Apeiría: Infinity in Platonic Thought & 3D LLM

Updated 13 July 2026

Apeiría is a dual-use term denoting regulated infinity in Platonic metaphysics and a neuro-symbolic 3D multi-modal LLM.
It models endless divisibility via anthyphairesis, where Division and Collection yield periodicity and self-similar unity.
The modern APEIRIA system integrates symbolic trace structures with object-centric 3D reasoning to enhance open-vocabulary spatial understanding.

Searching arXiv for the specified works to ground the article in current arXiv records. Apeíria denotes two distinct but technically structured notions in the material indexed under this name on arXiv. In Platonic metaphysics, apeíria, “the infinite” or “indefinite,” is identified with the endless divisibility of a Being under binary Division, and is interpreted as the philosophical analogue of the Pythagorean and Euclidean process of anthyphairesis, later “equalized” by Logos through Collection into a self-similar One (Negrepontis, 2012, Negrepontis et al., 19 Nov 2025). In contemporary machine learning, APEIRIA is a neuro-symbolic 3D multi-modal LLM that distills symbolic reasoning patterns into an LLM with natural-language chain-of-thought, using a three-stage curriculum to combine explicit spatial verification with open-vocabulary flexibility (Mo et al., 31 May 2026).

1. Apeíria in Platonic metaphysics

In Plato’s framework, as reconstructed from Sophistes 256d–257b, Parmenides 144b–e, and Philebus 14d–e, apeíria refers to the endless divisibility of a Being when it is subjected to binary Division. It is not treated as a mere aggregate of parts. Rather, it is the indefinite dyad of opposing “powers,” exemplified as $A$ and non- $A$ , which continually yields further species-parts under division (Negrepontis, 2012).

The interpretation advanced in the cited work traces this structure to the Pythagorean theory of anthyphairesis. Whenever two magnitudes $a>b$ are incommensurable, their mutual subtraction-division does not terminate; Plato is said to abstract this inexhaustible process philosophically as apeíria. On this reading, apeíria is the source of indefinitely many parts within a Form, while Collection supplies the principle by which those parts are brought into unity.

This suggests that apeíria is not simply “infinity” in an undifferentiated sense. It is a regulated infinity: an unending divisibility that becomes intelligible only when paired with Logos.

2. Anthyphairesis, continued fractions, and periodicity

The mathematical substrate of this interpretation is the anthyphairetic, or Euclidean, algorithm. Starting from two magnitudes $a>b>0$ , one applies successive divisions

$a = k_0\,b + r_1,\quad b = k_1\,r_1 + r_2,\quad r_1 = k_2\,r_2 + r_3,\dots$

with recurrence

$r_{i-1} = k_i\,r_i + r_{i+1},\qquad r_{-1}:=a,\;r_0:=b,$

where each quotient $k_i\in\mathbb N$ and each remainder satisfies $r_{i+1}<r_i$ (Negrepontis et al., 19 Nov 2025).

From this process one obtains the continued-fraction expansion

$\frac a b = k_0+\cfrac{1}{k_1+\cfrac{1}{k_2+\ddots}} = [k_0;k_1,k_2,\dots].$

When the process never terminates, the quotient sequence is infinite. The crucial periodicity condition is the Logos Criterion: if for some $n<m$ one has equality of successive remainder-ratios, then the anthyphairesis becomes periodic from that point onward. In one formulation,

$A$ 0

implies eventual periodicity of the continued fraction (Negrepontis, 2012, Negrepontis et al., 19 Nov 2025).

The 2025 reinterpretation further states that every real quadratic irrational has an ultimately periodic expansion. Within the Platonic application, that mathematical fact functions as the model for a Being that is simultaneously infinite in division and stable in structure.

3. Division, Collection, and the Logos Criterion

Plato’s method of Division and Collection is presented as the philosophical counterpart of anthyphairetic division and periodicity. Division corresponds to the unending dyadic unfolding of a Being into opposed species. Collection corresponds to Logos: the recognition that a ratio appearing at one level of division is equal to a ratio appearing elsewhere in the division tree, just as equality of successive remainder-ratios signals periodicity in anthyphairesis (Negrepontis, 2012).

