Richard George Davis
Revised author version 2 · 24 October 2026 · Exploratory discussion note
Submission status: Submitted by email to Philosophia Mathematica on 24 October 2026. Submission does not establish acceptance or peer review.
Version history: This revision clarifies the explanatory contribution, adds the later Wright and Pantsar source comparisons, and preserves the strongest objection and conditional reply. The original public author version remains available. The philosophical argument below matches the submitted revision; public presentation and disclosure formatting differ. Concept formation and the role of practices in establishing, justifying or acquiring standards remain open.
Learning to calculate and explaining why a calculation is correct are distinguishable questions. This note asks how shared human practices might help answer both.
Abstract
This note examines what an enculturation account must explain when invoking cognitive command for elementary arithmetic. Building on Shapiro’s distinction between excluding blameless disagreement and locating cognitive shortcoming, it separates explanatory convergence from warranted correction in Pantsar’s account. His later treatment of apparent objectivity sharpens the issue: explaining experienced objectivity need not establish the stronger warrant. Elementary checking supplies a conditional reply, but leaves the authority of its standards open. The contribution is a focused test of explanatory scope, not a refutation. Concept formation and whether practices establish, justify or transmit standards remain unresolved.
1. Acquisition and correctness
This inquiry began with the possibility that different bodies and counting practices might lead to different mathematical concepts. The present note addresses one part of that question: how learned practices relate to standards of correctness once elementary arithmetic is in use. How those practices shape the concepts that become available remains open.
The note also leaves open whether learned practices help establish correctness standards themselves, help us acquire or justify them, or contribute in more than one of these ways.
A change of numeral base illustrates one distinction: thirteen can be written 13₁₀ or 23₅ while retaining its value. Such a translation does not establish equal ease of calculation or identical concepts. It also supplies no conclusion about what makes a mathematical judgement correct. The question here concerns the step from the cognitive support of arithmetic to its standards of correctness.
Winter and Yoshimi [2020, pp. 7–11] argue that the embodied-cognition findings they review leave mathematical ontology—what mathematical things exist—underdetermined while remaining relevant to epistemology, the explanation of mathematical knowledge. Benacerraf [1973, pp. 661–679] frames the demand to reconcile mathematical truth with a plausible account of mathematical knowledge. Together, these considerations motivate a narrower inquiry: what must an account of acquiring arithmetic explain to support a particular criterion of objectivity?
The empirical findings have a bounded role. Frank et al. [2008, pp. 819–824] report that Pirahã adults could match visible items one to one more successfully than in tasks placing greater demands on memory or portable cardinal representation. Pitt et al. [2022, pp. 371–381] find an association between verbal count range and exact orthogonal matching in 30 selected Tsimane’ adults. They interpret verbal symbols as supporting large exact representations. These task-sensitive results support a role for learned numerical tools. Restricted samples and observational contrasts do not isolate language as the sole cause, establish absent concepts, or directly test mathematical objectivity.
The contribution is a test of explanatory scope. Shapiro already supplies the distinction between excluding blameless disagreement and locating a shortcoming; neither requirement is introduced here. Applied to Pantsar’s foundation argument, that distinction asks which explanatory task a proposed foundation completes. A foundation that explains convergent acquisition or experienced objectivity need not thereby justify correction. A foundation that supports correction must explain how its content constraints bear on justified application. The note makes this difference explicit without deciding whether a single enculturated practice can perform both tasks.
The present domain is familiar finite calculations such as 2 + 2 = 4, under shared numerical meanings and operations. It does not include every quantified proposition of arithmetic or every disagreement over mathematical proof. Within this domain, the relevant question is how cognitive foundations bear on justified application and correction.
2. Two requirements of cognitive command
Shapiro [2007, §3, p. 356] discusses Wright’s criterion of cognitive command: it must be a priori that relevant differences of opinion can be satisfactorily explained only through divergent information, unsuitable conditions or cognitive malfunction. The central demand is stronger than widespread agreement. A discourse must exclude the relevant possibility of disagreement that has no such explanation. Here the requirement calls for a warrant through reasoning, rather than simply evidence that people usually agree. This note uses the criterion as a lens for elementary arithmetic and leaves open whether locating error is necessary for objectivity generally.
Applied to an enculturation account, the first requirement is a warranted connection between arithmetic content and determinate standards of correct application. If shared capacities constrain the development of arithmetic, an account must explain how those constraints support the a priori exclusion of blameless disagreement over the same calculation. One example of a blameworthy error cannot by itself establish that general requirement. This is a point about the inference; it does not demonstrate that blameless disagreement is possible in elementary arithmetic.
Wright’s later primary treatment explains the stronger demand through a dispute about eliminating double negation: excluding the absence of shortcoming need not warrant a locatability claim [2001, pp. 85–86]. The present note takes no position on the general logical revision he proposes. It uses Shapiro’s application of the refinement to mathematical discourse, cross-checked against that later original text. The 1992 formulation itself has not been directly examined here.
