For the Love of Logic and Mathematics
One can teach any person to do mathematics, in theory. Given enough time, anyone of any age could learn about derivatives, infinity, symmetries, and distributions. Once one understands elementary functions of mathematics, these concepts are all simple enough to mentally comprehend and master when explained properly and fully.
Why then, did teachers of time gone by insist that logic be a prerequisite to mathematics? As a child, I would see boring strings of logic in a lesson on mathematics and sigh. Seemingly unintelligent patterns of logic such as this are often posed in the older mathematics curricula:
All dogs have four legs.
Susan is a dog.
Therefore, Susan has four legs.
All dogs have four legs.
Susan has four legs.
Therefore, Susan is a dog.
Why teach such elementary and seemingly common sense reasoning?
Because logic is foundational to a true mastery of the mathematical world. Mathematics never contradicts itself—just as He who created it never contradicts Himself. Therefore, proper logic enables us to not just perform mathematical calculations performed previously by others; but also to continue to discover more and more mathematical truth.
Let us take a closer look at what students are commonly taught in a mathematical logic class—or were taught, as logic seems to be a dying art. Students were taught to look for flaws in logic, formally called fallacies. A simple and well known mathematical fallacy is the division by zero fallacy. Though many variants are possible, here is a simple one:
1. Let a and b be equal, non-zero quantities-- a=b
2. Multiply by a-- a^2 = ab
3. Subtract b^2-- a^2 – b^2 = ab-b^2
4. Factor both sides, the left will factor as a difference of squares, on the right b will be extracted from both terms-- (a-b)(a+b) = b(a-b)
5. Remove (divide out) a-b from both sides-- a + b = b
6. Observe that a = b-- b + b = b
7. Combine like terms-- 2b=b
8. Divide by non-zero b-- 2 = 1.
Tragically, it seems that we have just proven that mathematics is not, in fact, universally sound and in agreement with itself. If true, this simple operation that we have just completed would be, quite literally, the death of science as we know it, and perhaps even cast doubt on the existence of a Supreme Lawgiver, who by nature must always be in agreement and unified within Himself.
However, take a closer look at line five. When we divided both sides of our equation by a – b, we, in fact, committed the fallacy of division by zero—mathematically unsound. We established in the beginning that both of our variables were, in fact, equal to one another. Thus, a – b is in fact zero.
Now, a person not properly schooled in logic may or may not be able to follow the strings of logic within the equation just reviewed, and be able to determine where the issue with our calculations lie. They may, in fact, attribute this seemingly unexplainable gap to anything—or just entirely give up on mathematics, deciding they are not good enough at math to master it.
All of this comes down to the slow death of logic in society as we know it.
The death of logic does not just impact mathematics. Indeed, it is only a small part of the full impact of logic’s departure. In the book, Not a Chance, theologian RC Sproul and Keith Mathison examine the nature of what we call chance. A properly trained statistician will correctly define chance as simply the mathematically unknown. When I roll a die, I will say that I have a one in six chance of getting any certain number on the faces of the die. I know, however, as a statistician and a mathematician, that there is no real chance in the equation here. The number that I roll is deterministically chosen; the air pressure, exact angle of the tabletop, the force at which I cast the die, and many other environmental pieces will determine the number that I will get from a die roll. All of these forces are too complicated for me to calculate and, through physics and mathematics, determine what number I will get—so I will simply call my die roll a one in six chance. What I call chance is in fact simply that which I cannot comprehend.
In some areas of pure science, however, some experts have come to personalize chance—attributing power to this label of the unknown. In quantum mechanics, there is a phenomenon commonly known as a quantum leap. It is currently one of the many unsolved problems of physics. This movement of a subatomic particle is commonly thought to defy science as we know it. A particle will cease to exist at one point, and simultaneously exist at another. The particle “leaps” through space without actually physically passing through it.
Sometimes, and by some scientists (through faulty logic), this power is attributed to chance.
Simple logic should tell these physicists that impotent labels cannot perform anything. It is a scientific impossibility. However, they continue to misapply the label of chance from a simple mathematic unknown, to actually having the power to move these particles through space without the physical journey through space that we attribute to normal physics.
Just like if our simple logic at the beginning of this article was flawed, we could think that perhaps four legged Susan was a dog, individuals have sometimes incorrectly attributed both the unknown and the cause of the unknown to chance.
Flawed logic is everywhere. Once one is aware of the death of logic, he will see it everywhere. Politics, math, science, philosophy, the workplace, and simple everyday life.
Science is unknowable without laws. Without scientific laws, the behavior of the natural world would be madness—helter-skelter and confused. Mathematics as a law is beautiful when understood from a viewpoint of the divine: it is always in total agreement with itself, in reflection of He who created it. Logic, a foundational law of the mathematical world, is the same—complete, unified, and thus reflective of its Creator.