Abstracta

Map or Terrain?

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Where is the number $7$?

It’s not in your cupboard. It’s not floating in the sky or etched into the bark of a tree. You’ve never bumped into it on the sidewalk. And yet, you use it daily. You rely on it. You trust that $3 + 4 = 7$ not because someone told you, but because something deeper holds it true. Seven seems invisible, untouchable — and yet absolutely real. But if it’s real, where is it?

This is the ontological puzzle of abstracta — concepts like numbers, sets, and propositions that explain our world yet remain elusive. They don’t age or rot. They don’t take up space. They are intangible in and of themselves and yet somehow they are utilized to explain everything. We speak of 'the set of even numbers' as if it exists and in one sense, it does. The set of even numbers exists by definition that it is literally 'that which is out of' the set of ordinary numbers. But is it real? Mental? Fictional?

This is more than a curiosity of logic or math. The question reaches to the heart of both ontology and theology. For just as we ask how the number $7$ is without being in or of the world, we might ask: How is God, if not in or of the world? If God as Creator is not 'that which is out of' anything and therefore does not exist but is instead ontologically prior to all that does exist, then is God, like $7$, not existent but abstracta? Not of existence but Principal to it? Not an invariant being which is a contradiction in terms but an invariant process?

In this article, I explore how abstracta — that which is ‘drawn away from’ the material world — functions to explain it. Abstracta are not existents but principal to explaining them. Abstracta are not the terrain itself but the map we use to navigate it. We ask not whether abstracta exist but how they function to give meaning to existence — without existing in and of themselves. And in asking that, we may find that the number $7$ and the idea of God as creator are not so far apart: not as things in the world but as constant invariants through which the world becomes intelligible.

Classical Positions

The philosophical debate over abstracta — entities like numbers, sets, and propositions — has unfolded over centuries. At stake is not just whether these things exist but what we mean by existence at all. The classical positions can be mapped across a spectrum of realism and anti-realism, with various attempts to explain how such elusive entities can be both indispensable and immaterial.

Platonism holds that abstract objects exist in a timeless, non-physical realm of Forms. The number $7$, in this view, is not invented by humans but discovered — eternal, unchanging and independent of minds or matter. Mathematical truths are true because they reflect these perfect entities, which we perceive dimly through reason rather than sense.

Nominalism denies the reality of abstracta altogether. Numbers are just names we give to repeated patterns. There is no 'sevenness' floating beyond language — just a convention for describing grouped objects.

Conceptualism offers a middle path: abstracta exist but only in minds. Seven is real because it is conceived. If no minds existed, neither would abstracta.

Formalism, Logicism, and Intuitionism each offer different accounts of how mathematical abstracta are grounded:

• Formalism treats mathematics as the manipulation of symbols within a formal system, where abstracta are defined by syntactic rules.
• Logicism reduces mathematics — and thereby mathematical abstracta — to pure logic.
• Intuitionism regards mathematical abstracta as mental constructions, dependent on the thinking subject.

Quine–Putnam indispensability argues we must accept the existence of numbers because they are indispensable to science. If numbers are woven into our best explanations, we are ontologically committed to them.

Across these positions lies a tension: abstracta are either strangely real or strangely useful. They seem nowhere and everywhere at once.

Current Flashpoints

The classical debate is now entangled with modern fields:

• Math and Physics: Is math invented or discovered? If it’s discovered, then abstracta must already exist. Physicists like Max Tegmark go further: the universe is math.^[1]
• Structuralism and Category Theory: Modern math often emphasizes relationships over entities. In category theory, numbers are not things but positions within patterns. This mirrors the ontological idea that identity emerges from relation.
• Modal Realism: If all possible worlds exist, so do all possible propositions — leading to ontological inflation.
• Cognitive Science: Numerical intuition appears in infants and animals, suggesting abstracta may not be purely linguistic.
• Computational Formalization: Tools like Rocq and Lean allow us to construct and verify abstract truths. This blurs the line between discovered truth and executed code.

