Did you know that math is about more than numerical values arranged in symbolic sentences? It has great value in itself for practical application, of course, but it has much merit as well in the general skills and virtues that its study encourages.
Recently, I have been reading a book by Ron Aharoni called Arithmetic for Parents. In it, he claims that:
[M]athematics is important not only for understanding reality. It offers much more than that—it teaches abstract thought, in an accurate and orderly way. It promotes basic habits of thought, such as the ability to distinguish between the essential and the inessential, and the ability to reach logical conclusions.1
There is a great deal of value being asserted about mathematics in this quote, and it is worth the time to examine it for the treasures it contains.
Mathematics is important for understanding reality:
Mathematics is the language of creation. It succinctly articulates the nitty gritty of chemical reactions, physical forces, and biological processes. It describes how reality ‘works.’ Just as we can describe reality in words, pictures, and even through music (in moods or feelings, for example), we can describe it via mathematics; in fact, mathematics is particularly powerful because it leads us into realities that we are, perhaps, unable to describe quite as starkly and clearly—getting to the physical heart of the matter in perhaps the most concrete ways—with any of those other mediums. Math seems to dive down into the absolutely unadulterated essences of things.
This gives us enormous power over nature; a sobering thought, because most of us—even highly trained mathematicians—will admit we do not fully understand all that our mathematical language describes. For example, consider Pi (π), or Euler’s ‘i’ (√−1), or infinity (∞). We cannot claim to truly comprehend any of these mathematical ‘entities,’ but we use them all the time in practical applications, placing them in our calculations and merrily going on our way…without much thought as a society about what it is we are doing.
It is important to remember, however, that simply because we do not fully understand these mathematical descriptors now, does not mean that we should conclude (as some quantum physicists have) that there is nothing ‘real’ out there to understand. Instead, as the history of mathematical and scientific discovery has demonstrated thus far, it is my hope that we will continue to grow in our understanding. In this increased comprehension, we will discover, ever more clearly, the nature of reality.
So what is a skill that studying mathematics perfects? That of being able to follow procedures which bring us to an understanding of reality! In itself, this is an amazing ability. What virtue can this impart? The realization that, in humility, we are able to pursue ever increasing knowledge, understanding, and wisdom with regard to creation.
Mathematics teaches abstract thought:
Counting, which is arguably the most rudimentary foundation of mathematics, involves an abstraction from the concrete. As children learn, they count up the apples, the crayons, and little plastic bears. Then, suddenly, they grasp the pure idea of number, divorced from physical existence. After that, it does not matter what ‘names’ you give the numbers, or on which ‘bases’ you build your sets (base 10, base 6…you name it). As long as you grasp what the names and bases represent, you can reason with them. Thus, mathematics uses abstract conceptualizations which are then put through processes governed by logical principles, and this leads us to conclusions we accept as accurate, even when we do not fully understand them or the concrete inventions (such as the many technologies we use daily) that proceed from them.
Mathematics is, in fact, a form of expression which, by all evidence, is capable of leading us beyond what we understand in the present moment; the rigor of mathematics guides us in an almost resolute way “To infinity and beyond!” (to quote the popular Buzz Lightyear of Toy Story fame).2
If we think about it for a moment, do we not find ourselves wielding a tool so powerful that it manipulates reality even when we do not completely understand either the tool or the reality we are influencing? For example, how many of us comprehend the nature of the Internet? How many know how e-mail works? How many of us, though we see—and in fact receive our very breath and bread of life—through the gift of light every day, can claim to understand what light is, or understand the mathematical equations which describe it? Or how about the fact that electricity, magnetism, and light are all possible manifestations of the electro-magnetic field?
Scientists themselves do not fully grasp these things. The real point to understand here, however, is this: look at how far our ability to abstract through mathematics has brought us! We travel into space, carry libraries on microchips, and perform surgeries with laser beams. It can be argued that in all of these cases, we do not understand the ‘how’ of it very well, let alone the ‘why.’
