Most folks expect mathematics to be more perplexing than playful. Stunning coincidences and spectacular synchronicity in the previous chapter were shared to shift our attitudes about math away from mumbled profanity and toward humbled proFUNdity.
Many of math’s delights remain largely unknown among the general population; this is unfortunate, though not ultimately devastating. On the other hand, within our common mathematical logic lurks a pervasive and deadly deficiency.
This chapter describes the need for a complementary counter-balance to offset the old limited logic and prevent its otherwise unlimited sway over our psyches.
An intuitive comprehension of this updated logic will result in breakthrough ideas; but even before that (while still learning this new logic), we improve our ability to anticipate, prepare for, and survive the harsh realities of the twenty-first century.
Our Logic Needs an Upgrade
How did we develop a deficient logic in the first place? Why does it persist?
Remember that number line we used in elementary school to help us learn how to add and subtract?
Because our first lessons in arithmetic were taught on this kind of number line, we developed a linear logic, focused on incremental movements toward either growth or depletion.
With this linear logic, we acquire a tendency to expect changes to be proportional to their cause. We are, therefore, prone to being surprised by sudden phase changes, bifurcations and disproportional* reactions to stimuli.
*Click this link to an easy little video that explains the difference between proportional/linear changes and disproportional/non-linear changes.
Without an expanded logic about exponential/non-linear effects, we are likely to make short-sighted decisions. Two examples follow:
- Offered the choice between a million dollars, or a penny plus a promise to double* it tomorrow, and continue doubling the pennies each day for a month, one might be swayed by a desire for instant gratification and choose to become an instant millionaire. As long as one fails to take into account what happens when effects are compounded, she is prone to make short-sighted decisions.
*Click the link for an example of how doubling works and to learn how much money the alternate decision is worth.
- When one imagines that something has made him feel good, he may have a tendency to think that having more of it will be even better. Will one or two more drinks, hits or snorts, girlfriends, purchases on borrowed money, etc., be the right amount to satiate? Or might the excess compound effects until they actually oppose pleasure (even to the point of being destructive or deadly*)?
*Everyday, hundreds of people in the U. S. die by accidental overdosing.
On the elementary school number line (reprinted below), units are counted sequentially; each unit is given equal spacing.
Adding and subtracting units is visualized on the number line as moving along opposite spatial trajectories: moving rightward on the trajectory increases a value, and moving leftward on the trajectory decreases a value.
The two examples illustrated above can be described as follows:
0 + 4 = 4 The purple arrow begins at ‘0’ and increases by moving four units to the right, arriving at the value +4.
4 – 6 = -2 The blue arrow moves from the +4 to the left by six units, arriving at -2.
Though -2 is the correct answer mathematically for the second example, we will never find in nature an example that matches what the number line indicates.
Below, a similar formation of unitary birds are spaced fairly equidistantly on a line. Seeing this, a child taught the number line above may be moved to count the birds, thinking the birds to the left of the central telephone pole are somehow negative birds.
Although we cannot know for sure the attitude of any of these birds, it is not mathematically logical to declare any of the birds ‘negative’.
Even if some birds fly away, that does not make them negative birds. We just have fewer (positive) birds left to count.
Our first introduction to a number line establishes a premise that breaks with nature, as it shows negative numbers (less than nothing) just as prominently as positive numbers. Showing negative numbers as a counter-balance to positive numbers belies the fact that, in the natural world, there is never an occasion to count things that are less than nothing.
This students’ number line also breaks with nature by centering on the number zero, which stands for nothingness (the absence of anything). In nature, even if all the birds fly away, there is no such thing as absolute nothingness; even the coldest darkest void of space is seething with energy.
Implied by the linear logic of their number line, students conceive of a constant tension between positive and negative values, having a tug-of-war across the center post of zero. The association of winning with advancing and losing with retreating is reinforced every time we play zero-sum games.
From the time we are little our performances are competitive: in pageants, sports, popularity contests and leadership elections. Eventually, it becomes the expected norm to compete for promotions, profits, property and power.
As long as we organize our lives around competitions we will not pause to consider that individuals depend upon group success for their own enhancement and survival, and that cooperation and altruism are the superior strategies for long-range group success.
Learning Math or Learning to Loathe Math?
When learning arithmetic on the number line the student begins with a value; then an algorithm tells him what to do with it, whether to add by pushing further to the right, or to subtract by pulling away from the right and toward the left.
The student is the agent who performs the algorithm. But is she apprised of her agency and the power that an algorithm gives her to discover an otherwise unknown outcome? It’s doubtful.
I am not, by any stretch of the imagination, a proficient mathematician; however, my self-taught mathematics come from the inside-out, from a desire to understand the meaning of perceived patterns. In contrast, grade school kids are too often force-fed the subject from outside-in.
When children are required to give math their attention under duress to pass tests, their experience can be painful and traumatizing. When this happens, the kids who grow more and more resistant to the subject will have no trouble finding others with whom to commiserate. Thus, it becomes rather fashionable to dish on the drudgery of doing math.
