A classic question in the philosophy of mathematics asks whether mathematics is invented or discovered. However, this question can be understood as a particular instance of a much broader problem: what does it mean to claim that something exists independently of whether any mind represents it, experiences it, or is even capable of detecting it?
It does not seem particularly problematic to claim that mathematical symbols are human constructions. The symbol “2,” for example, is part of a historically developed notation. Two strokes, two dots, a different symbol, or any other conventional system could have been used instead. Mathematical notation is a language and, in that sense, a construction.
The difficulty arises when the symbol is distinguished from what the symbol refers to. If two planets orbited a star in a universe with no conscious beings, would it still be true that there were two planets? Would the cardinality corresponding to two exist even if no mind possessed the concept required to represent it?
The question then ceases to concern mathematics alone. It can be extended to different dimensions of what is usually regarded as real. To analyse the problem, it is useful to distinguish, at least heuristically, three domains: 1) abstract structures and their conceptual representations, 2) physical reality, and 3) conscious experience.
This distinction is not intended as an exhaustive taxonomy of everything that exists. Its purpose is to allow the same question to take different forms in each domain: in what sense could something exist independently of any representation or experience?
1. Abstract Structures and Representations
Mathematics provides an especially clear case because it requires us to distinguish at least three elements: the symbols used, the concepts through which they are interpreted, and the relations or structures that those symbols and concepts allow us to describe.
Symbols are conventional. So, at least partly, are the definitions and formal systems chosen to work with them. However, once certain axioms, definitions, and rules of inference have been established, consequences emerge that do not depend on the will of whoever formulated the system. It is not possible to decide arbitrarily which theorems follow from a given set of premises.
This feature gives mathematics a peculiar combination of construction and discovery. A formal system can be chosen, but not all of its consequences can be chosen in the same way. In a certain sense, access to a structure is constructed, while relations that were implicit in the initial conditions are subsequently discovered.
Yet this feature does not by itself resolve the ontological question. The fact that certain consequences are necessary once a system has been specified does not demonstrate that the system, or the corresponding structure, exists independently of any mind.
Several possibilities can be considered. Mathematical structures might exist independently of any representation; they might be constructions dependent on conceptual systems; or there might be a space of possible structures whose relations are determined by logical constraints, while minds construct the languages used to identify and explore them.
The opposition between invention and discovery may therefore be too simple. Even where there is a clear element of invention, relations may emerge whose determination does not depend on those who study them.
2. Physical Reality
In the case of physical reality, the usual intuition seems different. Scientific realism readily allows us to describe a universe that existed long before human beings appeared, and even before any known form of life. Cosmological, geological, and evolutionary models describe precisely this kind of history.
Here, however, an epistemological question arises that is distinct from the ontological one.
Empirical access to physical reality takes place through perception, observation, experimentation, and measuring instruments. Instruments can record events without an observer being present at the time, but those records become part of knowledge when they are observed, interpreted, and related to models.
This mediation does not imply that physical reality depends on the existence of observers. To conflate these two claims would be to move unjustifiably from an epistemological thesis to an ontological one. The fact that knowledge of a reality is mediated by representations does not imply that the reality itself is produced by those representations.
What it does imply is that there is no standpoint external to all perception, measurement, or representation from which a model of reality can be directly compared with a reality entirely independent of any cognitive access. Physical theories are justified through their explanatory, predictive, and experimental capacities, not through unmediated access to reality.
This situation creates an interesting epistemological asymmetry. For the subject undergoing it, the existence of a conscious experience has a form of immediacy different from that of claims about an independent physical reality. By contrast, the structure and properties of that physical reality are established through inferences supported by observations and models.
But no automatic ontological conclusion should be drawn from this asymmetry either. The fact that experience may have a certain epistemological priority does not demonstrate that it has ontological priority. Likewise, the fact that physical reality is known through evidence and models does not imply that it is less real or that it depends on consciousness.
3. Conscious Experience
A widely held naturalistic view maintains that conscious experience depends on certain physical processes, particularly the activity of sufficiently complex nervous systems.
The available evidence provides very strong reasons to accept a close dependence between brain states and conscious states.
