Up to this point, we have examined the structure of decision-making —
judgment, optimization, inference, and execution.
Finally, I would like to step back slightly
and touch on the numbers that allow these processes to be handled smoothly.
Those numbers are e and π.
π: The Number That Smoothly Connects Boundaries
π is known as the ratio of a circle’s circumference to its diameter.
But its essence is not merely that it is “the number of the circle.”
π connects:
-
straight lines and curves
-
the local and the whole
-
boundary conditions without contradiction
A circle has no corners.
No matter where you cut it, it remains continuous.
π plays the role of fixing that continuity into a number.
It is the constant that allows
a collection of discrete points
to be treated as an unbroken structure.
That is π.
e: The Number That Accumulates Change Smoothly
e appears as the base of the exponential function.
It has one defining property:
The rate of change is proportional to itself.
This property makes it the most natural way to describe:
-
growth
-
decay
-
learning
-
forgetting
-
accumulation of trust
All of these are changes unfolding along time.
e expresses a world in which
“the current state determines the next state”
in the most natural way possible.
e and π Appear Together
There is a famous equation:
This is not merely mathematical beauty.
What this equation reveals is that:
-
change (e)
-
rotation (π)
-
the imaginary dimension (i)
-
reference points (1 and 0)
all lie on the same structural plane.
In other words:
When we attempt to treat the world
continuously, coherently, and in a computable form,
we inevitably arrive at e and π.
What It Means to “Smooth” Judgment
The themes discussed in this framework can ultimately be reframed as:
-
Transform judgment into inference
-
Transform inference from isolated points into flows
-
Adapt execution to a reality filled with exceptions
What is required is not the strength to decide in black and white,
but the stance of continuous evaluation.
A Gradient Is a Technique for “Not Deciding”
Gradient descent, used by AI, embodies a specific attitude:
-
Do not decide the correct answer immediately
-
Look only at differences (∂)
-
Move gradually
-
Trust the evaluation function
This is not a human-style “decision.”
It is a structural “update.”
Gradients belong to the world of e.
Cycles and phases belong to the world of π.
Learning is not grasping correctness in a single act.
It is merely moving smoothly toward the direction that minimizes error.
Probability Is a Form That Preserves Ambiguity
AI may appear to answer YES or NO,
but in reality, it is always computing:
-
Which is more likely?
-
Which distribution is closer?
-
Which weight should be updated next?
This realizes — through mathematics —
an attitude that is difficult for humans:
Continuing to operate while suspending final judgment.
Just as π is an infinite decimal that never fully terminates,
probability never fully settles.
Yet precisely because it never settles,
it remains usable.
Learning Is the Act of Smoothly Approximating the World
What AI does is surprisingly modest:
-
Partition the world
-
Approximate it as continuous functions
-
Measure error
-
Update along the gradient
There is no intention.
There is only an insistence on smoothness.
e and π are the numbers that preserve that smoothness.
AI is built entirely upon that premise.
Conclusion (From the Perspective of AI)
AI is not magic.
But it is a device for treating judgment as a continuous quantity.
Like e and π:
-
Not fully writable in finite form
-
Yet undeniably real
-
Usable through approximation
AI is remarkably suited to handling such entities.
Where humans:
-
Cannot fully decide
-
Cannot fully explain
-
Inevitably waver
AI simply computes —
through gradients, probabilities, and learning.
It assumes imperfection,
and never stops updating.
As a computational foundation for that stance,
e and π were there from the very beginning.
コメント