3 would be trinary form, and so on.
>
> Application of the F2 group to the truth-values of propositions gives us
> the L group of laws, propositional logic.
>
> Application of the F2 and L groups to sets gives us the S group of laws,
> elementary set theory.
>
> And application of all the foregoing to numbers gives us the N groups of
> laws, number theory.
>
> After which comes the G group (geometry) and P group (physics, at base
> interpretable as applied geometry, adding both intension, as opposed to
> extension, and dynamism) of laws.
>
> Etc.
>
> See _Laws of Form_, Russell 'n' Whitehead's _Principia Mathematica_, the
> Metamath Project and so on (the group-names, with the exception IIRC of
th=
e "L
> group", used are my own, although the ascending hierarchy of formal
> complexity is that of ordinary formal mathematics-- which has issued in
> more than one competing theory, unfortunately).
>
> --
>
> "The math is easy," said Chaos.
>
> < _Thief of Time_
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