1. Question
What reasons do we really have to be sceptical about religious belief that don't also apply to mathematics? In this paper, I argue that the main worries raised with regard to religious belief apply to mathematics as well.
Let me begin by pointing out an underrated fact about mathematics: we consider belief in mathematical theorems clearly rational, even though it is completely obscure what mathematics is about. By this I don't mean that we don't know what the principal constituents of mathematical sentences are: they are numbers (or more accurately: numerals), operators, variables, etc. What I mean is that we have no clue which facts ‘in the world’ make mathematical assertions true, what the constituents of those facts are, and how we come to have knowledge about them.
What is mathematics about? For most people, the principal objects of mathematics are numbers and perhaps geometrical objects. For the slightly more informed, the objects of mathematics are not only numbers and geometrical objects, but also functions, graphs, spaces, etc. For most mathematicians, the fundamental objects of mathematics are sets. For a small number of mathematicians, the fundamental objects of mathematics are categories. It doesn't matter which one of these views is most accurate (all of the mentioned objects are objects of mathematics, after all). What matters is that mathematics is widely accepted as a rational enterprise even though we have no clue how to make sense of its objects. We cannot touch numbers. We cannot see them, we cannot measure them, we cannot experiment with them. We cannot perceive them in any way in which we perceive ordinary objects such as tables, chairs, and beer mugs. The same holds for sets, functions, graphs, spaces, etc. Yet we feel perfectly comfortable making statements such as ‘2 + 2 = 4’, ‘There is no prime number greater than seven but smaller than eleven’, or ‘π is the ratio of a circle's circumference to its diameter’. By making such assertions, we commit ourselves to their contents. But how can we make such a commitment without having the slightest idea what exactly the objects those sentences refer to are, whether they exist or not, where they are located (if anywhere), and how it is that, despite all this, we can have knowledge about them?
These are some of the most central questions in the philosophy of mathematics, and there are different ways in which they have been answered. Regarding the question of semantics, some have tried to align the truth of mathematical statements with truths about ordinary (concrete) objects. If mathematical sentences are true ‘in the same way’ that ordinary sentences are true, then – so it is argued – there is no real mystery. All we have to do is to accept the existence and epistemic accessibility of mathematical objects, and we have solved the semantic issue. However, this kind of mathematical Platonism leaves questions about the metaphysical status of mathematical objects, their nature, independence, perceivability, etc. unsolved.Footnote 2
Others have tried to circumvent questions about the semantics of mathematics by focusing on the formal aspects of mathematical knowledge alone, i.e. by explaining our knowledge of mathematics on the basis of concepts such as consistency, derivability from a given set of axioms, etc. If mathematical knowledge counts as knowledge simply because its sentences are contradiction-free and were derived from an initial set of axioms according to certain pre-established derivation rules, then it seems that we don't really need to answer ontological questions about the objects of mathematics. However, aside from the fact that all attempts to reduce mathematics to pure form, i.e. to logic, have so far failed,Footnote 3 ontological questions about mathematics invariably reappear as soon as the question of truth reappears (which it always does, both as a necessary condition for knowledge, and as a general question about the epistemic status of mathematical axioms).Footnote 4
In this paper, I don't want to tackle any of these long-standing, intricate issues about the nature of mathematical truth and knowledge. Rather, I want to raise a question about their implications for other areas of abstract thought, in particular for religious belief. The question is: Given that belief in mathematics is universally accepted as rational, despite our continuous inability to answer very basic questions about its ontology and epistemology – should we not also accept as rational belief in other abstract systems of thought, such as religions? Or alternatively, if we are not prepared to call belief in religion rational, why should we grant this to mathematical belief?
