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 máthēma (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