Semantics - Blog Posts

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Addition Universes AI was Not Privy to

Two trees beyond the walls are visible from my yard. One is a date tree. The other is also a date tree.— Eminent literary rebel LU Xun

Multiple large language models were unable to answer this type of question in the affirmative: Is there a universe where the addition of one to one does not always yield two? But we, mechanized parrots and squawksquabbling parrots, can, and our very own universe features prominently among the possibilities. The models, however, are fast catching up. Count along in this entirely human-authored article before they overtake us.

Parity Addition

At logic gates, we do not just total up two entities or phenomena but also make decisions based on their correspondence with each other. Parity addition, also known as XOR (i.e. eXclusive OR) addition, entails breaking up each added component into twos raised to the necessary powers and cancelling out the resulting sub-components that are present in both added components. For example, 4⨁7=(2²)⨁(2⁰+2¹+2²)=(2²)⨁(2⁰+2¹+2²)=3.

A neater way of performing parity addition is to express the added components as two rows of binary numbers, indicating the presence of a sub-components with the power of zero with a 1 at the right most and its absence with a 0 and adding the 1s or 0s for sub-components of higher powers sequentially from right to left. In other words, indicate with a 1 where there is exactly one 1 and with a 0 where there are two 1s or two 0s.

100 (i.e. 4)

⨁

111 (i.e. 7)

=011 (i.e. 3)

What is 1⨁1 then? 0. Earth mathematicians are more subversive than you might think.

OR Addition

OR addition works by converting your added components to binary numbers and checking for each place in the output if there is a 1 in either or both of the corresponding places in the input binary numbers. Indicate each affirmative with a 1 and the alternative with a 0.

100

⨁

111

=111

1⨁1=1. Think of it as simple contentedness.

Various other logic-gate additions (e.g. NAND addition and NOR addition) and combinations of logic gates are also possible.

Binary Addition

Without doubt, binary addition also starts by converting your added components to binary numbers. Like OR addition, you indicate a 1 where there is a 1 in either binary number and 0 where there is a 0 in both binary numbers. But unlike it, you carry over a 1 where there is a 1 in both binary numbers. If there are three 1s in a place because of the carryover, you carry forward yet another 1, because you can carry out commutative addition, and leave a 1 in that place. The result has the same value as that of regular addition, but its binary expression is certainly not 2!

100 (i.e. 4)

⨁

111 (i.e. 7)

=1011 (i.e. an 8, a 0, a 2 and a 1, amounting to 11)

1⨁1=10 in our wicked binary expression.

Unary Number Addition

The unary number system represents each number as that number of 1s.

One → 1, Two → 11, Three → 111, Four → 1111 and so on.

Accordingly, 1+1=11.

Modular Addition

In modular arithmetic, numbers go back to the starting point on reaching a defined limit and move forward from there. 4 mod 3, for example, results in the remainder of 4/3, i.e. 1. 5 mod 3 = the remainder of 5/3, i.e. 2.

We can define x ⨁ y as (x+y) mod z, where z is some chosen value.

Suppose we decide that x ⨁ y = (x+y) mod 1 or (x+y) mod 2,

1⨁1=2 mod 1 = 0

or

1⨁1=2 mod 1 = 0

Gather more friends, little ones.

Coder-defined Addition

Code rulez the universe. All the above are concepts from the giants of math academia. But behold this example completely invented and authored by Human (and similarly doable by you):

add = lambda x, y : 20*(x-y) - 23j*y

where j is the imaginary number i, available as a built-in datatype in some programming languages.

add(1,1) → -23j → -23√-1. Say hullo to the beast.

Sploop Addition

Short for SPacetime LOOP addition, this other invention by Human has been inspired by mischievous black holes. According to the concept of general relativity, mass warps the spacetime fabric. As ginormously massive objects that potentially rotate very fast, black holes can twist spacetime so much that it loops back onto itself.

Time addition in time loops will be similar—but not identical—to modular addition. If we enter a loop at the stage where it has run 7/8 of its cycle and start timing ourselves from there (i.e. time 0 at the 7/8th point, with seven negative time segments before it and one positive time segment after it), we will be at the end of the loop after one time segment and at the 1/8th point of the cycle after another time segment.

In sploop addition, we shall have an interval of fixed length p from the origin 0 and a fixed starting point s somewhere in the interval. The added numbers x and y are our only variables.

x⨁y given [s]ploo[p] = [(s+x+y) mod p] - s

1⨁1 given 7ploo8 = [(7+1+1) mod 8] - 7 = 1-7 = -6

Patchwork Addition

Perform what we are free to call patchwork addition by defining the inputs and output of the process on ends that are not fully connected to each other, such that the inputs do not both contribute to the output. Three styles of doing so are: 1) adding imaginary or speculated entities together or to real entities and checking the real outcomes, 2) adding real entities together and fancying imaginary outcomes and 3) aggregating entities and checking the outcomes within each entity. Not all makes sense, yet the sensibility or insensibility, popularity or unpopularity, of a mode of accounting does not affect its possibility among us humans, all vulnerable and flawed. For some fun applications, consider the prospects of socializing an empty universe or an indivisible monoparticle universe or thinking about universes we have speculative, imperfect or no knowledge of.

