Hyperfinite graphs ~ manifolds

[Alok Singh]

Manifolds are better described as ‘shapes’.

Hyperfinite is an unlimited number, often an integer. A whole number larger than any standard n.

Imagine a graph with hyperfinitely many vertices and edges, where the edges have infinitesimal weights. If I were nearer my iPad I’d draw it.

The edges are the tangent vectors of differential geometry. Another way of thinking about it is that the vertices are each the zero vector of an infinitesimal space, because they are. differential equations become difference equations. this is also why so many techniques from riemannian geometry (which includes every relevant shape) make their way into graph theory, and why diffusion is fundamental.

A world in every grain of sand

Hold infinity in your hand and eternity in an hour. If your time scale was infinitesimal, an hour would be an eternity (hyperfinite).

Important visual: $\omega$ (spelled “omega”, pronounced “oh-mega”) represents a hyperfinite number. It’s our hyperfinite unit. This is exactly like picking how long a meter is. We fix it ONCE AND FOR ALL TIME. It carries no information other than its name and level of size. We may set it to a convenient value, such as the product of all (standard) natural numbers. Now we have a bit more info about $\omega$: it’s an integer and any standard number can divide it (handy for picking mesh sizes).

Example: Let’s take the grain of sand to be the point $2\omega +1$, on the right. I picked an infinite point to illustrate how these really are just ordinary numbers. Meditate on the phrase radically elementary. The world is the pink line around it, a 1D microvector space. The sand cannot be discerned in any finitary way from an infinitesimal displacement (sum/​difference) of it.

Note that the microvector space really is a vector space. infinitesimal + infinitesimal = infinitesimal. standard x infinitesimal = standard * 1 /​ infinity = standard /​ infinity = infinitesimal. The rest of the definition you should verify, dear reader in Christ. If you grok the Transfer Principle, this exercise is trivial.

POLL: microvector vs tiny vector as terminology?

Aside: For a radically elementary Feynman path integral, take the universe and put it in a hyperfinite 4D cube of edge length $\omega$. Then its 4D measure is ${\omega }^4$. Whether or not the universe is infinite, we can always do this embedding. Now cut it into 4D cubes of edge length $\frac{1}{\omega }\sim 0$, an infinitesimal.

This gives us a lattice, which is easy to realize as a graph. The formal technical issue of infinitely many paths to integrate over dissolves, though the problem is still hard. But now we can ask the question properly.

Aside Aside: Our notation for numbers is remarkable. It started with only whole numbers, but extending to fractions just required enlarging the exponent set from positive numbers to positive and negative numbers, plus a dot. Now we extend it again, this time to numbers big and small.

Example: $37\overline{0}.\overline{0}21$. Another way to write it is 37...0.000...021.

The first digit represents $3\times {10}^{\omega }$, the second represents $3\times {10}^{\omega -1}$, the second to last digit represents $2\cdot {10}^{-\omega +1}$ and the last digit represents $2\times {10}^{-\omega }$. This sequence is infinite, but it is COMPACT. So it has a beginning and end. On that note, $.9\overline{9}9\ne 1$.

This lets us use a discrete object (hyperfinite graph) to describe a continuous one (manifold). For ML fans, pixels are (hyper)tokens. This is because the hyperintegers are isomorphic to the limited reals.

To see how they are isomorphic, imagine zooming into an interval. At a hyperfinite zoom factor, the continuity of the line breaks, because there are infinitesimals in between the standard numbers. The infinitesimal neighborhoods are disjoint, so we can separate points by infinitesimal balls of size $\frac{1}{\omega }$. But we can zoom in again and separate those points by balls of size $\frac{1}{{\omega }^2}$…

Since this argument applies at each zoom level, this implies the hyperreals are totally disconnected! They sure don’t look disconnected. Zeno all the way down.

Here’s a picture to illustrate the concept of microdiscretization, or ideal discretization (I did not make the picture because I would never use the perverted cross product).

[Slider]

I have always thought that $\omega$ was the first transfinite number. The hyperfinite angle does take an interesting poke at the gap problem by kind of embracing the gaps.

I assume that the notation works by having more overlined numbers for each infinity level. That $25\overline{0}37\overline{0}.\overline{0}21$ is $25\ast {10}^{2\omega }+37\ast {10}^{\omega }+21\ast {10}^{-\omega }$. For ${\omega }^{\omega }$ start adding lines? $87\overline{\overline{0}}25\overline{0}37\overline{0}.\overline{0}21$

Atleast for surreals they are “too continous” to count as continous on the same axiom as reals get defined. As reals $\{1/2,1/4,1/8,1/16,...\}$ has a unique limit of $0$. But for surreals $\{0|1/2,1/4,1/8,1/16,...\}=ϵ$ gives birth to $\left\{0|ϵ\right\}=ϵ/2$ and with enough iteration you can get to $\{0|ϵ/2,ϵ/4,ϵ/8,ϵ/16,...\}=ϵ/\omega$ . So there are plenty of strong limits going on but they are not unique because a tighter limit is always possible but none of them is “the limit”.

I am not convinced that the limit procedure is the correct one. The “old” notion of smooth would only count up to real accuracy, so in surreals and I would imasgine in hypernumbers reals are actually disconnected. In what they try to build as surreal analysis they can get the real integral etc by only taking the standard part much like one takes the integer part from a real to get a result type compatible with N. I would expect a funtion that approaches $1+ϵ$ from the left and $1-ϵ$ from the right to be discontinous when one approaching 1 from left and right would have chances to be continous. I am not convinced that the approach treats the non-standard parts adequately. First times encountering this it hammered home that smoothness or “completeness” is a relative property. Natural numbers considered in themselfs are without gaps, it is only when we compare them to other systems that “unrepresented elements” are apparent.

The bit about diffusion being fundamental seem quite important but unfortunately I am not understanding that at all.

[Alok Singh]

Reread this and this is awesome: I didn’t think of the case of multiple bars at all. As for the surreal stuff, I’ve read about them and while the lexicographic ordering is nice, the lack of transfer principle hurts. Are you in the bay?

[Slider]

I think there are quite a lot of transfer principles going on. Atleast real to surreal and the hypernumber transfers have exact analogs (but might not have to the surreals that are not that kind of hypernumber). Would not be surprised if “hyperfinites” would be limited to ${\omega }^x$ where x is finite (so ${\omega }^{\omega }$ is actually too big to be a hyperfinite).

I am not in the San Fransico bay area and it is not convenient for me to physically meet in America.

[Alok Singh]

Think of diffusion as a micro random walk. A hyperfinite number of infinitesimal steps, sampled from a discrete set of vectors.

[Alexander Gietelink Oldenziel]

Beautiful chaotic math energy here Alok, keep it up! =)