Abstracta in Time: A Metaontological Reappraisal of Mathematical ‘Existence’

Abstract

This paper investigates how abstracta, such as numbers or functions, can be said to emerge at particular times, as in the claim that “complex numbers did not exist before the sixteenth century.” Standard accounts encounter well-known difficulties. Platonism posits timeless entities and struggles to explain historical emergence, while nominalism denies mathematical existence and fails to explain applicability. The paper develops a neutralist metaontology that interprets temporally indexed existence claims as statements about when a concept is available within a community’s inferential and representational practices. Unlike deflationary or thin-object views, neutralism distinctively formalizes these claims through the operator EX_t(x), which captures the historical emergence of abstracta without positing timeless entities. Formally, ‘x exists at t’ is true iff EX_t(x), where EX_t(x) specifies that x is licensed by the rules, resources, and acceptance conditions operative at t. This framework provides a systematic account of temporally indexed existence claims in mathematics, illustrated by historical cases such as zero and complex numbers, and improves on both Platonist and nominalist approaches.

1. Introduction

This paper investigates how mathematical abstracta—such as numbers, functions, or geometric objects—can be said to emerge or come into existence at particular points in history. Typical examples include claims that ‘zero did not exist in Ancient Greece’ or ‘complex numbers were unknown before the sixteenth century’ [1,2]. Such historically framed existence claims concern the abstracta themselves, not merely the concepts of them, and it is this feature that makes them difficult to reconcile with standard metaphysical accounts. Throughout this paper, ‘existence’ refers to the status of the abstracta themselves rather than merely to the availability of concepts of them; neutralism offers an account of how abstracta can be said to emerge historically without collapsing into mere conceptual change. Platonism, which posits timeless entities, renders historical emergence epistemically puzzling [3,4], whereas nominalism, which denies mathematical existence altogether, struggles to explain the applicability and apparent objectivity of mathematics [5,6].

This paper develops a neutralist metaontology, which interprets temporally indexed existence claims as statements about when a concept becomes available within a community’s inferential and representational practices. Neutralism shifts attention from metaphysical change to conceptual adoption. Its formal device, EX_t(x), specifies that ‘x exists at t’ is true if and only if x is licensed by the practices operative at t.

Unlike semantic deflationism, which treats existence-talk as merely context-sensitive semantics, neutralism provides a principled criterion for when temporally indexed existence claims are literally true. It grounds existence in historically embedded practices while preserving the truth-aptness of mathematics. For example, a statement such as ‘2 + 2 = 4’ remains truth-apt without positing timeless, mind-independent entities.

The paper develops this view in four steps. Section 2 situates neutralism in dialogue with Azzouni and Linnebo; Section 2.7 advances positive arguments for temporally indexed existence; Section 3 applies EX_t(x) to case studies; and Section 4 clarifies neutralism’s position between Platonism and nominalism.

On this account, temporally indexed claims reflect conceptual availability rather than metaphysical discovery. For example, ‘zero did not exist in Ancient Greece’ is literally true because Greek mathematics lacked the resources to employ zero [1,2]. The case of complex numbers likewise illustrates how abstracta can be said to ‘exist’ through discursive integration rather than metaphysical instantiation [7].

The problem of temporal existence raises broader questions about how entities are said to exist in time. Ordinary claims such as “no human societies existed in the Arabian Peninsula before European settlers arrived” function historically, not metaphysically. Likewise, “complex numbers did not exist before the sixteenth century” records their absence from mathematical practice, not their absolute nonexistence. Distinguishing temporal location, that is, when something exists, from temporal quantification, that is, how existence extends across time [8], is crucial for abstracta such as numbers, propositions, and geometric forms, which lack spatiotemporal location yet remain indispensable [9].

Neutralism addresses this puzzle by reinterpreting “x exists at t” as a claim about recognition, reference, and use, rather than intrinsic being. It is not a comprehensive ontology but a methodological tool for clarifying temporally indexed existence claims. On this view, abstract objects exist at a given time when they play a role in historically situated practices, unlike physical objects, whose existence is spatiotemporal.

Neutralism thus reconciles timeless mathematical truths such as “2 + 2 = 4” or “√2 is irrational” with the historical unfolding of conceptual resources. The operator EX_t(x) decouples truth from metaphysical being: theorems remain valid, while temporal existence depends on when communities acquire the resources to articulate them. This permits literal temporal existence claims without ontological inflation, preserving rigor while tracing the historical emergence of concepts.

This paper addresses a specific puzzle: ordinary and scholarly discourse contains tensed existence claims about abstracta (e.g., “imaginary numbers did not exist before the sixteenth century”). Platonism treats these as strictly false or paraphrastic; nominalism treats them as about use only. Neutralism introduces a formal operator EX_t(x) delivering literal truth-conditions: ‘x exists at t’ is true iff x is licensed by the inferential and representational practices operative at t. This is not paraphrase but a semantics with predictive force (Section 2.7 and Section 3): it yields the correct verdicts across historical/cross-cultural cases without auxiliary stories, while preserving the tenseless truth of theorems.

