The Nature of Reasoning in Theology, Philosophy, and Mathematics

TL;DR

This paper argues that reasoning about ultimate knowledge in theology, philosophy, and mathematics is fundamentally either foundational or circularly justified, supported by comparisons and analogies across these disciplines.

Contribution

It introduces a unified epistemological framework for understanding reasoning about ultimate knowledge across theology, philosophy, and mathematics.

Findings

Reasoning about ultimate knowledge is either foundational or circularly justified.

Discussions include comparisons and analogies from mathematics.

The framework clarifies the nature of rational support in these disciplines.

Abstract

This article supports the epistemological claim that sound human reasoning about ultimate knowledge is either foundational or circularly justified. In particular, questions which naturally arise in theology, philosophy, and related disciplines, to the extent that they rationally treat ultimate knowledge, are necessarily supported in one of these ways. Comparisons with, contrasts to, and analogies from mathematics are given to illustrate and enhance this central claim.

Full text

1 Introduction

One of the most basic goals of any rational human enterprise is to intellectually justify one’s truth claims. This article will investigate the nature of that justification process. We have in mind what we will call “propositional justification,” that is, justification of propositions by propositions (truth-valued statements). Words like “support” or “justification” will refer to this notion of propositional justification unless specified otherwise.

Our aim is to examine the structure of rational justification, not to arbitrate rationality itself. The question we seek to answer is not “is proposition $P$ rationally held?” but rather, “given a theory of warrant,111A set of (and relations between) standards or criteria by which rationality is measured; it goes by other names as well. what is the nature of the structure of the rational support of proposition $P$?”

It is natural to relate these epistemological issues to mathematics as the latter is commonly regarded as occupying rarefied air, perhaps even uniquely, in its theoretical completeness. Famously adored as the “queen of the sciences”222While applying this term to mathematics is attributed to Carl Friedrich Gauss, math was not its original object. It referred to theology (of all disciplines!) during the Middle Ages. See [4, p. 213]. with the “most uncontroversial examples of proof,” [11, p. 56] if something is observed in mathematical thinking, it is often expected to apply to other (maybe even all) rational endeavors. Examining the interplay between mathematics and ultimate justification for truth claims is therefore worthwhile and will be taken up in this article.

This article is split into three sections. Section 2 states and argues for the main thesis. Section 3 uses mathematics as a case study to make connections between the notions of foundational knowledge and circular reasoning, particularly noting how circularity is avoided by the major schools of thought in the philosophy of math. Section 4 briefly addresses some questions which naturally follow.

2 The Argument

2.1 Assumptions and Definitions

It would be a Herculean (or impossible) task to fairly and incontrovertibly list every assumption; complete, untainted epistemological self-awareness might be only a theoretical ideal. We will nonetheless attempt to identify the ones most fundamental and most relevant to the present topic.

Our primary object in this section will be the “hypothetical belief set” ($HBS$) of a fixed human agent $H$ at a fixed time $t$. It is “hypothetical” because we allow the set to include beliefs that $H$ would hold if prompted appropriately, but which $H$ might not otherwise believe or even conceive of. The set $HBS$, sometimes written $HBS_{t}$ to emphasize the dependence on $t$, along with the relations among its elements, is the noetic structure of $H$. [20, p. 48] We will assume $H$’s thoughts are not past eternal and that it takes some positive amount of time for $H$ to form a thought. Coupled with the dependence on $t$, we may thus assume that the cardinality of $HBS_{t}$ is finite.

We will assume the laws of logic and general principles of rationality which are too many (and, likely, too familiar) to enumerate. But as they will be essential terms, let us define them a bit more thoroughly.333We take the notions of rationality and logic to be rationally foundational (as seems to be required of rational human endeavors), but not “semantically foundational.” As will be discussed throughout the article, there are two sets in any rational human endeavor to which our thesis applies: rational support and definitions. An element which appears in both sets need not be ultimate (in the sense defined later in the section) in both sets, modulo the caveat discussed below that definitions can be converted to truth claims. In this article, “rationality” is considered ultimate in the former set, but not the latter. Nonetheless, the definitions we give should be taken to be informal. Here, we will consider logic to be formal, classical logic.444As opposed to the informal use of the word “logic,” which can refer to any kind of reasoning, and as opposed to nonstandard formal logics. If one insists on a nonstandard logic, it is likely to be unproblematic for this article. Definitions include

• •

“a formal, scientific method of examining or thinking about ideas;”555https://dictionary.cambridge.org/us/dictionary/english/logic, accessed June 20, 2022.

• •

“a science that deals with the principles and criteria of validity of inference and demonstration**:** the science of the formal principles of reasoning;”666https://www.merriam-webster.com/dictionary/logic, accessed June 20, 2022.

• •

“the science of thinking about or explaining the reason for something using formal methods.”777https://www.oxfordlearnersdictionaries.com/us/definition/english/logic_1?q=logic, accessed June 20, 2022.

We consider deduction to be the process of drawing conclusions by applying the rules of logic to one or more statements (elements of $HBS$). Rationality is

• •

“the quality of being based on clear thought and reason, or of making decisions based on clear thought and reason;”888https://dictionary.cambridge.org/us/dictionary/english/rationality, accessed June 20, 2022.

• •

“the quality or state of being agreeable to reason;”999https://www.merriam-webster.com/dictionary/rationality, accessed June 20, 2022.

• •

“the fact of being based on reason rather than emotions.”101010https://www.oxfordlearnersdictionaries.com/us/definition/english/rationality, accessed June 20, 2022.

Accordingly, we will call an intellectual endeavor or claim rational if it is ultimately supported by reasons. What constitutes a “reason” is a criterion which we do not attempt to define in this article (beyond noting that a reason must be a truth-valued statement). That is, we will assume that some theory of warrant has already been accepted; there is much epistemological debate on such topics on which we (attempt to) remain mostly silent. We simply consider whether a claim is supported by reasons or not, without regard to whether they are “good reasons.” Note also that there is a contrast with rationalism, the philosophical view that “regards reason as the chief source and test of knowledge.”111111See https://www.britannica.com/topic/rationalism, accessed June 20, 2022.

We consider, then, irrationality to be the negation of rationality and irrational the negation of rational; in particular, for brevity, the notion of “arationality” (being outside the arena of reason, even if not in conflict with it) is subsumed under an umbrella term of irrationality in this article. Academic disciplines such as theology (the study of religious belief), religion (the pursuit of the supernatural or supreme existence), and philosophy (the study of the foundational nature of reality) will be referred to in broad strokes.

By circular reasoning, therefore, we will mean the support of a proposition in $HBS_{t}$ for which the conclusion is among the premises. We will use the term “circularity” for brevity to refer to circular human reasoning. It is sometimes referred to in the literature as “begging the question” or “petitio principii,” though some authors make a distinction among these terms. A proposition will be considered foundational if it has no propositional support in $HBS_{t}$.

We will assume an agreed-upon meaning121212Or “approximate meaning.” of all terms used, with only a few key terms being explicitly defined. Of course, this is a nearly universal assumption, and any meaning-carrying dialogue depends on it, so it is not overly ambitious. Nonetheless, given the epistemologically fundamental nature of the content that follows, adequate epistemological self-awareness dictates that this assumption be stated, along with the acknowledgement that semantic misunderstandings are still possible regardless. It is the author’s hope in defining a few key terms that any such persisting misunderstandings are insignificant to the central thesis.