Thus it is true that studying mathematics not only teaches us to conceptualize abstract ideas, but it somehow mysteriously connects those pure abstractions with the physical world, endowing humanity with the power to influence reality:
There exists…a world which is the collection of mathematical truths, to which we have access only through our intellects, just as there is the world of physical reality; the one and the other independent of us…The synthesis of the two is revealed partially in the marvelous correspondence between abstract mathematics on the one hand and all the branches of physics on the other.3
Therefore, practicing mathematics imparts the skill of the ability to abstract. The virtue that comes from experiencing such abstraction is the humble gratefulness we can learn, as stewards of creation, for the gift of this power to impact and incarnate in the world.
Mathematics teaches abstract thought in an accurate and orderly way:
Human beings are naturally given to abstraction, and to being creative with abstract concepts. That is, we take abstract ideas and give them physical form: in technology, architecture, medicine, art, music, and literature, to name just a few. We give them form by following detailed steps, sound principles, and logical processes. A building will not stand if the mathematical foundations upon which it is built are logically faulty, just as a story will not convince us if its plot is full of holes and inconsistencies. Thus, consistently practicing mathematics helps us practice ordering ideas in sound ways. When we train our minds this way, we will apply this same procedure to other areas of thought as well. We become accurate and orderly readers (in terms of comprehension), writers, artists, musicians, builders, and leaders (as fathers, mothers, teachers, businessmen, farmers, statesmen, pastors, and so on); we also become more accurate and orderly speakers and, thereby, we become lucid, persuasive communicators.
Thus another skill that learning mathematics imparts is to think in sound ways. The virtue this instills is that we ourselves become better creators in the world, whether we are creating a nurturing home for our families, a successful business in the market place, or a beautiful sculpture.
Mathematics promotes basic habits of thought:
In 1892, the so-called ‘Father of American Psychology,’ William James, wrote that “Habits are due to pathways through the nerve centers.”4 Now, over one hundred years later—a century filled with vast advances in scientific knowledge—understanding about how the brain physically adapts itself to patterns of thought through learning is widely accepted (for a nice visual metaphor of this idea, go to http://youtu.be/BEwg8TeipfQ). As we practice mathematics, the habits of thought associated with this discipline (clear, concise, and logically dictated steps, the nature of which are revealed in the very word we often use to describe it: a ‘discipline’), our brains literally create physical pathways of orderly thinking, and our thoughts in general will follow in the footsteps of these mathematical pathways. Consider the implications of this in the following way: the Creator is a God of order. When we train ourselves to habitually think in orderly ways, we are more able—as the famous Christian astronomer Johannes Kepler said—to “think God’s thoughts after Him.”
William James had this insight long ago, recognizing the depths of the physical effects of learning not only on our brains but with respect to our education and the right judgment we hope to exercise in our lives:
Down among [the] nerve cells and fibres the molecules are counting it, registering and storing it up…Nothing we ever do is, in strict scientific literalness, wiped out…Let no youth have any anxiety about the upshot of his education…If he keep faithfully busy each hour of the working day, he may safely leave the final result to itself. He can with perfect certainty count on waking up some fine morning, to find himself one of the competent ones of his generation…Silently…the power of judging…will have built itself up within him as a possession that will never pass away.5
Note how James connects rightly ordered physical entities (the nerve cells, fibers, and molecules of the nervous system) to right judgment and discernment—asserting that this capacity, once acquired, is central to a person’s being. Furthermore, see how he connects all of this to education itself: “The great thing, then, in all education, is to make our nervous system our ally instead of our enemy.”6
Regular study of mathematics through drilling and problem solving according to right principles is, therefore, a wonderful way of obtaining exactly this kind of education, and acquiring exactly this kind of judgment—the ability to discern what is true from that which is false, thereby fulfilling the Scriptural mandate to “Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ” (Col. 2:8, KJV).
Another skill of studying mathematics is, therefore, that we will learn how to develop good habits in any circumstances and with regard to any subject or task. Is this not, undeniably, one of the greatest possible human virtues? Add to that the virtue of right assessment and discernment, and we see that “He that getteth wisdom loveth his own soul: he that keepeth understanding shall find good” (Prov. 19:8, KJV).
Mathematics promotes the ability to distinguish between the essential and nonessential:
Mathematics seems to present reality without any dress-ups. No fluff, no flair—it is somehow the description of elemental ‘being-ness.’ Take a look at a physics problem or chemistry exercise in stoichiometry and you will see this truth in action. Expression in this dressed-down, bare language is not, however, devoid of beauty. This expression can be accomplished eloquently. In some ways, there is nothing more beautiful than an elegant geometry proof that is executed in a graceful way (for an example of such elegance, see http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI30.html).
