In mathematics, there is a basic idea: If we have a smooth curve, we can zoom in, and keep zooming in. As we do this, the curve no longer looks like a curve, it begins to look like a line!
Now imagine we could keep zooming in infinitely to a point on the curve. In calculus the idea is that if we zoom in forever, we do get a straight line at that point, touching (tangent) to the curve. This is called the derivative.
The mathematics of calculus and its derivative have been incredibly useful tools in the modern world. Let's list a very few of its applications: It is used to build structures, to analyze them, to design airplanes. It plays a fundamental role in training a great many machine learning algorithms. It is used to model chemical reactions, turbulence in the atmosphere, the movement of the planets, the flow of underground water in geology. In physics it is used to study motion, electricity, heat, light, harmonics, acoustics, astronomy, and quantum mechanics.
In short, this derivative is incredibly useful to describe change over space and time. Many physical shapes, speeds, velocities, and rates of change of any kind, can all be understood and modeled by the derivative.
The list of its uses goes on and on... and on. Without the tools of calculus, a great deal of the modern world would be challenging, perhaps impossible, to study and develop. Possibly we would still be living with the technology of the 1700s or 1800s.
Perhaps we can begin to see the irony...
We often immediately dismiss people who believe the earth is flat, thinking ourselves better, because we are 'people of science'. But a great deal of our entire world, and more importantly, the underlying assumption that we can analyze phenomena scientifically, is based on the derivative, the idea that curves are locally flat, that we can take the derivative.
That is, as scientists, we are all, to some extent, flat-earthers (!!).
Indeed, pick anything you call 'curved'. Now imagine you are extremely, extremely small, standing on that curve. To you, the world is actually flat!
Let's try zooming in to a curve and see what happens.
Starting out, our curve is so large and our person so tiny, we barely see them.
Let's zoom in.
Perhaps you notice how the curve seems flatter?
Let's zoom in some more...
Wait, what happened? The curve disappeared!
If we look carefully, we can just make out a curve in the landscape, but it is hard to see.
Another issue, as always, is us. We humans seem to have challenges looking at things globally. We are intensely aware of our immediate surroundings, both in space and time. We know our room very well, we think a lot about what we are doing right now or about to do.
We are not as good at thinking about things that are far away. We do not often zoom out and think from a global perspective. We do not think about what we'll do 5 years from now, while we scroll on our phone and answer texts from second to second. We do not think about the state of the world while we eat our pizza. Our world is greatly dominated by local events, local information.
Another way to say this is that our thoughts are dominated by our local scale. Look back up at the figure on the zoomed-in curve. The horizon is just a few body lengths away, it is on a similar scale as the figure. That is the scale he or she navigates and perceives most of the time. And on this scale, the earth is flat.
Now look up at the un-zoomed curve at the top, with the tiny figure. That curve is most definitely not on the same scale as the figure.
So, from the perspective of the figure, the world is flat! The curve does not exist.
In fact, Aristotle had to use detective work to understand that the earth was curved, by observing the earth's shadow on the moon, and the way ships gradually descended over the horizon.
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| The impossible gunfight |
So, we can now begin to reconcile these two perspectives, that the world is flat, ...and also curved. It is globally curved... and locally flat. We can also begin to understand that the belief is about the scale of information, of perspective, we are operating on.
We also notice that some perspectives are very difficult to obtain. There may be many huge curves we are living on, far beyond our scale, that we are completely unaware of. For example, Einstein showed that space and time are actually curved, with the theory of general relativity. But this curve is so far beyond our local scale, experience of it is extremely rare in our normal daily lives.
Sadly, any disagreement about the flatness of the earth may be more of a statement about wealth and use of resources. We have seen that perception of the curvature of the world depends on the scale of our perspective. Well, as humans, in order to get this global perspective, we have to consume resources. If we fly over the horizon, we are getting this global scale beyond our normal human one. We see the curve of the horizon very clearly. We see the sun out of the window, almost in the same place during an 8 hour flight, we experience jetlag for days. All of these are evidence presented to us from our brief adventure into another scale. However, flights are expensive and they consume a lot of energy.
Education and surrounding oneself with highly developed scientific ideas is sadly also expensive. That pyramid, those shoulders of giants above do not come for free. It required a huge network of infrastructure over a long period of time. There are many entry costs to obtaining a non-local perspective that way as well. Those of us who have it can consider ourselves extremely fortunate, that we are able to understand and study phenomena beyond our local scale, due to our participation in education. We were very fortunate that a confluence of birth parents, region, country, educational system, friendships, belonging to a social network, etc. etc. came together to allow us to achieve these deeper understandings.
Further resources:
You can try drawing your own curve and zoom in at home (link) .
Draw any smooth curved shape, then keep zooming in!
Try using a circle and a figure height proportional to our height on earth.
(The average human height is approximately 0.0000267% of the earth's radius.)
More on the derivative (link).
