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The View from Hilbert's Hat

Sebastian Gil Rodriguez is proudly Costa Rican and currently based off Munich, Germany where he is pursuing graduate studies in logic and philosophy of science. This story is his first published work, but he intends to put his education in natural science to good use by producing more pieces in the immediate future. The main themes in his fiction correspond to the intersection between physics and metaphysics, mainly in terms of how characters respond to the emergent inconsistencies in their surroundings and consequently strive to make sense of their underlying reality. In this way, he is greatly indebted to the writings of Philip K. Dick, whom he considers as his primary influence.

"I'm telling you, none of this is real. Just go down to the END to find the bottom of our world."
"Wait, what could that possibly mean?"
"All of this," he says waving his hand to an invisible audience, "this field of rolling grass with hills like white elephants, you and me and the car that brought us here, and the half empty bottle of wine, it's all a dud. We've made it to the surface of the set. The stars above our heads comprise the topological closure. If they're really stars, that is."
"There you go again with your abstract mathematical nonsense." She shudders. "Look, I'm well aware that we've only known each other for a short time, but you gotta let loose and live a little. You just go off like that at the drop of a hat and suddenly it's almost as if our whole lives were some sort of conspiracy theory cooked up in some guy's head."
"Well," he retorts musingly, "our whole space is either infinite or closed, but we're too small to tell the difference with just our senses. For all we know, we could live in the surface of Hilbert's hat and the entire weight of the world is determined by the curvature in the differentiable manifold.
"Don't try to educate me about the nuances of Hilbert Space, before we get all tied up again in geometry. Just have more wine and let us be merry."
The conversation segues into silence as the man and woman look up to the night sky filled with stars. They feel the cool breeze brush against their faces and let the chorus of cicadas lull them into a mild stupor. Presently, the man opens a bottle of wine.
"Still, what do you think it would be like if we found out that nothing but us is real? Do you think there would be any way for us to find out what kind of space we really live in?"
"I don't know what you're talking about, my friend. I know perfectly well that I'm real."
"But do you think you can produce a general proof for it? Of you being real, that is?"
The woman pauses to think. "I have something much easier than that," she retorts. "I can prove that I have two hands as follows: here is one hand," she says showing her palm, "and here is another."
"That's pretty good, I think, but it opens up to logic other questions about the boundary of our world. We would have to ascertain whether our spatial relations to the hills and to the car are reflexive, and if so whether they are symmetric and transitive."
"I will just assume that our mutual existence and the reality of our world is guaranteed. In fact, I put my full faith on it like my ability to see from here the car that drove us to this field."
"It would be nice to find out once and for all what it's really like on the outside," he says and sighs. "Imagine if someone could tell what space we're in just by reading it off from a story."
"What makes you bring this up again, when we are so close to the END?"
"The readers could read every paragraph again, but this time in reverse, starting from the other end and working their way backwards. That way, they could determine whether there are any symmetries or not."
"Wait, what could that possibly mean?"
"I'm telling you, none of this is real. Just go back to the top to find the surface of our world."
The End
This story was first published on Monday, August 12th, 2019

Author Comments

Shortly after discovering Daily Science Fiction, I began to brainstorm ideas for my first submission to the site. At the time, I was taking a course in formal logic and realized that, although we tend to consider mathematics as a science, very little science fiction is written based on mathematical ideas. My intent in writing this story was to represent the kind of inferences made by logicians and mathematicians when developing proofs. To wit: when a theorem involves the expression "if and only if," we mean to say that its proof is biconditional. This means that one can start from the premise and formally derive the conclusion or start from the conclusion and formally derive the premise. This is inherently symmetric, and I began to wonder what it would be like for a pair of characters to live inside such a closed structure. The result is a story with a symmetric plot: the same meaning is conveyed whether it is read from top to bottom or from bottom to top! Finally, we have the looming figure of David Hilbert: an unparalleled thinker whose contributions extended beyond pure mathematical logic to both Quantum Mechanics and General Relativity and also knew how to rock a straw hat.

- Sebastian Gil Rodriguez
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