Mysterious though it may be, and precisely because mathematics does seem to delve down into the bare essences of things, whatever is inessential is removed. Lovely as they can be, all the distractions and bunny trails of other forms of expression (such as rhetorical devices in writing or flourishes in art and music) that seek to enhance reality are eliminated. Indeed, the best mathematical proofs are the ones which come to the concluding point in the fewest steps, using the most appropriate laws and principles; the same is true for proofs in Formal Logic. This could even be argued to be true in many practical areas, such as in cooking; the best cakes are those in which the ingredients come together well in perfect accord, with no extraneous, distracting flavors or textures. Anything that does not speak precisely to the end-goal—even if it may have intrinsic value and even, in fact, add value—is unnecessary to the unadulterated task at hand.
Repeatedly practicing this bare-bones way of expression teaches us to recognize what is vital as opposed to what can be left out of an argument, an essay, a painting, a poem, or a musical composition. Perhaps ironically, it also teaches us to place a higher premium upon less essential items that, when they are then introduced into an existing composition, truly add something profoundly beautiful to a creative work. Thus studying mathematics not only teaches us how to distill minimal truths, it also teaches us to esteem and use wisely any methods of ‘dressing up’ those truths.
Yet another skill resulting from practicing mathematics is, therefore, the ability to distill things to their quintessence, and then, if desired, to embellish them appropriately. A virtue derived from this is an ability to perceive and appreciate the essences of things, as well as an increased awareness of all the things we can add to essences to increase their beauty while not disguising their natures, and without confusing, blurring, or smothering reality.
Mathematics promotes the ability to reach logical conclusions:
We use the term ‘logical’ often. We say things such as, “That isn’t logical!” when talking about everything from actions we think ought to be governed by common sense to lines of argument we are struggling to understand.
What do we really mean by this? Merriam-Webster tells us ‘logical’ means using reason in an orderly, cogent fashion. Of course, the term ‘logical’ derives from the word ‘logic,’ whose etymology reaches back to ancient Greece: Middle English logik, from Middle French logique, from Latin logica, from Greek logikē, from feminine of logikos of speech, argumentative, logical, from logos meaning word, reason, speech, account.7 I think perhaps we can amalgamate the term a little, understanding ‘logical’ to convey something like: ‘using orderly reasoning to give a cogent account.’
Now let us widen our discussion and consider the word ‘judge.’ Again, Merriam-Webster gives a definition: to form an opinion through careful weighing of evidence. Remembering our definition of ‘logical,’ it becomes clear that these two activities are very similar, doesn’t it? In fact, is it not possible to say that reaching logical conclusions is the same as exercising judgment? Thus:
Using orderly reasoning to give account = forming an opinion through careful weighing of evidence
Consequently, reaching logical conclusions is akin to thinking well, to exercising good discernment, and to using sound judgment. It is here that perhaps the greatest skill endowed by studying mathematics becomes obvious: inculcate sound judgment in your students by making sure that they diligently study and practice mathematics. Do not allow the frustrations, difficulties, and seeming irrelevance of it to their lives convince you to set it by the wayside. The virtue gained from studying math will be a life lived wisely and gracefully, giving glory to the Creator. This is of inestimable value.
1 Aharoni, Ron. Arithmetic for Parents: A Book for Grownups about Children’s Mathematics. El Cerrito, CA: Sumizdat, 2007. 8
2 This phrase was used in similar context in discussion about math between Leigh Bortins and Lisa Bailey in a video shown at Classical Conversations Parent Practicum, Summer 2013
3 Nickel, James. Mathematics: Is God Silent. Vallecito, CA: Ross House Books, 2001. 225: quoting Charles Hermite, mathematician who proved the transcendence of ‘e.’
4 James, William. Psychology: The Briefer Course. Notre Dame, IN: University of Notre Dame Press, 1985. 3
5 ibid. 17
6 ibid. 11
7 See Merriam-Webster Unabridged online
Image: Maxwell Equations – http://www.goshen.edu/physix/204/gco/maxwell.html
Image: Proverbs – @2005 Heartlight, Inc.