3Blue1Brown is one of my favorite YouTube channels,
and in “the paradox of the derivative” Grant does a great job at presenting derivatives and the paradox.
Watch that video.
In this video, I’ll quickly go over Grant’s arguments related to the paradox
and I’ll present a different perspective on the issue suggesting that his resolution was incomplete.
But first: The paradox.
This is a plot describing a car’s journey.
It’s distance travelled as a function of time.
For brevity, I’m not going to mention units.
Ok, the derivative of the function at time t=0 is exactly zero.
So at time t=0, is the car moving?
And if not, when does it start moving?
Okay, so here’s my 1 minute summary of Grant’s argument.
Velocity is a measure of change,
change in distance over change in time.
It only has meaning across a span of time.
Velocity at a single moment makes no sense.
The derivative is what that the velocity function converges to as delta t approaches 0.
For this function, if we do the algebra, the derivative looks like this,
where the change in time, represented by dt, approaches 0.
If we assume that dt is non-zero, we can simplify this.
And if we ignore the vanishingly small terms,
we can simplify it further, giving us a derivative of 3t-squared.
The derivative is an approximation for velocity around a point.
These highlighted words are important,
because they mean that a derivative of 0 at t=0, does not imply the car is initially static.
And the question “when does it start moving?”
references the idea of change in a moment, which doesn’t exist.
So the question makes no sense.
From the usual perspective Grant’s argument makes perfect sense.
But is there a different way to look at the paradox where the questions are valid and have exact answers?
Consider this: this plot can be constructed using 2 approaches.
One approach is to construct the whole from the parts, assembling points to form a continuum.
Now, if we interrupt the construction,
we’re left with a finite number of points and a trivial continuum,
so to produce the original plot we must complete the construction, and place infinite points.
The other approach is to construct the parts from the whole,
cutting up a continuum.
With this approach, vertices, edges, and faces emerge as cuts are made.
If we interrupt this construction, we’re left with something non-trivial.
If we complete the construction, cutting up this continuum in every possible way,
we’re left with something indistinguishable from the complete point-based construction.
So let’s set this complete continuum-based construction aside.
The complete point-based construction is how we do math.
Points are fundamental.
So if you ask me “when does the car start moving?”
I’m going to interpret that as “at what point (or at what instant in time) does the car start moving?”
And as Grant explained, that question makes no sense since change doesn’t happen at a point.
But what about the incomplete construction?
Change doesn’t happen at a vertex, so the car is static at the vertex corresponding to t=0.
Change happens between vertices, along the edges, so the car starts moving at the first edge beyond 0.
And if that edge is too long for our liking, we can cut it, and cut it, and keep on cutting it such that it’s length approaches 0.
With this construction, that’s what a derivative is.
It’s an endless, incomplete process.
Now it’s worth repeating that the complete point-based construction is how we do math.
The incomplete construction simply offers a different perspective on the paradox.
We should not discard established ideas just because a different view might offer a more appealing resolution to a single paradox.
All I am saying here is that Grant’s resolution to the paradox needs a qualifier.
When does the car start moving?
The question makes no sense
when using a complete point-based construction of the car’s journey.
Can we even do math with incomplete constructions?
Or are there insurmountable problems with that approach?
Let’s talk about it.
Throw your thoughts in the comments and thanks for watching.
