Continuum and Points

Ksana
Buddhist schools about ksana and Zeno
Ksana and mathematics
Physics and Planck length
The root of the paradoxes
How real is *ℝ?
Back to Buddhism

Ksana

In Buddhism there is an interesting term kṣaṇa (or khaṇa, I will use "ksana"). It means the smallest time unit, approximately one seventy-fifth of a second. Within one ksana, there are nine hundred instances of arising and ceasing. There are 32,820,000 ksanas in one day. It comes from the idea of the mind stream: the mind stream consists of small quantums - psycho-physical factors called dharma which has a duration - ksana. Initially, Buddhists imagined it as a period of time, with a beginning and an end. But very soon they hit old and good known problem of continuity of time (continuity of space is related philosophical problem, I think the first philosopher who got the infinity depth of this problem was Zeno):

Buddhist Abhidharma often describes conditioned dharmas as arising and ceasing in a single ksana. But then philosophers faced two apparently contradictory requirements:

a ksana cannot have zero duration. If a dharma existed for literally no time at all, it would be impossible to say that it arose, existed, or ceased. There would be no temporal interval in which it could exist.
but a ksana cannot have any positive duration either. If the dharma lasts for even a tiny interval of time, then during that interval it has to remain present. But that seems to mean it has multiple temporal stages - a beginning, a middle, and an end. It therefore isn't truly instantaneous/momentary.

It was important for early Buddhists because consciousness and karma consisted of elements of cause-and-effect relationships - this is where Buddhism's love of classifying everything manifested itself. Accordingly, thoughts and experiences were analyzed, and it was established that they consisted of smaller proto-thoughts (you can do this yourself, managing to "catch" the emergence of a thought or mood, and even stop it from arising!). This is where the idea of small, indivisible quanta of consciousness and experience (karma) arose: this "piece", quantum of time forms samtana (the mindstream) which is that something that is reborn - as is known in Buddhism, a human is not reborn, he dies once and for all...

Buddhist schools about ksana and Zeno

There was another man who also thought about similar problems, but much later, it was Leibniz.

Zeno asked himself: can a continuum consist of infinitely divisible intervals?
Buddhist also sked himself: can reality consist of instantaneous events, without duration, and still possess continuity?
Leibniz asked himself: how can one analyze a continuous quantity in terms of infinitesimal changes without treating the infinitesimal quantity as an ordinary finite interval?

These questions actually touch on the same conceptual problem: how to ensure continuity, process from "points" or ultimate entities that in themselves seem to have no span, duration?

Buddhist schools are divided on the question of determining the duration of ksana.

Sarvastivada school viewed the ksana as the smallest determinate unit of time, and dharmas were described as existing during a very short phase of time.

Sautrantika school, however, leaned toward a much more radical instantaneity: dharma does not exist at all during any period of time; it arises and immediately ceases.

if a ksana has a duration, then dharma continues for this duration and is therefore not strictly fleeting. But:
if a ksana has zero duration, how can there be a sequence of ksanas that constitutes temporal continuity?

This was not just a modern objection. The Buddhist schools themselves experienced difficulty defining the relationship between fleetingness, continuity, and causal efficacy.

Today, we know about biology, neurology and we can comment a dharma in the context of a brain activity (not in general) as a process having a duration which depends on a brain state, on health, etc but from the point of view of the mind it can be close to zero, like:

a brain cannot work faster, so this is the smallest unit of a change of the state of consciousness, but:
the consciousness is limited (especially untrained) and it cannot see the duration, it sees it only as 1 undivisible "event"

So, the true is that Zeno of Elea was shocked by these questions before Buddhism and, later, Buddhism came to the same paradoxes in 2-3 century BCE. And after it, mathematics returned to these paradoxes - further development of mathematics was impossible without some adequate understanding, if not solution, of these problems.

Ksana and mathematics

Again, if to look at this more generally, we have a time (mind, space, individual experience, etc) and the idea to:

analyze it: so we divide it to fragments (units)
then we switch from instrumental, artificially introduced fragments (units) to fundamental smallest undivisible units - kind of quantums
and we have to connect, assemble back these 2 situations: something is continuous, but divisible to so small units that it seems they are close to ZERO!

The idea of infinity is very close to this: zero and infinity are connected as 1 divided by 0 - the essence of Zeno's question boils down to this: how can one perform an infinite number of "steps"?

The problem of instantaneity in Buddhism has a roughly reverse structure: how can an infinite/indefinite set of moments without duration represent a real temporal sequence?

