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Background. Jorge Luis Borges was a post-modern short-story writer of the 20th century, whose stories often invoke a healthy dose of surrealism. One of his works is called The Aleph. In this book, there exists a point in space, aptly called the Aleph, that contains all other points. Anyone who gazes into it can see everything in the universe from every angle simultaneously, without distortion, overlapping, or confusion. I quote:

All language is a set of symbols whose use among its speakers assumes a shared past. How, then, can I translate into words the limitless Aleph, which my floundering mind can scarcely encompass? Mystics, faced with the same problem, fall back on symbols: to signify the godhead, one Persian speaks of a bird that somehow is all birds; Alanus de Insulis, of a sphere whose center is everywhere and circumference is nowhere; Ezekiel, of a four-faced angel who at one and the same time moves east and west, north and south. (Not in vain do I recall these inconceivable analogies; they bear some relation to the Aleph.) Perhaps the gods might grant me a similar metaphor, but then this account would become contaminated by literature, by fiction. Really, what I want to do is impossible, for any listing of an endless series is doomed to be infinitesimal. In that single gigantic instant I saw millions of acts both delightful and awful; not one of them occupied the same point in space, without overlapping or transparency. What my eyes beheld was simultaneous, but what I shall now write down will be successive, because language is successive. Nonetheless, I’ll try to recollect what I can.

On the back part of the step, toward the right, I saw a small iridescent sphere of almost unbearable brilliance. At first I thought it was revolving; then I realised that this movement was an illusion created by the dizzying world it bounded. The Aleph’s diameter was probably little more than an inch, but all space was there, actual and undiminished. Each thing (a mirror’s face, let us say) was infinite things, since I distinctly saw it from every angle of the universe. I saw the teeming sea; I saw daybreak and nightfall; I saw the multitudes of America; I saw a silvery cobweb in the center of a black pyramid; (...) I saw the coupling of love and the modification of death; I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon --- the unimaginable universe.

I felt infinite wonder, infinite pity.

Question. How can we put the concept of The Aleph on a solid mathematical footing?

What I tried. To simplify discussion, let's forget about time, and start with a Riemannian $3$-manifold (whether it's compact I'll leave to the cosmologist) $M$, which serves as a model for our universe. Let us model The Aleph as an additional point $\aleph$.

A priori it is not connected to the universe; however, we may endow $M \cup \{\aleph\}$ with a topology extending the one on $M$ that has, as its only neighbourhood of $\aleph$, the entire space. Topologically speaking, this implies that $\aleph$ is 'near every other point'. Upon doing this, however, our space is no longer Riemannian, and as such, much of the relevant notions, such as geodesics, start breaking down. So I'm not sure if this could get us any further.

This also does not take into account relativity in any serious way. I am vaguely aware that general relativity views the universe as a Lorentzian $4$-manifold such that light curves travel along suitable null-geodesics, but I am otherwise unfamiliar with the details; thus to prevent me from saying anything stupid I'll leave it at this.

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    $\begingroup$ Is there a mathematician that does not know JLB? $\endgroup$
    – fosco
    Commented Nov 4, 2018 at 21:25
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    $\begingroup$ (Also, is there a mathematician who didn't try at least once to turn Borges' short-stories into real mathematics?). I've been an avid reader of Borges for half my life, and I believe that if we want to follow the text verbatim, the passage "[...]and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth" contradicts the regularity axiom ($\aleph\in\aleph$). $\endgroup$
    – fosco
    Commented Nov 4, 2018 at 21:33
  • $\begingroup$ I have tried to do this type of thing with Cortázar, too. La Noche Boca Arriba, for example, can be studied as a reflexive Banach space. $\endgroup$ Commented Nov 4, 2018 at 21:49
  • $\begingroup$ A different idea: $\aleph$ is a category. We all know that there is a way to re-interpret mathematical structures internally, and thus there is the category of groups, but also the category of groups internal to groups (=abelian groups). Probably $\aleph$'s internal language is sufficiently expressive to reproduce "the universe" (somehow then, I'm led to think $\aleph$ as a topos...). Problem is that even though in $\aleph$ there will be an $\aleph$, the second Aleph will be "smaller" and less expressive...[cont] $\endgroup$
    – fosco
    Commented Nov 4, 2018 at 22:10
  • $\begingroup$ [cont] because for example groups in $\bf Ab$ remain abelian groups. There will be then a sequence $$ \aleph_0 \prec \aleph_1 \prec \cdots$$ (terrible but unavoidable clashes of notation)... $\endgroup$
    – fosco
    Commented Nov 4, 2018 at 22:13

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I think there is an easy way to do this with a standard idea from projective geometry!

First imagine a 2D flat universe, an infinite plane, full of flatlanders if you'd like. And specifically I'd like you to imagine it embedded in 3-space as the z=1 plane in R3.

Now take a unit sphere at the origin. Project it onto the z>0 side. It is now the case that if you put your eye at z=-1, you will be able to see the whole of the surface of the sphere, allowing you to see the whole of the 2D universe! You'll be able to see the internal organs of the flatlanders for example, which the flatlanders themselves would consider god-like. You already get that ability for free being a 3D being, but now you can see ALL the flatlanders and everything in their universe - assuming your eye has infinite resolution!

There is some angle distortion with this projection. It would get you a single, circular picture. I think there are real world lenses that could do this, eg the lenses of the Ricoh theta 360 camera. A functionally identical thing happens with the Poincaré disk model of hyperbolic space.

Just add 1 to the dimensions to see how this is possible with 3d :) so evidently the aleph:

  1. Turned the guy's retina from a 2d layer of cells that can receive light into some kind of volume of cells that can receive light. He also made it so there were way, way more cells than usual.
  2. Created a 4d equivalent of a panoramic lens. It is the shape of a 4-ball cut in half. Let's say it was 2m in diameter. The aleph then sets up that 4d lens 1m outside our universe, so that our universe is tangent to the lens at the zenith (topmost point)
  3. It put this guy's weird volume-eye 2m outside the universe and faced him towards the lens, so he could see the whole universe.

A description of something similar appears in Olaf Stapleton's "star maker", which I think comes before Borges. I think the "you can see a creature's surface and internal organs simultaneously" thing has been known since flatland. Flatland didn't try to make a person see the whole universe at once though, which is cool. I wonder if Borges saw Escher's Poincaré disk artworks?

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