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Bijectivism

Imagine you throw a ball into the air. Physics tells you exactly where it will land. Now imagine you see a ball on the ground. Can you tell exactly where it was thrown from?

In the first case (throw \(→\) landing), you have one answer. In the second case (landing \(→\) throw), you might have several possible answers — maybe it was thrown from here, or there, or over there.

Traditional mathematics has a strange rule: A process is called reversible only if every outcome has exactly one possible cause.

But why? If you see a ball on the ground, the truth is: it could have come from several places. That set of possible origins is the reverse of the throw. It’s not “no reverse” — it’s a set‑valued reverse.

Bijectivism says: Every process has a reverse — just sometimes the reverse is a set of possibilities, not a single point.

The Problem with “Invertible”

In math, a function is called invertible only if it’s bijective — one input gives one output, and different inputs always give different outputs.

That leaves out most functions in the real world:

Squaring:

\(2^2 = 4\) and \((-2)^2 = 4\) .

Different inputs, same output.

So math says:

“Squaring has no inverse over all numbers.”

But clearly, the reverse of 4 is both 2 and -2 .

Measuring temperature:

Many micro‑states of molecules give the same thermometer reading. The reverse of “37°C” is a huge set of possible molecular arrangements.

Becoming a parent:

Two people together make one child. The reverse question — “Who are the parents?” — naturally expects two people, not one.

Why do we say “no inverse” when what we mean is “the inverse is a set”?

Bijectivism in One Slide

Let \(f\) be any function. Define its Bijectivistic inverse \(f^{\dagger}\) as:

\[ \boxed{f^{\dagger}(y) = \{\text{all } x \text{ such that } f(x) = y\}} \]

That’s it.

It always exists. If \(y\) never comes out of \(f\), the set is empty. If only one \(x\) gives \(y\), the set has one element. If many \(x\)’s give \(y\), the set has many elements.

Now every function is reversible — because the reverse is precisely that set.

What Changes? Almost Nothing — and Everything

You can keep using ordinary inverses like \(\sqrt{y}\) or \(\arcsin(y)\) . Those are just choices from the full set \(f^{\dagger}(y)\) :

• \(\sqrt{4}\) chooses \(2\) from \(\{-2, 2\}\) .

• \(\arcsin(0)\) chooses \(0\) from \(\{0, \pm\pi, \pm 2\pi, \dots\}\) .

Bijectivism doesn’t break math — it reveals the hidden step: We first compute all possible reverses, then we pick one according to some rule (like “take the positive root”).

That picking step is useful, but it’s secondary. The primary truth is the full set of possibilities.

A New Way to See Old Friends

Injective (one‑to‑one) functions These already have single‑valued inverses — their Bijectivistic inverse sets are just singletons. So injections are already “good” in the old sense — they’re just special cases of Bijectivism.

Surjective (onto) functions Every output has at least one input. The Bijectivistic inverse sets are never empty. If we accept set‑valued answers, surjections become reversible too.

Non‑injective, non‑surjective functions These have some empty sets and some multi‑element sets. Bijectivism handles them without any patches.

Suddenly, the landscape is unified: Every function has a well‑defined reverse map. The reverse is a set. Choosing a single element is an extra, optional step.

Why This Matters

It mirrors how we think naturally

When you hear a sound, you might think of several possible sources. You don’t say “sound has no inverse” — you consider all possibilities.

It respects information

The forward process \(f\) may lose information (two inputs give same output). But the set \(f^{\dagger}(y)\) keeps all inputs that could have led there. Nothing is truly lost — it’s just grouped together.

It simplifies teaching

No more “you can’t invert this unless you restrict the domain.” Instead: “Here’s the full reverse set. Now if you want a single answer, here’s how to pick.”

It connects to physics

Many physical laws are reversible at the microscopic level (like Newton’s laws), but appear irreversible when we ignore details. The Bijectivistic inverse keeps all details — it’s the microscopic reverse.

Try It Yourself

Think of any process:

• Making a shadow (many objects can cast the same shadow).

• Mixing paint (many color combinations can give the same final color).

• Writing a summary (many detailed stories can yield the same summary).

The forward direction is clear. The reverse direction isn’t “undefined” — it’s the set of all possible origins.

That’s Bijectivism.

The Quiet Revolution

Bijectivism doesn’t ask you to change calculations. It asks you to change what you call the inverse.

Instead of saying “this function has no inverse,” say: “Its inverse is a set. If you want a single value, pick one — and tell me your picking rule.”

That small shift —

• from inverse as a single‑valued function

• to inverse as a set, with single‑valued choices as special cases — unifies injections, surjections, and everything in between.

It makes mathematics a little more honest, a little more complete, and a lot more reversible.

In one sentence:

Bijectivism is the view that every function has a well‑defined inverse — you just have to be willing to accept sets as answers.

Next step:

If we want to make this set‑valued inverse as natural as single‑valued ones, we might need numbers that carry their possible origins with them. That’s a story for another day.