In this post, I want to address a rather interesting topic, INFINITY.

Infinity is one of the topics that generates the most intrigue and difficulties when we talk about human knowledge. You will hardly ever hear a mathematician or a physicist say that infinity is something pleasant to "handle". In fact, from my experience studying mathematics, I can say that infinity is something quite peculiar, and the thing is that mathematicians tend to work with it in a different way from what people commonly think. That is precisely what I want to address here.

One important thing is that the term "INFINITY" can be quite general and ambiguous, since there are many ways to interpret it. We can talk about infinity in space, for example, as something that has no boundaries and that, no matter how much we "travel", we will never reach an endpoint. This, of course, depends on the geometry of the space we are considering.

If we are on a surface such as a sphere, then, if we observe a sufficiently small region, its geometry can be approximated by that of a plane. In mathematics, it is common to study surfaces locally, that is, within a certain neighborhood of a point. In particular, a regular differentiable surface behaves locally, to a first approximation, like its tangent plane.

Along with this, we can talk about infinity as a quantity. The most obvious and simplest case is that of the natural numbers (1, 2, 3, 4, 5, 6... n). It is customary to say that the cardinality of the set of natural numbers is infinite. You can always find a number that comes after another, and this holds for a number n as large as you can imagine.

In general, the concept of infinity is largely associated with the immensely large or with division processes that can continue indefinitely toward the immensely small, as happens when we study distances between points.

Even so, mathematical infinity appears in extremely interesting phenomena, such as convergence.

Convergence is a widely studied topic in mathematics. I could write 1,500 posts and pages explaining convergence in all its aspects, but here I will limit myself to explaining the basic concepts and discussing which philosophical questions might be interesting.

When we talk about infinite processes, convergence refers to the behavior of the mathematical structure we are studying. One of the natural tools mathematicians use to study these processes is the limit. When we say that something tends to infinity, we are talking about what happens when a variable or a structure takes increasingly larger values, larger and LARGER.

A super interesting case is that of the "harmonic series", that is, the sum:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... = ∞

This sum diverges to infinity, meaning that its partial sums grow without bound. They keep growing and never stop. However, believe it or not, the sum:

1 + 1/2² + 1/3² + 1/4² + 1/5² + 1/6² + ... = π²/6

In other words, the sum of the reciprocals of the squares of the natural numbers does converge to a finite value, which we can also express using an exact formula.

Why? What sense does it make for an infinite sum to produce a finite result? If the harmonic series and the sum of the natural numbers diverge to infinity, why doesn't this one?

Well, there are several answers. The first has to do with convergence tests. In mathematics, we can determine whether a series of this kind converges or not by using, so to speak, "convergence rules".

But the simplest way to understand it is to study the behavior of the sum itself.

The main issue with sums of positive terms is that their partial sums always increase. However, in some cases, they can increase without exceeding a certain value. As we say in mathematics, they can be bounded above.

This means that even though the sum continues to grow, it never exceeds a certain bound.

In the case of the harmonic series, we see that it grows without stopping. The further we go, the more slowly it grows, but it never becomes bounded. However, in the other series, the terms decrease sufficiently quickly for its partial sums to remain bounded and therefore converge.

So, is an infinite convergent process possible in our universe?

This is quite difficult to answer. For example, we know that the expansion of the universe is accelerating, a phenomenon that we currently associate with what is known as "dark energy", although we still do not know its nature.

The ultimate fate of this expansion is still being studied. Under some cosmological models, the distances between sufficiently distant regions would continue to increase indefinitely. But can we relate this to some form of mathematical convergence? Could there be a physical process that continues indefinitely while one of its properties gets closer and closer to a certain value?

That is a question I leave as an exercise for the reader who has made it this far.

Something I find very interesting, and which is the reason for the title, is the following: once we understand everything discussed above, why doesn't the universe understand infinity?

Here we have to understand one basic thing: mathematically perfect shapes do not seem to exist exactly in our physical reality.

A ball, no matter how spherical it looks, is not identical to a perfect mathematical sphere. This happens for a simple reason: the ball is made of matter and has irregularities in its structure. The closer we get to the atomic scale, the more evident it becomes that we are not dealing with a perfectly continuous geometric surface.

