Is It Time for Really New ‘New Math’?

Remember the ‘old math’ - ‘rithmetic, taught with a (hopefully metaphorical) hickory stick?  Meet the ‘new math’ (scourge of Gen Xers) – same as the old math! (The Who) Boomers can  at least make change, albeit with ‘fear and trembling’, while Xers just shrug and say  whatever! 

In this article, I hope to demonstrate that the purported differences between new and old math are superficial at best, imaginary at worst, and that neither has any applicability in the  Real World.  

Today’s math, new and old, is based on premises that I contest. Let’s start with something  simple: 1 + 1 = 2…not! 1 + 1 can be more than 2 or less than 2 but it can never equal 2. In  fact, 2 is the only value for 1 + 1 that we can exclude ab initio. As in Carroll’s Looking-glass  World, there’s jam yesterday (< 2) and jam tomorrow (> 2) but never jam today (= 2). 

Of course, according to the system known as old/new math, 1 + 1 = 2 just because we say  so in our best imitation of a 1950’s parent. And we’re perfectly entitled to do just that! And  we have built a magnificent intellectual edifice based on it! And that’s A-Okay, as long as we don’t imagine we’ve created anything relevant to anything real. Math (old or new) is  Minecraft! 

So what’s wrong with 1 + 1 = 2? How could any sane person even question it? (Please don’t  answer that, at least not until you’ve read what’s coming.) 

First, ‘one’ is not a number; it’s a way of referring to an atomic entity, i.e. a quantum of the  Real World. It’s roughly equivalent to ‘that’ or ‘this’. There’s nothing in the Real World that  corresponds to the number one in math. How could there be? There’s no such thing as a  being in isolation, not even (trinitarian) God.  

Think about quarks in a sub-atomic particle. They cannot exist in isolation. “Three quarks  for muster mark.” (James Joyce) And even if an isolated unit of being were possible in some  real world, which it is not, ‘1’ would still not properly belong in any number system with  aspirations to apply to real events.  

Case in point: an Amazonian tribe, the Piranha, have no math…and they do not  have the concept of one. Instead they use demonstrative pronouns and adjectives (like this or that) to designate real entities.

The concept of ‘number’ only applies once two or more atomic entities are considered as a  group. We can measure such ‘groups’ by their size, i.e. by the number of elements in each  group. So it’s possible to imagine, again ab initio, that a group might have a value of ‘2’ if it  groups two atomic entities together. R. Buckminster Fuller was fond of saying, “Universe is  plural and at minimum two;” but as we shall see, even ‘2’ is not a stable state. 

To make our so-called ‘number system’ work, we postulate that a group can have a ‘value’  (size) of 1, or 2, or 0, or even ø, but there are no such groupings of elements in the Real  World.  

Math, old and new, is based on the premise that atomic entities, even if they are totally  unrelated, can nonetheless be grouped together without impacting in any way the  contribution of each element to the ‘size’ or ‘measure’ of that group. 

Of course, this is impossible in any real world. Any real process of ‘grouping’ must impact  the value of the group’s elements. If it doesn’t, there’s no process and there’s no group.  

Why? The real world Law of Recursion: Every ‘process’ (by definition) must alter the values of its elements. The definition of a ‘process’ is an act that modifies the values of its constituents. Of course, here is where professional mathematicians will part company with  us, and that’s ok, so long as they don’t expect us to believe that their ‘ice cream castles in  the air’ have anything to do with the Real World. 

If ‘grouping’ is real, then it must be a ‘process’ applicable to real elements and it must then  alter the values of those elements. Therefore, 1 + 1 could never equal 2. Here we are  borrowing a concept from Gregory Bateson and extending it: A ‘difference’ is a difference only if it makes a difference! 

The process of grouping two elements must impact those elements in some way;  otherwise ‘the process is inert’, meaning that the ‘alleged process’ is not an ‘actual process’  at all. I can fantasize about a grouping of 2 or more entities where the process of grouping  has no impact on the elements grouped…but that’s exactly what it is, a fantasy.  

Once grouped, the elements co-modify one another. So the process of recursion is self perpetuating and interminable. Fortunately, however, the process tends toward a limit. But  what limit? 

When two atomic elements are grouped, they interact until they result in a stable, concrete  value, a ‘steady state’. That interaction can take either of two forms: Fission or Fusion. (No,  this is not a category in Jeopardy.) In the case of fission, the grouping of two atomic units  automatically generates a third: 1, 1, and (1, 1). (In Trinitarian theology, the Holy Spirit  ‘proceeds from the Father and the Son’; likewise, (1, 1) proceeds from 1 and 1.) 

