I have just said goodbye to a visitor from The Universe Next Door.
When I write "said goodbye", I write figuratively. I wasn't able to say "Goodbye Zork" or "Goodbye Wxpltharg", because my visitor has no name. Names are phonemes concatenated, and my visitor's universe doesn't have concatenation. It doesn't have concatenation because, when you concatenate sequences, you can work out the result of adding their lengths: just count the elements in the concatenation. But my visitor's universe doesn't have addition.
My visitor's universe doesn't have addition because its elementary physical processes don't implement addition. As you might say, they don't count. Ours do: put one rock next to another rock, and you get two rocks. Put one autumn-fallen apple onto the teacher's desk next to another autumn-fallen apple, and you get two autumn-fallen apples and two pats on the head. One pat on the head because you were courteous; and one because you showed the teacher you realise that juxtaposing apples implements addition. She replies by juxtaposing pats. Rocks and apples and pats on the head are "things"; and the essence of "things" is that they have boundaries, which shield. But my visitor's universe doesn't have boundaries. It has ramifications, which penetrate. We draw circles and potato shapes; my visitor would draw spidery daddy longlegs and jellyfish with radiating fractal tentacles.
I was going to write that my visitor's universe doesn't have mathematical proofs. Because proofs are sequences of transitions between axioms and propositions derived from axioms, and my visitor's universe doesn't have sequences. If it had sequences, it would have to have concatenation.
But, in fact, my visitor's universe does have proofs. Although they work differently, by a process akin to stacking holograms on a hot plate and melting them into one. Because my visitor's universe has proofs, we could share mathematical knowledge. True, we needed several superfast computers and a bucketful of hyperFourier transforms on the algebraic syntax domain, thereby implementing an adjunction which mediates between the category of discrete shieldings and the category of indiscrete penetrations. But with our translator set up, my visitor showed me the reef of texturings, explained viscid and rrhoeic as well as the operations of clathration and exspumation and isthiation, and proved the existence of a maximally rrhoeic texturing which penetrates every texturing. I showed my visitor the ring of integers, explained positive and negative as well as the operations of multiplication and division and exponentiation, and proved the existence of a minimally positive integer which multiplies unchanged every integer. We symbolised my visitor's maximally rrhoeic texturing by a splat , and my minimally positive integer by a stick . The splat and the stick have utterly different properties, and yet are equally central and indispensable within their own axiomatic systems.
I showed my visitor practice as well as theory. I placed apples on my desk and counted them: one, two, three, 4, 5. I placed one apple next to two apples and totalled them. And I placed no apples next to two apples and totalled those. My visitor found this confusing, being unable to decide which patch of desk I had put the no apples onto.
It took an exhausting day, in which the sun had set and the street lamps come on by the time we finished, but I then went on to a real-world example: adding the three Jammie Dodgers left in my biscuit tin to the fifteen in the packet I had freshly bought from Oxford's Summertown Co-op. My visitor was sceptical. The Jammie Dodgers in my tin were softer than the ones in the packet. Four in the packet had dimples of missing jam, and on two next to one another, the edges had crumbled away. Perhaps a beetle had nibbled its way in, and was now taking a contented siesta in the bin of Kit Kats behind the biscuit display, replete with shortbread and plum jam. "How do you know," my visitor asked, tentatively counting a lamp post, "that addition will work when the 'things', as you call them, are not all the same? And why doesn't it add the crumbs?"
Because I research into spreadsheets, I showed my visitor Excel. I explained money, bookkeeping, accountants, and debt. I demonstrated multiplication and division, and the varieties of addition. The plus. Excel's COUNT function. Excel's SUM function. Autosum, symbolised by Σ on the "Standard" toolbar. Adding values in a column by using an outline. Subtotals in pivot tables. The SUBTOTAL function. COUNTIF, which counts only cells that meet some condition. DCOUNT, which takes the condition from a range of cells rather than an argument. COUNTA, DCOUNTA, and COUNTBLANK, for when cells are blank. SUMIF. And even SUMX2PY2, which Excel Help tells me "returns the sum of the sum of squares of corresponding values in two arrays. The sum of the sum of squares is a common term in many statistical calculations." Not to mention SUMX2MY2, which "adds the difference of the square of corresponding values in two arrays". And SUMSQ. Then too, you have addition in other guises. Run a finger along the cell grid and watch the column addresses climb; run a finger down the cell grid and watch the row addresses grow. Such a simple physical process: mere movement, and yet it adds numbers to addresses.
But why doesn't Excel have DIVA, or PRODSQ, or MULTIF? Or subdividends in pivot tables? Or an Autoproduct, symbolised by Π on the "Standard" toolbar? It is remarkable, addition's pervasiveness and dominance. The Martian Poets never wrote poems about addition; but that's because Mars has the same physical laws as Earth. You'd need to be a Universe-Next-Door Poet.