1.1. The Neutralist Alternative

Before developing the view, it is important to clarify how this use of ‘neutralism’ relates to existing positions. In current philosophy of mathematics, ‘quantifier neutralism’ is most closely associated with Azzouni, who argues that quantification need not carry ontological commitment. My neutralism differs from Azzouni’s: it is not a thesis about the meaning of quantifiers but a metaontological framework for interpreting temporally indexed existence claims. Likewise, although Craig’s ‘neutral logic’ is mentioned later for structural analogy, the present view is not derived from Craig’s logical programme. The neutrality at stake here concerns ontology, specifically, how abstracta can be said to exist-at-t relative to historically situated practices, rather than neutrality at the level of logic or syntax.

Neutralism rejects both Platonism and nominalism by shifting the locus of existence claims from timeless metaphysical domains to historically situated conceptual frameworks. On this view, abstract objects have significance only once they are embedded within the inferential rules and representational resources of the relevant community (for example, mathematical or musical), rather than existing timelessly (as Platonism holds) or not at all (as nominalism holds). Platonism faces Benacerraf’s access problem, while nominalism faces the indispensability problem. Neutralism offers a third option: it explains how existence claims can track the historical availability of mathematical concepts without invoking either timeless abstract realms or wholesale denial of mathematical ontology. It treats existence as framework-relative—that is, relative to the conceptual and inferential norms of particular historical communities—using the operator EX_t(x) to specify when a concept is licensed by the inferential practices operative at time t (terminological note: “exists/does not exist (at t)” targets the tensed copula in ordinary language, not the untensed existential quantifier of formal logic [10,11]). This allows historical phenomena, such as the emergence of complex numbers, to be explained in terms of shifts in symbolic resources and inferential practices, rather than metaphysical discovery.

Neutralism may also extend beyond mathematics; for instance, the existence of a musical composition at time t could similarly depend on whether the relevant community possesses the representational resources to recognise and employ it, suggesting a promising direction for further research.

Neutralism, as used here, is best understood as a metaontological framework, distinct from Azzouni’s quantifier neutralism and Craig’s neutral logic, rather than as a compromise position. It preserves the truth-aptness of mathematical discourse while minimizing metaphysical commitments. This distinguishes it from Azzouni’s non-committal quantification and Linnebo’s abstractionism, although it remains compatible with insights from both. Its key innovation lies in the explicit temporal indexing EX_t(x), which allows statements such as ‘zero did not exist in Ancient Greece’ to capture conceptual development rather than metaphysical absence.

Under neutralism, abstract objects are considered ‘thin’: their existence is fully determined by the conceptual roles they occupy within discourse, which accounts for why statements such as “2 is an even prime” are literally true within a given practice without invoking a Platonic realm [12,13]. The view further resists Quine’s ontological interpretation of quantifiers: quantification can remain ontologically neutral while conveying temporal information, since the statement “there are imaginary numbers” is true relative to the inferential rules operative at a given time t [4,14,15].

1.2. Temporal Dimensions of Abstract Objects

Neutralism distinguishes between the timeless validity of mathematical truths and the historical conditions under which abstracta are recognised and deployed within mathematical practice. Importantly, this distinction does not render mathematical truths mind-dependent: neutrality applies only to existence-at-t, not to the truth-values of theorems themselves. Arithmetic statements such as “2 + 2 = 4” remain tenselessly true across all t; what changes is the availability of the conceptual framework that licenses their articulation.

Platonism reads untensed statements such as ‘2 + 2 = 4’ as evidence of timeless ontology, while nominalism denies the existence of abstracta altogether. Neutralism instead maintains that mathematical propositions are tenseless in form, but their existence-at-t depends on when conceptual resources allow their recognition [16]. The operator EX_t(x) formalises this distinction: truth remains stable, while existence is indexed to historical practice.

For example, Greek mathematics lacked the conceptual resources to recognise zero, which emerged in India in the fifth century and later spread to Europe. Thus, ‘zero did not exist in Ancient Greece’ is literally true under EX_t(x) [1,2]. Neutralism interprets this absence not as a failure to apprehend a timeless object but as genuine conceptual non-existence. This avoids Platonism’s unexplained epistemic gaps and explains emergence in terms of available inferential and representational practices.

Historical cases illustrate how abstracta emerge through conceptual integration. Zero became meaningful only once it was embedded within mathematical frameworks, and imaginary numbers gained acceptance after initial resistance when new inferential needs made them indispensable. Their emergence marks the point at which EX_t(x) is satisfied, rather than the discovery of pre-existing objects. Likewise, geometric ideals such as perfect circles are intelligible within structured reasoning, not located in space and time.