So the minimum quantitative value of a group assembled via the procession of fission is 3. ‘3’ is a fundamental unit of quantity – ‘1’ and ‘2’ are not. 

But what of groups created by the process of fusion? If Trinity is a model for ‘grouping by  fission’, Quantum Entanglement (John Bell) is the model for‘grouping by fusion’. When the solution of 1 + 1 < 2, 1 and 1 are grouped by fusion. 

According to Bell’s Theorem of Non-Locality, two independent entities, ‘entangled’ by  fusion, have the quantitative value we know as √2. The process of fusion eliminates ‘redundancies’ inherent in the concept of two. The resulting ‘simplest possible quantity’ is  neither 1 nor 2 but √2. 

We can think of such a ‘complex unit’ as akin to ‘heavy hydrogen’, deuterium. While  ‘normal’ hydrogen consists of 1 lepton (electron) and 1 hadron (proton) and ‘normal’  helium consists of 2 leptons (electrons) and 2 hadrons (protons), heavy hydrogen consists  of 1 lepton and 2 hadrons (proton + neutron). 

To summarize, ‘one’ does not exist in any real world and ‘two’ is inherently unstable. Any  two entities must either collapse into a single, complex entity with a quantitative value of  √2 or they must generate a third, resulting in a single entity with a quantitative value of 3. 

In biology, the 30 Trillion cells that make up your body are all descended from a single cell  that merged with another cell and formed a symbiotic relationship (cell + nucleus). The  resulting organism is clearly ‘more’ than a single prokaryotic cell but ‘less’ than two  completely independent cells. Similarly, in math one entity and another entity merge  creating a single entity with a ‘heightened value’.  

As noted above, in the Real World, all process, including the process of grouping, is  recursive. Therefore, there are no linear equations. Mathematics begins with equations that  have at least two solutions, e.g. √2x or 3x. 

Note that we seem to be relying still on ordinary ‘natural’ numbers like 1, 2, and 3 to tell our  story. But these, obviously, as just placeholders. To avoid massive confusion, we’ll need  two new symbols to represent our two quantitative minima. I suggest Δ for the fundamental unit of fission and ꓦ for the fundamental unit of fusion. We also need symbols to represent  the two basic arithmetic processes, fission and fusion, replacing addition and multiplication.  

Instead of + and * from linear arithmetic, I propose ↗ for fission and ↘ for fusion. And for the inverse operations, currently represented by – and /, I propose ↙ and ↖. Armed with these basic symbols we can now build a new algebra. 

Note that the familiar identities of linear arithmetic 1 + 0 = 1 and 1 1 = 1 disappear since +  and are processes and no process in the Real World leaves its elements unaltered; the  result of a process can never be the same as any of its elements (except perhaps in a rare  case of accidental coincidence). 

Counting works quite differently in the new new math. There are two different counting systems depending on whether we’re in fusion or fission mode. In fusion mode, instead of n = 1, 2, 3, 4… we count like this: n = 2^((n-1)/n) < 2.  

I’m looking forward to playing a game of hide and seek with my great grandchildren (my  grandchildren have aged out) and asking the cherubs to count using new new math in  fusion mode. Where a muggle-child might begin “1, 2, 3, 4…”, my great grandchildren would  say “2^(0), 2^(1/2), 2^(2/3), 2^(3/4)…” In muggle-speak that corresponds (approximately) to  “1.0, 1.4, 1.7, 1.8…”  

Bonus: I’ll never have to worry about being found because in the entire history of the  cosmos, they’ll never even get to 2. I think a little frustration can be character building,  don’t you? Plus, I can head into the house and make up a pitcher of margaritas, no salt…for  me, of course, not for them. (They’re still counting, remember?) 

But take heart, children. Counting in the fission mode is a bit easier…at first; but it gets progressively harder as we climb the number ladder (rather than slither along the Real  Number line). Instead of n = 1, 2, 3, 4, we count “Δ^1, Δ^2, Δ^3, Δ^4…” Again, in muggle speak, we’re talking 0, 0, 3 … 9 … 27 … Wait, there’s already a name for these numbers;  they’re called p-adic.  

Obviously, I am limited by space and by ability to this simple outline of new, new math  principles. You, dear reader, are better qualified to expand this sketch into a full algebra and to explore the nooks and crannies of this system and I, for one, look forward to the  results.