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Thread on convexity and antifragility.These are the main points I made to 18 year olds in beginning Calculus, so the math is quite basic, yet elegant in its simplicity.
From Antifragile, by @nntaleb 1/
From Antifragile, by @nntaleb 1/
We associate convexity/concavity with shapes of things - a geometric feature of objects.
Like, today I tried on contacts for the first time, and I got so frustrated because they kept flipping from convex to concave (relative to my finger) as I was trying to put them in 2/
Like, today I tried on contacts for the first time, and I got so frustrated because they kept flipping from convex to concave (relative to my finger) as I was trying to put them in 2/
But graphs of functions have this feature too!Because they *are* geometrical objects, right?
Now functions are actually relations between quantities.And the convexity feature of a function's graph says smth about the relation between input and output that function captures 3/
Now functions are actually relations between quantities.And the convexity feature of a function's graph says smth about the relation between input and output that function captures 3/
What does this have to do with antifragility?
If a function is non-linear (assume an increasing f first), then a given variation in its input gives *different* variations in output.
If convex, output variation increases.
If concave, it decreases 4/
If a function is non-linear (assume an increasing f first), then a given variation in its input gives *different* variations in output.
If convex, output variation increases.
If concave, it decreases 4/
So convexity actually corresponds to benefiting more and more from shock, which is what the antifragile does!
Think Benefit on the y-axis, f(x), and the Event size on the x-axis. For every *same* bit added in Event size, the Antifragile benefits *more* (the fragile - less)5/
Think Benefit on the y-axis, f(x), and the Event size on the x-axis. For every *same* bit added in Event size, the Antifragile benefits *more* (the fragile - less)5/
Now if we're talking Harm from shock - and this is the beauty in the simplicity of the underlying math - say f is decreasing, for every *same* bit of added shock, the Antifragile is harmed *less* (the fragile - more: do the corresponding concave graph to convince yourself! ) 6/
And calculus tells us if a function is convex or concave. Convex can be characterized as increasing slope, concave decreasing slope. In other words increasing f' or decreasing f' (1st derivative of f is actually the slope of its graph - always changing if f is nonlinear) 7/
Finally to determine increasing/decreasing behavior of f', it suffices to calculate its own slope: its derivative, which is f prime prime, or f"! 8/
(I'd go off on a tangent now about the beauty of describing geometry of shapes with algebraic calculations via the coordinate system, e.g., convexity via calculus of 2nd derivative). The awe I experience again and again when my students first grasp Descartes's invention...
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