Haueis and Colaço rightly point out that resource considerations constrain the space of possible and plausible models (viz. Blokpoel, Reference Blokpoel2018; van Rooij, Reference van Rooij2008; van Rooij & Baggio, Reference van Rooij and Baggio2021; van Rooij et al., Reference van Rooij, Blokpoel, Kwisthout and Wareham2019). Yet, multiple realizability (MR) remains an intrinsic feature of explanatory models. We offer a complementary but different view on MR to demonstrate that it does not deal in vagaries, nor exist solely at the computational level, and is a productive part of modeling.

First, notions of MR have existed for millennia. Voula Tsouna (Reference Tsouna2023) explains: “Epicurus’ conception of the pollachos tropos [method of multiple explanations] is rich, nuanced, fairly coherent, and strongly motivated by methodological and scientific considerations” (p. 240). Epicureans contemplated celestial phenomena, but the argument applies to cognitive neuroscience: “there are in-principle reasons why explanatory pluralism is to be expected[:] the computational description is no more and no less realistic and fundamental than the mechanistic/implementational one” (Chirimuuta, Reference Chirimuuta2018, p. 406).

Second, we define four distinct notions of function: $\varphi, T,G$ , and $\tau $ . A function $F:I\to O$ maps elements in the set $I$ to elements in the set $O$ . Importantly, functions are not guaranteed to be computable. And for any given computable function, there exist many algorithms. This is an instance of MR at the algorithmic level to which we return below. Before that, what could a function be?

• Functional capacity $\bf{\varphi }$ : A system $\mathbb{S}$ (modeled by $\mathbb{M}$ , see Figure 1) may be held to have some purported functional capacity, denoted by $\varphi :{I}_{\varphi }\to {O}_{\varphi }$ . For instance, a person, brain, or component may be believed to have the capacity for addition, $\varphi $ .

  Figure 1.

  Relationships between the system under study $\mathbb{S}$ and models $\mathbb{M}$ . The single solid line is relationships within the neurocognitive system. Double solid lines represent the theoretical mapping proposed between $\mathbb{S}$ to $\mathbb{M}$ . The dashed line links $\mathbb{M}$ to its digital substrate. Dotted lines represent relationships to lower substrates.

  

• Functional theory T: A theorist may specify a functional theory for a purported capacity $\varphi $ using a well-defined function, $T:{I}_{T}\to \mathrm{}{O}_{T}$ (Egan, Reference Egan2017). For instance, $T\mathbb{}:\mathbb{}\mathbb{N}\times \mathbb{N}\mathbb{}\to \mathrm{}\mathbb{N}$ with $T(a,b)\mathrm{}=\mathrm{}a+\mathrm{}b\mathrm{}$ where $a,b\in \mathbb{N}$ may model the functional capacity for addition $\varphi $ .

• Functional goal G: An algorithmic model A may be designed or tuned (e.g., so-called “training” of artificial neural network models; Guest & Martin, Reference Guest and Martin2023,Reference Guest and Martin Reference Guest and Martin2026) to perform a given goal capacity, denoted $G:{I}_{G}\to \mathrm{}{O}_{G}.$ We say A (approximately) computes functional goal G if and only if $A\left(i\right)\mathrm{}\approx \mathrm{}G\left(i\right)\mathrm{}$ (cf. Blokpoel, Reference Blokpoel2018; van Rooij & Wareham, Reference van Rooij and Wareham2012). When $G=T$ , then A could be an algorithmic-level explanation for functional theory T.

• Functional role τ: A functional capacity $\varphi $ (modeled by T) may be hypothesized to play a functional role $\mathrm{\tau }:{I}_{\mathrm{\tau }}\to \mathrm{}{O}_{\mathrm{\tau }}$ . For instance, adding may aggregate synaptic inputs or determine the cost of groceries. A capacity $\varphi $ (modeled by T) fulfils a functional role if and only if it can behave consistently with the teleological specification, i.e., $\varphi \left(i\right)\equiv T\left(i\right)=\tau \left(i\right)$ , for all $\mathrm{}i\in {I}_{\varphi }={I}_{T}\supseteq {I}_{\tau }$ .

All models $\varphi $ , T, G, and τ, are defined by the scientist. Knowledge obtained from experiments may be used to abduce candidate theories T of $\varphi $ , but not to deduce them. In contrast, τ is definitionally prima facie assigned to $\mathbb{S}$ ’s $\varphi $ and only unassigned if the system is deemed to have broken down (Egan, Reference Egan2017, Reference Egan2025). An example from biology for τ is that the function of the heart is to pump blood; perhaps less useful for cognitive science under computationalism as it results in truism, such as the purpose of vision is to see.

Next, we offer MR formalisms.

• MR ${_{\varphi}}$ : A functional role τ is said to be multiply $\varphi $ -realizable if and only if there can in principle exist multiple distinct functional capacities ${\varphi }_{1},{\varphi }_{2},\ldots $ (modeled by ${T}_{1},{T}_{2},\ldots $ ) that fulfil that role τ in some possible world. We furthermore say τ is multiply $\varphi $ -realized if there exists more than one such capacity in our world. For example, if one wants to soft-boil an egg for 6 minutes, and knows there are 6 minutes left on a video one is currently watching, then one can use the video player as a timer.

• MR ${_{m}}$ : A functional capacity $\varphi $ is said to be multiply $m$ -realizable if and only if there can, in principle, exist multiple distinct mechanisms ${m}_{1},{m}_{2},\ldots $ that have that capacity $\varphi $ in some possible world. We furthermore say $\varphi $ is multiply m-realized if there exists more than one such mechanism in our world; recall Figure 1 and bear in mind that substrates do not (straightforwardly) cut it as mechanisms under computationalism (Guest & Martin, Reference Guest and Martin2025). For example, the capacity for addition can be performed in the head, using a calculator, or on an abacus (Guest, Reference Guest2026). Here, m may refer to the algorithm, the physical implementation, or both.

By abuse of terminology, we may also say that a functional role τ can be multiply m-realizable if there exist mechanisms ${m}_{1},{m}_{2},\ldots $ that can realize one or more $\varphi $ that can realize functional role τ. For example, a timer can be instantiated in a wall clock, a software stopwatch, or as above by the video player.

Finally, these meanings of MR demonstrate that it need not be vague nor operating only at the computational level (also see: Chirimuuta, Reference Chirimuuta2018; Egan, Reference Egan2017; Figdor, Reference Figdor2010; Hardcastle, Reference Hardcastle1995; Litch, Reference Litch1997; Polger & Shapiro, Reference Polger and Shapiro2016; Ross, Reference Ross2020). MR operates at each level. That is, a particular functional role can be performed by different functional capacities. Functional capacities can be realized by different algorithms. A particular algorithm can be physically realized in different ways.

Any serious proposals for metabolic additions under computationalism will need to contend with MR – at the level and type the modeler defines – not because MR and metabolism have something specific to say about each other but because both are relevant. We thus urge and support theorists to take seriously their commitments to MR by providing these formal tools, which help think through what our formal models imply.