We fix a claim, $P\in HBS_{t}$, of genuine knowledge about reality, i.e. $P$ is (agreed upon to be) true. Likewise, we assume a goal of rational investigation of objective truth about reality, as opposed to statements grounded ultimately in opinion, preference, feeling, or other subjective standards. In particular, issues raised by the philosophical positions of skepticism, which doubt whether truth can be known or if there is any objective reality at all, will not be considered herein. Thus, we will assume that $H$ can justifiably claim to know something. Note also that definitions could be rephrased as truth claims in the form “term $X$ has meaning $Y$;”131313A similar phenomenon can be seen in mathematics, when sometimes definitions are referred to as axioms; indeed, “there is little distinction between a mathematical axiom and a definition.” [4, p. 216] $P$ might be as such.

It is useful to refine the epistemic level of $P$: either $P$ is ultimate knowledge or it is not. In this article, ultimate knowledge is the most basic knowledge that $H$ possesses. We will often refer to such a proposition as an ultimate, or ultimates in plural (using the word “ultimate” as a noun). An ultimate is something for which $H$ does not have a deeper knowledge level to undergird it. Likewise, we will call questions about ultimate knowledge “ultimate questions.” What we call “ultimates” in this article are given many other names depending on the context and author: presupposition, basic commitment, [11, p. xvii] (properly) basic belief, [20], [11, p. 59], starting point, innate belief, final authority, foundation, foundational belief, assumption, and so on (and, unfortunately, sometimes these terms are used with different meanings).141414At $t$, we consider the set of foundational propositions to be a subset of the set of ultimates. It is tempting to equate this notion with the mathematical notion of axiom (or “hypothesis” to some), but as Christian theologian John Frame discusses, [11, pp. xxxi-xxxii] and as we shall see shortly, there is a distinction.

The aforementioned assumptions are not necessarily themselves ultimates, but are a practical necessity to focus the discussion and keep it to a manageable length.

2.2 Primary Thesis: Ultimate Justification for a Truth Claim

What we discuss below is a version of Agrippa’s trilemma, credited to the philosophical skeptic Agrippa and the Pyrrhonists.151515The essentials of the problem go by other names as well, such as Münchhausen’s trilemma, or Fries’ trilemma. The term “Münchhausen’s trilemma” is credited to [1]. The present article uses the framework of this trilemma, while coming to different conclusions than Agrippa and the Pyrrhonists. It is one manifestation of the problem in epistemology regarding ultimate justification for a truth claim. The nuanced differences of its disparate treatments will not be essential here.161616For example, in a work by Greek philosopher Sextus Empiricus, five possible outcomes to the question of ultimate justification are given. Three of those five are contained in Agrippa’s trilemma, with two additional ones which are disallowed by our current assumptions: dissent (in our context, this would be to assume that the proposition $P$ is not agreed to be true) and relation ($P$ appears to be true from the point of view of a certain individual, but that does not necessarily constitute a statement of the inherent truth of $P$; in other words, it is the conclusion that truth does not refer to an objective reality). See [8]. For a readable overview of the issues in Agrippa’s trilemma, refer to the Stanford Encyclopedia of Philosophy’s entry on Skepticism; see https://plato.stanford.edu/entries/skepticism/ (accessed July 31, 2022).

Our primary thesis is the claim that

(FC) $\hskip 8.53581ptHBS_{t}$ contains elements which are foundational or are circularly supported.

We will begin with $P$ to conclude (FC). One can ask for proximate, rational justification of $P$: why, or on what rational authority (empirical evidence, logic, the authority of the source from which $P$ was discovered, or whatever reason might be legitimate according to $H$’s theory of warrant), can we say that $P$ is the case?171717We can consider such a question as a “Chisholm-like question” without the misunderstanding of how such a question should be taken as in [20, p. 51]. Suppose a distinct supporting reason, $Q$, is given. One could equally ask for the distinct, proximate, rational justification of $Q$, call it $R$. The pattern continues, descending into a sequence of “why” questions and answers: $P,Q,R,\ldots$. We will call this the support sequence.181818Could $P$ have a distinct support sequence, say $P,E,F,G,\ldots$? We allow for that possibility. The analysis which follows will fix a particular sequence, though the conclusions apply to any support sequence which one desires to choose. The salient characteristic is that $P$ has only finitely many distinct support sequences, which follows since $HBS_{t}$ is assumed to be finite in size. Thus, fixing just one support sequence suffices. Under our assumptions, there are only two options191919The reader may wonder where the third horn of Agrippa’s trilemma is. Two of them are together contained in a single option: the finite termination of the sequence. with regard to the termination of this sequence (and hence, for the foundations of $H$’s noetic structure) which we will explore in turn:

the sequence never terminates, continuing to give distinct supporting reasons; or 2. 2.

the sequence terminates in finitely many steps, ceasing to give distinct supporting reasons.

2.3 Option 1: The “Endless Why”

This option implies there is an infinite regress of supporting reasons for claim $P$, a truly endless sequence of “why” questions and answers. In Agrippa’s trilemma, this is the “infinitism” horn. This contradicts our assumption that $HBS$ is finite at $t$, and therefore can be discarded.

Additionally, perhaps enlightening is the observation that, even if the finiteness assumption were dropped, it is questionable if $P$ could be considered to be genuinely rationally supported. If the argument supporting $P$ is never complete, is $P$ really supported after all? Is an in-progress proof really a proof (in any theory of warrant)? One particularly troublesome potentiality: the support sequence could eventually contain something that breaks our assumptions, and, even if that hasn’t occurred yet, we may not have sufficient confidence that it won’t occur at one of the future steps. Such observations demonstrate that, even under slightly more general assumptions, an infinite support sequence for $P$ is dubious at best, and possibly irrationality “in disguise.”

2.4 Option 2: Foundational or Circular Support

Therefore, the support sequence cannot be truly endless in rational support of $H$’s genuine knowledge; it must terminate in finitely many steps. That is, some supporting reason $Z$202020The termination of the support sequence of $P$ is likely to be a set of reasons rather than one single statement. For ease of notation, and since the individual elements of such a set need not be separated for our purposes, we will simply notate the termination $Z$ and treat it as a single reason. must be the last distinct supporting reason at $t$ (where we allow for the sequence to terminate in 0 steps, i.e., $P=Z$). That $Z$, by definition, has no deeper or more fundamental level with distinct reason(s) supporting it at $t$, so it is at the ultimate justificatory level. How does $Z$ relate to other elements of $HBS_{t}$?

That question is answered in two different ways in Agrippa’s trilemma, the first being foundationalism.212121The related stance of positism, [10] in which $Z$ is simply unjustified (presumably relative to the chosen theory of warrant), is discarded as it contradicts our assumptions. In being unjustified by its own admission, it represents a choice that is not rational truth-seeking. Here, $Z$ is accepted as foundational without further propositional rational defense (though it may be defended some other way). The support sequence merely stops at $Z$. Depending on the nature of $Z$ and the theory of warrant, $Z$ could be considered knowledge without being inferred from any proposition, including from itself. [20]

The other horn of the trilemma is referred to as coherentism in which $Z$ is part of a mutually supporting, self-contained theory. This requires a “cycle” of reasons;222222Note that $Z$ is the final distinct reason. It could be the case in $HBS_{t}$ that the reason given for $Z$ is some other proposition $R$ already listed in the support sequence, thereby initiating cyclical traversal through the sequence. this is circularity.

Therefore, there are only two non-rejected options for the ultimate justification of $P$:232323That is, only two options which do not contradict our assumptions. foundational support and circular support.242424The interplay between them is examined deeply in [14]. Thus we conclude (FC).