The second requirement concerns error location. In Shapiro’s discussion of Wright’s constructive refinement, excluding blameless disagreement is insufficient: the cognitive shortcoming must be locatable (pp. 359–360). An account must warrant identifying a relevant failure in a disputant’s information or application, rather than merely asserting that somebody is wrong. Cognitive blame here is a technical attribution of shortcoming, not moral fault or a diagnosis of personal or cultural deficiency.
The constructive refinement strengthens the exclusion requirement; the two are not claimed to be logically independent. Their warrants nevertheless need distinguishing in an argument. A reason to think that a correct answer is determined does not automatically give a way to identify the mistaken judgement. Conversely, explaining why one familiar answer is mistaken does not yet warrant the general exclusion of relevant blameless disagreement. Keeping both requirements visible prevents a successful error example from carrying more weight than it has earned.
3. Pantsar’s proposed foundation
Pantsar’s main project is to explain apparent objectivity and mathematical applications; his conclusion also considers whether basic arithmetic is objective [2021, §§2 and 7]. His account connects proto-arithmetical capacities—early quantitative abilities that are not yet developed arithmetic—with developed arithmetic through enculturation and cumulative cultural evolution (§6). Enculturation concerns the development of cognitive abilities through participation in cultural practices; cumulative cultural evolution concerns their development and transmission across generations. Crucially, this is a claim about constraints on arithmetic content as well as a history of learning. Treating it as a purely causal explanation would weaken the view before assessing it.
In §7, Pantsar conditionally argues that if basic arithmetical truths are determined by proto-arithmetical abilities, the belief 2 + 2 = 5 conflicts with that foundation and with experiences of subitizing—recognising small quantities without counting—or estimation. He takes this to support cognitive command. The charitable reading is that his foundation premise is intended to explain a general constraint, with the example illustrating it. The argument should therefore be tested at that premise, rather than dismissed simply because a particular error cannot prove a general conclusion.
The closest later development qualifies the target. Pantsar [2023, §2] again uses cognitive command, but §5 explains experienced objectivity through maximal intersubjectivity and leaves open whether the latter counts as objectivity. That distinction must be respected. The present demand therefore applies when the developmental account is asked to warrant cognitive command, rather than merely explain the experience of objectivity. It is not an objection that an account of appearance fails to deliver a stronger conclusion it expressly leaves open.
A strong reconstruction needs two further connections. The proposed constraints must give standards sufficiently definite to explain why relevant conflicting results involve a cognitive shortcoming. Those standards must also support a justified practice of locating the error. These are requirements identified by the reconstruction here, not premises demonstrated by the matching studies. With both justified, the account could support cognitive command in the stated elementary domain while leaving mathematical ontology open.
4. The strongest objection
Even granting constraints on arithmetic content, the account must explain why their contribution has the required normative force: the authority to distinguish correct from incorrect application. A constraint may restrict the practices people acquire without uniquely fixing every standard needed to assess their judgements. Shared capacities and a strong tendency to agree do not themselves provide an a priori warrant that every relevant disagreement must have one of the permitted explanations.
The refinement makes the issue more precise. A conflict between a quantitative experience and a numerical judgement does not automatically show which should govern correction. The explanation needs a warranted checking practice. Calling the false believer cognitively deficient would bypass that need; Shapiro’s discussion explicitly resists the shortcut from being wrong to displaying an identifiable shortcoming (pp. 359–360).
The scope test yields two conditional assessments. If the conclusion is explanatory convergence, the account can succeed without settling the full authority of checking standards. If the conclusion is cognitive command, convergence must be connected to both a warranted exclusion of relevant blameless disagreement and justified error location. That connection cannot simply be supplied by the fact that everyone has learned the same procedure: agreement about the procedure is part of what needs interpretation. Nor does the demand require standards to come from outside practices. It leaves open whether their justificatory resources are internal, external, or interdependent.
The objection therefore asks whether enculturation explains the authority of the standards and their correction, or chiefly explains how people acquire standards justified through additional reasoning. It does not require the empirical studies to establish the entire philosophical account. Nor does it infer that concepts acquired through experience cannot subsequently support a priori reasoning. It identifies an argument still needed for the proposed foundation to do its full explanatory work.
5. The strongest reply
An enculturation account can distinguish these explanatory roles while allowing the same practices to contribute to both. Biological and cultural facts can explain how exact concepts and practices become available; conceptual and procedural reasons can justify particular applications. A cumulative account need not derive every mathematical warrant directly from the evolutionary history of quantitative abilities.