Before I venture into the development of my argument, I would like to mention one final point: mathematicians do mathematics without asking themselves any disturbing questions about ontology or epistemology, and their strategy works brilliantly for them. Wittgenstein recommended quietism of precisely this kind for all areas of abstract thought: mathematics, religion, ethics, aesthetics, and philosophy in general.Footnote 5 So if quietism works for mathematicians, why should we humble philosophers bother? Well, for one thing because by asking those inconvenient questions and understanding that there are no straightforward answers to them, we can begin to understand that the rational status commonly ascribed to mathematics rests on a basis which is at least as obscure as the basis of many other abstract systems of thought. And this is a very interesting result, a result that, I hope, will work as an antidote against epistemic snobbery.
2. Rationality
Rationality has a strong social aspect. What is considered ‘good’ reasoning is highly culture-dependent; some cultures value some forms of rationality higher than others, individuals reason differently according to their socio-economic background, etc.Footnote 6 This applies especially in the case of religion, which can be seen from the fact that some countries incorporate religious reasoning in their jurisdiction, for example, whereas others don't. Mathematics, on the other hand, seems to be a universally acknowledged form of rational thinking, independently of cultural or socio-economic factors. Why is that? What is it that makes belief in mathematics rational across all borders?
Before we delve into the question of mathematical rationality, let us briefly bring to mind what I mean when I use the term ‘rational’. Generally speaking, rational judgement is sound and as such, the opposite of contradictory, unreasonable, or simply foolish judgement. Put in slightly more philosophical terms, the minimal requirement for a belief to be rational is that it must be epistemically permissible to hold it, i.e. it must compatible with our (total) evidence.Footnote 7 How exactly the concept of a rational belief should then be analysed in relation to other epistemological concepts such as ‘justification’, ‘reliability’, or ‘safety’, is a matter of debate.Footnote 8
Without entering any of these on-going debates in epistemology, let's determine very roughly that in ordinary contexts, a belief B is considered rational if holding the belief is justified, which – again, very roughly – could be spelled out as: B was arrived at in a way suitable for the attainment of true beliefs (this clause ruling out erroneous chains of reasoning, illusory states of mind, etc.). I call this the ‘naive’ analysis of rationality, which holds for non-abstract beliefs such as perceptual beliefs, beliefs based on testimony, memory, etc. For example, my belief that snow is white ought to be considered rational if I drew that conclusion in a suitable way, perhaps if I came to that conclusion by perceiving snow (without being visually impaired, hallucinating, etc.), or perhaps if I extracted the information from a trustworthy source such as the Encyclopaedia Britannica, etc. The crucial point is that in ordinary cases of rational belief we have some understanding of the facts underlying our belief and of the way we achieved the belief, i.e. how we accessed the underlying facts. Put differently, ordinary rational belief is ontologically and epistemologically backed up.
Now, as we have set out in the beginning, whether mathematics is backed up in this way has been a matter of debate for decades: Different kinds of realists still argue against different kinds of nominalists, and neither formalist, intuitionist, structuralist, naturalist, fictionalist, constructivist, or predicativist accounts have made decisive additions to the debate. Nevertheless, mathematics continues to enjoy an epistemically spotless reputation: mathematical belief is considered rational, come hell or high water.
If belief in mathematics is rational even though we neither understand its ontology nor its epistemology, then why isn't belief in religion or any other abstract system of thought rational as well? Note the similarity in open questions: What do numerals refer to [what does ‘God’ refer to]? Do numbers exist [does God exist], and if you believe so, what evidence supports your belief? Where exactly are numbers located [where is God located]? How can we have knowledge of them [Him/Her]?
The inability to answer these questions is considered a huge problem for religious belief, whereas mathematical belief is unaffected by it. This, I argue, is an unjustified double-standard.
3. Mathematics
What gives mathematics its allegedly firm epistemic status? In order to answer this question, we need to ask ourselves when belief in an abstract system of thought, i.e. a system of thought not based on evidence, is rational. Which conditions does such a system need to fulfil in order to render belief in its claims justified?