Damage to particular brain regions can alter perceptual capacities; neurochemical changes can profoundly modify subjective experience; anaesthesia can temporarily eliminate certain conscious states; and stimulation of particular neural systems can produce or modify perceptions, sensations, and emotions.
These observations establish systematic relations between the brain and experience. However, they do not by themselves necessarily determine the correct metaphysical interpretation of that relationship.
One possibility is that certain physical processes produce conscious experience. Another is to identify conscious states with particular physical or functional states. Views have also been proposed according to which the physical and the experiential are different aspects of a more fundamental reality. There are also more speculative hypotheses in which the brain would act by constraining, selecting, or modulating experiential possibilities.
These alternatives do not enjoy the same degree of empirical support, nor do they necessarily offer the same explanatory or predictive capacity. Some may be difficult to turn into testable theories. Listing metaphysical possibilities therefore does not amount to assigning them equal plausibility.
The relevant issue is different: the observed relations between brain processes and experience do not directly reveal what kind of ontological relationship exists between them. Evidence can show dependence, correlation, and certain causal relations, but moving from those relations to a general theory of what consciousness is requires additional assumptions.
We do not directly observe a transformation of something devoid of experience into experience. What we observe are physical processes, changes in the brain, behaviour, subjective reports, and regularities among them. On the basis of this evidence, theories are constructed about the relationship between physical reality and consciousness.
Three Domains and a Crucial Distinction
The three domains considered raise different problems.
Abstract structures appear to contain relations whose validity may not depend on the particular decisions of those who represent them. Physical reality appears capable of existing independently of any representation, although knowledge of it requires observation, measurement, and models. Conscious experience, by contrast, appears necessarily to require that there be something for which that experience is occurring.
There is no obvious reason to assume that all three domains must have the same kind of existence. Physical reality could be fundamental, with the other phenomena depending on it. Experience could occupy a more fundamental role than conventional physicalism assumes. Abstract structures could possess some form of independence from both domains.
It is also possible that all three categories are manifestations of a more basic structure, or that the division itself reflects limitations in the conceptual tools used to describe reality.
At this point, it becomes particularly important to keep two questions separate:
- The first is ontological: what exists?
- The second is epistemological: what reasons justify the claim that it exists?
The inability to observe or directly detect something does not demonstrate that it does not exist. It can only show that its existence is difficult, or perhaps impossible, to justify empirically. Likewise, immediate access to an experience does not demonstrate that experience constitutes the ultimate foundation of reality.
This distinction makes it possible to reformulate the initial question about mathematics. The problem is no longer simply to determine whether numbers were invented or discovered. It also requires distinguishing symbols from what they represent, models from what they are intended to model, and the conditions of knowledge from the conditions of existence.
Broader questions then emerge. Did mathematical structures exist before there were beings capable of representing them? Would physical reality exist in the absence of any conscious experience? Could experience exist without some kind of physical substrate? Each of these questions requires a careful distinction between what exists and what can be known about its existence.
There is also a limiting case that puts this distinction to the test. One can imagine an entity that was never experienced or represented, that did not causally interact with anything else, that left no observable trace, and whose presence made one universe empirically indistinguishable from another universe in which that entity did not exist.
Such indistinguishability would not necessarily imply that the entity does not exist. It would imply, however, that no empirical procedure could decide between the two possibilities. The problem would then cease to be exclusively ontological and would also become a question about the meaning and justification of existence claims.
The question raised by mathematics thus leads to a deeper problem. It is not merely a matter of determining which things are invented and which are discovered, but of clarifying what it means to claim that something exists, what distinction can be drawn between an entity and its representation, and what kind of evidence could justify a claim about a reality independent of all representation.
Methodology
This article was produced using a model of human conceptual leadership and editorial execution delegated to artificial intelligence. The author provided the thesis, examples, objections, revision criteria, and editorial decisions. The AI developed the prose, proposed reorganisations and alternative formulations, searched for and cross-checked references, and maintained the master version of the text—that is, the documentary reference to which successive changes were applied.
The author critically reviewed the successive versions and approved the final version.
Summary of the method: conceptual leadership, human; drafting and master version, AI; reference search and verification, AI; critical review and final approval, human.
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