*NOTE: I am posting this installment separately and in sequence in the first blogpost of this series so that you can read it here or in context below.***FIVE;
PARADOX AND PROGRESS**

I happen to share the birth date of Danish
physicist Niels Bohr. Not the same year, but the same month and day. He was
born in 1885 and died when I was 26 years old. Bohr was one of the main players
in the drama surrounding the birth of quantum physics, and the primary
spokesperson for quantum mechanics in the famous Einstein-Bohr debate over
quantum uncertainty. Albert Einstein, the theoretical physicist who inspired me
to become a scientist when I was fourteen, sparked the debate by proposing an
experiment that challenged the validity of the concept at the heart of quantum
mechanics known as the Heisenberg uncertainty principle. That experiment, based
on the belief that elementary particles were tiny, localized particles of
matter, became known as the EPR paradox because Einstein, Podolsky, and Rosen
were the authors of the paper. The EPR experiment presented physicists with a real
paradox because it demonstrated that, if elementary particles were what physicists
thought they were, then the expected outcome of the experiment disproved the
uncertainty principle. To Einstein, that meant that quantum mechanics was, at
best, an inconsistent or incomplete theory. Niels Bohr understood this, but the
following quote illustrates his attitude toward paradox:

"*How
wonderful that we have met a paradox. Now we have some hope of
making progress."*

This is the proper attitude for a scientist.
Science only advances when paradoxes within its logical structure are explored
and resolved. If it is a real paradox, it can only be resolved by expanding the
existing model of reality. A real paradox reveals the incompleteness of the
existing theory. All other so-called scientific advances are just the filling
in of the details of an existing model, which requires no original thinking. Problem
solving within the established paradigm is not science, it is engineering. Engineers
are practical thinkers who solve day-to-day problems by applying science-based
technology. I am not denigrating engineering. I made a living for myself and
family for many years as a registered professional engineer (PE). Engineering is about using science to solve
technical problems, while science is about thinking outside the box and testing
hypotheses about the nature of reality. Scientists should be looking for
problems that cannot be solved using the engineering methods existing within
the current paradigm. About problems that lie beyond the reach of engineering
thinking, Bohr said:

*“Every
great and deep difficulty bears in itself its own solution. It forces us to
change our thinking in order to find it.”*

There is a wealth of insight in these two sentences.
They contain deep truths about problems that can be addressed using *mathematics*,
the language of science. The first sentence is true about any problem. However simple
and easy, or deep and difficult a problem may be, if it is properly stated in
the language of the paradigm within which it has meaning, then the solution,
i.e., the answer to the question that it poses, is actually contained within
the question itself. Any meaningful problem in an algebra text will demonstrate
this fact.

The information needed to solve a problem is
always contained within the statement of the problem. All that needs to be done
is to translate the statement of the problem from English, German, or whatever,
into mathematical expressions, so that you can use well-defined operations of
calculation to transform the statement, through a series of simple steps, into a
new form that is recognized as an answer. The second sentence relates specifically
to the “great and deep” problems that Bohr concerned himself with as a
scientist. If the problem is truly deep and difficult, we will have to “change our
thinking” i.e., we have to think outside the box by expanding the paradigm, or
developing new methods, to solve it.

I realize that there may be people reading this who
already understand the points I am making. This post is not for them. It is for
the millions of people out there, the average citizens of planet Earth, who
shrink away in horror anytime they see the word “mathematics”. If you are one
of those who hates mathematics, I am quite sure that it is because you were never
taught what mathematics really is. What I want to do here is change the way you
think about *mathematics*. Let’s start with the meaning of the word. It
doesn’t mean “difficult stuff with numbers and abstract symbols”, as most
people think. The word mathematics
comes from the Ancient Greek word *máth**ē**ma* (μάθημα), meaning “that which is
learned” or “what one is able to know”. Mathematics is not just “the language
of science”, it is the heart and soul of science.