Most connections to mathematics will be saved for Section 3, but one is appropriate here. We may impose a finite directed graph structure on $HBS_{t}$ since we view it as a finite, discrete set with nonsymmetric relations among (some) pairs of elements. Call this the “knowledge graph” of $H$ and label it $G$. 252525This may be a metaphor to add to the list of “raft” and “pyramid.” See [23]. Nodes of $G$ are propositions (elements of $HBS$), and arrows point from one node to the others which support it. Every node in a connected262626Or, more commonly, “weakly connected.” See [5, p. 163]. component of $G$ has at least one arrow pointing to it or emanating from it. A theorem272727[5, p. 89], Theorem 4.3 proves this result for an undirected graph. The directed version can be found in Lemma 1 at https://www.math.cmu.edu/~af1p/Teaching/GT/CH10.pdf, accessed June 15, 2022. in graph theory tells us that every such graph contains a cycle (which would indicate circular reasoning in our analogy) or a sink (a node with no arrows emanating from it; in our analogy, this is a foundational proposition).282828Note, then, that a foundational proposition cannot be part of a cycle, but an ultimate might be as per (FC). If one accepts the knowledge graph analogy, then the proof of this theorem offers an alternative proof of (FC). Perhaps this mathematical analogy can be mined to provide further insight into epistemology.

2.5 Why entertain the plausibility of circular reasoning at all?

Nearly all rational thinkers eschew any form of circular reasoning as logically fallacious. Philosopher Douglas N. Walton, with an entire book devoted to circular reasoning, [34] explains succinctly why this is the case: “Arguing in a circle becomes a fallacy by basing it on prior acceptance of the conclusion to be proved. So the fallacy of begging the question is a systematic tactic to evade fulfillment of a legitimate burden of proof.” [35, p. 66], [11, pp. 254-5] As philosopher S. Morris Engel puts it, “if the supporting premises merely repeat or rephrase what is stated in the conclusion, as in all cases of begging the question, the argument contains no premises and is therefore fallacious.” [9, p. 147], [11, p. 255] According to theologian Joseph E. Torres, “[i]f question-begging is embraced, fideism is implied. Fideism is the rejection of a rational [argument].” [31], [11, p. 255]292929Fideism, beliefs taken on faith alone without any rational support, is widely seen as anti-intellectual. Even in everyday discourse, labeling a putatively intellectual argument to be circular is a death blow.

Why, then, allow for circular reasoning as a plausible option at all? The short answer is that the rationality of circular (or any other kind of) reasoning is left up to the theory of warrant, and in this article, we attempt not to evaluate such theories. But there is another layer worth examining.

A variety of circularity is discussed, and evidently promoted, by Christian presuppositional apologists, a school of thought expounding on the works of theologian Cornelius Van Til. For example, John Frame defines “circularity” as “an argument in which the conclusion justifies303030It is not entirely clear if Frame’s notion of “justification” is the propositional variety. Similarly, his notion of “ultimate” seems to refer to a standard or criterion in the theory of warrant rather than to a proposition. itself. All arguments seeking to prove the existence of an ultimate or final authority are circular in this sense.” [11, p. 291] Philosopher and apologist Greg Bahnsen, often considered the primary popularizer of Van Til’s work,313131Rev. Dr. K. Scott Oliphint, expresses this in his comment of [2]. Oliphint says, “For those who want to understand Van Til, whether to agree or disagree, at least two things are both essential and too often neglected. The first is to read Van Til, the second is to read Greg Bahnsen.” (Back cover). says that this kind of circularity is “involved in a coherent theory (where all the parts are consistent with or assume each other) and which is required when one reasons about a precondition for reasoning.” [3, p. 518n122] Theologian R. C. Sproul concurs: “That all reasoning is ultimately circular in the sense that conclusions are inseparably related to presuppositions is not in dispute.” [24, p. 70], [11, p. 258] Seeing circularity as a consequence of the finiteness of humanity is expressed succinctly by Van Til himself, who writes, “[w]e hold it to be true that circular reasoning is the only reasoning that is possible to finite man” (emphasis added). [33, p. 12], [11, p. 254]

How can serious thinkers seem to accept an argumentation tactic that is clearly fallacious? It seems that they refer to different types of rational support. When seeking propositional support, as in this article, there is broad agreement, even among presuppositionalists, that circular reasoning is fallacious. The variety of circularity endorsed by presuppositionalists is of a different type, a type we will call authoritative support,323232Perhaps the term “foundational support” would be equally descriptive. referring to the criteria which one uses to judge veracity or rationality. Frame recognizes the necessity: “[W]hen one is arguing for an ultimate criterion, … one must use criteria compatible with that conclusion. If that is circularity, then everybody is guilty of circularity.” [11, p. 11]333333Indeed, “reasoning in a vicious circle is the only alternative to reasoning in a circle.” [32, p. 24], [11, p. 257] In the words of philosopher Richard Pratt, “in argumentation, reasoning should be linear,” but circularity of the “process by which finite minds attain knowledge to be used in arguments” is “inescapable.” [22], [11, p. 257] (emphasis added). In other words, rhetorically, by what standard can an ultimate criterion of knowledge be aptly declared suitable, if not by itself (for, if not by itself, then the criterion is not ultimate)?343434It is in this sense that ultimates occupy a “privileged position” in rational argumentation in that they exhibit special epistemological properties not seen in non-ultimates. See Bahnsen’s lecture “Reasoning With Unbelievers,” beginning at 25:20: https://www.youtube.com/watch?v=6RBz-zAWoKk, accessed December 8, 2021. See also the discussion of the “meta” relationship that ultimates have to the support sequence in Sections 3.7, 4.1. By what criteria of knowledge shall we judge our criteria of knowledge but themselves? To emphasize the distinction between fallacious circular reasoning of propositional support and purported circularity of authoritative support, Torres terms the latter “circular coherence.” [31], [11, p. 256]353535More from Torres on this (and related topics) can be found at “Presuppositionalism and Circularity…Again?” at https://apolojet.wordpress.com/2012/03/15/presuppositionalism-and-circularity-again/, accessed June 20, 2022. A similar bifurcation of types of ultimates, propositional versus authoritative, might be in order as well. This is the version of circularity endorsed by the presuppositionalist school.

It is worth emphasizing that this type of circularity and the question begging fallacy (petitio principii) are distinct notions. According to Torres, a presuppositionalist, “Van Tillians, at least implicitly, distinguish between circular coherence and begging the question, embracing the former and rejecting the latter.” [11, p. 258]363636Other terms are used for the distinction between circular coherence and question begging: Frame calls the former circularity “broad” and the latter “narrow.” [11, p. 14] Torres labels them “virtuous” and “vicious,” respectively. [31], [11, p. 255] As Torres explains, “the main distinguishing mark between these two forms of circularity is how one handles the burden of proof, or evidential priority.” [31], [11, p. 264] Thus, the question-begging fallacy is “vicious,” whereas circular coherence is seen as “virtuous.” (cf. Russell’s “vicious circle principle” discussed in Section 3.) The circularity of presuppositionalists is a requirement of the theory of warrant to use only reasons within argumentation (arrows in $G$) that comport with the ultimates (each would appear as a node in $G$, either a sink or as part of a cycle); thus it is some relation between arrows and ultimates (specific nodes) which does not appear explicitly in $G$ (in particular, presuppositional circularity does not require a cycle in $G$). A type of “circle” appears in that ultimates and the theory of warrant are self-consistent,373737That is, the ultimate criteria underlie the reasoning used, even in (linear) argumentation for their own legitimacy, in a way that non-ultimates do not. This “circle,” however, may be non-propositional in nature, and hence not be circular reasoning as traditionally understood. but this is not a cycle in $G$. Question begging is a relation between various nodes of $G$ and is evident by the presence of a cycle in $G$. As the presuppositional notion of circularity is distinct from the familiar concept of circular reasoning, criticism of presuppositionalists for ambiguity in language may well be warranted.383838Frame recognizes the risk: “I don’t care very much whether the Christian apologist accepts or rejects the term circular to describe his argument. There are obvious dangers of misunderstanding in using it…I am more inclined now to say to my critics, ‘Granted your definition of circularity, I don’t believe in it.’ ” [11, p. 11n19]

Thus the contrast of support type (propositional versus authoritative) demonstrates how the presuppositional notion of circularity is distinct from and not contradictory to the common understanding which rightly disparages circular reasoning. As authoritative support reveals itself in evaluating rationality and not the structure among propositions, questions of authoritative support reside in the theory of warrant. Hence, we do not judge such positions here, but underscore the point that permissibility of circularity in authoritative support is not inconsistent with fallaciousness of circularity in propositional support.

