Philosophy and Science for the Third Millennium

Mathematics in Ten Minutes

An Essay by Christopher Bek



Summary—Mathematics in Ten Minutes summarizes the branches of mathematics including geometry, arithmetic, algebra, analytic geometry, trigonometry, calculus, fractals, number theory, group theory, and probability and statistics.


They deem him their worst enemy who tells them the truth.


The gateway to universal knowledge may be opened by the unified field theory upon which Einstein has been at work for a quarter century.  Today the outer limits of man’s knowledge are defined by relativity, the inner limits by quantum theory.  Relativity has shaped all our concepts of space, time, gravitation, and the realities that are too remote and too vast to be perceived.  Quantum theory has shaped all our concepts of the atom, the basic units of matter and energy, and the realities that are too elusive and too small to be perceived.  Yet these two great scientific systems rest on entirely different and unrelated theoretical foundations.  The purpose of Einstein’s unified field theory is to construct a bridge between them.

—Lincoln Barnett


Restricting a body of knowledge to a small group deadens the philosophical spirit of a people and leads to spiritual poverty.

—Albert Einstein


Albert Einstein discovered that even the most complex notions could be reduced to a simple set of fundamental principles.  

—Paul Strathern


It is a wonderful feeling to recognize the unifying features of a complex phenomena which present themselves as quite unconnected to the direct experience of the senses.

—Marcel Grossman



The Bernoulli Form elucidates the notion of Platonic Forms in describing how a motley crew of Forms—including Delphi, forecasting, integration, utility, optimization, efficiency and complementary—come together to form The Bernoulli Model.


The Method of Moments elucidates the notion of Platonic Forms in describing how a motley crew of Forms—including Delphi, forecasting, integration, utility, optimization, efficiency and complementary—come together to form The Bernoulli Model.


The Efficient Frontier examines the notions of God, option theory, portfolio theory, faith, reason and Arab mathfinally arriving at the inescapable conclusion that all roads of sound decisionmaking lead to the efficient frontier.


The Unpardonable Sin charges all honourables and doctors in Canada with heresy, child abuse and the unpardonable sin that Christ spoke of—which is the deliberate refusal to follow the light when seen.


The Uncertainty Principle contrasts Einstein with Heisenberg, relativity with quantum theory, behavioralism with existentialism, certainty with uncertainty and philosophy with science—finally arriving at the inescapable Platonic conclusion that the true philosopher is always striving after Being and will not rest with those multitudinous phenomena whose existence are appearance only.


A Formal Patient congratulates Alberta Health and Wellness for insisting on the accountability of due process in declaring individuals to be formal patients—and argues that I am being considered a formal patient as the result of an absence of due process elsewhere in Canada—and that I should not be considered a formal patient but that I should be declared disabled on account of being outside the cave of behaviorism.



Singularity identifies the trigger of the looming paradigm shift from the three-dimensionally conscioused Everyman to the four-dimensionally conscioused Superman as the 1935 Schrödinger's Cat though problem—which proves that consciousness is real.


The Great Cosmic Accounting Blunder compares the two physical fixedpoints in the universe—lightspeed and Planck’s constant—and argues that we have been guilty of double counting up until now and that in fact there is but one fixedpoint—which, as it turns out, is the boundary of the universe.


The Unified Field Theory counts down the Euclidean hits from five to one in categorically nailing the vast majority of this little thing I like to call cosmic pi.  At this point in spacetime I would like to pay special tribute to my excellent wingman Albert Einstein (18791955).


Closing the Liars Loophole identifies the malignant cancer within the healthcare system and society as the outwardly focusing behavioural psychological model, which denies the existence of consciousness—while the inwardly focusing existential model makes consciousness and the soul primordially important.

The Greek Plato’s (427-347 BC) theories of knowledge and Forms holds that true or a priori knowledge must be certain and infallible, and it must be of real objects or Forms.  Thales and Pythagoras laid the foundation for Plato by founding geometry as the first mathematical discipline.  Mathematics is the systematic treatment of Forms and relationships between Forms.  It is the science of drawing conclusions and is the primordial foundation of all science.

Geometry.  Geometry is the division of mathematics that deals with space and time.  In its simplest outward appearance geometry is concerned with problems such as determining the areas and diameters of multi-dimensional figures and the volumes of solids as well as the areas of surfaces.  Other foundations of geometry include analytic geometry, topology, fractal geometry (ie. fractions of dimensions) and non-Euclidean geometry (ie. based on less than the five Euclidean axioms or rules).  The goal of geometry is to calculate properties such as area, boundary and diameter.  The Greek mathematician Pythagoras (582-500 BC) laid down the foundation for analytic geometry by showing that the various erratic and disconnected laws of geometry could be proved to follow the logical endings of a limited number of axioms.

Arithmetic.  Arithmetic is the knack of counting.  Numbers used in counting are called positive integers.  They are created by adding one to each number unto infinity—so that each number in the sequence is one more than its predecessor.  Civilizations through history have developed different types of mathematical systems.  One of the most common usages is that used in modern cultures in which components are counted in groups of ten.  This so-called decimal system is used in modern mathematics.  Arithmetic is concerned with the ways that numbers are joined via the operators of addition, subtraction, multiplication, division and roots.  Numbers also includes negative numbers, fractional numbers and irrational numbers (eg. p = 3.1416).  The base-ten integers are represented by numbers that are articulated by the powers of ten.