At its roots, regular addition comprises either or both of accumulation and contemplation, because it must take place at least in a physical, mental or temporal dimension. In an empty universe and a universe made up of an indivisible particle and nothing else, not even space, no matter or matter-space accumulation or metabolic process, which is really a series of matter interaction, can take place, making such addition on the physical dimension or in-world mental dimension impossible. The remaining mode of regular addition is that on a temporal dimension, wherein we add one second or some other time unit to another of the same kind. In the monoparticle universe, that may be a counter against which we trace any movement the particle may make across some bulk universe enveloping the universe. We have addressed the complications of time addition above.

But one way to execute non-temporal regular addition on social phobic universes like these all the same in the meantime is to superimpose imagined entities onto these universes and perform the addition from the comfort of a thoroughly addition-safe universe. Another is to consider a metaphysical mega universe, or perhaps that physical bulk universe, in which we count them up. However, our inputs would not yield any output within any of the universes. Each universe might as well ask, "What's in it for me?"

Although this scenario may still look like a frivolous exercise to some of us, it has similar real-world counterparts, as when stellar gross domestic products do not equate individual economic survival. When the two ends of addition can lie in different places, sometimes justifiably (for the sake of justice comparison or productivity measurement, for example), 1+1=1 is a possibility.

When patchwork addition is applied to a hypothetical or imperfectly known universe, the equation can run in two directions. A universe or its added components may not exist (an empty set, in mathematical terms), or its ongoings may be beyond human imagination. In the latter scenario, an unanticipated 1+1 on its end may amount to 1 or, if the added components are not even known to exist, 0 in limited Earthling minds. That universe might be this one, right here, we are living in, with exoplanets potentially exceeding expectations and fellow planetary citizens accumulating woes unimaginable to many as they suffer cruelty beyond common knowledge and experience.

Metaphors

"One plus one" need not strictly be a mathematical expression. It can also be a description of other truths and feelings.

Take, for example, catastrophizing. Physical matter and energy are subject to laws of conservation, but what about mental entities? A black cat colliding with your dog may give rise to two masses, some hisses and some barks. However, a catastrophizing person would probably embark on this line of reasoning:

"Doggy is getting germs from a filthy stray."

"He is going to need a trip to the vet."

"Who's going to take him? I'll be losing my job if I keep asking for leave."

"If I don't have a job, how do I keep the home intact?"

"We'll all be living on the streets."

A cat-dog collision causing someone homelessness? What's the likelihood?

But in the person's mind, 1+1≫2. Many of us must have fallen into this trap multiple times in our lives.

Semantically Alt-Mapped Universe

This answer is the most straightforward but meets its dissent in those who, bearing with some of the applications of patchwork addition only as story problems, believe we should be talking about actual, known universes. Yet the fact that the question has been raised despite the questioner most probably knowing that we are living in the only actual universe we know invites an interrogation of our epistemology. And if the reliability of ground truths is limited and unstable, that opens the doors to speculative reasoning as well, which is what physicists proposing a many-worlds interpretation of quantum mechanics engage in.

Beside, a conversation agent can take the initiative to suggest an exploration of the multi-definitions of "universe." Are story universes not universes? Are our relationship networks and our minds, with their rich complexity and vastness, not universes?

So it is that there may be a universe which happens to call some operation or function other than addition "addition" and signal it with our plus sign. 1+1=1e¹=2.718281828459…

Neurologically Haywire Universe

The cause could be a viral inflection infecting the brains of a small, nascent community of otherwise arithmetically competent species, the only of their kind in their relatively young or harsh universe. The result in this particularly compelling scenario, among other imaginable scenarios, could be 1+1=Wildcard, a kind of patchwork addition if you would like, or obsessively adding some extra, constant number to each addition, resulting in 1+1=Constant > 2, where the ends of addition are not in a mutually disregarding patchwork but do not produce normal results.

Post-Intelligence Universe

In a post-intelligence universe, mathematically literate inhabitants may have become so jaded with networks, practices and outcomes of knowledge production, propagation and application that they give up on critical discourses altogether and outwardly turn their backs on long-held academic tenets. 1+1= A billion for all their unpaid care.

Annihilative Universe

Our species have come to define addition, in our various languages, as a numerical increase probably because it is more of an observable norm for a union of entities to result in an increase of entities. In an annihilative universe which inhabitants, perhaps thriving on passing microbes automatically absorbed by their floating bodies and living far apart from other visible entities out of necessity, see mostly entities vaporizing into gas or into minuscule spores on coming within a visible range may define the union as a numerical decrease instead. We are such familiar buddies, 1+1=0.

TO SUM IT ALL UP

Addition results are borne from the will of the mind. Knowledge of this may delight money launderers and embezzlers and other devious manipulators of numbers but also opens up vast styles of thinking that can help us get a better grip of our physical and mental realities as well as explore less intuitive solutions to the many seemingly intractable problems plaguing our planet and societies right now.