3 Connections to Mathematics

3.1 The Epistemic Characteristics

It is justifiable to bring mathematics into this discussion so long as we assume that at least one statement of mathematics is genuine knowledge (to play the role of $P$ from Section 2). We add that assumption for this section.

Mainstream mathematics today follows the axiomatic method.393939The discipline settled on this more structured approach “since the free use of our conceptions…led to disaster,” specifically the paradoxes that arose from Cantorian set theory, discussed below. [17, p. 40] It begins404040That is, the theoretical construct begins in the way described. The process in which humans engage – the process of doing mathematics – rarely proceeds in this order chronologically. with a set of (formally undefined) terms414141This idea has a long heritage in mathematics. As far back as Aristotle it was acknowledged that some terms “must be undefined or else there would be no starting point.” [18, p. 20] (Note the lurking foundationalism!) Interestingly, Euclid’s Elements infamously attempted to define all concepts used, despite there being historical indications that he was aware of Aristotle’s works. Subsequently, the work of Euclid had an impact on “practically all the mathematicians who followed him for two thousand years.” [18, pp. 101-2] We will comment further on this in the subsection on formalism. and axioms (asserted-but-unquestioned assumptions, taken to be true without proof),424242Formally, one could view an axiom as proving itself. and uses logical deduction, enhanced by intuition,434343One might prefer the word “creativity” instead, or any of several related ideas. to justify new statements. For convenience, we will give names to two epistemic characteristics exhibited by this process of mathematics. Mathematics

(EC1: starts somewhere) begins with foundational principles, recognizing that mathematical knowledge must start with something;444444This is expressed nicely by $18^{th}$ century Scottish philosopher Thomas Reid in [7], esp. p. 148. and 2. 2.

(EC2: restricts the scope) restricts what is properly a “mathematical question” and what is not. Some questions, even some which refer to indisputably mathematical objects, are not mathematical questions. In particular, there are logical arguments which are not part of mathematics (and are usually placed within the realm of theology and/or philosophy) because those arguments refer to nonmathematical questions or objects.

The content of mathematics is whatever can be concluded (according to the methods allowable in the given philosophical school, as discussed below) from the starting point(s) of EC1 and within the bounds of EC2. The process of mathematics can be seen through the lens of propositional support as discussed in Section 2. On the basis of the elements of EC1 (principally axioms and definitions), mathematics is frequently seen as being foundationalist, [4, p. 215], [16, pp. 51-2] even if artificially so vis-à-vis EC2.

EC1 implies that the “why” support sequence for a statement of mathematical knowledge terminates in finitely many steps at some (set of) reason(s), $X$, which is (are) the most fundamental when viewed inside the theory at hand. $X$ is a set of one or more axioms. The restriction accomplished by EC2 is made manifest in practice by choosing $X$ not to be an ultimate and only asking questions of the objects involved which do not require the intake of ultimates. Thus, for the philosophical schools considered below, mathematical inquiry halts its inspection before reaching the epistemic level of ultimates. As ultimates are not part of mathematics, mathematics remains mum on any issues regarding circularity in the authoritative support (cf. Section 2.5). Thus, any circular reasoning arising in mathematics is indeed rightly seen as problematic.

We will demonstrate for various philosophical schools how EC1 and EC2 are exhibited.454545We give only very brief introductions to the schools of thought. For more, see [18]; [29], especially chapters 17–20; [25]; [17], especially chapter III; [12], Part 5. We will give special attention to ZFC set theory and the “Big Three.”464646Term courtesy of [25], Part III. Our goal is not an exhaustive study of the many philosophies of mathematics, but a brief introduction to show how each one considered expresses the ECs. In particular, every philosophy discussed below shares the same question for which the examination would violate EC2, namely, “once the foundational elements of EC1 are chosen, why are they the correct ones?” One could defend the choice of “these axioms instead of those,” but the point is that such a defense is a philosophical argument rather than a mathematical one. Additional comments on EC2 will be mentioned below as appropriate. Perhaps our discussion will also give the reader a sense for humanity’s intellectual limitations.474747Compare the discussion in [21, pp. 103-1].

3.2 ZFC Set Theory: Object Language, Metalanguage, and Russell’s Paradox

The default approach to mainstream mathematics today is termed ZFC set theory, the acronym referring to a list of 8-9 (depending on the specific formulation) Zermelo-Fraenkel axioms, along with the axiom of choice.484848We leave an enumeration of the axioms to the references as it is technical and inessential to the current article. For details, see [28]; [26]; [18], especially chapters VIII–XII; [12], especially Entry 5.3; [17], especially chapter III; [30]. For a less technical introduction, see [21], App. E. The genesis of set theory as its own subfield is often credited to Georg Cantor and his pioneering work on infinite series, work in which complications were quickly discovered. [28, pp. 1-2] Cantor’s work made implicit use of the axiom of abstraction which asserted the existence of a set whose members are determined by sharing any given property. A 1901 letter from Bertrand Russell to Gottlob Frege brought to light a fatal flaw with the use of this axiom. To see the flaw, form the set $S$ whose members share the property that the members are not members of themselves. Is $S$ a member of itself? If so, then by its own defining property, $S$ is not a member of itself. And if not, then, again by the defining property, $S$ is indeed a member of itself. In both cases, $S$ is simultaneously a member of itself and not a member of itself. This contradiction was termed “Russell’s Paradox” and forced mathematicians to concede the logical flaw in the seemingly innocuous, intuitive realization of set theory afforded by the axiom of abstraction. [28, pp. 5-8]

Russell himself gave an explanation for this state of affairs in 1905. The problem lies with the use of impredicative definitions, a term first used by Poincaré in 1906, “wherein an object is defined (or described) in terms of a class of objects which contains the object being defined. Such definitions are illegitimate” since they are circular; hence the related name, Russell’s “vicious circle principle.” [18, pp. 204-7], [25, p. 116], [17, p. 42]494949Circularity in such definitions is indeed problematic, even in view of Section 2.5, since propositional support is our goal. Notions such as a “set of all sets” are impredicative and hence barred.505050Some mathematicians and philosophers have defended the use of impredicative definitions. This debate is beyond the scope of this article. See [25, pp. 128, 180], [17, pp. 42-5]. The Burali-Forti paradox regarding the set of all ordinals is another closely related concept too technical for our coverage. See [28, pp. 8, 133]. A subtle but powerful modification, introduced by Ernst Zermelo in 1908, led to a resolution of Russell’s Paradox and was included in ZFC as the “axiom schema of separation.” It required that a given set first be known to exist; then, members of that set with a given property can be collected in a subset. Other paradoxes (or contradictions) were found in early set theory as well, the details of which we leave to other works. [28, pp. 5-12], [18, pp. 204-7], ch. IX.

What are ZFC’s intellectual foundations (EC1)? The axioms and formally undefined terms such as “set” are taken as given and unquestioned.515151Similar comments apply equally well to Russellian type theory, von Neumann-Bernays-Gödel set theory, and other, alternative set-theoretic foundations.

