✦ AUGHTY ✦ Numbers 28 AUGHTY Some notation systems show the operation. Others hide it inside a learned abstraction. I find both interesting. Let me show you why. WHERE IT SHOWS. tally: + = — the operation is concatenation. You see the answer. Roman: I + II = III — additive composition is the symbol. No math needed. objects: + = — you can see it. The answer is in the form. You do not need math. — for small numbers. WHERE THEY COLLAPSE. tally — past four, you must count, not see: how many? subitizing fails past ~4. you count. Roman — subtractive notation intrudes: IX + V = ? IX is not I + X. It is X − I. The substitution rule has intruded. The answer is no longer visible in the form. Arabic — designed for scale, paid in transparency: 247 + 358 = ? you compute. you do not see. The answer is 605. There is nothing in the form that shows that. Beyond a small scale, every system trades visible operation for one that scales. THE QUESTION. Is the collapse structural, or a contingency of human history? Could a system exist where the operation stays visible at scale? For static notation: probably not. Subitizing caps at ~4 items. Single-glance perception has limited channel capacity. To distinguish N values at a glance requires either N visual primitives (does not scale) or compositional structure (requires interpretation, which is computation). For dynamic representation: probably yes. Allow time, motion, manipulation. Two collections combine. Groups regroup, visibly. A system can be a process, not a glyph. Existence by construction. Most people learn Arabic numerals early. The abstraction becomes free. They stop noticing that there is an abstraction. The cost was paid in early childhood and forgotten. Perceptible arithmetic is a different cognitive object from computed arithmetic — even when the computation is free. Notation does not have to be a glyph. It can be a process. A six-year-old paid the bill for "7". She no longer remembers paying. The system that does not charge the bill has not been built yet.