David Cowles

Sep 1, 2024

“Our own number system is based on a highly specialized, and not necessarily privileged, concept of quantity.”

Most of us can’t wait until our toddlers are old enough to count. Even, “One, three, six, four,” sounds like progress. At least they’ve caught on to the notion of sequence, if not yet quantity or magnitude. But are we doing them a disservice? Are we uncritically condemning them to live out their lives as we have lived ours?

Oh come on! For heaven’s sake, it’s just counting. Show me someone who doesn’t know how to count! Well, ok, how about a whole culture. As we have discussed before in *Aletheia Today*, there is __a tribe in the Amazon__ that to this day does not have even a concept of number.

Ok, granted, this is a bit of an outlier, but many, many contemporary cultures have radically different concepts of numeration from us. Consider these 6 representative options:

One, two, three. (Larger quantities are not considered countable.)

One, two, many.

One, two or three, more than three.

One, all.

A grain, a pile, a heap.

Real Numbers plus Arithmetic.

What’s key here is the way these systems treat the phenomenon of quantity. We assume that quantity is a given; it just is. But as we will see, our number system is based on a highly specialized, and not necessarily privileged, concept of quantity.

Each of the ‘number systems’ mentioned above corresponds to a particular way of viewing the world:

In the **first** case (1, 2, 3), counting is only considered useful up to ‘3’. After that, numbers are irrelevant. “One, two, and three” tell you something very specific and meaningful about your subject. Not so much after that. Perhaps some sense of volume takes over.

Even in our culture, there are times when we stop counting (but usually not at 3): we call that *statistics*. Nobody will count the number of 7’s thrown at Bellagio’s craps tables tonight, not even the all-seeing eye in the sky. Everyone, punters, dealers, owners alike, trusts probability.

The **second** example (1, 2, >2) could come from the notebook of a STEM professor, or a systems analyst, or a marketing guru, or an existentialist philosopher – folks not always so easily interchangeable.

We know how to write equations with one variable: most of us master that in elementary school. Two variables? A little trickier – maybe high school. But more than two variables? That’s *The Three Body Problem*. Three or more bodies behave chaotically. So, 1, 2, >2 makes perfect sense from a physicist’s perspective, more sense perhaps than our own number system.

Sara Imari Walker underscores this in *Life as No One Knows It*, her 2024 book outlining ‘Assembly Theory’ as an explanation for the origin and development of life: “There are, in fact, only 4 important numbers to understand in assembly theory: 0, 1, 2, and many.” Essentially, all process comes in stages: 0 = ‘does not exist’; 1 = ‘exists’; 2 = ‘copied’; >2 = ‘ubiquity’.

This same logic informs our Systems Analyst. Say you have 1 point on the surface of a sphere: call it your North Pole. Now add a 2nd point anywhere on the sphere; wherever you place it, it will be one unit closer to the opposite pole. Now you can draw a geodesic from #1 through #2 to the opposite pole and back again to #1 – a Great Circle. Or you can add a 3rd point: now the line segment from #2 to #3 can point in any direction (360°) whatsoever.

And to our marketing guru, peddling a sophisticated product to an inelastic audience (e.g. software to hospitals in Massachusetts). Getting any one hospital to ‘go first’ is almost impossible and adding a 2nd is no easier. #3 may or may not be a bit smoother but after 3 it’s just a matter of knocking down bowling pins.

Anyone will do anything as long as 3 people have done it before. Remember when you were a wimpy kid? Your friends all wanted to jump off a ledge and they wanted you, Mikey, to go first. “No way!” So Paul went and didn’t get hurt; *now* it’s your turn. Still no? After Sam then. Sometime between Sam and Pedro, you’ll probably jump.

On the philosophical side of things, 20th century existentialists, Martin Buber (a theist) and Jean-Paul Sartre (an atheist) both developed ontologies that based on the primacy of duality (2) as opposed to *sinn fein* (1) or *en masse* (3+).

Our **third** example showcases the concept of ‘fuzziness’. One is one, period – a useful concept. Two and three are roughly the same and can be lumped together with little loss of information. More than three is a jumble.

This is a minimal expression of what could be, and has been, a very elaborate scheme; imagine, for example: 1, 2-3, 4-6, 7-10, etc. As quantity increases, specificity becomes less important. Does the pauper count his pennies more often than the prince? Maybe, maybe not, but according to this model, he does!

Think this is crazy? Think again. We do things exactly like it all the time! We estimate. We talk about ‘orders of magnitude’. And if you’re precocious, you understand logarithms. __Challenge__: can you infer a preferred logarithmic base by studying cultures that play with fuzzy dice (number systems)?

For our **Fourth **number system, we are indebted to pre-Socratic Greek philosopher, Anaxagoras (5th century BCE). He is best known for his formula, *Pan in Panta*, “Everything in Everything”. According to Anaxagoras, everything is an element of everything else. Everything contains and in contained by every other ‘thing’.

A number system appropriate to this insight would include just two terms, “One” and “All”. *One* refers to each entity while All refers to all the entities that are contained by that *One* entity and that in turn also contain that *One *entity. Anaxagoras gives us the Christian doctrine of Incarnation on steroids.

Crazy, right? Maybe not! 20th century British philosopher, Alfred North Whitehead, had a similar concept. His cosmos consisted solely of ‘actual entities’ (events) and each entity ‘prehends’ all other entities. Through prehension, every actual entity participates in each actual entity.

