For the Love of Logic and Mathematics

An Essay By Sarah Liz // 2/20/2017

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:

Good logic:
All dogs have four legs.
Susan is a dog.
Therefore, Susan has four legs.

Bad logic:
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.

Comments

Sadly, I have not cultivated

Sadly, I have not cultivated the art of understanding mathematics. :'(

Damaris Ann | Wed, 02/22/2017

"It is the small temptations which undermine integrity unless we watch and pray and never think them too trivial to be resisted."
-Luisa May Alcott

This is awesome! I loved

This is awesome! I loved getting your input on statistcs topics - the nature of chance, the quantum leap - as math has long been a favorite subject, but thinking in terms of statistics hasn't come quite as naturally. It's still really cool though, and as a result I'd love to hear more from you on the subject! And thanks for mentioning those books, who wrote "Redeeming Mathematics" and "Love and Math"? One book I read on the subject was "Truth & the Transcendent" by Larry Zimmerman and it was soooo interesting . . . I think I'll pick it up and read through it again soon . . .

I really didn't get the logic-foundations to my math curriculum until I hit geometry. Actually, I did Math-U-See geometry, and was given the impression that geometry just requires you to memorize a bunch of volume formulas. Then we switched over the Teaching Textbooks, and that geometry curriculum gave me the whole lessons-in-logic background. I had just heard some of Jason Lisle's logic/apologetics lectures so was instantly captivated. The only other logic-heavy math I took was Linear Algebra - good heavens, some of that stuff got super abstract! :P

I love how you talk about the beauty of mathematics - I feel the same way about it! Especially when I got to use calculus in physics, or anywhere, really, where my math curricula applied the subject to real-word phenomena. There's something about setting up and evaluating an integral to describe tangible objects and how they behave. It's really just amazing that mathematical rules reasoned through in some random guy's head actually can be used to describe and understand the real world. It's such an incredible testimony to the supreme, rational God who created this universe. : )

Keep writing stuff like this! : ) I'm totally captivated by your story right now but it's always nice to read some nerdy essays on here too. :D

Hannah D. | Wed, 02/22/2017

"Reason itself is a matter of faith. It is an act of faith to assert that our thoughts have any relation to reality at all." - G. K. Chesterton

Thanks ladies!! <3 Yes I love

Thanks ladies!! <3 Yes I love love love LOVE mathematics. Especially statistics and data analysis types, but all types of math are so beautiful! I absolutely love nerdy stuff too, haha! I will have to read that book as well that you mentioned. Redeeming Mathematics is more of a theology book, and less of a math book (although everything in there is predicated on mathematics). It is for mathematicians and non-mathematicians alike. Love and Math, however, is by Frenkel and is definitely not a theology book at all. It tells the story of the authors life, and talks about everything from symmetry groups to braid patterns to quantum physics. OH my word the way he describes symmetry groups.....I actually cried a little, it was so beautiful!!! Anywhoo yes I can get very carried away with my nerdy talking! :)

Sarah Liz | Thu, 02/23/2017

"Knowledge is knowing that a tomato is a fruit; wisdom is not putting it in a fruit salad." ~ Anonymous

I was never a big fan of

I was never a big fan of math, but loved how logical it was, just struggled with the numbers (one reason I love music theory so much is it reminds me of math, but without numbers!). Before I was allowed to be "done" with math in school, my dad made me read the book "Mathematics: Is God Silent?" and it gave me further appreciation for it. I haven't read the others you mentioned, but highly recommend the one I read if you're looking for further reading!!

Kyleigh | Thu, 02/23/2017

Oooo just got that book. It

Oooo just got that book. It looks amazing!!

Sarah Liz | Tue, 02/28/2017

"Knowledge is knowing that a tomato is a fruit; wisdom is not putting it in a fruit salad." ~ Anonymous

I really want to get all the

I really want to get all the books mentioned here ("Love and Math" and "Mathematics: Is God Silent?" look sooooo cool) but for now I just wanted to share this lecture on math and Christianity from astrophysicist Jason Lisle ( https://www.youtube.com/watch?v=nkwCl0ymTfg ). I really want to write an essay on what he's talking about here (namely, imaginary numbers and fractals) but I'd almost certainly need pictures, so probably not. ;) But still, it's a really interesting part of the Math, Logic, and Christianity discussion. :)

Hannah D. | Tue, 03/07/2017

"Reason itself is a matter of faith. It is an act of faith to assert that our thoughts have any relation to reality at all." - G. K. Chesterton

Yayyy!!!

Oooo I'm gonna listen to that when I get a sec. I'm about to post an essay on general revelation through mathematics within the next week or so. I've been really interested in that after reading Redeeming Mathematics. Oh yes yes yes YES read Love and Math. It is one of the most beautiful books I have ever had the pleasure of reading.

Sarah Liz | Wed, 03/08/2017

"Knowledge is knowing that a tomato is a fruit; wisdom is not putting it in a fruit salad." ~ Anonymous

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