I have had
several years of experience teaching mathematics, from practical applied math
to mathematical physics, and advanced mathematical modeling, so I am not just
blowing smoke rings. My first job, after earning a degree in physics and mathematics,
was teaching math in the Newburg Missouri Public High School, where I was the
complete Math Department. In my second year of teaching there, my students
swept every category in a regional math contest where we were competing with
several much larger schools. Even though I used the textbooks supplied by the
state, I didn’t follow the state lesson plans. I always related mathematical
principles to common-sense ideas that my students could relate to and understand.
Over the years, I have had several very gratifying success stories teaching,
but the first one happened at Newburg.

The
first day of school in my first year at Newburg, a young man from a rural
farming area south of Newburg, walked up to my desk as students were filing into
the classroom and said:

“Mr.
Close, my name is Gary Haven. I just want you to know that I don’t like math. I
am only here because general math is a requirement for all Freshmen.”

“OK
Gary,” I replied, “Thank you for being so honest! I just want you to know that I’m
going to see what I can do about changing your mind about math.”

The
second year Gary took every math class he could cram into his schedule, and
eight years later, he graduated from one of the best engineering schools in the
county with top honors in electrical engineering.

Mathematics
should not be about showing others how smart you are. It should NOT be about
learning how to manipulate abstract symbols to solve complicated problems, even
though you may learn to do that. It should be about learning how to think rationally
in a way that will increase your understanding of reality and make your
thinking more efficient and effective. In my opinion, the way math is taught in
our schools today, from grade school to grad school, is really stupid, and
almost criminal. The way math is fragmented and turned into mindless memorization
of detail, alienates and repels students, robbing them of an opportunity to
develop critical thinking skills.

Learning
to use mathematical tools to solve practical problems is important; but using
calculators and computers to solve numerical problems without understanding the
underlying principles, is a recipe for disaster. Pure mathematics should be
understood to be what it actually is: a simple reflection of how your mind and
the universe works. Science is about what we can say about the reality we
experience, and math is about how we can say it in the most efficient way.

Before I get into my explanation of a system of logic,
I call the Calculus of Dimensional Distinctions (CoDD), a calculus that describes
reality more precisely and efficiently, I want to introduce you to some key
concepts needed to understand non-physical reality. They concern the axiomatic
nature of science and the threshold between every-day consciousness and *Turiyananda*,
the joy of experiencing pure consciousness.

First, an axiom is a self-evident truth. All systems
of mathematical logic have two or more axioms upon which they are based. Within
any consistent finite logical system of thought, an axiom is a question that
cannot be answered within the system that allows it to be stated as a
meaningful statement. I like to think of an axiom this way:

*The Question that cannot be Answered
is the Answer that cannot be Questioned***.**

We have been told by some very intelligent
people, specifically British mathematical philosophers Bertrand Russell and
Alfred North Whitehead, that there are only three types of statements: 1) True,
2) false, and 3) meaningless. Sounds rather obvious, doesn’t it? If we accept
this declaration, then we can ignore and discard meaningless statements
because, well, because they are meaningless! If statements that appear to be
meaningless in the context of our current understanding of reality are
eliminated from consideration and we accept the statement of Gottfried Wilhelm
Leibniz (another *very* intelligent person) that for every meaningful
question there must also be a meaningful answer, then every meaningful
statement is ultimately either true or false.

This binary thinking is in fact, the basis of the
logic, philosophy, science, mathematics, and computer technology of Western
Civilization. It turns out, however, that despite the fact that these axiomatic
statements were pronounced by some of the most highly intelligent people of the
world in recent history, none of these statements are true.

In the next installment of this series, I will explain
how and why there are actually four types of statements in any language, including
mathematics, not three, and I will also explain how this fact leads us back to
the threshold of Pure Consciousness and the doorway to extra dimensionality where
we will see how pure number theory relates to consciousness.

ERC – 12/29/2021