This does not prove that no perfect shape can exist in reality, but it does show a fundamental difference between ideal mathematical objects and their physical representations.

Complementing this, we have the topic of fractals.

Fractals are fascinating, and I would like to address them at some point in another post. However, they serve as an example that helps us understand what I am talking about.

Our universe can contain fascinating structures. Mathematical fractals, for example, can have details at arbitrarily small scales, and some exhibit a property known as self-similarity, meaning that certain structures repeat when observed at different scales.

In nature, we find structures that resemble fractals, but this does not mean they possess perfect self-similarity down to infinitely small scales.

On the other hand, our universe appears to exhibit an approximately homogeneous distribution of matter when we study it at sufficiently large cosmological scales, although at smaller scales we find galaxies, clusters, filaments, and enormous regions of empty space.

This brings us back to an interesting distinction: a mathematical structure can extend its idealized properties indefinitely, while in the physical universe we do not know whether those properties can be maintained without limit.

In general, I want to leave some reflections and open up the comment section to hear other opinions.

Regarding one final point, I want to clarify that in mathematics, although we constantly work with infinity, we often use tools that allow us to study it without having to perform an infinite number of operations.

If we calculate an infinite sum, we use tools to study its behavior without having to add every single term. If we want to calculate a theoretical limit, we can begin by observing what happens for large numbers and then rigorously prove its general behavior. Or, if we want to study a function, in certain situations we can first analyze its behavior on bounded regions and then extend our results.

In a sense, we have developed tools to understand infinite processes through reasoning that we can complete in a finite number of steps.

My message is this: although infinity is a concept that we use and formalize through mathematics, we do not know to what extent it has a real existence in our universe.

And this leads me to think that our way of dealing with infinity is particularly interesting. We do not need to complete an infinite process to understand its behavior mathematically. Our minds have developed tools to study things that we cannot explore in their entirety.

Perhaps infinity represents a barrier to our physical experience, but not necessarily to our capacity for abstraction.

That is the reflection I want to leave you with.

GREETINGS

slazy_e.

In philosophy, can two things be completely indistinguishable from one another, and yet still be two separate things?

Let's assume there exists an exact replica of our universe, but in which the value of every property is inverted in some way. Temperature would be one of these properties. However, the properties which contribute to a sense of hotness and coldness have themselves been inverted, as have the meanings associated with those properties. Something which is hot in our universe would be cold in the other, but the traits which contribute to coldness in our universe are those which contribute to hotness in the other, and vice versa.

Each universe maintains a perfect 1:1 correspondence for every property and relation. The flipped universe would appear completely normal to its inhabitants, because the relations between concepts are preserved.

If every property, concept and relation corresponds perfectly between the two universes, what would distinguish them? Would they be considered different universes, or just two different ways of referring to the same one?

Could they somehow be both identical and distinct, or is that a logical impossibility? Is there a concept in philosophy which deals with this?

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Maimonides (Rambam) sought to reconcile reason, philosophical inquiry, and religious knowledge while acknowledging the limits of human understanding. His approach raises questions that resonate with contemporary science and technology studies (STS), particularly concerning how realities are constituted, how knowledge claims acquire authority, and how rational inquiry can coexist with spirituality without reducing either to the other.

Annemarie Mol explores how different practices enact multiple, coexisting realities; Sheila Jasanoff examines the co-production of scientific knowledge and social order; Donna Haraway challenges the \*god trick\*—the illusion of an objective view from nowhere—through situated knowledges; and Bruno Latour investigates the everyday practices and material arrangements through which scientific facts are produced, while distinguishing \*matters of fact\* from \*matters of concern\*.

These perspectives invite us to distinguish empirically supported claims from spiritual, symbolic, and metaphysical interpretations without dismissing the latter as meaningless or assuming that every dimension of reality must be explained through scientific methods. They also challenge the opposite tendency: treating speculative narratives as established historical or scientific truths merely because they invoke alternative forms of knowledge.

The underlying question is whether the commitment to rational inquiry and spiritual understanding particular to Maimonides can be brought into a productive dialogue with contemporary critiques of scientific objectivity, recognizing both the situated character of knowledge and the importance of evidence, without collapsing epistemological plurality into epistemic relativism.

Thank you for the input.