Imagine 0 + 0 + ... = ? Leibniz found an answer: mathematics teaches us a very important lesson - the problem arises partly because we intuitively imagine as an infinite moment = a tiny segment of time.

Mathematical analysis doesn't work quite that way.

A real number, for example, 1 (maybe a point on a time axis), has no space (or temporal) span. But the interval (a segment) [1,2] contains an infinite number of points. These points are not simply small fragments of an interval. An interval is not constructed by physical addition point + point + ..., instead, an interval is a continuum, and the points are locations within it.

That is, the problem with kshanas is much deeper. Essentially, it's not even about whether space and time are continuous or not, and if they are pixelated (discrete), then what is the size of a "pixel"? The problem comes from very wrong idea of Leopold Kronecker, who once wrote that "God made the integers; all else is the work of man." - the God did not make integers (natural) - people invented them and they lead to this problem...

Over the course of evolution, the human brain developed certain models for understanding the reality around it. It invented words, recognized colors and gave them names, sizes, tastes, and came to understand the concept of quantity. Quantity is calculated in its simplest form using whole numbers. Natural numbers were the first human invention in this area. Natural numebrs was simple, obvious and natural, because without microscopes and telescopes, people saw objects around them with clear boundaries, and this made it possible to count the number of objects united by some characteristic.

3 apples. 5 fruits: 2 apples, 3 oranges, etc. People lived for thousands of years without the idea that there could be "objects" so small that they could not be seen, as well as "objects" so huge that they were significantly larger than the entire planet.

Without a microscope, there is no microcosm. The microscope has made it possible to see microbes. But is this the limit? Perhaps, further down the scale, lie discoveries that humans will not make anytime soon, or perhaps never. The life of microbes is not like the life of forest animals. Their world is more like a 2D world. It is a world of organic molecules. A world of natural nanomechanics: it is not gears and belts, programs and microprocessors that operate there, but molecules and atoms.

This world contains not stones, sticks, and dogs, but electrons, protons, photons... And their nature is very different from anything we are familiar with. But intuitively, we assume that everything else - space, metrics, fields - is the same. Only now it contains electrons and protons, and no dogs. But it's possible that's not true. And then that means the idea that everything can ultimately be counted, divided by two, counted again, and so on and so forth - may be a big mistake. The idea that everything can somehow be expressed through the simple and ancient concept of the number of apples and oranges - may be very wrong. The idea that space is piecized (like time) is obvious and simple. And we find analogies all around us - we grew up among things that can be counted: they are isomorphic to natural numbers. Let's imagine how things could be different.

The border of our world

On very small scales, physics is different. There's something there that has no analogue in our world. We can't imagine the objects there, we can't imagine how they move, their trajectories, their shape (and they probably don't have a shape or trajectory), their color - they don't have one! The same is true for very large scales. Our world is enclosed in a capsule of scale; beyond it lies another reality, where the laws of physics are completely different! Think about it: without any new dimensions, parallel universes, or anything else, simply by changing size, we enter another reality, where everything is so unique that it's impossible to even imagine. Scale - its change - is a door to another reality. Thus, the scale of our reality has certain boundaries, beyond which lies another reality and another physics.

Some philosophers argue that this is not true and that the world (space itself) is pixelated, discrete, time and space are made up of tiny quantum pieces, and there is nothing smaller. This could mean that the size of the universe is not infinite but limited. When you move, you don't glide, you bounce: you disappear from one position and magically reappear in another, because you can't occupy a position between two pixels. In a way, this violates conservation laws. It looks a little strange, but only a little: in old computer games, characters moved exactly like this. So, it's familiar to us. It's also easier for our brains to imagine: then we're left with our familiar integers, which we can use to express the exact number of pixel steps - in space and time.

The alternative is frightening.

Continuum and continuous space and time

Now this is quite difficult to imagine. Continuity of space does not mean that the scale of space is physically infinite, but if there are no laws limiting it, it may be so. Although we are talking about an imaginary space that exists in philosophy, in mathematics.

Can you catch what always slips through your fingers? CAN YOU COUNT WHAT ALWAYS DIVIDES?

Now think about this: if space is infinitely divisible, then you can move from a certain point to a very small distance, and always to an even smaller distance. If you can move an incredibly small distance, you can always divide that distance by two and move half that distance. And so on ad infinitum. Infinity is the key concept here: infinity, infinity small (ksana, dharma), infinity large - these 3 concept are connected and expressed from each other.