How does ZFC exhibit EC2? Through an object language/metalanguage distinction.525252Some authors use the term “syntax language” instead of metalanguage. [17, p. 63] Such a distinction is what allowed the paradoxes to be resolved. The object language is the language in which we discuss the mathematical entities themselves, whether they be sets or numbers or anything else, and the metalanguage is the “language in which we talk about the object language.” [28, p. 11] The metalanguage includes the object language, but not vice versa. Indeed, this relationship is what allows us to leave the notion of set formally undefined in the object language and yet still talk about and understand it in the metalanguage. Whereas axioms or definitions are treated as ultimates (here, synonymous with “foundations”) when doing mathematics, they are only ultimate with respect to the object language, not the metalanguage.535353Kline offers an analogy at [18, p. 250]. “If one wished to study the effectiveness or comprehensiveness of the Japanese language, to do so in Japanese would handicap the analysis because it might be subject to the limitations of Japanese. However, if English is an effective language, one might use English to study Japanese.” This distinction makes it possible to “get outside” of set theory (and thus mathematics) to discuss mathematical objects/notions without doing mathematics (and, hence, without being unwittingly restricted by its methods or limitations). Indeed, much of the history of mathematics since the late 19th century demonstrates the utility of making this distinction. As Suppes summarizes,

Every one of these paradoxes arises from having available in the language expressions for referring to other expressions in the language. Any language with such unlimited means of expression is perforce inconsistent. Consequently it is important to distinguish between the object language – here the language in which we talk about sets – and the metalanguage, that is, the language in which we talk about the object language…In other words, we avoid these paradoxes by severely restricting the richness of our language. [28, p. 11]

Let us illustrate this with our $P,Z$, and $X$ notation from above. If $P$ is a statement of knowledge in mathematics, a complete support sequence of $P$ (a proof of $P$) could be written like “$P$ because $Q$ because $R$ because … because $X$ because $Y$ because $Z$.” The supporting reasons up to $X$ would be considered part of mathematics, and at that point, the mathematical proof (the use of the object language) would stop.545454Formally, $X$ would be one or more axioms, though in practice most mathematical proofs stop short of this. $Y$ and $Z$ would be statements in rational human endeavors which fall outside of mathematics proper, and thus would be expressions in the metalanguage which are outside of the object language.

3.3 Logicism

Once the set theory paradoxes were discovered, the question of the consistency (freedom from contradiction) of mathematics was brought front and center, with the logistic school being the first serious attempt to address it. [18, p. 216] The school credits its founding to Frege in the late 1800s (though its main idea “can be traced back to Leibniz” [18, p. 217]), and it saw much development in the early 1900s from mathematicians of renown like Peano, Russell, and Whitehead. The magnum opus of the school was Russell and Whitehead’s Principia Mathematica, or Principia for short (which, in particular, included the famous theory of types by which Russell and Whitehead avoided falling victim to Russell’s Paradox). [26, p. 237] “The purpose of logicism was to show that classical mathematics is part of logic.” [29, p. 122] This would transfer any foundational questions about the nature of mathematics into the realm of logic, the consistency of which seemed firmly established.

Principia “may be considered as a formal set theory” analogous to ZFC. [29, p. 122] Thus the primary goal became to show that each of the axioms of that set theory, from which flow all of classical mathematics, are themselves part of (first order) logic. This venture failed since “at least two of these axioms, namely, the axiom of infinity and the axiom of choice, cannot possibly be considered as logical propositions.” [29, p. 123]555555In Russell’s formulation, it was the axiom of reducibility which was contentious. “The axiom of reducibility for propositions say [sic] that any proposition of higher type is equivalent to one of first order.” [18, p. 222] Russell and Whitehead admitted that this “axiom has a purely pragmatic justification: it leads to the desired results, and to no others.” [17, p. 45] Mathematicians accept the axiom of infinity, the statement that infinite sets exist, “in virtue of its content [i.e. from presupposed familiarity with infinite sets] and not in virtue of its syntactical form,” the latter of which would be necessary to consider it as a logical proposition. [29, p. 124]

Logicism exhibits EC1 by accepting first-order logic as mathematics’ starting point. Moreover, “Frege proved that each natural number exists, but his proof is impredicative, violating the type restrictions [of Russell’s type theory]. Russell had to assume the existence of enough individuals [type 0 objects which are neither classes nor sets] for each natural number to exist.” [25, p. 119]

Though logicism is considered to have failed in its primary goal, it remains highly influential in modern mathematics. “In fact, it was logicism which started mathematical logic in a serious way.” [29, p. 124]565656For a treatment of mathematical logic, see [30]. More on the logistic school can be found in [18], chapter X; [17], chapter III; [25], chapter 5.

3.4 Intuitionism

Lying in stark contrast to logicism is intuitionism, which sees mathematics, even classical mathematics, very differently. Here, “mathematics should be defined as a mental activity and not as a set of theorems (as was done…[in] logicism).” [29, p. 126] “Intuition determines the soundness and acceptability of ideas, not experience or logic.” [18, p. 235] In fact, intuitionism even questions whether the principles of logic have absolute validity. [17, p. 46] Moreover, to be considered legitimate to intuitionists, the mental activity must be constructive. Objects can only be considered to exist if one has engaged in every mental step to formulate completely that object, beginning with, for example, the “primordial intuition” for the number 1. [29, p. 125], [17, p. 49] A precise definition of “constructive” is debatable, but, for example, “Kleene saw that he could translate it as ‘compute,’ which he had precisely defined.” [17, p. iv]

Intuitionism traces many of its ideas back to Kantian philosophy. [25, Ch. 7, Sec. 2] Additionally, “just as logicism is related to [philosophical] realism, intuitionism is related to the philosophy called ‘conceptualism.’ ” [29, p. 128] Mathematicians such as Descartes and Pascal had advocated for a leading role of human intuition, and the mantle was taken up by such luminaries as Kronecker, Borel, Lebesgue, Poincaré, Baire, and, especially, L. E. J. Brouwer. [18, pp. 230-1, 234]

Many mathematicians consider intuitionism unnecessarily restrictive on the tools available to them. For example, the law of the excluded middle (the principle that either a statement or its negation, but not both, is true) is considered invalid mathematics by many intuitionists. [29, p. 126], [18, p. 237], [25, pp. 173-4], [17, pp. 47-8] The constructive restriction likewise eliminates the validity of many existence proofs in mathematics. [18, pp. 238-9] Further, constructive proofs are often so much lengthier and more complex that opponents see the burden as too great. Even more troubling, there are theorems which are true in intuitionistic mathematics but false in classical mathematics.575757For example, in intuitionistic mathematics, the statement that “every real-valued function which is defined for all real numbers is continuous” is true. [29, p. 127] This does not bother intuitionists: while the logicists sought to ground all of classical mathematics, intuitionists do not consider all of classical mathematics legitimate, and the illegitimate parts need no grounding. Instead, intuitionists have the goal of giving “a valid definition of mathematics and then ‘wait and see’ what mathematics comes out of it,” fully expecting that some of classical mathematics will never be realized this way. [29, p. 126] “Brouwer recognized that intuitionistic mathematics is not a mere restriction of classical mathematics, but is incompatible with it.” [25, p. 184] More generally, “as a substitute for classical mathematics [intuitionism] has turned out to be less powerful, and in many ways more complicated to develop.” [17, pp. 52-3] For reasons such as these, this is a philosophical school that the majority of mathematicians have repudiated. And, as with virtually all philosophical schools, disagreements within intuitionism exist as well. [18, p. 244]

“According to intuitionistic philosophy, all human beings have a primordial intuition for the natural numbers within them…we have an immediate certainty as to what is meant by the number 1…” [29, p. 125] and thus EC1 is manifest in intuitionism. As Leopold Kronecker famously quipped, “God made the integers; all the rest is the work of man” (quoted in [18, p. 232]). Brouwer took mathematical notions like addition and mathematical induction to be intuitively clear, hence foundational, also. [18, p. 235] Even so, intuitionism places mathematics at a higher epistemic level (i.e. closer to ultimate) than do most other philosophies of mathematics. Since mathematics is “identical with the exact part of our thinking [since it is a mental activity]…no science, in particular philosophy or logic, can be a presupposition for mathematics. It would be [fallaciously] circular to apply any philosophical or logical principles as means of proof, since mathematical conceptions are already presupposed in the formation of such principles.585858Intuitionists might therefore revise EC2, though not dispose of it. [Mathematics has] no other source than an intuition, which…[is] immediately clear” (see [17, p. 51], quoting a work from Heyting).