Neutralism’s framework further aligns with ordinary language analogies. References to “holes in a shirt” or “edges of a shadow” denote context-dependent features without requiring independent existence. Similarly, abstract objects acquire meaning and indispensability through their role in practices. Unlike metaphorical absences, however, existence claims about abstracta retain literal truth-aptness because EX_t(x) fixes their truth-conditions at a given time.

In sum, neutralism reconciles timeless mathematical validity with the historical contingency of conceptual uptake, offering a more parsimonious alternative to Platonism and nominalism.

2. Discussion

2.1. Neutralism, Existence and Abstracta

Neutralism challenges Quine’s dictum that quantification entails ontological commitment (“to be is to be the value of a variable”; [17]) by arguing that quantificational truth does not require metaphysical existence [4,18,19]. Linnebo develops a ‘thin’ conception of mathematical objects grounded in abstraction principles [4]. Although Linnebo positions this view within a lightweight realist framework rather than a neutralist one, his account of thin objects nevertheless supplies tools that neutralism can appropriate without adopting his metaphysical commitments. Neutralism builds on these insights by introducing EX_t(x) as a formal mechanism for evaluating temporally indexed existence claims. Unlike purely therapeutic approaches, neutralism treats such claims as literally true within temporal and conceptual frameworks, while avoiding commitment to timeless entities. This provides a clearer explanation of the historical emergence of mathematical objects than either Platonism, which postulates timeless existence, or nominalism, which denies existence altogether. Linnebo’s abstractionism is compatible with, but not fully equivalent to, neutralism; differences are discussed in Section 2.6.

2.2. Framework-Dependent Existence and Ordinary Claims

Framework-dependence is a familiar feature of ordinary discourse. For example, rules of chess or national borders ‘exist’ only within social and institutional practices, yet they have real and wide-ranging consequences. Neutralism extends this insight to mathematics: mathematical objects exist through their role in reasoning and application, without invoking Platonic ontology. Just as chess rules structure gameplay and borders organise social reality, mathematical objects structure mathematical practice and scientific understanding. Their existence is robust within frameworks, even if not metaphysically independent. This analogy supports the neutralist claim that existence-at-t can be understood in terms of conceptual and institutional uptake rather than metaphysical discovery.

Everyday Framework-Existence

The same schema applies beyond mathematics. Chess rules: EX_t(rules) is true when, at t, the constitutive regulations are operative within the institution, explaining why a player can be disqualified for violating them. Borders: EX_t(border B) is true when administrative and legal practices at t license B’s demarcation. Race and gender: EX_t(K) is true when social-institutional practices at t license K’s classificatory and normative roles. These cases show that EX_t(·) captures a general pattern of practice-dependent yet truth-apt existence claims.

2.3. Context-Dependent Existence

Azzouni refines neutralism by showing that “exist” is not inherently ontologically committing; its force depends on context [20]. We can truthfully talk about numbers, fictional characters, or theoretical entities without positing metaphysical reality. Neutralism interprets “x exists at t” as conceptual recognition within practices at t. The formal notation EX_t(x) captures this: the claim is true when the relevant inferential and representational resources are operative at t.

This approach imposes constraints on linguistic expressions for ontological claims. Statements like “Prime numbers exist” or “The number seven does not exist independently of mathematical discourse” function as ontological assertions. In contrast, claims like “There are 20 students in the classroom” are descriptive, emphasizing situational concerns rather than ontology. Expressions like “exist,” “there is,” or “entity” are suited to ontological discourse; for example, “There are uncountably many real numbers” or “Set-theoretic universes are not concrete entities” engage ontological commitments [20].

While mathematics treats abstract entities as if they exist, this does not commit us to independent reality. Statements such as “Category theory provides a foundation for mathematics” or “There are functions continuous but nowhere differentiable” are ontologically neutral: “exist” and “there is/are” function linguistically without implying mind- or language-independent reality [14]. This linguistic contextualism undermines the Platonist claim that existence statements necessarily refer to timeless entities, and challenges nominalism’s eliminativism by preserving literal truth-aptness. Neutralism integrates this insight into a formal framework, EX_t(x), which explains how existence claims can vary over time while maintaining stable truth conditions for mathematical theorems.

2.4. Reference Without Existence

Neutralism distinguishes between reference with ontological commitment (reference^R) and reference without it (reference^E) [20]. This allows discourse about abstracta, fiction, or mathematics to be meaningful without presupposing independent entities. For instance, “the square root of two is irrational” is true within number theory because the concept is embedded in the inferential system, not because √2 exists timelessly [21].

Reference^E underpins EX_t(x): temporally indexed existence claims can be literally true without committing to Platonic or Meinongian objects. “Exists” in ordinary language is context-sensitive and flexible, reflecting speaker intention rather than metaphysical commitment [20]. Adams similarly observes that everyday references to rocks, numbers, or cities are sincere but do not require deep ontological claims [16].

