We sat back to back and face to face, each of us hunched over a desk or a formica-topped table.
Outside, the rain pelted down in sheets and waves, the wind seemed like it would break in at any moment. No work for us until the sunshine returned.
Lotte had handed round her puzzle magazines and books. Euan and I hadn't yet finished the Sunday papers.
We'd been at it for a while and silence prevailed.
I had reached the stage where my brain hurt, well, sort of, if you know what I mean. I stared at the page and nothing was happing upstairs, in here.
"Look," said Acey, "I've got eleven fingers." He held up his hand and began to count. "Ten, nine, eight, seven, six," he changed hands, "one, two, three, four, five. Five plus six makes eleven. See? I've got eleven fingers." Acey gave us his famous 'mischief' grin.
Brenda and Euan gave me a funny look, I had let out a sigh like a gasp of relief, I was glad of the excuse to take break.
"That reminds me," said Arri, "mathematicians and scientists are wrong."
I laughed, shook my head and turned back to my newspaper.
"They've made a fundamental error" Arri continued, "in the way they handle calculations."
"If that were true," said Acey, "the results of their calculations would be conspicuously wrong."
"Correction," said Arri, " I should have said, 'in the way we count'. Our numbers are wrong."
"If that were the case," Acey said, "everyone would see that there is a fatal flaw."
"Not necessarily," Arri stood up and stretched his shoulders and neck, he walked round to face us. "The numbers we use are mere tokens, markers if you like. What we call the digits hardly matters. As long as our tokens, markers, whatever, match up, everything fits together and works."
"Coherence," said Lotte.
The Bottom Line
"As long as the digits add up," Brenda grinned. "It's called, 'The Bottom Line'." Brenda beamed.
Eta entered with a tray of drinks and biscuits, not a moment too soon.
We were holed up in a rambling mansionhouse several miles outside Ardnamurchan on Scotland's West Coast. Across the water, when the weather was fine, we could see the Isle of Mull.
Eta frowned and looked at each of us in turn as she carefully put down her tray.
"Mathematics," Lotte said.
"More like primary-one 'Learn To Count,' class" said Tosh
Eta pulled a face, "hmm", she said.
Cup after cup whizzed under my nose and missed my grasping hand as Eta dutifully allocated the drinks.
"You're weird," Igvarts, slide-rule in hand, tried to return to his drawing but Arri exuded such confidence he put down his slide-rule, scrunched his ginger snap and listened.
"Reality is weird," Arri countered.
I clasped both hands around my hot cup as I supped.
"First of all," Arri continued, "we write our numbers backwards." Arri stopped to see our reactions. Igvarts frowned as only Igvarts can.
Arri suppressed a smile.
"We say, 'Four-Teen' yet we write, 'One-Four'. Arabs write from right to left."
A draught of cold air told us someone had just come in from outside. It was Saleh el Moharbi, rarely did he grace us with his presence.
"We adopted their numerical system," Arri said, he looked at Saleh as he quietly closed door. "We adopted Arabic numbers without reversing the written order, as we should have, according to our left-to-right protocol."
"Protocol," Acey mimicked, "ge’in’ pre’y technical here." [getting pretty technical here]
"That's nonsense." Tosh waved his hand in dismissal.
"Now now, children, let's not get personal about this," said Brenda, "professor! Explain thyself."
Saleh el Moharbi quietly made himself comfortable in the darkest corner, his glasses glinted.
Arri held his ground, "logic says we ought to have the units on the left building up to thousands and millions on the right."
"He has a point," said Euan.
"Well so what, that's no big deal," Tosh said.
Arri maintained his professorial stance.
"Second mistake," he continued, that mischievous look in his eye once more, "our numerical sequences are a nonsense." Arri looked pleased with himself.
"A, 'Nonsense'? Who the devil are you to make a pronouncement like that?" Tosh scowled at Arri.
"You're the 'Number One Nutter', ha ha ha." Acey could be a bit cruel at times.
"Patience, please, we've got two hours till dinner time," Eta used her school teacher tone, it was highly effective.
"Seriously, children," Acey put on a falsetto voice, "let the loony splutter!"
There was a trickle of laughter, he was funny but Arri was not.
"You must excuse our scepticism," Lotte said, "but you have to admit, it's a bit of a tall order to expect us to accept your proposal without a hint of question or challenge."
Several of us nodded in agreement.
Arri smiled, unpeturbed.
Tosh rose and, with a flourish of his hand, said, "prey, continue, my dear friend."
"One," said Arri, finger raised. Acey interrupted him, he raised his middle finger, again we laughed in spite of ourselves at his rude variation.
Arri, like a pro, stuck to his guns; "is the first in the sequence, not 'Zero'."
Arri held up four fingers and tapped the tips as if counting.
"Our number 'Ten'" he continued, "is a misnomer."
Arri perched his forefinger on his raised pinkie.
"Ten is not the last in the sequence," he moved his forefinger to the empty air above a non-existent finger, " it is the first of the next sequence."
"Meaning what?" said Tosh.
"Meaning," said Arri, wide eyed, "we - do - not calculate to 'Base-Ten' but to 'Base-Nine." he paused. "That is a big error."
There was a moment's silence.
"Rubbish," Igvarts, the nearest to an engineer among us, twanged his slide-rule against the desk. "Everybody knows we work to base-ten."
Arri shook his head, confidence intact.
"Base nine," said Arri, "we are missing a digit."