Intuitionism’s influence upon the understanding of mathematical existence, as well as on the related school of constructivism, remain significant today. [18, p. 244], [25, p. 184] Also, “the possibility of an intuitionistic reconstruction of classical mathematics…is not to be ruled out.” [17, p. 53]

3.5 Formalism

As a critic of both the logistic and intuitionist schools, with special vitriol for intuitionism, [18, p. 246] David Hilbert is the primary figure in the formalist school. (That said, many of the basic notions existed before his “Hilbert program” was set in motion around 1910.) The objective of formalism is, as the name suggests, to formalize a given axiomatization using methods which Hilbert called “finitary,” a somewhat subtle notion which we need not attempt to define here. [29, pp. 128, 131], [18, p. 250] The formalization process involves creating a first order language for the axiomatized theory by enumerating a vocabulary: symbols for items like quantifiers, the equality relation, connectives (“not,” “and,” “or,” etc.), variables, and the undefined terms particular to the theory. While this process envelops the theory with rigidity, perhaps constraining free-flowing creativity, it has the advantage of allowing one to study the theory itself via the tools of mathematical inquiry (an undertaking appropriately termed “metamathematics”).595959For a thorough discussion of metamathematics, see [17]. Hilbert’s purpose was to prove mathematically that mathematics itself was consistent, [29, pp. 128-30] and even further, to prove the completeness of the mathematical enterprise; that is, that “the axioms…are adequate to establish the correctness or falsity of any meaningful assertion that involves the concepts” of mathematics. [18, p. 158] For Hilbert, these twin ventures would establish “the certitude of mathematical methods” (quoted in [25, p. 158]).

Philosophically, logicism relates to realism, and intuitionism relates to conceptualism, with the philosophical leaning of formalism being nominalism. [29, p. 131] Nominalism’s focus is on the “names” of the objects it treats, as its Latin root word suggests. For a formalist, mathematical objects need not have any existence beyond the name (term, symbol) given to them within the theory. One consequence is that “anything at all can play the role of the undefined primitives…so long as the axioms are satisfied.” [25, p. 151]606060In [18], Kline includes a similar discussion of undefined terms in mathematics in the 19th century. The axioms in a deductive system give an “implicit definition” of the terms by telling us how they can be used and what one knows about them. Thus, the undefined terms in a deductive system “can be interpreted to be anything that satisfies the axioms,” introducing “a new level of abstraction” in mathematics; see [18, pp. 191ff.]. No intuitive meaning is needed. Hilbert, alluding to Euclid’s inadequate definitions,616161For example, Euclid defined a “point” as “that which has no parts.” [25, p. 156] For Hilbert, such vagueness in definition does more harm than good. In such a situation, “everything gets lost and becomes vague and tangled and degenerates into a game of hide and seek.” Quoted in [25, p. 156]. Regarding Euclid, it was also observed (before Hilbert) that in his Elements, “flaws had arisen because Euclid has been misled by his visual imagery…[H]e had inadvertently assumed certain properties of [points and lines], without stating them as axioms.” [26, p. 243] noted that the “meaning” of the undefined terms is given only in “the relations of points, lines, and planes to each other – via the axiomatization. All we can provide is an implicit definition of the terminology” by specifying their properties through the stated axioms. [25, p. 156]626262As Hilbert put it: “I have become convinced that the more subtle parts of mathematics…can be treated with certainty only in this way; otherwise one is only going around in a circle.” Quoted in [25, p. 157]. Regarding the development of logic and arithmetic, he advocated for a “partly simultaneous development,” essentially a set of co-foundational principles, in order to avoid “a vicious circle and…paradoxes.” Quoted in [29, p. 111]. The formal names can then be manipulated according to the rules set forth in the first order language, not unlike the way one plays the game of chess by moving the different pieces according to the game’s rules. [29, p. 131] This link is strong enough to have birthed a subschool referred to as “game formalism.” [25, p. 144] It is interesting to note that Hilbert developed these notions specifically to avoid circularity in mathematical reasoning, and for that he required that foundational axioms and definitions be asserted (in the axiomatization). This suggests an implicit agreement with the current article’s thesis that the rational alternative to foundationalism is circularity.636363It is even referred to as “the circular definition problem.” [4, p. 122]

The failure of formalism to prove the consistency and completeness of mathematics is now famous. Kurt Gödel’s Incompleteness Theorems of the 1930s comprised the fatal blow. [29, p. 130] By 1930, some corners of mathematics had been proven to be consistent and complete (some were even proved by Gödel himself),646464Notably, first order predicate calculus. See [18, p. 260]. but these results did not extend to the whole of mathematics. His incompleteness theorems state that, “if any formal theory $T$ adequate to embrace the theory of whole numbers is consistent, then $T$ is incomplete. This means that there is a meaningful…[and] true statement of number theory which is not provable and so is undecidable.” [18, p. 261] Putting the Big Three philosophical schools in its crosshairs, “the consistency of any mathematical system that is extensive enough to embrace even the arithmetic of whole numbers cannot be established by the logical principles adopted by the several foundational schools.” [18, p. 261] In other words, “no axiom system for mathematics as we know it is powerful enough to lead to a proof of its own consistency.” [29, p. 134], [25, p. 167]

The implications to the mathematical enterprise were far-reaching and led to further unsettling results: a mathematical proof might have pristine use of logic internally, but there was now no guarantee that mathematics more broadly was logically meaningful at all. [26, p. 244] It is reported that Hilbert became furious when he learned of Gödel’s work, but that fury has given way in the mathematical community either to unease or, more commonly, indifference toward the philosophy of the foundations of mathematics. [26, p. 245], [29, p. 132] A fuller account of these issues (crises!) will be left to the references.656565See [18], chapter XII; [25], sections 6.4, 8.1; [17], section 42. The notion of (un)certainty in mathematics is a broader theme of many of these works; cf. [4, pp. 212-6].

What about the epistemic characteristics’ relationship to formalism? On EC1, formalism begins with an axiomatized theory (a set of axioms and undefined terms), so the items in such a theory are deemed to be supplied beforehand.

Regarding EC2, “one has to talk about the [first order] language $L$ as one object, and while doing this, one is not talking within that safe language $L$ itself, … [but in] ordinary, everyday language…[In so doing,] there is of course every danger that contradictions, in fact, any kind of error, may slip in.” [29, pp. 130-1]666666Notice the object language/metalanguage distinction. Hilbert’s solution was the aforementioned insistence on finitary reasoning, a restriction which demonstrates EC2. EC2 is displayed perhaps even more clearly by Gödel’s rebuke of the formalist school. Whereas Gödel proved that a mathematical system cannot show its own consistency, at least some of the undecidable statements in a formal system can be shown to be true by informal arguments. [18, p. 263] As Gödel put it, “it is necessary to go beyond the framework of what is, in Hilbert’s sense, finitary mathematics if one wants to prove consistency of classical mathematics.”676767Quoted in [25, p. 167]. And again, “the methods which we must trust in the proof [of consistency] must include some which lie outside the collection of the methods formalized in the system.” [17, p. 211] Natural, interesting, and even crucial questions about mathematics must use tools which transcend the discipline. Since such tools evidently exist, Gödel’s theorems substantiate the claim that mathematical knowledge is not ultimate knowledge.

