(R.1) Mathematics is the use of logic to
create an abstract representation of is in the
mind. It is a nimbus for which we have created
an abstract language of its own for use in communicating the
knowledge that results from its use.
(R.2) The most commonly produced abstraction of mathematics is
numbers that we use for assessing magnitude or size which we
shall call quantity. But the number itself
does not describe any other aspect or property of the objective thing or
things which are being quantified other than
their quantity. And further, numbers are
ineffable. They can quantify but cannot be described. Nature
itself has no need of mathematics as it quantifies
empirically.
(R.3) The language of mathematics enables quantification
of abstract numbers. Examples include the operations
of addition, subtraction, multiplication, division, vectors, matrices, calculus
and statistics as well as many other even more abstract operations.
That is, mathematics is a system of logic to quantify
something that can be measured or counted as
a number.
So how is it that we quantify? The base is
what is called sets of things. As an example, let us suppose that
{*} is a set that abstractly represent an apple or an elephant or
another other thing. Similarly {***} would represent a set of
{*} and another set {*} and yet another set {*} of
apples or an elephants or another other things or even a set of
an {*} apple and a set of an {*}elephants and a set of
a {*} anything else. That is, {*} is a set and {{*}{*}{*}} is a set
of subsets of {*}. And that is mathematics in
a nutshell. We have abstractly quantified the existence of
one or more things.
As a shorthand for sets we give each
possible set a token symbol. The set {*} we
designate as 1 (Hindu-Arabic) or א (Hebrew) or I (Roman) or many other
different token in other language. But they all represent the set
{*}. Similarly, for the set {***} we designate as 3
(Hindu-Arabic) or ג (Hebrew) or III (Roman), etc. In the language of
binary code used with computers we represent {***} as 11. In the decimal
notation we use in our daily lives to quantity, say money, we use $3.00 to represent
the set {$$$} of dollars. The $ being the set of 100 ¢,
the set for $3.02 would then be the set including the subsets {{$}{$}{$}} and {{¢} {¢}} or {{$$$} {¢¢}} which we would use spoken and
written language to communicate the quantity three
dollars and two cents of money in terms of material
dollars bills and pennies.
More abstractly, three dollars
and two cents can be stored in a computer as the binary number 100101110
representing credit for the buying power of 302 cents. Both the dollars
bills and pennies and the binary number stored as credit in the
computer have the same quality, which is power to used for the action of
acquire something, say a cup of coffee. In electronic transactions, the
binary number 100101110 is subtracted from one account and added
to the "account" owned by someone else even though nothing
material changes hands. Only the abstraction of value was
exchanged as measured in pennies resulting in the transference of
buying power by the action of two
equivalent changes in the material memory
of computers. Thus, the mind perceives the nonmaterial concept
of credit which exists only as a number stored
in the memory of a computer in a bank (or written on a promissory note like
Both the sets and their token names
are nnimbus and can take no physical actions.
They cannot add {*) plus {*} plus {*} to equal the
set {***}. We can only logically communicate that they
do and equate them in memory as such. We must remember that
1 plus 3 equals 4. Show them to a tribe of natives deep in the Amazon jungle
who have had no knowledge of numbers and they would
be meaningless. The only quantities they know are 0, 1
and more than 1
to which they ascribe no tokens.
We can also perform mathematical operation
with sets. We can subtract one set from another to obtain a
remainder set: {****} minus {*} is the {***} set. We can multiply
sets to obtain a set of sets: {***} sets
of {**} set is the {******} set. And we can
divide sets of sets: {******} set divided
by {**) set is the set {***}. We can do logic of
equality of sets: {***} is the same as {***}: true; (**} is the
same as {***}: false; {***} is true and {***} is true: true; {***} is true but
{**} is not true: true: etc. All of mathematics, however
sophisticated, are based on these simple concepts. It is only a matter of
learning the mathematical systems they employ. And, yes, even the most advanced
computers use these simple concepts and depend upon tables sets to know that 2
plus 3 equals 5 and upon logic hardware to know that 2 plus 3 is not equal 6.
Alas, it has been logically proven that a
complete and consistent set of axioms for all mathematics is
impossible and even mathematics has it unknowable. Considering
that all sets are subsets of sets (save
the null set which contains nothing at all), it begs the question of what is
the set of all sets?
(R.4) The language of mathematics also enables
definition of special shapes or forms which we call geometry.
Example of geometry include points, lines, areas and volumes
in the three dimensional world with which we are most sentient.
But geometry also includes not only these dimensionless points,
one dimensional lines, two dimensional lines, and three dimensional
shape of the sentient world but also the
abstract four and higher dimensional geometries used primarily by
those who study the world of the very small that lie beyond our direct observation
with our sensory abilities. As an example, a the shape of a
donut has a four dimension geometry.
The mathematics of these abstract geometries is
beyond the experience and, thus, knowledge of
other than a small number of mathematicians and scientists.
Only the knowledge gained using them will be commonly known by
most of us. The discovery, for example, of the Higgs boson was widely published
but the mathematics and geometries used to
predict its existence was not.
(R.4.1) Geometries other
than three dimensional geometries may exist in sentient three
dimensional space. Common examples are the two-dimensional geometry of
the Mobius strip that exists in a three-dimension space and of the
four-dimensional donut that exists in three-dimensional space. But humans are
limited to the sentience of three-dimensional space. We only experience
"up", "down", and "back". It challenges
our mind to observe that a Mobius strip has a geometry
with only one surface or that a donut has four dimensions.