In the Sophistes, two canonical examples are used.

The Angler: the abbreviated division proceeds from all art to acquisitive, coercive, hunting, water-animal hunting, and finally to tridentry and angling. The decisive equality is

$A$ 1

This recurrence of ratio is treated as the analogue of periodicity and “collects” the division into the unity called Angling.

The Sophist: the division proceeds through human productive arts, images, phantastic, self-mimetic, opinion-based imitation, dissembling, and sophist. The cited equalities are:

Step 1 ratio = Step 4 ratio: real / image = knowable / opinable
Step 2 ratio = Step 5 ratio: likeness / phantastic = simple / dissembling
Step 3 ratio = Step 6 ratio: instrumental mimetic / self-mimetic = demagogue / sophist

Each equality is presented as a Platonic Logos analogous to

$A$ 2

thereby collecting the infinite division into the One called the Sophist (Negrepontis, 2012).

A plausible implication is that Collection does not cancel plurality; it stabilizes it by showing structural recurrence within the plurality.

4. Self-similar One, dialectic number, and associated doctrines

In a periodically anthyphairetic pair, every remainder segment generates, by the same infinite sequence of quotients, a sub-process identical to the whole. The philosophical interpretation translates this into a doctrine of self-similar Oneness: each part “participates in Logos” and is equalized to the whole. Plato’s Forms are therefore described as self-similar Unities, infinitely divisible yet one (Negrepontis, 2012).

The 2025 paper extends this structure to several difficult Platonic themes. It identifies Division and Collection with Name and Logos, and with True Opinion plus Logos. It also introduces the “plus one” rule for dialectic numbers, stated as: $A$ 3 For an anthyphairetic sequence $A$ 4, there are $A$ 5 successive ratios $A$ 6 but $A$ 7 terms $A$ 8 (Negrepontis et al., 19 Nov 2025).

The same reinterpretation uses periodicity to explain the “Indivisible Line,” the statement that “not-Being is Being,” and the Third Man Argument. Its claim is that once periodicity is established, no genuinely new unit appears beyond the period; hence an Idea can be “indivisible” despite infinitely many parts. Likewise, one term of a dyad may be designated “not-Being,” yet by equalization still counts as a genuine Being. The Third Man regress is said to be neutralized because, after one full period, every new “third” form coincides, up to anthyphairetic equivalence, with the original (Negrepontis et al., 19 Nov 2025).

These are not uncontested historical conclusions; they are components of a mathematically driven reconstruction of Platonic ontology.

5. Zeno, de-mathematization, and interpretive controversy

The 2025 work frames its project as a “radical re-interpretation” of Plato against a long trend of de-mathematization in modern Platonic scholarship (Negrepontis et al., 19 Nov 2025). According to that argument, Plato’s intelligible Being has the structure of the philosophical analogue of a geometric dyad in philosophic anthyphaeresis, and this structure is visible not only in the Sophist and Parmenides but also in the Theaetetus, Statesman, and Meno.

A central extension concerns Zeno. The paper argues that Zeno’s “true Being” and Plato’s intelligible Being essentially coincide, and that Zeno’s arguments and paradoxes have mathematical content grounded in anthyphairesis. In this reading, Zeno’s description of beings as both infinite in multitude and finite in number corresponds to infinitely many remainders together with a finite period; sensibles, by contrast, are associated with non-periodic anthyphaireses (Negrepontis et al., 19 Nov 2025).

The paper explicitly presents this as running against Burkert’s claim that “ontology is prior to mathematics.” More broadly, it disputes what it describes as the dominant interpretation of Zeno’s paradoxes as devoid of mathematical content. The significance of the dispute lies in methodology: the debate is not only about Plato’s doctrines, but about whether late archaic and classical Greek ontology should be reconstructed through mathematical structures such as incommensurability, periodicity, and continued fractions.