Constructivism and intuitionism

Intuitionism

Instead of treating

N = {0, 1, 2, 3, ...}

as a completed infinite object, one can regard the natural numbers as generated by the rule

0, S(0), S(S(0)), ...

where S(n) = n + 1. There is always a next number, but there need not be a final completed collection containing "all of them" in the same sense as a finite set.

The strongest historical connection is intuitionistic mathematics, associated especially with L.E.J. Brouwer.

For Brouwer, mathematics is fundamentally a mental construction. Mathematical objects exist insofar as we can construct them according to legitimate mathematical procedures.

This produces an important distinction:

Potential infinity: you can continue the construction indefinitely.
Actual infinity: a completed infinite totality exists as an object.

Brouwer was very skeptical of treating actual infinities as if they were ordinary completed objects: infinite mathematical constructions can exist, yes, but we shouldn't automatically treat them as completed classical objects. So:

1, 2, 3, 4, ...

can be understood as a never-ending construction, a process, rather than as an actually completed thing containing infinitely many elements.

Constructive mathematics

A constructive mathematician generally doesn't accept: "There exists an x satisfying P(x)", merely because assuming that no such x exists leads to a contradiction. Instead, one wants some kind of construction/witness of x.

For example, classical mathematics can prove:

∃x, P(x)

without actually producing x. We just declare some relations. A constructivist may say: that's not enough. Show me how to construct x. Talking about programming, Prolog programmer writes declarative programs close to standard math, but a constructivist is like a imperative programmer. But this is also close to languages like Agda: if you can write it, construct it, then it can exist, it is treated as proven, as true.

Finitism

Finitism goes further. The most famous example is David Hilbert's finitistic program, although Hilbert's position is more complicated than simply "infinity doesn't exist."

A strict finitist wants mathematics ultimately grounded in finite symbolic manipulations.

There are no completed infinite objects in the basic ontology.

However, Hilbert still allowed mathematicians to talk about infinite structures as useful idealizations, provided their use could ultimately be justified finitistically.

His goal was much more ambitious:

Preserve essentially all of classical mathematics while giving it a secure finitist foundation.

This became the famous Hilbert program.

Hilbert accepted ordinary classical mathematics - including infinite sets, real numbers, analysis, etc. - as a useful formal system. He did not want to throw it away. Instead, he proposed roughly this division: The "ideal" mathematics. This includes things like:

infinite sets
real numbers
abstract mathematical objects
classical logic
nonconstructive existence proofs

Hilbert called such objects ideal elements, somewhat analogous to how imaginary numbers can be useful in algebra. Hilbert's strategy was: "We can use infinitary mathematics, provided we can finitistically prove that it doesn't lead to contradictions."

In particular, he wanted a finitistic proof of the consistency of classical mathematics. Gödel broke these hopes, but this is another story...

Ultrafinitism

After Hilbert's failure more radical guys came: they looked more closely at the idea of scale and how quantity (movement along the scale) can lead to a change in quality - and they gave a birth to ultrafinitism.

Consider a number 10¹⁰:

Classical mathematics has no problem with it.
A conventional finitist probably has no fundamental problem either.
Ultrafinitist would say: "What does it actually mean to say that this number exists if we could never physically write out its decimal representation?"

Look at this number: 2.23606797749979... its representation as a decimal fraction is infinite. IS it OK? No, it is not, because I don't know what comes after the "last" 9! And it can be important in some algorithms. In general what is it?! I can imagine its continuation as 2.236067977499790001000100010001... and if it's true, then it looks very strange. Or something else, actually it is √5 (the square root of any prime number is irrational). And pay attention: now we know another, very short representation of this number, it's formulae, a "process generating it"! Actually we can have very huge numbers (see Hyperoperations), and an ultrafinitist might ask: "What justifies asserting that such a gigantic object exists if we could never actually construct or survey it?".

But ultrafinitist is reasoning: the fact that a short formula describes something doesn't necessarily mean that the described object is mathematically available. Is it true that finite description ⇒ finite object exists? Imagine:

N = 10^10¹⁰

To construct its decimal representation, we would need an absurd number of digits. An ultrafinitist may say that the ordinary mathematician is quietly treating a gigantic object as existing merely because we have a compressed description of it. But perhaps the real mathematical object is supposed to be the actual finite string:

77777...777777
<- N digits ->

We cannot produce, survey, or manipulate that string. So one ultrafinitist intuition is:

A finite object should be grounded in a possible finite construction, not merely in a symbolic abbreviation.