Abstracta have never been more functionally central — or ontologically mysterious.

Abstracta within the Conference of Difference

In the Gospel of Being, existence is not a substance but a condition: a 'process of declaring together' revealed as the conference of difference $\{\Delta\}$, a constant expression that functions to express: 'press out' existence.

In this ontological model, the process itself $\{\Delta\}$ is invariable and thus deterministic; but what issues through that deterministic process is variable by virtue of probability.

There is that which is doing (unfinished) and that which is done (finished).

Numbers belong to the realm of the done. They are not doing or acting. They are invariant constants. As invariant constants, they are not created, they are revealed. Their identity depends on the integrity of relational distance within the numbering system. If $3$ and $7$ are not separated by $4$, the relationship that defines the real numbering system collapses. Thus, each number stands not on its own but through invariance of its relation to others.

When we perform operations like $4 + 3$, we are conducting an abstract process — an invariant relation within the conference of difference. The result, $7$, is not created or constructed but revealed: an invariant disclosed through the formal coherence of $3$ and $4$. Because both operands and outcome are abstract constants, the entire operation remains in abstracta and thus not of existence. To bring numbered abstracta into existence, the process must be applied to existent variables — apples, cars, people — where number becomes count and abstract coherence enters the world as physical relation.

In this sense, the conference of difference processes both invariant abstracta and variant existents, revealing one and transforming the other:

• With abstracta, the conference of difference $\{\Delta\}$ is predetermined in both process and outcome. The process is fixed and so is the result.
• With existents, the conference of difference $\{\Delta\}$ is predetermined in process but probabilistic in outcome. The process is invariant but what the process expresses varies.

This means that in the conference of difference, operands are not just populated by existents but also by abstracta through which existence can be unveiled. Thus the conference of difference supports both abstracta and existents but where the conference of difference of existent variables transforms existence, that of abstracta reveals it.

We might then say:

The purpose of abstract systems is to distil those principles that are metaphysical: 'originating behind' of existents. Gravity is real but the equations that model it are not. They are abstracta — conceptual tools to explain, predict and map reality. The map is not the terrain. It is a system from which to navigate, predict and understand the terrain.

So, abstracta do not exist. Instead, they help us to reveal the probability of existence. They make existence intelligible by alluding to it's conditions, processes and relationships.

Convergence & Divergence

Convergence:

• Platonism and the Gospel of Being both affirm that abstracta are not reducible to matter.
• Structuralism and the Conference of Difference both see relation as prior to substance.
• Conceptualism and the Gospel of Being both avoid treating abstracta as detached entities.
• Intuitionism and the Gospel of Being treat knowing as participatory and generative.

Divergence

• Platonism posits a separate realm; the Gospel of Being locates abstracta within the same process that transforms existence.
• Nominalism treats abstracta as fictions; the Gospel of Being treats them as true to the extent that they explain existence.
• Formalism sees numbers as symbolic games; the Gospel of Being sees them as bearing real conceptual weight.

Summary

Abstracta do not exist in the world. They are not of substance nor spatially/temporally existent in and of themselves. Yet they are the means by which we explain everything. The number $7$ is not existent like a rock but without it, rocks could not be counted or songs metered or the seasons calculated.

Only that which is expressed in probability can claim to be 'that which is out of' the conference of difference and thus claim to exist. That which is, outside of probability is determined by a system other than existence i.e. one that predefines the exact product of every relationship e.g. number systems.

In the Gospel of Being, this is not a mystery but an ontological necessity that informs us that:

In the realm of ist: 'that which is', there is that which is revealed through the conference of difference i.e. abstracta and that which transforms through the conference of difference i.e. existence.

Next week: Grounding & Metaphysical Dependence. If abstracta only reveals existence and is not causal of it then what grounds being? Does reality rest on a base layer — or is grounding itself a reciprocal structure?

The Gospel of Being

by John Mackay

A rigorous yet readable exploration of why the universe works—and how you fit inside it.

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