Arri looked Igvarts in the eye.
"Ten is Zero-One," Arri continued, "or, 'One-ty', the start of the Teens." He waved his finger in the empty air beside his raised fingers.
"And it's wrong?" said Eta.
I think I heard a sound from Saleh, I could barely see more than his specs.
"We don't count to base nine or to base ten," Arri said, "we're out of sync. One to nine, then, ten to nineteen. Whoever heard anyone start counting with, 'zero, one, two'? It can't happen because our first digit, nothing, is exactly what it says, nothing. 'Not there', 'no thing'. Geddit?"
Brenda smiled, "the first sheep doesn't exist."
Eta rose, she clasped her hands like a school teacher about to address her children.
No Business Like Show Business
Tosh roared with laughter, "just like this business we're in," he said, a little too loudly. "One project, ten million quid, multiply by nothing, as in, no bums on seats and 'Whoosh!' it's all gone."
Arri waited patiently as we laughed.
I thought Arri would tear down his poster and give up in a huff but it seems he had anticipated resistance. I guessed he might be repeating stuff he had heard from his uncle.
"The number 'Ten'" said Arri, forefinger raised, "should be a single digit."
Igvarts practically exploded.
"You," he shouted, "are deranged!"
Euan slapped his thigh, shot to his feet and pointed skyward. "Base thirty-six!" he exclaimed, "imagine how science would differ if we learned to count to Base 36." With that, he sat down.
For the first time, Arri began to look like the wind was being taken out his sails.
I wasn't at all sure I agreed but I did feel sorry for him and I admired his tenacity.
Arri kept his nerve, it seemed he had anticipated the response he would get.
Fly Stops Train
"Physicists tell us," said Arri, "that a fly can stop a train. When a fly flaps along and hits an oncoming train, there is a moment when the fly and the train are both in the same place. Before the fly can travel at the same speed and in the same direction as the train it has to be momentarily motionless. If the fly is motionless and it is attached to the front of the train, then the train, too, must be stationary. Hence, fly stops train."
Acey put on a stage-yawn and waved his hand in front of his mouth to let us all know he knew that one.
"In reality," Arri continued, "the fly is stationary for zero time, therefore, the train, too, is stationary for zero time, that is, never."
Eta made a most curious sound, I think she may have agreed with him.
"Zero, or 'Nothing'," said Arri, "is therefore exactly what we say it is, nothing."
Tosh nodded gravely.
"One to ten," Arri's fingers flew up and down, "where 'Ten' is a single digit numeral. Next, eleven to 'Oneten' or Tenteen, then, twenty-one to twenty-ten, and so on. The revised 'Tokens' we use to represent objects - "
"Abacus beads," said Lotte.
"Thank you, exactly - the abacus beads remain the same," said Arri.
"That's amazing," I said.
"What a piece of pish," Acey laughed. "Ye're aff yer dafty wee heid, yoos toos!" Acey pointed at me and Lotte.
There was laughter mixed with consternation.
Arri spread out his hands, "you can't be expected to grasp the consequences on first introduction." He knitted his fingers together. "Zero does not disappear. Zero still exists and it is still relevant."
"Just as infinity is relevant," Saleh nodded and spoke softly.
"Mathematicians use infinity in their calculations," Euan's hand shot heavenward, "they know how infinity works and how it fits in."
"Exactly," said Arri. "Recognising that zero is not a number but a 'gap' does not invalidate it."
"An absent counter, a non-existent abacus bead," said Brenda.
Tosh stroked his chin, "it's bound to have an impact."
Arri smiled, "computing," he said, "hard to imagine our computational geniuses could overlook such an obvious and conspicuous misnomer."
Igvarts decided it was time to be disruptive. He twanged his slide-rule, scraped his chair and clattered his empty mug back onto the tray. The expression on his face would have inspired even Rembrandt.
"It's not binary but unary," Arri explained.
No Such Thing As 'Binary'
1 1 1
One or nothing, blank, not there, absence.
but it can still be written:
exactly as at present, all that has changed is the understanding & interpretation.
Igvarts slapped his slide-rule against his table and stood up. He pointed at Arri.
"Bonkers," he said. "First you say 'Zero does not exist,' then you tell us nothing, ha ha, has changed."
"Zero exists," said Arri, "but you show me where it sits on the Abacus."
"Bit hard sliding an absence," I observed wryly.
Igvarts, still standing, glared at me, glared at Arri, looked at the others, huffed and puffed. He seized his slide-rule, snapped it across his knee and threw the pieces at Arri. He dropped into his chair with such a force I thought it might break. He folded his arms and stretched out his feet, his mouth pulled down in a grimace as far as it could go.
"Oh ma Gawd," Acey groaned, "ah'm a-gonna haf ta re-lurn all mah shkool-taeybles." He slapped his hand across his forehead as he stifled a smile.
0 = ∞
"Zero and Infinity," said Euan, "must therefore be virtually the same."
"In a way, yes but then again, no, or maybe not." Arri prised his paper away from the Blu-Tac.
"Looks like everything is rolling up into a virtual reality lost somewhere between those three rings," said Tosh, bemused.
Lotte held up her notepad, she had drawn a variety of symbols.
"Ten?" she asked, "any good?"
That was it, we were now on the hunt for the perfect tenth symbol.
Nothing had changed. I smiled to myself as the thought crossed my mind.
- o0 ∞ 0o -