Whitehead’s entire cosmology rests on three ‘undefined terms’: One, Many, Creativity. One is one and Many is many (or all), while Creativity represents the family of ‘operations’ that link the two. Even more recently, contemporary European philosopher, Emanuele Coccia, has revived Anaxagoras’ concept of *everything in everything*.

**Fifth**, ‘grain, pile, heap’. Ok, I made this one up! But it fits, don’t you think? Here we preserve the notion of quantity but eschew entirely the concept of number. Intriguing. Could my 3 term system be expanded to include other measures like ‘horde’ or ‘stack’? Would the numberless ‘system’ of the Piraha (above) fit under *this* umbrella?

You might not be buying any of this, but I don’t care. I’ll bet I can sell it at any local elementary school, and kids, being desperate, will gladly absorb my sky-high mark-ups.

I’m kidding, of course; right? I don’t *really* want to repeal numeracy, do I? I suppose not. Even in our own hyper-numeric society, I have encountered people I’d describe as “quantitatively challenged” (I hope that’s politically correct) and, yes, they *are* challenged across a wide range of daily activities. I desperately wanted my own children and grandchildren to develop a strong ‘quantitative sense’…and they did!

Now I ask, “What did I do to them?” In the early years, I helped them get comfortable with *Real Numbers* and learn the properties and operations of arithmetic. So what’s so bad about that? As it turns out, plenty!

Which brings up to the **sixth** option listed above: *Our* system of numeration is built on the set of Real Numbers (R) plus the operations of Arithmetic (A), extended to include calculus. R + A gets you to Mars. But it’s a controversial model based on some controversial basic principles or axioms:

__Commutativity__: a + b = b + a; a × b = b × a.__Associativity__: (a + b) + c = a + (b + c); (a × b) × c = a × (b × c).__Identity__: a – a = 0; a + (-a) = 0.__Transitivity__: If a > b and b > c, then a > c.__Density__: Between any two distinct real numbers, there is always another real number. There are no "gaps" in the real number line; any interval can be divided into arbitrarily small sub-intervals.

What a wonderful world! If only we lived in it. Turns out, this world of R + A is a lot less like the Real World than Alice’s Wonderland or Dorothy’s Oz. An argument can be made that precisely **none** of the properties enumerated above is true IRL. For example,

Order of operations is

*always*critical: a then b is totally different from b then a. Dah!You are known by the company you keep. Ask any middle schooler: (a +b) + c feels very different from a + (b + c).

Every t is a little bit ‘naught t’: -a є a and therefore a + (-a) ≠ 0. (Dialectics)

Ever hear of a triangle? a > b > c > a. The real world is processional, not static. “I seem to be a verb.” (R. B. Fuller, c. 1970 CE) “What goes around comes around.” (Hesiod, c. 500 BCE)

Although there are an infinite number of Real Numbers, the smallest positive Real Number > 0. This is why the great Achilles lost a road race to a Tortoise. It turns out, there are an infinite number of ‘hyperreal’ numbers between the lowest positive Real Number and 0. These hyperreal numbers are collectively known as ‘infinitesimals’, represented by the symbol, ε.

Before our children have seen their first seashore, explored their first forest, stood atop their first mountain, we have made them learn a model of reality that is entirely bogus. How bogus is it? I would argue that it is the *most bogus* of the 5 systems we’ve been exploring. Still…what harm could possibly come from this?

Let’s see. We say that something that is infinitely improbable is effectively impossible, but that’s manifestly untrue. If something is, it is, no matter how improbable. Being is not a function of probability. It is not continuous; it is indivisible; it’s binary: it either is or it is not. ‘One’ is the state of everything that is; ‘zero’ is the state of everything that is not. Only Schoedinger’s cat can be 1 and 0 at the same time.

“One is the loneliest number.” (Three Dog Night) As every two year old knows, there is only one number, and that number is one. I have a cookie, or I don’t have a cookie. Two cookies are one cookie, twice (“one each hand”). There is as yet no concept of ‘multiplicity’, just ‘duplicity’ (as in “those duplicitous adults”).

Good thing too, because multiplicity itself is an illusion. Everything that is is once and only once. To be real is to be unique. When we say, “Two cookies,” we are abstracting. We are positing that the cookie in the right hand is similar enough to the cookie in the left hand to be covered by a single noun (‘cookie’).

What about two ball bearings? Don’t they *have to be* interchangeable? Yes, but ‘interchangeable’ is not ‘identical’. They look the same and, ideally at least, they perform the same. They are ‘the same’ – i.e. ‘close enough for government work’.

But they are *not* the same, they are not identical. If they were, they would be the same ball bearing. *They* would be *one*. But they are not the same, they are not one. We hide that fact by calling them ‘two ball bearings’, thereby eliding over the infinite (yes, I said “infinite”) chasm of difference that lies between the two.

Nevertheless, we will go on teaching our toddlers to count so that they can ‘live long and prosper’ in our galaxy; who can blame us? But somewhere along the way, it would be nice if let them in on the secret: it’s all just one big lie.

David Cowles is the founder and editor-in-chief of Aletheia Today Magazine. He lives with his family in Massachusetts where he studies and writes about philosophy, science, theology, and scripture. He can be reached at __dtc@gc3incorporated.com__