Applied to abstract objects, temporal existence indicates recognition and use within a conceptual framework rather than spatiotemporal presence. For example, “At t, the number seven exists” signifies that the concept of seven is licensed by operative inferential practices at t, making the claim literally true in a framework-dependent sense.

This distinction between reference^R and reference^E strengthens the neutralist position by showing how meaningful reference can occur without metaphysical commitment, thereby addressing a key Platonist objection.

2.5. Intentionality and Abstract Reference

Neutralism also aligns with the phenomenological tradition: intentionality does not require its objects to exist [22,23]. We can direct thought toward numbers or strategies without metaphysical commitment. Thus, temporal claims about abstracta are best read as about their role within intentional and discursive frameworks, not about their independent reality. Under the neutralist model, EX_t(x) formalizes this insight, translating intentional and discursive uptake into clear conditions for when such existence claims are literally true.

By formalising this phenomenological insight through EX_t(x), neutralism provides a precise account of how intentional reference can support literal existence claims without invoking timeless abstracta.

2.6. ”Thin” Mathematical Objects and Neutralism

The concept of “thin” mathematical objects is central to contemporary philosophy, framing abstract entities like numbers and functions as being individuated through abstraction principles and inferential roles, rather than mind-independent intrinsic properties [4,20]. These entities are ontologically minimal, or “thin,” serving primarily to facilitate deductive reasoning and formal representation [4,24]. Neutralism embeds this thinness in a temporal framework: EX_t(x) allows abstract objects to have framework-relative existence without postulating ontologically weighty existence.

Quantificational expressions like “there exists” are context-dependent, governed by mathematical practice rather than metaphysical reality [19]. Assertions such as “there are infinitely many prime numbers” are understood as deploying internal quantifiers governed by mathematical practice, grounded in inferential frameworks rather than metaphysical assumptions about external reality.

This context-dependence applies equally to reference and epistemic access. Consistent with Azzouni [24] and Hodes [25], reference is evaluated by internal standards—coherence and deductive strength—rather than metaphysical correspondence. This ensures that mathematical discourse maintains integrity without ontological inflation. Furthermore, abstraction principles, such as those used by Frege to define numbers through equivalence classes, are treated as intra-theoretical, linguistic devices, not metaphysical posits. While Linnebo’s work builds on this logicist legacy and is compatible with the minimal metaphysical commitments of neutralism, his abstractionism does not fully endorse neutralism’s comprehensive context-dependence [26].

2.6.1. Conceptual, Epistemological, and Historical Foundations

To clarify, the neutralist framework rejects the metaphysical import of “x exists at time t.” Temporal existence for abstracta is reinterpreted as framework-relative conceptual activation: the point at which an abstract object gains meaning within a historically situated discourse, distinct from spatiotemporal occupation. Saying “Imaginary numbers did not exist in ancient Greece” reflects a discursive absence—the lack of formal tools needed to operationalize them—rather than a metaphysical void. Temporal predication reflects historical shifts in knowledge. Existence is an epistemic status, determined by theoretical uptake, rather than a binary metaphysical fact.

The thinness of mathematical objects yields significant epistemological consequences. Knowledge of these objects is grounded in conceptual frameworks that admit abstraction principles. As Linnebo argues, these principles enable “easy reference” without demanding deep metaphysical explanation, functioning as epistemic tools for structured reasoning rather than uncovering external reality [4]. Neutralism rejects the assumption that quantification implies robust metaphysical existence, allowing statements like “There is a prime number between 2 and 4” to be true without metaphysical commitment [19]. This metaontological framework supports the coherence of thin objects without invoking metaphysical realism.

Historically, the concept of thin objects was systematically developed by Øystein Linnebo [4], securing the existence of mathematical entities through logical abstraction principles rather than robust ontological commitments. This project originates in Frege’s logicist tradition [26]. Abstraction principles state that a unique abstract object corresponds to an equivalence class under a given relation. For instance, Hume’s Principle, central to neo-Fregeanism, asserts that the number associated with concept F is identical to the number associated with concept G if and only if F and G are equinumerous [27]. Formally, this can be expressed as:

∀F∀G(#F = #G↔F ≈ G)

Linnebo argues that mathematical entities are ‘thin,’ existing solely by virtue of the internal coherence of these abstraction principles, without requiring metaphysical substance. While his abstractionism shares this commitment to thinness, it aligns with, yet remains distinct from, the broader neutralist position.

2.6.2. Methodological Commitments and Scope

Neutralism adopts three minimal commitments: (i) Quantifier neutrality—truth in mathematics need not entail robust ontological commitment; (ii) Framework-relativity—existence-at-t for abstracta is fixed by licensing within historically situated practices; and (iii) Agnosticism about extra-framework existence—none of the arguments require a “thick” ontology or reference outside those practices. From Azzouni I take the notions of quantifier neutrality and the reference^E/reference^R distinction; from Linnebo, the idea of ontologically thin abstraction. EX_t(x) adds explicit temporality without importing any thicker metaphysical layer. Thus the six arguments in Section 2.7 remain valid even for readers who deny extra-framework existence altogether.

