My five-year old grandchildren, Evan and Brianna, began kindergarten recently. They’ve entered the brave new world of primary school and it’s fascinating to watch the rapid infusion of knowledge as they process information and conceptualize its meaning. Of particular interest to me is arithmetic.
While we might consider mathematics to be grouped in the sciences, it isn’t. Math is not based on science, nor is it a science in itself, per se. Math is more closely aligned with principles of philosophy, specifically logic. There is a unique quality to numbers and their manipulation that set them apart from other forms of mental reasoning.
Consider the concept of numbers themselves. Numbers are unchangeable. The number 3 (and all numbers) are never modified or changed into anything else. Nor is the number 3 interested in the scientific method or the concept of true and false which that approach implies. One can never “prove” or “disprove” the number 3. The idea itself seems silly because there is no need for verification of its truth value. It is an “a priori” truth, a valid expression that is known without reference to any external experience. And numbers have the unusual property of being independent of everything else. The number 3 will remain the same for all eternity, regardless what changes occur in the universe at large. Even if all of humanity ceases to exist, the number 3 will remain. How can this be? Aren’t numbers a human convention, simply a way that we’ve developed to explain certain aspects of the world around us? To suggest that numbers transcend human thought is to give them a kind of independent reality of their own.
In one respect, that is exactly what numbers are. Consider how we conceptualize arithmetic, specifically counting. It “appears” that young children learn to count by associating a series of objects with corresponding words. If there are three balls on the table, the first ball is labeled “one” the second ball is labeled “two” and the third ball is labeled “three”. But it’s more complex than that, because removing balls “one” and “three” doesn’t mean that ball “two” represents two balls. Ball “two” has now become ball “one” if there are no more balls on the table because it’s not about labeling objects at all. Although outwardly it appears to be a verbal association, what is actually happening internally is quite different. The child is really applying two concepts that already exist prior to any experience of the world around them: the concepts of space and time.
When any of us has an empirical encounter, what we would describe as an “everyday experience”, (let’s suppose our attention is directed to three balls on a table) that encounter presupposes two underlying concepts even as the experience itself unfolds.
Firstly, we experience all objects as they relate to space. Everything that is outside of ourselves is known as it appears to us in a three-dimensional reality. Even if only one object is before us, it is qualified in our mind asexisting in space. But what about “space” itself? How do we come to know that property? Or can it properly be called a “property” of our experiences at all? Let’s remove all objects from space. There is now nothing before us except emptiness. There is nothing to experience. Has space disappeared? Has it ceased to exist? Surely not, because as soon as any object again appears before us, we experience it immediately and empirically as it is known to us in space. So, while space itself does not constitute an experience, it is a necessary prerequisite in order to HAVE an experience. We never experience objects outside of space. If that is so, if the concept of space is a prerequisite in order to have an experience, it must be known prior to experience in some essential way. Concepts that are known prior to experience are given the label “a priori” (before experience) to differentiate them from concepts known after experience, or “posteriori”. These two terms are important because they determine how reality works and what we can truthfully assert that we can know.
Secondly, our experiences are given to us as they relate to time. Everything outside of ourselves is known as it is arranged “before”, “now” or “after”. Time itself is conceived as past, present or future. Objects that are not static experiences are perceived with properties of motion, which is a linear representation of the concept of time. Even the primary act of arranging objects of our experience requires us to place them in some kind of order, either by counting or by assembling them in a procession of “before” and “after”. Since we cannot perceive all objects of our experiences simultaneously, we utilize the concepts of space and time that we know “a priori” to give meaning to an experience.
We see three balls on a table. Our ability to define the experience as “balls on a table” requires that we have an inherent concept of space. Our ability to define the experience as “three balls on a table” requires that we have an inherent concept of time, since what are “three balls” except one ball, then another ball, then another ball – in other words, a linear progression of three objects in a row.
Now, back to kindergarten.
My grandkids can count three balls on a table. (Even Abby, at younger than two years, can perform this feat). But counting doesn’t rely on objects to have validity. Remove the balls and my grandkids can still count (and they love to show how “high” they can count) with no objects at all and the meaning hasn’t changed because the underlying principles do not require an external world of objects. They are realized internally, without reference to experience. Arithmetic is a tangible expression of the properties of space and time and as such, it gives substance to these two concepts which are difficult to know on their own.
All of this is nice and extremely boring, you say, but so what? What does it matter? Well, it matters because of the postmodern world in which we live. Science and the scientific method have been elevated in our culture to the level of a demigod, with the assumption that only what we can show through experiment and conclusion is regarded as truth. Mathematical truths reveal an opposing reality, one that presupposes a component to human reasoning that is not dependent on the empirical method or scientific inquiry at all.
If we understand the concepts of space and time as “a priori” truths and express their meaning through the arithmetic of counting - independent of material objects - the question arises as to the origin of these concepts. If we have knowledge of these concepts as independent of empirical encounters, if we intuitively “know” these concepts prior to our experience, if our experiences are dependent on these concepts in order for our senses to give meaning to our impressions, then we must search for their origins outside of our daily experiences. Clearly, we cannot utilize the scientific method, since that method, by definition, is limited to only what we can conclude from our experiences.
If we know a truth prior to our experiences, its origin must be prior to our experiences. And that is the crack that reveals the weakness of materialism, and the reason why our search for meaning will always extend beyond the physical. We may not be aware of the underlying reasons why we aren’t content with the limits of science to explain reality, but the nagging sense that there are truths beyond what science can tell us weighs on our mind. Perhaps the concepts of space and time and their creative expression through one of our most elementary ideas – counting – ruminate just below the surface of our collective experiences, tugging at our consciousness and whispering faintly that our existence begins not as a tabula rasa but with a set of tools ready for our purpose.
Any kindergartener can help point us in the right direction. Just listen to them count.