This is sometimes described as a constructive or epistemological motivation for ultrafinitism.

It looks strange, but lets try to defent this position. Imagine super big number 10^1,000,000. What would be larger? In classic math we answer: 10^1,000,000 + 1. Finitist may accept it because this is a finite construction. But an ultrafinitist might ask whether the notation and construction genuinely give us a legitimate number at all. He can do it, really, look: if we always can do +1 then we declare that no any boundaries, limits, it's true for space, university, for time, for scale, for everything and quantity does not turn into quality. Never. Space can be divided infinitely! And... we come to Zeno's paradox and ksana. Is it correct? Is it true? Or do we always have some boundaries for "infinite processes"? Physics teaches us that there are limits, and as we approach them, physics and its laws change---a different reality emerges. For example, such a lower limit is the Planck length. This is a fundamental question of logic and philosophy: one can declare that any infinity is only apparent, that it has boundaries, upon reaching which it transforms into something else, but the primitivism of the recursive representation, the model of infinite summation, does not reflect this. In other words, the model is simply incorrect. One cannot grow to infinity. One cannot mature to infinity. One cannot run forward to infinity. Although kinematics (a branch of physics) allows this---it doesn't take energy into account!

What does it mean to say that such numbers exist if there is no possible procedure by which its relevant finite structure could ever be realized? And here ultrafinitists introduce a notion of feasibility: instead of merely saying n < ∞, we might require something like n < B, where B represents some bound on physically, computationally, or mathematically feasible finite constructions. The precise choice of B is highly controversial.

For the example with 10^1,000,000 it may look short (due to short notation) but ultrafinitist would say: short description ≠ short construction.

That distinction becomes fundamental for ultrafinitism:

finite describability ⇏ mathematical existence
finite ⇏ legitimately constructible.

where ⇏ means does not automatically lead to (it is crossed out).

We can even see some success of ultrafinitism. Or think about grains of sand. Standard mathematics can give a rough estimate of the number of grains in the Sahara Desert and even an estimate of the number of atoms in the universe! It sounds like a hoax! But they do it without even batting an eye! But ultrafinitism declares that this is meaningless and that no calculations using this number (calculations of trajectories, total forces, friction forces, etc.) can lead to anything real. It's simply speculation and unprovable claims. And then they turn out to be right, and another approach, another theory, comes into play - the "theory of chaos".

And as you might suspect, ultrafinitists even defy the principle of mathematical induction!

OK, now we are returning back to Leibniz to switch to an alternative to his math after it.

Leibniz, Cauchy, Heine

Great mathematician Leibniz thought about these problems. And another one mathematician - Newton also did it. As well as Heine and Cauchy. While exploring the limits of sequences and functions at points, they came up with the idea of a delta neighborhood of a point. As the function's argument tended toward the desired point, they had to imagine infinitesimal increments of the function's argument. From history, we know that Leibniz reasoned precisely in terms of infinitesimal quantities and only later arrived at the epsilon-delta notation.

 The idea of limit expressed using
      epsilon-delta notation:

    🡑                 __f(x)
    |               ／  
    |              ∕              lim f(x)=Y
Y+ε |- - - - - - -⹊- - - -        x→0
    |           ／:
  Y |- - - - - o - - - - -        ∀ε>0 ∃δ>0: ∣x∣<δ ⇒ ∣f(x) − Y∣<ε
    |        ／:  :
Y-ε |- - - -⹊- - - - - - -
    |     ／:  :  :
    0-------:--:--:------⟶
          X-δ  X  X+δ

Well ε is an ordinary positive real number. It can be arbitrarily small, but it is never an actual "infinitely small" number, ie, ε is finite - just potentially as small as you demand. But if they would, the relations will be correct, true anyway.

Again, ε and δ are not infinitesimals. But they can be. What are infinitesimals then?

Leibniz's monads

Thinking about limits, differentials, the reality, God, Leibniz came to the new idea - the idea of monad. He defined monads as the fundamental, INDIVISIBLE, and animate building blocks of the universe. They do not physically interact with each other, and their existence is pre-harmonized by God, etc... Look again: indivisible. Like dharma. Like its duration: ksana. What is common?

The common is the trick of the human logic: we can continue to live with countable objects like apples, and oranges --

TO DEMAND THAT CONTINUOUS THINGS CONSIST OF SMALL, ALBEIT INVISIBLE, BUT COUNTABLE PIECES, COUNTABLE LIKE APPLES.