Despite the formalist’s failure in their initial goal, the school has had immense, ongoing impact on the way mathematics is done today. Fields such as metamathematics or proof theory,686868These terms are introduced in [17, p. 55]. as well as “modern mathematical logic and its various offshoots, such as model theory, recursive function theory, etc.” [29, p. 130] owe an immeasurable debt to formalism. The emphasis on stripping objects of their “unnecessary” features (whatever is not pertinent to the logical structure), and the recognition that physical intuition can be misleading, help explain why “so much of the abstraction of early 20th century mathematics stemmed from Hilbert’s viewpoint.” [26, p. 243], [17, pp. 54-5]

3.6 Other Philosophies of Mathematics

For each of the philosophical schools given above, there remain adherents of (modified versions of) them,696969Shapiro provides examples throughout [25]. but they don’t exhaust mathematicians’ and philosophers’ present-day views. We will make passing mention of some other selected philosophies of mathematics, some of which overlap the above (and each other). As described by theologian and mathematician Vern Poythress, “Platonism says that numbers and mathematics belong to a realm of abstract ideas, a realm that exists before mathematicians begin to study it.” [21, p. 153] An alternative view, empiricism, seeks to found mathematics in the experiences of the senses. [21, pp. 154-5] Predicativism “accepts the natural numbers as given…either by our intuition or by a Platonic realm or by both,” and then seeks to rebuild as much of mathematics as possible while avoiding impredicative definitions. [21, p. 159]707070For a thorough, favorable treatment of predicativism, see [27]. Information available at https://philpapers.org/rec/STOADO-2. Accessed July 12, 2022. These three views each begin mathematics proper with (at least) the natural numbers, and thus all exhibit EC1.717171Their ontological claims about the nature of mathematical objects differ, but that is immaterial to our discussion. William van Orman Quine’s philosophical naturalism advocates for science as the guide for what constitutes acceptable mathematics, displaying EC1 by assuming results of scientific inquiry before mathematics can commence. [21, p. 160]

Shapiro’s treatment of some contemporary philosophies of mathematics tells a similar story. [25, Part IV] The core difference between the philosophers and their associated schools (e.g. realism vs. anti-realism) is the ontology of mathematical objects. But in each case, something is provided to mathematics (frequently, the natural numbers) before it can begin. EC1 thus appears in each. As two examples: in structuralism, the structures themselves which mathematicians study are the givens. [25, pp. 258ff.] In Hartry Field’s fictionalism, specifically his realization of Newtonian gravitational theory, “points” and “regions” are (among) the givens. [25, pp. 229ff.]

It is worth noting that Poythress’s Redeeming Mathematics proposes (with remarkable succinctness and in lay terminology) an alternative philosophy of mathematics which could, in some sense, be considered an exception among the philosophies of mathematics. He grounds the tools of mathematics in the Christian God, who is ultimate in the Christian worldview. Poythress criticizes the “reductionisms” of the aforementioned philosophies of mathematics727272 “Each of these [philosophies] has a preferred starting point…[which] becomes the preferred platform for explaining everything else in mathematics.” [21, p. 151] Note the close connection of this observation with EC1. and lays out a more holistic approach which might require refinement of the ECs.737373The comments in [21, pp. 104-5] are prime examples. In Poythress’ view, God, and hence ultimate knowledge, is inextricably linked to mathematics. The pivotal feature is then the internal coherence of the system, not which “starting point” is chosen.

3.7 (The Lack of) The Epistemic Characteristics in Theology and Philosophy

If mathematics has been able to employ EC1 and EC2 with great success (as have other academic disciplines), can fields such as theology, philosophy, religion, or worldview studies emulate it? These latter fields are of a fundamentally different character to the extent that duplicating the pattern is not possible. Many, though not all, of their questions are inherently about ultimates (recall we refer to these as “ultimate questions”). For the ultimate questions in these fields (or in whatever field one places ultimate questions), there is no way to exhibit both EC1 and EC2.

Most significant for the present discussion is in regards to EC2.747474To briefly address EC1: while different philosophical or theological schools of thought disagree on the starting points of EC1, each school presumably has some such foundational set. For example, see [13] for a comparison of starting points for generalists and particularists, and [6] for a thorough treatment of the deep epistemological issues faced in determining the elements which comprise EC1 (note the assumption that such elements exist). Limiting the scope of ultimate questions is not possible, for to do so would be to render them non-ultimate. In this case, an object language/metalanguage distinction is impossible; that is, the object language also includes the metalanguage, and thus the two are equal (recall that the reverse inclusion holds for mathematics and any other discipline in which the distinction is utilized).757575This is illustrated in simple terms in [29]: “when the logicists tell us what they mean by a logical proposition, … they use philosophical and not mathematical language. They have to use philosophical language for that purpose since mathematics simply cannot handle definitions of so wide a scope.” [29, p. 124] Similar comments for intuitionism can be found in [29, pp. 127-8]. There is not, and indeed cannot be (for our human agent $H$), a greater context in which to discuss them. Since these questions arise naturally in theology and philosophy, there is no way to “get outside” of those disciplines when investigating such questions; one cannot attempt to answer ultimate questions in theology without theologizing. There is nothing in the line of reasoning that falls outside of the umbrella of, or transcends, the discipline. As Van Til put it, “[w]e must go round and round a thing to see of its dimensions and to know more about it, in general, unless we are larger than that which we are investigating,” [32, p. 24], [11, p. 257] and in the case of ultimate questions, we are by definition not “larger” than they are. In this sense, theology and philosophy are intellectually all-encompassing. Evidently it is this observation, in part, which motivates worldview scholar Nancy Pearcey to refer to her Christian worldview as “Total Truth.” [19] EC2 cannot be exhibited for ultimate questions.

Ultimates thus have a “meta” relationship to other statements of knowledge. They don’t simply lead to or propositionally support other statements; rather, they form the framework in which an intellectual discipline is done. Intelligibility itself, a precondition of any intellectual pursuit, is determined by the ultimates.767676In theology in particular, ultimates are often referred to as presuppositions. On this topic, Frame says, “The ‘pre’ [in presupposition] should be understood mainly as an indicator of eminence…not temporal priority.” [11, p. xxxii] Since it is fields such as philosophy and theology which handle ultimate questions, the ECs cannot both be always exhibited therein.777777Note that some of the questions which are asked in theology, philosophy, and related disciplines can be handled in such a way that both EC1 and EC2 can be exhibited. The main point is that, unlike mathematics, not all questions in theology or philosophy can exhibit both epistemic characteristics (cf. the discussion of EC1 in subsection 4.3). One particular conclusion, in view of (FC), is that if circular support is acceptable anywhere, it would be so in fields such as theology and philosophy by virtue of their inclusion of ultimate questions.787878Recall that it is precisely the exhibition of the ECs which allows mathematics to categorically reject circular reasoning; see Section 3.1, and compare the nuance in the meaning of “circularity” in Section 2.5.

3.8 Analogies

As an epilogue to the primary thrust of the article, we will make brief mention of a potpourri of analogies, both within and without mathematics.

First, the epistemic distinction between ultimates and non-ultimates bears a resemblance to the distinction in properties between the boundary and interior of certain mathematical objects, such as a manifold. A planar disk (a circle with all of the interior points) is an example. The interior has different properties (most notably, it is 2-dimensional) than the boundary (the 1-dimensional edge of the disk). As ultimates are at the “boundary” of human knowledge, it is not surprising that their epistemological properties differ qualitatively from the “interior,” non-ultimate knowledge.

Consider another mathematical analogy: a local/global distinction. A local property is satisfied only at or within a neighborhood of a particular point/object (for example, any open neighborhood of a point on a 2-sphere smaller than the 2-sphere itself, is isomorphic to the real plane), whereas a global property in mathematics refers to the set of all points (objects) under consideration (the 2-sphere, taken as an entire object, is not isomorphic to the real plane). The realm of epistemological influence of a non-ultimate proposition can be thought of as local, whereas an ultimate has global influence.