Algebra.  Algebra is the branch of mathematics which employs letters to correspond to essential arithmetic relationships.  As with arithmetic, the basic operations are addition, subtraction, multiplication, division and roots.  However, as with arithmetic, algebra cannot generalize mathematical relations such as the Pythagorean Form—which states that the sum of the squares of the sides of any right triangle is also a square.  Algebra is concerned with solving equations by using symbols instead of numbers—and using arithmetic operations in order to establish ways of handling symbols.  Modern algebra has evolved from classical algebra by increasing its focus on the structures of mathematics.  Modern mathematics considers modern algebra to be a set of objects with rules connecting them.  In its basic form algebra may be depicted as the language of mathematics.

Analytic Geometry.  It was Descartes (1596-1650) who first realized that arithmetic, geometry and algebra are the same thing—ie. analytic geometry.  In analytic geometry lines, curves and geometric figures are represented by algebraic expressions using a set of coordinates.  Analytic geometry has contributed to the advancement of mathematics because it has unified the concepts of analysis and geometry (both spatial and temporal).  The study of non-Euclidean geometries of space and time having more than three dimensions would not be possible without analytic geometry.  As well, the modus operandi of analytic geometry made possible the representation of numbers and algebraic expressions in geometric terms which have cast new light on calculus and other problems in mathematics.

Trigonometry.  Trigonometry is the division of mathematics concerned with the relationships between the sides and angles of triangles. The two branches of trigonometry are plane trigonometry (concerned with figures lying in a single plane) and spherical trigonometry (concerned with triangles that are spherical sections).  The initial purpose of trigonometry was in the playing fields of astrophysics, surveying and navigation.  The main problem was to determine the distance such as that between the moon and the earth.  The notion of the trigonometric angle is fundamental to the study of trigonometry.  Other purposes of trigonometry include engineering, physics, and chemistry—particularly in the field of episodic phenomena such as the study of bridge-building and the stream of electrical current.

Calculus.  Calculus is the mathematics of motion.  It is the division of mathematics concerned with the study of such notions as the slope of a curve, the rate of change, the computation of the maximum and minimum values of functions, and the calculation of area bounded by curves.  Calculus is used in engineering, biological, material and societal sciences.  It is also used in the physical sciences by studying falling bodies, the rate of decomposition of radioactive substances, and the pace of transformation of chemical reactions.  Calculus is broadly used in the study of probability and statistics.  It can also be used in many problems involving the concept of acute quantities such as the fastest, largest, slowest and roots.

Fractals.  A fractal is a geometric phenomenon that maintains a detailed structure under any level of scaling.  They are self-similar in that they have the property such that each small portion of the fractal can be viewed as a reduced-scale replica of the sum total.  Examples of fractals include broccoli snowflakes, mountains, broccoli, lightning and galaxy clusters.  In 1975 the French mathematician Benoit Mandelbrot adopted an abstract definition of fractional dimensions that were used in Euclidean geometry.  Mandelbrot posed the question—How long is the coastline of Britain?  Appealing to relativity, Mandelbrot pointed out that the answer depends on one’s perspective.  From space, the coastline is shorter than it is to someone who is walking.  That is because on foot the observer is exposed to greater detail and must travel farther.  Ultimately, according to Mandelbrot, when the shape of each pebble is taken into account, the coastline turns out to have infinite length.

Number Theory.  Number theory is the division of mathematics that deals with the belongings and relations between numbers.  The theory of numbers consists mainly of mathematical analysis.  It is concerned with the study of integers.  A perfect number is a positive integer that is equal to the sum of all its positive divisors.  For example, 6 equals 1 + 2 + 3—and 28 equals 1 + 2 + 4 + 7 + 14.  A positive number that is not perfect is imperfect and is deficient or abundant according to whether the sum of its positive, proper divisors is smaller or larger than the number itself.  Thus 8 with proper divisors (1 + 2 + 4 = 7) is deficient—while 12 with proper divisors (1 + 2 + 3 + 4 + 6 = 16) is abundant.

Group Theory.  Group theory is the fundamental configuration of algebra consisting of a set of elements and an operator.  The operator uses two elements of the set and forms another element of the set in such a way as to meet with certain conditions.  Group theory is the subject of powerful study contained within the field of mathematics.  In relativity theory (1905—ie. the fundamental law of space and time) group theory plays a pivotal role—also known as the Lorentz contractor from relativity—which is also coincident with the Pythagorean Form.  A case in point of group theory is found in the set of all numbers with the operator of addition.  The operator + takes two numbers (eg. 3 and 5) and forms their sum of 3 + 5 = 8.

Probability and Statistics.  Probability is the division of mathematics that deals with the quantitative measuring of likelihood that an incident or an experiment will have a specific outcome.  It is based on the study of permutations and combinations and is the necessary foundation for statistics.  Permutations and combinations are the arrangement of elements and are the basis of probability and statistics.  Probability and statistics are used in actuarial science and began in an attempt to answer questions arising from games of chance.  Statistics is the division of mathematics that deals with the set, association and analysis of numerical data.  It deals with problems such as decisionmaking and experiment design.  Simple forms of statistics have been in use since the beginning of civilization when symbolic representations were used to record numbers of people and animals.

Conclusion.  Ancient mankind lived in the fear and dread of natural events because he could not explain them.  Fables and enchantments dominated his thought process.  Increasingly, man began to understand the laws of nature.  A second illumination is needed so that we can live in peace with our unearthing and constructions.  A movement must take place in the emphasis from an old-style humanity to a new-style science in our educational system.  This shift cannot take place through normal educational means.  The scientific material available is only read by a small group of people.  We clearly require material with sufficient appeal and persuasive power to include the enlighteningly as the scientifically uninformed multitudes.  There is promising evidence that the new enlightenment may come to pass—that the cultural gap between our new technology and its meaning in terms of human values that in the end may be bridged.


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Last Updated—28 March 2009.
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