But such indivisible "thing" can be not invisible monad, but the "infinite" space itself.

Nonstandard analysis

In nonstandard analysis, ε is a completely different mathematical object - an infinitesimal.

The point is that non-standard analysis works not with the set of real numbers ℝ, but with the set of hyperreal numbers *ℝ: these are real numbers + non-standard numbers and the "main" of them are infinitesimals - non-zero quantity that is closer to 0 than any non-zero real number is. It is smaller than ANY real number, in other words, reminds ksana, dharma problem, right? And If you divide 1 by this number, you get an infinitely large number - a number greater than any other real number... So, mathematics is close to the Buddhists problem, very close, historically it seems to be:

Zeno ... Buddhism ... Leibniz/Newton/others ... Non-standard Analysis ... ???

Physics and Planck length

Physicists asks a different question: does physical space actually have a scale smaller than Planck length? You could be an ultrafinitist and still have no opinion about Planck length, because, ultrafinitists think mostly about mathematical objects, so they can reject it or accept it, it depends.

What is Planck length? It is 10^-20 times the diameter of a proton. "...It is comparable to Schwarzschild radius of a particle though whether those concepts are in fact simultaneously applicable is open to debate." Can something be smaller? Can something move to a distance shorter than the Planck length?

There is a such device called Fermilab's Holometer. And in 2015 they performed some experiment.

In 2015, after about a year of measurements, the Holometer reported that it did not detect the predicted holographic noise:

has ruled out Hogan's theory of a pixelated universe to a high degree of statistical significance (4.6 sigma). The study found that space-time is not quantized at the scale being measured

To be accurate, it did not prove that space is continuous: it showed that one particular class of models predicting a certain kind of spacetime pixelation/fluctuation was not supported by the experiment.

And this is still an active question. For example, the newer GQuEST experiment is specifically being developed to search for other possible quantum-spacetime fluctuations; a 2025 paper described its interferometric approach, see this and this.

There's also another observation: NASA observations of extremely distant gamma-ray sources were used to constrain models in which spacetime has quantum "foaminess." Those observations likewise found no evidence for the predicted effect within the tested models.

So, from the point of view of the modern/last experiments, the space is still "continuous".

The root of the paradoxes

Can you imagine the consciousness of an animal?
Can you imagine the nervous system of a tardigrade? Its world?
Can you set a contact with bacteria colony?
Can you imagine a silicon life in a volcano?

...There are more things in Heaven and Earth, Horatio, than are dreamt of in your philosophy.

-- Hamlet, Act 1, Scene V

We think we already underdtand the reality, at least the SPACE. But the space - is the most mysterious, enigmatic something. The problem with the space is that it is the MAIN, BASE of the reality, the main category. And it challenges us because it is the same as to define zero without 1. Zero is "no one, no 1+1, no 1+1+1...". No, try it without 1, 2, 3...

Natural numbers. When we are little, we learn to count. What's behind the idea of counting? It's based on a property: there might be one apple, there might be no apple at all, or there might be three apples, four, and so on. Counting is based on the idea of grouping objects by a common property and determining the size of the group. The number of apples in something. In a basket. Main concepts of countable objects are:

a metaphorical "basket" containing countable objects
zero objects
one object
more objects

It is monoid: an algebraic structure with neutral element and an commutative operation of summing. Now look at the space:

no any "basket" (frame, boundary) when we talk about space. So, modern physicists try to invent such boundary (close space, we are inside "black hole", etc - their brains are not able to work with a space without any limits. INFINITE SPACE, it is outside of our experience, we, former children, cannot imaginee it)
no "zero" spaces - space always exists. So, modern physicists try to violate even this. Imagining a time when space didn't exist! Not matter. But space. It didn't exist. Before the Big Bang. How true is that? Don't forget: the modern theoretical physics is very disconnected from the reality and experiment - it's pure math.
one space. Comfortable. Yeah, we are here.
More spaces, 2 spaces, 4, 10... Physicists and science fiction writers talk about multiple universes, parallel universes, etc. But this still cannot be proven, likely it will not be ever.

It reminds the concept of Allah: He is the one, no place outside Him, no time before Him, He has not relatives, parents, friends...