This thin approach reconciles the utility of mathematical entities with ontological parsimony [28,29]. Abstract entities function as conceptual tools within context-dependent linguistic and conceptual conventions.

In this broader context, neutralism plays a crucial role by providing a means of discussing the temporal existence of abstracta without committing to their deep metaphysical existence. Abstract entities, such as numbers, legal concepts, or strategic frameworks, can be treated as conceptual tools that serve specific purposes without requiring them to exist in the same sense as physical objects. Existence itself is understood as context-dependent, shaped by linguistic and conceptual conventions rather than fixed metaphysical conditions. Statements involving abstracta are thus ontologically non-committing, reflecting their utility within a framework rather than claiming independent reality.

Neutral logic, as Craig argues, maintains that quantificational locutions like “there is” or “there exists” are not inherently ontologically committing but are merely conventional means of structuring discourse [19]. This perspective integrates seamlessly with abstractionist and structuralist views of mathematics, reinforcing the idea that mathematical objects are thin epistemological constructs rather than deeply metaphysical entities. By adopting neutralism, we can meaningfully discuss the temporal dimensions of abstract objects, their discovery, development, and refinement, without burdening ourselves with problematic ontological commitments.

Neutralism builds on this thin ontology by adding a temporal dimension, thereby explaining both the ontological minimalism and the historical variability of mathematical existence claims.

2.7. Positive Arguments for Temporal Existence

The central contribution of this paper is to show that temporally indexed existence claims are not merely convenient reformulations but are literally true under a neutralist account. The neutralist framework formalises these claims through EX_t(x), providing a structured means of evaluating temporally indexed existence. The following six arguments offer independent philosophical, epistemic, and historical reasons for preferring the neutralist interpretation over Platonist and nominalist rivals.

Argument 1: Practice-First Indispensability

P0. (Principle of Explanatory Preference) If Theory T explains phenomenon X at least as well as rivals and avoids problems that are as severe or more severe than those of the rivals, then (ceteris paribus) T is to be preferred.

P1. Any adequate philosophy of mathematics must explain the applicability and predictive success of mathematical practice (X).

P2. Platonism explains X but incurs the serious epistemic access problem.

P3. Nominalism avoids access worries but struggles to explain the indispensability and apparent objectivity of mathematics.

P4. Neutralism explains X by appeal to historically stable inferential practices and does not incur problems as severe as those in P2–P3.

C. Therefore, (ceteris paribus) neutralism is to be preferred to Platonism and nominalism.

Argument 2: Explanatory Minimality

P0. (Parsimony Constraint) Between theories that are equally explanatorily adequate with respect to X, prefer the one with fewer/metaphysically thinner commitments.

P1. Neutralism accounts for mathematical agreement, stability, and applicability at least as well as Platonism and nominalism.

P2. Neutralism avoids commitment to robust, timeless abstracta.

P3. Platonism posits robust timeless abstracta; nominalism sacrifices (or strains) the explanation of applicability/stability.

C. Therefore, by P0, neutralism is preferable on grounds of parsimony.

Argument 3: Epistemic Access Without Mystery

P0. If a theory of knowledge requires that agents stand in an appropriate epistemic access relation to entities E, and no such access relation is available, then knowledge of E is epistemically problematic.

P1. On Platonism, mathematical entities are non-causal and timeless; humans cannot stand in any epistemic access relation (such as perception, interaction, or causal connection) to them.

P1*. Platonism explains mathematical knowledge by appeal to some form of access to abstracta (e.g., grasping, intuition, or apprehension), but offers no mechanism consistent with their non-causal nature.

This reflects the standard Platonist commitment that mathematical knowledge requires some epistemic relation, however construed, to abstracta, even though their timeless and non-causal nature precludes such relations.

C1. Therefore, Platonism renders knowledge of abstracta epistemically problematic. (from P0, P1, P1*).

P2. On neutralism, the truth of mathematical claims and existence-at-t claims are grounded in historically situated inferential and representational practices to which agents do have causal access.

C2. Therefore, neutralism renders knowledge of mathematics epistemically intelligible (no analogous mystery). (from P0, P2)

C. Hence, neutralism is preferable to Platonism on the dimension of epistemic access.

Argument 4: Conceptual and Historical Availability

P0. (Ontological Conservatism) In theoretical contexts, and ceteris paribus, prefer a theory that preserves the literal truth-conditions of central target sentences without requiring paraphrase, since preserving surface structure avoids unnecessary semantic revision and respects competent speaker judgments.

P1. On neutralism, “zero did not exist in Ancient Greece” is literally true because EX_t(zero) was false given the practices operative then.

P2. On straightforward Platonist readings, the same sentence is not literally true (since zero exists timelessly); preserving it requires treating the sentence as elliptical for “was not recognized/used.”