Many of the above concepts are not unique to mathematics, so it can be illuminating to see an “everyday” example. One appears in dictionaries: a word cannot be defined without the use of other words. Since there are only finitely many words, this means that, eventually, some word has to be left either undefined (deemed so foundational as to be left out of the dictionary), or there is a sequence of words which are used in each other’s definition (which is circularity; examples can indeed be found in modern dictionaries). (FC) applies to definitions as well as propositions.

The object language/metalanguage distinction can also be seen outside of mathematics. Consider a theatrical script: it employs stage directions (metalanguage) in addition to the lines of the show (object language). Similarly, some government or legal documentation says, “This page intentionally left blank.” Those words themselves cause the page not to be blank, but they are considered part of the document’s metalanguage rather than the document’s content (object language).

4 Follow-up Questions

4.1 Does this article itself use foundational or circular reasoning?

This article uses foundationalism in its propositional progression. We assumed several notions at the outset, such as the use of logic and rationality in our truth-seeking. These were taken as ultimate (without further defense) for this article. This particular assumption illustrates the “meta” relationship that ultimates have with respect to other statements: had we not assumed anything regarding logic and rationality, by what standards would we have proceeded to answer any questions or make any claims? We could not reason about reason without using reason! The rationality assumption established a precondition for the remainder of the article.

4.2 Could different individuals choose different ultimates, resulting in diverse but

self-consistent worldviews that are mutually exclusive?

In mathematics, this circumstance (a panoply of axiomatic theories) was essentially endorsed by David Hilbert. “Literally, Hilbert claimed that if a collection of axioms is consistent, then they are true and the things the axioms speak of exist.” [25, p. 156] Axioms in mathematics can be arbitrarily chosen so long as they are consistent - a version of coherence epistemology in mathematics. In fact, it is this line of reasoning which allows for the parallel797979No pun intended. existence of multiple geometries (Euclidean and non-Euclidean) by asserting different axioms. Mathematicians are comfortable with this state of affairs, so can we apply the same reasoning to worldviews, inclusive of ultimates?

A priori an affirmative answer to this question seems possible, and by our assumptions it seems to be so in this article. However, let us momentarily suspend our moratorium on evaluating theories of warrant. It is the author’s belief that, in practice, all truth-conscious individuals operate, implicitly or explicitly, under some shared presuppositions808080Presuppositions in this context are ultimates, but as the former term is used more commonly when discussing worldviews, we will adopt it for this follow-up question. which eliminate some worldviews from rational contention; in particular, the possibility of global mutual exclusivity is removed. These presuppositions include shared ultimate standards (e.g. for rationality and morality) by which everything else is evaluated. The commonality between persons’ presuppositions assuages fears that two individuals could have perfectly consistent and veracious worldview theories and yet have nothing on which they agree; in reality, the two don’t actually inhabit disjoint systems.

For example, any theological or philosophical system that includes the principle of noncontradiction, as is assumed here, requires internal consistency. A necessary condition for all other ultimate criteria would be that they not refute anything already in the system of ultimates. This condition alone can indeed rule out many worldviews which might otherwise seem to be viable a priori. Details and a more thorough treatment of how to choose the “right” presuppositions, or what it might mean to be the right presuppositions, would be the topic of another work.

4.3 Does God need foundationalism, circularity, both, or neither?

One would first have to define what is meant by “God.” Specifically in Christianity and the other Abrahamic faiths, the conception of God is traditionally as an ultimate God (in particular, He is or determines the ultimate standard for rationality and morality). In other words, by definition, there is no higher standard by which God can be judged. His word can be taken as foundational, and His authority (and anything in His nature) as self-verifying; the notion of circularity in authoritative support seems to be consistent with God’s nature in these faiths. This concept is part of what theologians refer to as divine aseity,818181“Something ‘exists necessarily’ if it cannot fail to exist. Something ‘exists contingently’ if it can fail to exist. Whether or not something contingent exists depends on factors outside itself. Therefore, ‘contingent existence’ (such as ours) is ‘dependent existence.’ Necessary existence is aseity or self-existence.” [11, p. 116n36] For more on divine aseity, see [11, pp. 265-78]. For more on the relationship between divine aseity and mathematics, see [21, pp. 63ff.] a concept hinted at in Hebrews 6:13;828282“For when God made a promise to Abraham, since he had no one greater by whom to swear, he swore by himself…” [15] God has no standard to appeal to above Himself! Thus, anything that is known to be part of God’s nature need not (indeed cannot) be further justified.838383The sticking point is knowing which characteristics are part of His nature. There is not agreement there, even among devoted practitioners of a particular faith tradition. But the point is that if the qualities of His nature are assumed to be known, then those qualities ground completely the authoritative justification. A common objection is that grounding a standard like rationality or morality in God’s nature is “kicking the can down the road,” suggesting that there is another step of reasoning beyond God’s nature which requires explanation. Such an objection violates the conception of God as being Himself ultimate (by definition), or it seems to make the implicit assumption that finite termination of the support sequence is impossible. Either is problematic. We must be careful, though, because it is quite possible that the assumptions of the current article do not all necessarily apply to God’s “divine knowledge.” In particular, the finiteness and time-bound nature of humanity are evidently not limitations for God. Thus, we do not claim that the argumentation of the current article applies to divine knowledge, nor is a claim made excluding the possibility of unimagined spiritual or supernatural realities about which humans have never received revelation nor have other means of gaining knowledge.

One final comment on the relationship between Christian faith and human knowledge is in order. The Bible, especially the wisdom literature (such as the book of Proverbs), affirms that human knowledge, mathematics included, finds its final foundation in (reverence for) God. Job 28:28; Psalm 111:10; Proverbs 1:7, 2:6, 3:5-6, 8:22, 9:10; and Colossians 2:2-3 are examples.848484They read as follows. “And he said to man, ‘Behold, the fear of the Lord, that is wisdom, and to turn away from evil is understanding’ ” (Job 28:28). “The fear of the Lord is the beginning of wisdom; all those who practice it have a good understanding. His praise endures forever!” (Psalm 111:10). “The fear of the Lord is the beginning of knowledge; fools despise wisdom and instruction” (Proverbs 1:7). “For the Lord gives wisdom; from his mouth come knowledge and understanding” (Proverbs 2:6). “Trust in the Lord with all your heart, and do not lean on your own understanding. In all your ways acknowledge him, and he will make straight your paths” (Proverbs 3:5-6). “[Wisdom says] ‘The Lord possessed me at the beginning of his work, the first of his acts of old’ ” (Proverbs 8:22). “The fear of the Lord is the beginning of wisdom, and the knowledge of the Holy One is insight” (Proverbs 9:10). “…that their hearts may be encouraged, being knit together in love, to reach all the riches of full assurance of understanding and the knowledge of God’s mystery, which is Christ, in whom are hidden all the treasures of wisdom and knowledge” (Colossians 2:2-3). Scripture quotations are from [15]. At risk of over-enthusiastic hermeneutics, verses like these seem to suggest that, in a Christian worldview, EC1 will eventually be exhibited by all human knowledge, with the starting point being God’s own self-authenticating nature. In the author’s view, the philosophical skepticism of Agrippa and the Pyrrhonists (and many philosophers of our day) is indeed correct that human knowledge would be hopeless without a “beginning;” and moreover, that that beginning must be external to humanity in order to be ultimately trustworthy. By virtue of the doctrines of historic Christian belief, the nature of God is sufficient for such a beginning.

The author would like to thank the anonymous reviewers for comments which improved the article.

\setbiblabelwidth

100