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A Brewers fan wore a wooden hat to the game, but is it actually a hat?

What is a hat? It may sound like a dumb question but let's indulge it for a second. Beginning with Descartes' meditations on a piece of wax, philosophy has used eidetic reduction to arrive at the essence of different things. Basically, eidetic reduction is a thought experiment where one strips away properties of an object that are not essential qualities of it.
Take a hat for instance. What properties can be taken away and still allow it to retain its hat-ness? We can adjust its height as both baseball caps and top hats are equally hats. We can change color. We can even take away a brim in the case of winter hats.
But what about the material? Does a hat have to be made out of some sort of fabric? Is a hat made out of wood actually a hat? Put another way: is this a hat?

As this fan informs us, his hat -- his term for it -- is made of ash and comes from what seems to be a rich tradition of making wooden objects in the shape of hats. This creation certainly looks like a hat and acts like a hat. Whether it is a hat is a question we'll leave to the philosophers.