P3. On straightforward nominalist readings, the sentence is either false or must be paraphrased as a statement about usage only.

P4. Platonism, nominalism, and neutralism are otherwise on comparable footing with respect to accounting for mathematical truth and applicability (as established earlier), so ceteris paribus applies.

C. Therefore, by P0 and P4, neutralism enjoys a semantic advantage regarding historically indexed existence claims.

Argument 5: Tensed Semantics

P0. (Linguistic Fit) Ceteris paribus, a theory that preserves ordinary tensed uses of “exist(s)” while accommodating tenseless mathematical theorems is preferable.

P1. Ordinary English uses “exists” tensely; mathematical theorems and quantifiers are tenseless in formal presentation.

P2. Neutralism, via EX_t(x), preserves the ordinary use of tensed existence claims while analysing their truth-conditions in a tenseless, temporally indexed form; mathematical theorems themselves remain tenseless truths. In other words, EX_t(x) is tenseless in logical form even though it grounds the correctness of everyday tensed utterances.

P3. Platonism can preserve tenseless mathematics but treats tensed “exist(s)” talk as derivative (needing paraphrase); nominalism preserves tenseless theorems but typically treats tensed existence claims about abstracta as strictly false or as paraphrase about usage.

C. Therefore, by P0–P3, neutralism better fits the combined linguistic data (ceteris paribus).

Lemma (Non-Relativism) 1.

EX_t(x) does not introduce tense into mathematical truth-conditions; it provides tenseless temporal indices for ordinary tensed assertions. Mathematical truths remain invariant, while only the licensing-conditions for “exists at t” vary with t.

Argument 6: Cross-Cultural Variance

P0. (Anti-Ad Hoc Principle)

When competing theories accommodate the same range of historical and cross-cultural data, prefer the theory that does so without relying on recurring ad hoc auxiliary assumptions.

P1. The historical uptake of mathematical concepts—zero, negative numbers, imaginaries, non-Euclidean geometries—varies widely across times and cultures.

P2. Neutralism accommodates this directly: EX_t(x) varies with the inferential and representational practices operative at a given time and place, so sentences of the form “x did not exist at t in C” come out literally true or false without further narrative supplementation.

P3. Platonist and nominalist accounts can fit the same data only by repeatedly appealing to auxiliary explanations (e.g., epistemic-access stories, paraphrase-and-usage strategies) in order to preserve their ontological commitments.

P4. The conditions for existence-at-t invoke sufficiently human-like cognitive frameworks (Here ‘human-like cognitive framework’ is intended broadly, referring to any sufficiently human-like conceptual and inferential system, including the possibility of non-human or alien agents), that is, any cognitive system capable of sustaining the relevant inferential and representational practices, not exclusively human ones.

C. Therefore, by P0–P4, neutralism is preferable with respect to cross-cultural and cross-framework variance.

Neutralism provides a coherent, parsimonious, and epistemically accessible account of temporally indexed existence. Unlike Platonism, which often explains historical absence by appeal to epistemic limitations, and unlike nominalism, which denies abstracta entirely, neutralism grounds existence claims in the historical and conceptual availability of practices. By doing so, it avoids both the explanatory opacity of Platonist appeals to inaccessible realms and the eliminative inadequacy of nominalism, offering a distinctive middle path anchored in practice-dependent criteria. In summary, in the neutralist account, ‘x exists at t’ is true iff EX_t(x).

Building on the positive arguments outlined in Section 2.7, I now explore the broader implications for abstract object theory and historical conceptualization.

3. Locational Uses of EX_t(x)

In ordinary discourse, existence claims are often qualified by temporal or spatial parameters. For example:

• There existed no potatoes in Asia in the 1450s.

This claim does not assert that potatoes were absent everywhere, only that they were absent within a specified region throughout that period. Formally, the existential quantifier can be indexed both temporally and spatially. Using EX_t(x), where x is an entity and t is a temporal parameter, “x exists at t” expresses existence relative to temporal or contextual conditions. Each case below follows a single format—Prediction → Evidence → Verdict—to test whether EX_t(x) yields correct temporal truth-values. For every historical episode, the framework predicts that “x exists at t” is false when the licensing rules for x are absent and becomes true once those rules are adopted. This procedure replaces anecdotal survey with a structured empirical test of the framework’s explanatory power. While this formalization is abstract, it clarifies how temporal reference interacts with existence claims, including for abstract entities such as numbers or sets.

The locational analogy motivates—but does not replace—the neutralist account: EX_t(x) converts the analogy into a literal truth-conditional rule specifying when temporally indexed existence claims about abstracta are true, thereby connecting linguistic observation to formal semantics.

3.1. Temporal Existence and Abstract Objects

Consider:

2.

At time t, the number seven exists.