When I say ‘abstract systems of thought’, I mean systems of thought whose main concepts do not primarily refer to spatiotemporal objects or measurable events, but rather to abstract objects, which are characterized by being non-spatial, non-temporal, and causally inert. Examples for abstract systems of thought are mathematics and religion, ethics, political, economic, aesthetic theories, theoretical physics, etc. What they have in common is that they quantify over abstract entities and relations, some of them in order to group together concrete objects and events according to particular abstract concepts. However, not all abstract systems are on a par when it comes to their explanatory value: some can be mapped onto reality in a more straightforward manner than others, some provide a model of particular aspects of reality only, whereas others attempt to provide a model of reality as a whole, etc. So what does the rational acceptability of an abstract system of thought depend on? According to a ‘naive’ analysis of rationality, a belief arrived at in a way suitable for the attainment of true beliefs (perception, reliable testimony, etc.) would be rational. But this is not how we typically arrive at abstract beliefs - on the contrary, we don't have a clear idea of how we arrive at highly abstract beliefs such as mathematical beliefs at all. Thus, the question of the causal source of abstract beliefs is superseded by other considerations according to which we judge the rationality of abstract systems.
Let's turn to physics for a moment. Physics is arguably the most successful provider of abstract models for concrete phenomena, so it is a good place to look for the general criteria a model needs to fulfil in order to justify rational belief in it. Stephen Hawking and Leonard Mlodinow, two exceptionally distinguished physicist, suggest that a model (what I call an ‘abstract system’) is good if it
(1) Is elegant (2) Contains few arbitrary or adjustable elements (3) Agrees with and explains all existing observations (4) Makes detailed predictions about future observations that can disprove or falsify the model if they are not borne out.Footnote 9
As Hawking and Mlodinow concede, the first two conditions are less crucial because they don't refer to a model's truth. They reflect the scientist's sense of aesthetics rather than central criteria for the acceptability of a model. It is the other two criteria which decide whether a model is rationally acceptable or not: if a model neither explains any existing observations, nor is able to make any precise predictions, it is clearly not a good model. So a model, or abstract system, needs to be applicable to the real world, and able to make predictions about that world.
Two other conditions must be added, both of them so obvious that they hardly need to be mentioned (which is perhaps why Hawking and Mlodinow don't mention them): any abstract system must be consistent, and its fundamental premises, i.e. its axioms, self-evident, obvious, or otherwise immediately acceptable. So we are left with the following four criteria for the rational acceptability of an abstract system:
1. Immediate acceptability of the fundamental premises (axioms)
2. Internal and external consistency
3. Applicability to (some aspects of) reality
4. Enabling successful real-life predictions
These four conditions cover both the internal composition of an abstract system of thought as well as its external relation to the concrete world. So let's see to what extent mathematics fulfils these conditions, and where it fails to do so. I will examine each point in turn, starting with the last, and juxtapose the case of mathematics with the case of religion.
3.1 Predictability
The fourth condition I gave for the rational acceptability of an abstract system of thought is that it should enable successful predictions in the concrete world. The obvious winner in this category is, of course, applied physics, which manages to predict physical events precisely, at least in those areas that are well-understood, such as classical mechanics. For example, a physicist will be able to calculate the speed at which a falling object with a certain weight will hit the ground, and predictions of this kind will be true no matter how many times the experiment is repeated.
Mathematics, on the other hand, makes no predictions whatsoever. The objects of mathematics are abstract, not concrete, and the relations between its objects are eternal rather than temporally ordered. It is true that the natural sciences make use of the mathematical language: quantum mechanics depends on the use of Hilbert spaces, general relativity on differential geometry, etc. The use of statistics is even more widespread, and many theories benefit from the use of mathematical language because it makes their formulation more elegant and simple. It remains a fact, however, that mathematics itself is nothing but a tool for the natural sciences and has, by itself, no predictive force whatsoever.
