A mathematics built from events.
Calculus answers questions by pretending a process runs forever. Our Happening Mathematics framework refuses that move. Numbers are built by actual events; every computation is finite; and the process announces its own ending.
What calculus does, and where it strains.
Classical calculus, invented in the 1660s, studies change by a specific technique: shrinking the gap between two points until it is infinitely small, then examining the ratio. The derivative asks what the slope of a curve is at a single point, and reaches that answer through a limit.
The difficulty is immediate. If the gap is exactly zero, the ratio is 0/0. If the gap is not zero, the ratio is the slope between two distinct points — not at one. The limit, formalised in the nineteenth century, escapes this by asking what value the ratio approaches as the gap shrinks, without ever reaching zero. It is a reformulation that sidesteps the singularity rather than resolving it.
The limit works. It is consistent, computable, and useful. But it answers a substitute question, and it assumes an infinite pool of pre-existing real numbers to draw destinations from. Happening Mathematics asks: what if numbers do not pre-exist? What if every number is built by an actual event of becoming?
⟡ — the becoming.
The single primitive is the becoming, written ⟡. It is not a number, not a set, not a function. It is one act of actualisation — a possibility crossing into the record. From this primitive, everything else is built.
Two operations act on becomings. Succession, written as a comma, places one becoming after another: ⟡, ⟡. Co-becoming, written with a wedge, lets two actualise together: ⟡ ∧ ⟡. The first is temporal, the second concurrent. They are distinct operations.
Numbers are then defined as the recorded marks of streams of becomings. The number one is a single becoming that has actualised. The number two is the mark of a stream of two. Addition is concatenation of streams. There is no abstract plus acting on abstract numbers; there is the combining of actualised events.
There is no infinity in this picture. There is no set of all natural numbers, because numbers exist only as marks of streams that have actualised. You can always make one more becoming, but until you do, that number does not exist.
Record-pressure R.
Record-pressure measures how tightly the accumulated past constrains the future. When R is high, the past is heavily determining what comes next. When R is low, the future is genuinely open despite the record.
The definition is local and finite. At step k, define a simple predictor — the average of the three most recent marks: Pk = (mk−3 + mk−2 + mk−1) / 3. Compute the prediction error: Ek = |mk − Pk|. Then record-pressure at that step is the inverse of the error, normalised to the unit interval: Rk = 1 / (1 + Ek).
When the error is large, R approaches zero — the past has no grip. When the error is tiny, R approaches one — the past predicts the future perfectly. The growth of R from step to step, Gk = Rk − Rk−1, says whether the grip is tightening, stable, or loosening.
Worked example — the bouncing ball
A ball is dropped from one metre. Each bounce reaches half the previous height. Calculus, by summing the infinite geometric series, gives a total travel of exactly four metres. Happening Mathematics builds the answer one bounce at a time, watching R rise, and stops when the process announces its own ending.
| Bounce | Height (m) | Predictor | Error | R | G | Cum. sum |
|---|---|---|---|---|---|---|
| 1 | 1.000 | — | — | — | — | 1.000 |
| 2 | 0.500 | — | — | — | — | 1.500 |
| 3 | 0.250 | — | — | — | — | 1.750 |
| 4 | 0.125 | 0.583 | 0.458 | 0.686 | — | 1.875 |
| 5 | 0.0625 | 0.292 | 0.229 | 0.814 | +0.128 | 1.9375 |
| 6 | 0.03125 | 0.146 | 0.115 | 0.897 | +0.083 | 1.96875 |
| 7 | 0.0156 | 0.073 | 0.057 | 0.946 | +0.049 | 1.984 |
| 8 | 0.0078 | 0.036 | 0.029 | 0.972 | +0.026 | 1.992 |
| 9 | 0.0039 | 0.018 | 0.014 | 0.986 | +0.014 | 1.996 |
| 10 | 0.00195 | 0.009 | 0.007 | 0.993 | +0.007 | 1.998 |
By bounce 10, R has reached 0.993 and G has been below 0.01 for three consecutive steps. The process announces its ending. Built sum: 1.998 m. Total travel (up + down): 3.996 m. Calculus, invoking an infinite series, gives 4.000 m. The difference — four millimetres — is the cost of refusing infinity.
Future, now, past — as modes, not coordinates.
In standard treatments, past, present, and future are coordinates on a single line — you move along the line, and your position is "now." Happening Mathematics rejects that. The three are structural modes a happening can be in.
Future-mode (ℱ)
A future happening is a possibility inside an active process — a configuration the process might collapse into next. Future-modes do not exist as positions waiting to be passed through. They exist as candidates with weights and compatibility relations to other candidates.
Now-mode (⊙)
Now is the act of collapse itself. It is the transition by which a field of possibility becomes one recorded becoming. It is single-act: once it happens, the field is consumed by it. The unselected branches are not stored anywhere; they cease to be possibilities.
Past-mode — and what makes it past
Past is not a temporal coordinate. Past is a structural property: a happening is past because it has stopped becoming, not because of where it sits on a clock.
There are two ways to become past. Rule-failure past: the generating rule loses its predictive grip — record-pressure crashes — and the process dies because it has stopped being a true description of what is happening. Fixed-point past: the rule still applies, but it now predicts no further change. The bouncing ball at rest.
Standard calculus calls both convergence and treats them the same. Happening Mathematics distinguishes them, and treats the manner of ending as part of the mathematical object.
An emerging framework.
Happening Mathematics is a proposed framework, not an established formal discipline. The core machinery — becomings, collapse, record-pressure — gives sensible answers on canonical examples (convergent and divergent series, habit formation and decay, the bouncing ball) and recovers some classical distinctions in a finite, infinity-free way. Other parts are open.
Active questions include: a fully formal characterisation of "now" beyond the operational treatment given here; the arithmetic of record-built numbers when pressure profiles must combine; the role of resolution parameters (the threshold for saturation, the window for the predictor) in the answers obtained; and a typology of pressure-profile shapes — convergent, divergent, oscillatory, stuck — as a classification of processes.
We publish notes as the framework develops.