This does not imply metaphysical independence but conceptual activation. Temporal existence for abstracta is understood as framework-relative conceptual activation: the moment an object becomes meaningful within a culturally located discourse, rather than spatiotemporal occupation.

Formally, let S be an entity. S has non-ontological existence if and only if:

3.

S is conceived by some thinker(s) at time t.

4.

S lacks spatiotemporal location.

5.

S depends on human conceptual, linguistic, or cultural frameworks for its existence.

This captures abstract objects like numbers or fictional characters while excluding entities posited as independent of human cognition (e.g., deities).

3.2. Temporal Aspects of Existence in Mathematics

The absence of an abstract object in historical records reflects conceptual non-recognition rather than metaphysical non-existence. For instance:

6.

The number zero did not yet exist in ancient Greece.

Here, “exist” signals that the concept had not been formally developed. Neutralism frames this as a temporal and practice-dependent phenomenon: EX_t(zero) is false in Greece but becomes true once the concept is introduced and integrated into mathematical practice. Importantly, the falsity of EX_t(zero) at that time does not entail absolute nonexistence, but only the lack of conceptual availability within the Greek framework, preserving truth-aptness without metaphysical commitments. Similarly, imaginary or negative numbers acquire temporal existence through recognition and formalization rather than timeless discovery.

3.3. Temporal Presence and Conceptual Emergence

Physical objects and abstract entities differ in temporal existence. “No potatoes existed in Asia in the 1450s” concerns physical absence in a spatiotemporal region. “Imaginary numbers did not exist in ancient Greece” addresses conceptual absence at a historical juncture.

Abstract objects lack independent existence; their recognition, conceptualization, and relevance emerge through human intellectual activity, communication, and cultural practices. We may define “to be present” as shorthand for “to be located at the present time.” This applies to fictional characters, mathematical objects, or other abstractions. Their presence signals engagement in discourse rather than independent ontological status [30].

Thus, in common discourse, “to exist” can often be replaced with “to be present,” emphasizing temporal engagement over physical or metaphysical location. Within neutralism, however, this presence is not merely sociological; EX_t(x) ensures that such presence corresponds to structured truth-conditions fixed by inferential and representational rules, preserving mathematics’ objectivity despite its temporal indexing.

3.4. Contextual and Conceptual Existence

Abstract objects emerge in diverse ways:

• Fictional characters have identifiable points of origin (e.g., Sherlock Holmes).
• Mathematical objects emerge through formal systems, logical necessity, and theoretical development.

For example, zero was unknown in ancient Greece but formalized in India around the 5th century CE and later integrated into Western mathematics [1]. This demonstrates how mathematical entities can transition from non-recognition to central theoretical status, supporting a neutralist, temporally indexed view of existence. Such transitions mark the point at which EX_t(x) becomes satisfied, i.e., when the inferential and representational resources required for the concept are in place, distinguishing genuine existence-at-t from mere cultural familiarity.

3.5. Application of Temporal Existence to Mathematics

Building on the temporally indexed existence claims in Section 2.7, this section examines how abstract mathematical entities acquire “temporal existence” through human recognition, theoretical development, and practical application. Historical episodes such as negative numbers, imaginary numbers, and set theory illustrate how mathematical objects emerge within conceptual frameworks, supporting the neutralist perspective. Neutralism treats existence as framework-relative and historically situated, rather than as a timeless, mind-independent property.

1. 

Negative Numbers: Temporal Emergence Through Conceptual Development

Greek mathematicians largely rejected negative numbers, considering them meaningless within geometric and magnitude-based paradigms. Yet, historical evidence shows their use in practical arithmetic contexts long before formal acceptance. Brahmagupta, in 7th century CE India, incorporated rules for negative quantities, and medieval Arabic and European merchants employed negative balances to represent debts in trade ledgers [31]. Full theoretical legitimization arose only with the development of symbolic algebra in the 16th–17th centuries, particularly in the work of Cardano and Descartes [32].

From a neutralist perspective, negative numbers did not gain existence by metaphysical discovery; rather, they emerged due to practical and conceptual necessity. In other words, EX_t(negative numbers) became true once mathematical frameworks incorporated rules and inferential practices capable of licensing their use, formalizing the role of necessity in terms of practice-dependent conditions.

2. 

Imaginary Numbers: Transformation from Conceptual Obscurity to Essential Theory

Initially, the square root of −1 was regarded as a formal device or nonsensical expression. Early algebraists, including Cardano, used it reluctantly in problems lacking real-number solutions [33]. Euler later standardized notation (i = √−1) and related imaginary numbers systematically to real numbers via the exponential function, while Gauss provided a geometric interpretation using the complex plane [34].

Neutralism interprets this trajectory as the temporal emergence of imaginary numbers. They gained existence when theoretical frameworks recognized their utility and coherence. What was initially dismissed gradually became essential to advancing complex analysis and other fields, highlighting how abstract entities are shaped by intellectual and practical contexts.

3. 