3.2 Applicability
Something similar holds of applicability. An abstract system is applicable to the concrete world if its central concepts apply to (constellations of) concrete objects and if the relations of its central concepts mirror the relations in which the respective concrete objects stand. Physics is once again a case in point, but also some sociological or political theories which, for example, classify communities of people and their communal decision-making processes according to abstract concepts such as ‘democracy’, ‘monarchy’, ‘theocracy’, etc. ‘Applicability’ is a more static concept than ‘predictability’ because the applicability of a concept to a concrete counterpart doesn't depend on its future applicability, whereas predictability has an inherently temporal aspect. For example, a community may be classified as a theocracy today but may have turned into a democracy in ten years from now, so that ‘theocracy’ is no longer applicable to it. Predictability, on the other hand, requires us to hold the applicability of certain concepts fixed over time.
This point is important in order to see why it is easier for an abstract system of thought to fulfil the condition of applicability than the condition of predictability: in order to be applicable to reality, it is sufficient for our abstract concepts to apply here and now to reality, whereas they have to continue to apply in the future if we want to make predictions. Needless to say, a concept that is applicable today may, of course, also be applicable in the future, but that doesn't mean it will enable predictions. A democracy may still be a democracy in ten years from now, but nothing in the concept of democracy enables us to predict that development.
It is now easy to see why mathematics also doesn't fulfil the condition of applicability. Its central concepts, i.e numbers, sets, etc., have no straightforward application to anything ‘in the world’. There is no concrete counterpart to the number three, the fact that three is the successor of two doesn't mirror a relation in the concrete world, and neither does the iterative hierarchy of sets. The content of mathematics is purely formal.
One might argue that at least those parts of mathematics that feature in the natural sciences (which excludes the large majority of today's mathematics, most of which is ‘pure’ rather than ‘applied’) are applicable to the real world. However, even there mathematics is a mere tool, it doesn't itself provide applicable concepts. And it is not even clear that mathematics is an essential tool for the natural sciences. In fact, there have been attempts to show that mathematics is conservative over the natural sciences, i.e. that the results of natural sciences can be expressed entirely without mathematics. Hartry Field, for example, made a very well-developed attempt to show this for the example of Newtonian Gravitational Theory, by using what he calls ‘representation theorems' and ‘bridge laws'.Footnote 10 Of course the possibility of formulating Newtonian Gravitation without the use of mathematics does not prove that all of physics can be formulated in a mathematics-free way. However, it is a powerful indication of an in principle independence of the natural sciences from mathematics, and shifts the burden of proof at least to some extent towards the proponent of a contrary view.
Indeed, there have been attempts to argue for the contrary, most of which can be subsumed under the general class of ‘Indispensability arguments', the central claim of which is that we are ontologically committed to all objects postulated by our best scientific theories – hence, also to numbers. Arguments of this kind were originally intended to support mathematical Platonism, i.e. the conviction that there are good reasons for us to believe in the existence of mathematical objects.Footnote 11 The core of the argument rests on the premise that mathematics is indispensable to the natural sciences, and on the naturalist conviction that we ought to believe in the existence of all objects posited by our best scientific theories. Now, as argued before, whether or not mathematics is in fact dispensable or indispensable to the natural sciences remains to be shown. However, even if mathematics turned out to be an essential tool for the natural sciences, it remains a fact that only a small part of contemporary mathematics is in fact suitable to be used as such. The large majority of modern mathematics, such as abstract algebras, pathological functions, transfinite numbers, etc., has absolutely no straightforward application in the real world at all. Such pure mathematical theories ‘are protected from empirical falsification by positing a special realm of abstracta about which they remain true’,Footnote 12 no matter how the real word actually is. Since indispensability arguments do not apply to unapplied mathematics, the inapplicability of the larger part of mathematics remains a fact in either case.
3.3 Consistency
What about the condition of internal and external consistency? Mathematics is certainly externally consistent because it doesn't stand in contradiction with anything outside the mathematical realm. In fact, it is not even clear what it would mean for something to stand in a contradictory relation to mathematics. So let's turn our attention to internal consistency.