APEIRIA is also the name of a 2026 model for 3D spatial reasoning that aims to bridge neuro-symbolic 3D concept learners and end-to-end 3D multi-modal LLMs. Its stated objective is to distill symbolic reasoning patterns into MLLMs with natural-language chain-of-thought while preserving transparent reasoning and modular interchangeability of planning and perception components (Mo et al., 31 May 2026).

Its architecture is object-centric. A ScanNet/Multi-view scene is first segmented by Mask3D into $A$ 9 object instances. Each instance $a>b$ 0 is featurized by a 3D geometric descriptor $a>b$ 1 via Uni3D, a 2D appearance descriptor $a>b$ 2 via DINOv2, and a learnable “lattice” positional embedding $a>b$ 3 for center $a>b$ 4 and size $a>b$ 5. The final object tokens

$a>b$ 6

are interleaved with text-tokenized instructions and fed into a frozen or LoRA-tuned LLM backbone such as Qwen3-VL-8B. A symbolic program module provides primitives including scene(), filter(·), and relate(·), whose execution traces are serialized into a CoT string of the form

$a>b$ 7

During supervised fine-tuning, the model is trained to reproduce plans, executions, and answers from the same object tokens and query; at inference, external plans from GPT-4 or Claude, or stronger detectors such as SegDINO3D, can be injected through the explicit interface (Mo et al., 31 May 2026).

The training procedure is a three-stage curriculum.

Stage 1: 3D Perception Alignment teaches object recognition, localization, and captioning with cross-entropy on token generation.

Stage 2: CoT-SFT injects query decomposition and stepwise verification distilled from symbolic programs. The reported data consist of approximately 78K single-step traces from ScanNet/MMScan and approximately 66K two-step traces from Sr3D.

Stage 3: CoT-RL extends reasoning to open-vocabulary concepts and deeper nestings on ScanRefer and Multi3DRefer. The reward is

$a>b$ 8

with

$a>b$ 9

and a binary format reward enforcing valid $a>b>0$ 0 tags. Optimization is performed with Group-Relative Policy Optimization (GRPO) using a clipped surrogate objective (Mo et al., 31 May 2026).

The reported quantitative results are as follows.

Benchmark	APEIRIA	Comparator(s) in data
ScanRefer grounding	[email protected] 58.4; [email protected] 51.2	LLaVA-3D 50.1/42.7; Inst3D-LMM 57.8/51.6
Multi3DRefer grounding	[email protected] 59.2; [email protected] 53.8	LLaVA-3D 49.8/43.6; Inst3D-LMM 58.3/53.5
APEIRIA† (+mod)	ScanRefer 60.5/53.2; Multi3DRefer 60.9/55.2	† denotes feeding in SegDINO3D proposals at `scene()`
Nr3D zero-shot	36.5%	NS3D zero-shot 7.3%; supervised 33.9%
Scan2Cap / SQA3D	[email protected]/.5 90.6/84.1; Exact-Match 58.6	prior $a>b>0$ 1; prior best $a>b>0$ 2

Within its own problem domain, APEIRIA is therefore positioned as a system that transfers the syntax of symbolic reasoning rather than concept-specific detector knowledge. The paper’s central claim is that this yields open-vocabulary flexibility without abandoning explicit intermediate structure (Mo et al., 31 May 2026).

7. Relationship between the two uses

The philosophical apeíria and the machine-learning system APEIRIA belong to different disciplines, but both are organized around the relation between an unbounded process and a structure that renders that process intelligible. In the Platonic case, the relevant pair is Division and Collection, with Logos supplying periodicity and self-similar unity (Negrepontis, 2012). In the machine-learning case, the relevant pair is open-ended language modeling and explicit symbolic trace structure, with the plan/execution interface preserving modularity while allowing open-set reasoning (Mo et al., 31 May 2026).

This suggests a merely analogical commonality rather than a historical or doctrinal continuity. The ancient term names a theory of intelligible infinity structured by periodic form; the modern acronym names a 3D reasoning architecture structured by distilled symbolic syntax.