The root of the problem is our commitment to believing that everything can be counted. Well, mathematicians go further and introduce uncountable sets:

∀ a, b ∈ ℝ, a ≠ b, ∣[a,b]∣ = |ℝ|

i,e the number of numbers between a and b on real axes is infinite - as the number of real numbers. Its number is the cardinality of the continuum. We have options:

If we ask: "If I keep dividing a segment, will I reach the final smallest piece?" then answers of

classical mathematics: "There is no smallest piece. The real continuum is infinitely divisible."
Constructive/intuitionistic mathematics: "Be careful about what it means for all those infinitely many divisions/points to exist as a completed object."
Finitistic/ultrafinitistic thinking: "Perhaps we shouldn't regard an actually completed infinite divisibility as a legitimate mathematical object in the first place."

The true is, however, that there are indivisible "things", there are singletons (one item sets), and other exotic "things". And the Universe is either:

self-sufficient, eternal and so complex that it is capable of reproducing within itself complex living organisms, their technological civilizations and many other things that we will never be able to comprehend
or the Universe has some "foundation", "base", it can be categories (know, and unknown), God, something else - and they are very untypical for the logic of beings inside the Universe.

We love to count, but here the counting breaks down.

How real is `*ℝ`?

Leibniz asks: "How can continuous change be mathematically represented by infinitesimal differences?"

In Buddhism, the question of transience is posed as follows: "How can a world constantly experiencing events be formed from radically transient events?"

These are not the same problem, but they belong to the same philosophical family: how can continuity/process arise if its "elements" themselves seem incapable of ensuring this continuity?

It is a problem in a logic, in our logic, in our imagination, not in the reality. Can we imagine hypercube (a cube 4D)? Actually no, absolutely no, because we have not such experience att all. "Imagination" is just a connection of one "thing" with another one. And this is not hypercube: File:8-cell-simple.gif

it is just our attempt to imagine it, how could our, limited brain and eyes see it:

but we can research the properties of hypercube in our 3D world. This is the mathematical approach: we research properties of objects - we don't know them, but we know their properties. It allows us to find a way how to work with it.

The world of elementary particles and quanta is unlike anything we've experienced. Everything is different there! But it's precisely there that incredibly small scale exists. And to understand what's going on there, mathematicians come up with a strange kind of mathematics, unlike anything else: you can't say a particle is here and moving there at this speed. They only talk about the probability that it's here.

Imagine a particle like a photon - it has no mass and never rests. How can you even imagine such a thing? And does the photon really exist? Perhaps it's simply an effect of something else---like a wave from a passing boat?

Working with such objects may require other abstractions.

Imagine continuous space. This means that motion consists of smaller motions, but (if we stay within the framework of kinematics, forgetting about friction, molecules, etc.) the truth is that any smaller motion consists of smaller motions.

And we are faced with a choice, as with the hypercube: either forget about it and deny its possibility, or come up with models, abstractions, simplifications that will allow us to study its properties. Our choice is this: either we imagine that space is pixelated, or we consider it to consist of a "floating," "variable" quantity that can always be made even smaller, divided yet again. Newton used term (his own term): fluent. No such things in this, second case, it is just our abstraction, like quantum of space, a pixel. But a pixel has a fixed size. For the second imagination, approach, case we can think about:

"floating" units, when you measure it, it becomes smaller (reminds "Alice in Wonderland")
or fixed size units (like pixels), but their size is not comparable to other, normal sizes - it is always smaller. It reminds "floating units" but it is ignore "floating nature" of them, we just say - we don't know their size but it is immeasurably small. So immeasurable that we can't determine their size (unlike a pixel or the Planck length) - it's fundamentally undeterminable (it will always be smaller). But we can simply identify a separate size class for such dimensions and say that they are all the same, smaller than anything imaginable, and agree to work with them in a special way.

This is the idea of infinitesimal. And these approaches happen only in our head, infinitesimals neither do not exist nor exist - we cannot know it - the space is unknowable! But we can invent such abstractions or models that will allow us to describe some its properties. Infinitesimals are such abstraction.

Back to Buddhism

The Stanford Encyclopedia explicitly notes that it is precisely transience that gives rise to these problems: the interplay of continuity and conditioning between fleeting dharmas, as well as the question of how a fleeting event can have a causal relationship after it has ceased.

And it's very important question for Buddhism because ksana is the mechanism of illusory continuity, which is in the base of mindstream which leads to rebirth.

What exactly ensures continuity?

This is where Buddhist philosophy becomes extremely interesting - and this is where analogies with information, memory, Leibnizian metaphysics, and modern theories of continuity begin to bear genuine fruit.

And it all started with Zeno