For a familiar finite addition, the extra connections may be modest. Fix the ordinary meanings of the numerals and addition, and suppose a justified checking practice is available. Successive increments from two by another two reach four. A person asserting five can display their working or repeat the calculation; comparison can expose a mismatching input or application. That supplies an error diagnosis rather than a bare allegation of falsity.
Shapiro expressly treats elementary calculation as a positive cognitive-command case: checking or a calculator can identify the mistaken party (p. 361). This supports the reply within the present domain. His later discussion of idealised mathematicians concerns broader disputes about purported proofs (pp. 361–366); it is not a necessary premise of the elementary-calculation example.
This reply answers a local challenge without by itself completing the developmental explanation. Once a checking standard is warranted, its use can expose a slip. Whether the developmental account explains that warrant is a further question. Conversely, an explanation of why people converge on checking practices can remain valuable even if it is offered only as an account of experienced objectivity. Keeping the conclusions separate avoids both an unfair refutation and an unearned vindication.
The reply is conditional on shared content and warranted checking standards. It does not derive their full authority from proto-arithmetic, require everyone to accept correction immediately, or demand a complete neurological explanation of a slip. Simply defining every dissenter as incompetent would conceal the issue. What matters is a justified diagnosis under the relevant conditions.
The cases considered here do not establish a genuine disagreement over the same elementary content that defeats this reply. Changed numeral meanings, merging physical objects or noisy estimation can change the proposition or fall under the criterion’s permitted explanations. Conflicting conjectures about unsolved problems fall outside this calculation domain. These boundaries block easy counterexamples; they do not prove either requirement satisfied in general.
6. What remains unresolved
The reconstruction applies two existing requirements to distinguish the explanatory work assigned to an enculturation account. Its proposed contribution is this scoped assessment: a developmental explanation of apparent objectivity should be assessed on that task, while an appeal to cognitive command requires the additional connections identified above. Elementary checking offers a plausible conditional reply. The account’s value as an explanation of convergence is not defeated by an unresolved claim about normative authority; neither is normative authority established merely by that explanatory value. The matching studies inform acquisition; they do not directly establish the proposed foundation or its normative force. The open issue is the explanatory role of enculturation in the authority of the checking standards themselves.
An account may contribute to both acquisition and justification. The present comparison has not determined how those contributions connect. A mathematical or conceptual defence independent of the developmental explanation need not assume mind-independent mathematical objects; independence of justification and a claim about ontology are different matters. The result is a bounded analytical clarification, with no established refutation of Pantsar or claim of theoretical novelty.
Unresolved question: How, if at all, do learned practices help establish or justify elementary arithmetic’s checking standards, beyond helping people acquire them?
AI assistance disclosure
AI assisted with drafting and editorial development.
OpenAI GPT models, accessed through Codex, materially assisted the research, argument development and finalising of this note. Assistance included literature searches, source and reference checks, generation and revision of substantial portions of the text, and model-based examination of objections. No new empirical study, raw-data reanalysis or computational validation of the philosophical argument was conducted.
The author reviewed the argument and scope in plain language, affirmed that the qualified position reflects his view, retained the stated uncertainties and approved submission of the revision. The author is responsible for the work. Model-assisted checks do not constitute independent human specialist review or journal peer review.
Recorded model labels for task-specific checks include GPT-6.1 Sol, GPT-6 Astra and GPT-6 Luna. Exact model versions for all earlier drafting sessions are not available in the retained record; no complete model-version history is claimed.
References
Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679. https://doi.org/10.2307/2025075
Frank, M. C., Everett, D. L., Fedorenko, E., & Gibson, E. (2008). Number as a cognitive technology: Evidence from Pirahã language and cognition. Cognition, 108(3), 819–824. https://doi.org/10.1016/j.cognition.2008.04.007
Pantsar, M. (2021). Objectivity in mathematics, without mathematical objects. Philosophia Mathematica, 29(3), 318–352. https://doi.org/10.1093/philmat/nkab010
Pantsar, M. (2023). From maximal intersubjectivity to objectivity: An argument from the development of arithmetical cognition. Topoi, 42, 271–281. https://doi.org/10.1007/s11245-022-09842-w
Pitt, B., Gibson, E., & Piantadosi, S. T. (2022). Exact number concepts are limited to the verbal count range. Psychological Science, 33(3), 371–381. https://doi.org/10.1177/09567976211034502
Shapiro, S. (2007). The objectivity of mathematics. Synthese, 156, 337–381. https://doi.org/10.1007/s11229-005-5298-y
Winter, B., & Yoshimi, J. (2020). Metaphor and the philosophical implications of embodied mathematics. Frontiers in Psychology, 11, 569487. https://doi.org/10.3389/fpsyg.2020.569487
Wright, C. (2001). On being in a quandary: Relativism, vagueness, logical revisionism. Mind, 110(437), 45–98. https://academic.oup.com/mind/article-abstract/110/437/45/974279
No published comments yet.