Graduates who do not go on to become scientists, engineers, financiers or pure mathematicians will begrudgingly perform only the simplest calculations necessary to their jobs or personal finance. Should they someday need to assist their own young’uns with math homework, they are likely to express their distaste for, and/or confusion by, the subject of mathematics, thereby promoting their prejudice downline.
If math is seen by students as nothing more than pushing and pulling irrelevant numbers around on a line, or graphing them on Cartesian coordinates, the boring exercises are done only for the goals of pleasing the teacher and avoiding parental punishment.
If a teacher gives a bad grade to a student who has improperly manipulated the meaningless numbers, the humiliation and anticipation of adverse consequences may put this student into a negative state (qualitatively worse than neutral, but not quantitatively less than zero).
Knowing what makes us feel bad, or what gets us into a negative state, gives us something/someone to blame. Assigning blame generally makes us feel better because it isolates the bad feeling; however, blaming also builds a strong association. The next time a child who blames math for feeling bad about himself is forced to do math anyway, he will have anxiety to overcome, piled on top of the math problems to solve.
Myth of the Quick Fix
The opposite of having someone to blame is when someone steps forward offering to move our circumstances in a positive direction, toward gains and pleasure.
A positive outcome is the desired effect and it is a big relief when a causal agent just happens by and makes a helpful offering. It’s incredible, right? It should be.
Advertisers and politicians count on us to adhere to hope, and to believe in quick fixes — at least long enough to buy what they are selling.
Those who are paid to say the sound-bites we want to hear and paint the promises we wish will come true assure us that help is on the way. In the meantime, they ask us, pretty-please, to:
- give them our votes;
- buy their products;
- perform their rituals;
- and fight their wars.
Adding and subtracting on the old number line gave us the impression that causes and effects are directly related. The logic intuited from lateral moves, one unit at a time, across a number line is that causation is local, straight-forward and fairly easy to identify.
A positive value/state is caused by a move in the right direction. A negative value/state results from a move in the opposite/wrong direction.
Our old number line exposed us to lateral relationships only. Nothing about it prepared us for how things are actually embedded in hierarchical structures, like atoms in molecules, molecules in cells, cells in tissues, tissues in bodies, etc.
If it had, we would have developed a logic that would warn us. Fake ‘fixers’ are embedded hierarchically in an intricately-webbed system that is motivated more to protect and profit its overlord(s) than to assist those who are ‘lorded over’.
The very pretense of quick fixes belies the fact of inertia in complex organizations.
Growing Accustomed to Misconceptions
The use of our traditional number line accustoms kids, fresh out of the start-gate, to develop and operate from chronic misconceptions. A few examples are listed:
- being debt free is something to strive for (having net-zero assets);
- having less than nothing (debt exceeds assets) is typical of most working families;
- competing for essentials (a job, promotion or limited resources) is expected;
- compromising to have time with friends and family is necessary;
- decision-making is based on an expectation that one’s actions and non-actions have proportional effects;
- looking only for external, local cause/blame for disappointing experiences is common-sense;
- playing the lottery is considered a valid financial strategy; and
- being convinced that zoning and building standards are expensive, unnecessary burdens on land-developers, major cities are engulfed by rising tides and storm surges, drowned by failing dams and levies, and decimated by damaging winds.
The fact that we passively accept a debt-based economy, perpetual war, racism, religious intolerance, xenophobia, class and cultural bigotry, etc., can all be traced to the linear, either-or, win-lose, zero-sum concept so deeply absorbed into minds exposed only to the linear logic of a number line centered on zero.
Although teachers may be obligated to prepare students for the near inevitability of counting their losses and calculating their debts, this kind of number line is not age-appropriate for first and second graders. These innocents are more likely concerned with making someone “count” who stands alone, and accounting for group behavior so they can decide legitimately whether they would rather be singled out or mingled in.
Linear thinking is passé; it misleads us intuitively, and it intellectually inhibits our progress.
Consider the benefits of adopting and instituting a complementary Nature-Based Number Line into the school systems everywhere. It would not replace the original number line, just be taught alongside it, so that the linear-logic can be compared to and contrasted with a non-linear logic. This broader view will enable us to comprehend the patterns of the natural world and the complex interconnected systems increasingly encountered in the twenty-first century.
Currently explored, studied and reported world-wide by scientists of various fields are topics such as these:
- chaos and sensitivity to initial conditions;
- disproportional effects and non-linearity;
- non-local causality;
- inter-connectivity and complex dynamical systems;
- scale symmetry and fractality;
- epigenetics and psycho-neuro-immunology;
- feedback between INformation and OUTformation;
- storing and accessing holographic memories; and
- how cooperation counter-balances competition in biological evolution.
Quadernity leads us to form a Natural Number Line that will develop in our youth an innate logic and adept aptitude for breakthrough ideas in the above-mentioned areas of research. Our children deserve to grow up prepared to make contemplative contributions to these and other fields of study. Moreover, their future and ours depends on it.
A case has been made for a new-world logic, and for a new number line to augment and update the old standard one; so now we are ready to proceed to the next chapter: Building Quadernity’s Number Line, Step-by-Step.