(R.5) The language of mathematics enables the
representation of the distribution of actions in time
intervals both in time past and time
future. This we shall call this the mathematics statistics.
We would say the probability of a coin landing heads is 1 in 2
and similarly 1 in 2 of landing tails for a total probability of
2 in 2 or 100%. But this is true only for a two-dimensional coin. The
coins in our pockets are three-dimensional coins with a heads
surface, a tails surface and an edge surface. The
total probability is the sum of the probabilities on landing on each of the
three surfaces, that is the probability of heads or
tails is less 1 in 2 because it can land on its edge and
occasionally does. The same can be said for all geometries of any dimension.
The determination of actions at the quantum
level at time present is impossible because they are
entirely random. We can only say that the probability of an action
at the atomic level in a time interval has determined
value of probability and that the sum of all
probabilities in all time intervals, past, present,
and future, has a value with the absolute certainty
of 1.
(R.5.1) With sufficient knowledge of
cause and effect, probabilities are
calculable. The mathematics of the Boltzman equation for the
statistics of the relation between entropy and
probability is an important example in physics. In an application
of everyday experience, it quantifies the flow of heat from a hot
stove to your hand when you touch the stove and the change in
temperature that you sense. Without the statistical probability
of heat flow from heat source to stove would not get "hot"
and without the statistical flow of heat from the stove to your hand,
you would not sense its hotness. And without the statistical
flow of heat there would be no change in entropy, no
change in anything, and no perception
of time.
(R.6) Mathematics is not essential for our existence
because nature does its own calculations empirically
according to the laws of nature. Indeed, the earliest know mathematical
systems are only about 3000 years old. But mathematics is
very helpful in gaining our knowledge and understanding
of what we call nature.
(R.7) Our perception of quantity diminishes
as the numbers get larger. We are very good at perceiving the
quantities of 0 and 1. A thing is either there or
not there. With 2 we perceive it is more than one. Beyond
that we begin to count to know the quantity.
Perception of a quantity of 10
is difficult. Quantities of one hundred, one thousand
or one million are beyond our perception of a thing that
exists only as a collection of things; that is to
say, as a set of things. Similarly, our ability to
recall the abstractions that are numbers stored in memory is
limited. About the best we can reliable do is about 7 numbers but
even them we divide the telephone numbers for each individual
telephone into two sets, one of three numbers and
another of four numbers. And as the area code, we do not
associate it with a specific telephone number at all but with a
specific area where individual telephones are located.
However, we frequently use perception of
quantity. We might only need look at an article of clothing to assess whether
it is too large or too small in size for the intended wearer.
The ability of groups to quantify by
perception can be somewhat accurate. As an example, 73
students were asked to guess the number of jelly beans in a jar. The actual
number was 1116. The guesses ranged from 250 to 4,100, with an average
error of estimation of the 73 guesses made by each of the individual students
of 700 or 62% of the actual quantity. But the average of all 73 guesses of the
group was 1,151 with an average group error of estimation of only 35 or 3% from
the actual number. And, two of the students got closer than the average error
of 35.
Even spiders use their primitive brains to
quantify the size of its captured prey by perception.
This is done to quantify the amount of venom needed to be
injected to kill its prey because replenishment of venom by the venom gland can
take days if all is expended in a single bite.
Humans do the same thing by quantifying the weight
of an object to be thrown and the distance to the waste basket in order to quantify of
the force be to exerted by the muscles in the arm and hand that the object
might fall into to the trash can. And, indeed, basketball players do exactly
the same when shooting a basketball. In all cases the action is
performed with the intervention sensory information with the brain.
(R.7) Numbers can exceed comprehension by the
human mind. Infinity, designated by the token ∞ is boundless, endless, or larger than any natural number (1, 67, 194674,
etc) we can count. Infinitesimal calculus developed the late 17th
century is the mathematics used in every branch of the physical
sciences, including computer science, statistics, engineering, economics,
business, medicine and demography is base around the concept of the infinitely small.
Imaginary numbers created by multiplying a
real number that an imaginary unit i which, when multiplied by
itself is equal to -1. Although imaginary numbers have no material association
as do other numbers used for counting, imagine numbers are essential in
understanding and quantifying the material properties of
subatomic particle in quantum physics.
Irrational numbers are real numbers that cannot be written as a simple
fraction. The best know and most used of these is the ratio of the
circumference of a circle divided by its diameter. Designed as π, this
number is used extensively throughout all of physics and mathematics. It begins
as 3.14159265359… and continues without repeating a pattern of numbers to
what is believed to be an infinite series of number of fractional numbers. Or,
at least, π has been calculated up to series of 62.8 trillion numbers
without finding a repeating pattern.
Another irrational number φ is known as "golden
ratio". Equal to 1.618033988749... , it is found in numerous cases in
mathematics and geometry often associated with the aesthetics in the visual
arts and has been used by artists and architects for that purpose.
Approximations of the "golden ratio" are to be often found in
nature. Examples are the is spiral shape of a nautilus shell and the ratio of
the lengths of the adjacent phalanx bones in the hands and feet of vertebrates.
In all cases, these incomprehensible numbers always relate to some aspect of reality for
which we can comprehend.
(R) Logic is a system used to create abstractions of relationships between things and/or actions in the mind. Mathematic is a system of logic to create abstractions of quantity in the mind. Both are used in reasoned thought to reach logical conclusion upon which to base knowledge and action. Neither is complete can lead to unsolvable paradoxes. Yet both serve us well in meaningful ways to obtain scientific knowledge.