Set Theory and the Infinite: Conceptual Innovation in Mathematics

Cantor’s development of set theory in the late 19th century exemplifies conceptual innovation. His theory of transfinite numbers, ordinal hierarchies, and proofs such as the uncountability of the real numbers expanded mathematical understanding of infinity [35]. Initially controversial, set theory became foundational to formal mathematics within a few decades.

From a neutralist standpoint, mathematical entities such as ℵ₀, ℵ₁, or the power set of ℝ emerged through conceptual innovation and theoretical necessity, not through metaphysical discovery. Their existence is indexed to their coherence, utility, and recognition within evolving mathematical frameworks.

Temporal Emergence and Intellectual Practice

Historical patterns do not by themselves establish a metaphysical conclusion, but they do place explanatory pressure on views that treat mathematical entities as fixed and timeless. What the historical record shows is that mathematical concepts appear, disappear, and change their inferential roles across cultures and periods. Neutralism offers a natural explanation of this temporality: shifts in intellectual practice alter which abstracta are available within a community’s conceptual framework, without requiring any change in metaphysical status. On this view, what “emerges” over time is not the object itself but the community’s capacity to deploy it.

Early rejection of imaginary numbers reflected limited utility within contemporary frameworks, while their later indispensability arose from evolving theoretical needs. The application of abstract objects in the natural sciences—Hilbert spaces, vector fields, wave functions—demonstrates how entities acquire practical relevance through their conceptual integration. Under neutralism, such objects exist within theoretical systems contingent on their recognition and use, rather than as items inhabiting an independent metaphysical realm [36].

In sum, neutralism provides a coherent metaontological framework: abstract mathematical objects gain existence-at-t through historically situated conceptual roles within intellectual practice, while mathematical truths themselves remain stable and tenseless.

3.6. Reinterpreting ‘Existence’ for Abstract Objects

To accommodate abstract objects, existence can be understood contextually rather than as a binary property:

i.

Existence through Conceptual Reference: Abstract objects exist when recognized or applied within theoretical systems; the Pythagorean theorem “exists” insofar as it functions in geometry, engineering, and related disciplines.

ii.

Existence through Theoretical Necessity: Some objects exist due to their indispensability in formal systems. Complex numbers exist because certain mathematical formulations require them, even if historically contested.

iii.

Existence through Cultural and Intellectual Engagement: Fictional or theoretical entities persist through ongoing reference and interpretation; literary characters and theoretical constructs are sustained via engagement.

iv.

Existence through Physical Application: Mathematical objects gain epistemic significance when applied in the physical sciences, e.g., differential equations in physics and engineering.

This approach reconciles the temporal dimension of mathematical existence with its abstract nature and aligns with debates in philosophy of language regarding discourse and ontological commitment. “Exist” signals the pragmatic role of objects in reasoning and application rather than mind-independent reality.

Physical entities exist in specific times and places, while abstract objects exist through conceptual and cultural recognition. Within an anti-realist framework, temporal existence depends on acknowledgment within intellectual contexts [37]. The number 5 “exists” in mathematics contingent on its recognition and application within that framework.

Neutralism formalizes this via EX_t(x), specifying conditions under which engagement yields literally true existence claims, distinguishing it from broader constructivist accounts. Existential claims about abstracta have tensed truth-conditions fixed by the rules and resources available at the relevant time; when rules change, the truth of “x exists now” can change while tenseless theorems remain stable [30]. This framework clarifies mathematical existence and enriches understanding of ontological commitment, epistemic justification, and the evolution of abstract knowledge.

4. Conclusions

The paper has moved beyond restatement to offer positive arguments for temporally indexed existence claims about abstracta. In the neutralist account, ‘x exists at t’ is true just in case x is available for representation and inference within the mathematical practices operative at t. This explains why historically accurate sentences such as “Zero exists now but did not exist in Ancient Greece” are literally true, while mathematical theorems remain tenseless and stable. The view improves on Platonism by explaining the timing of conceptual emergence and avoiding Benacerraf’s access problem, and on nominalism by effectively accommodating mathematics’ applicability without ontological inflation. The resulting picture advances the “invented/discovered” debate: mathematical truths are not invented, but the existence-at-t of mathematical entities is practice-dependent. This situates neutralism within an anti-realist tradition while distinguishing it from both semantic deflationism, which reduces existence to shifting linguistic contexts, and structural Platonism, which preserves timeless entities. Neutralism instead grounds existence in historically contingent practices via EX_t(x), reconciling timeless truths with temporal emergence.

This paper’s contribution is twofold: it clarifies the anti-realist thesis with a precise metaphysical and semantic package, and, more importantly, it offers positive arguments that show how the view explains core features of mathematical practice better than its rivals. Its originality lies in formalizing temporally indexed existence claims through EX_t(x), a mechanism absent from rival accounts, thereby offering a unique synthesis of anti-realism, thin ontology, and temporally situated practice.