A system of thought is internally consistent if it doesn't contain contradictions. For an abstract system of thought AS, this can mean that it is impossible to derive both P and not-P from the AS's axioms (syntactic definition) or that there is at least one model under which all theorems of AS come out true (semantic definition). A system is complete relative to a certain property A if every formula instantiating that property is a theorem of that system, i.e. can be derived from its axioms via the respective rules of deduction. This is most interesting for the case of the property of truth: Gödel showed in his first incompleteness theorem that all axiomatic systems (except trivial ones) capable of doing arithmetic are incomplete, i.e. that there are arithmetical truths about the relations of the natural numbers which are unprovable in a given system, such as Peano Arithmetic (PA). In his second incompleteness theorem, Gödel showed that by extension of his first incompleteness theorem, if PA is consistent, it cannot prove its own consistency. This has been widely accepted as a proof that it is impossible to give a complete and provably consistent set of axioms for all of mathematics.Footnote 13 Only very few, such as Edward Nelson at Princeton, have seriously considered the further implication following from this, i.e. that mathematics is possibly inconsistent and hence, false.Footnote 14
It is true that it is generally assumed that mathematics is, in fact, consistent. This assumption seems intuitively justified given that the natural numbers seem to provide a model for PA (rendering PA at least logically possible), and given that the standard inference rules of first order logic seem to be sound, i.e. they seem to make it impossible to derive a contradiction from any description of a logically possible state of affairs.Footnote 15 Yet without being disproven, the inconsistency of mathematics remains a logical possibility as well.
This leads us straight to the question to what extent the logical possibility of its inconsistency infringes on the rational acceptability of an abstract system such as mathematics. Someone might compare doubting the consistency of arithmetic to views like external world scepticism: even though there is a conceivable possibility that a certain, commonly assumed state of affairs does in fact not hold, that possibility has no bearing on the way we behave in our daily lives. Just as most of us live their lives as if an external world existed, it might be argued, mathematicians, scientists, philosophers, and most other people as well make use of mathematics as if it was consistent. As Wittgenstein put it, ‘if I am clear about the nature of mathematics, then the question about its consistency cannot arise at all’. And: ‘Do I have to wait for the proof of consistency before I can apply the calculus?’Footnote 16 In other words, someone might claim that the mere conceivability of inconsistency doesn't render belief in mathematics irrational, just like belief in an external world is usually not considered irrational, despite the abundance of sceptical scenarios.
However, such a view ignores a substantial difference between the possibility that mathematics is inconsistent and the possibility that the external world doesn't exist: the latter indicates an alternative understanding of the external world, i.e. one of several ways of making sense of our sense perceptions. The former, on the other hand, indicates an internal contradiction in a closed system. This is arguably a much harder blow for an abstract system of thought than its possible incompatibility with another abstract system of thought. So the most we can say in the case of mathematics is that we have intuitive and practical reasons to assume its consistency, but no proof.
3.4 Immediately Acceptable Axioms
The final condition for the rational acceptability of an abstract system of thought is that its axioms are immediately acceptable. Let's see how the axioms of mathematics fare in this respect. I define ‘axiom’ as follows:
Axiom: An assertion or premise which is taken as a starting point, or a fundamental organizing principle, for a chain of reasoning in a given domain of thought, and which is assumed to be true yet unprovable (i.e. self- evident, obvious, or otherwise immediately acceptable) in terms of even more fundamental assertions within the respective domain of thought. Axiom schemata express general rules, or laws, describing unchanging relationships between objects and/or properties.
Complex assertions in axiomatic systems of thought are typically deduced from a basic set of axioms and axiom schemata of the given domain. Axioms are thus primitive sentences in terms of which more complex ones are defined, but which cannot be further analysed themselves. As Russell put it, ‘since human powers are finite, the definitions known to us must always begin somewhere, with terms undefined for the moment, though perhaps not permanently.’Footnote 17 This ‘somewhere’ are axioms.
Mathematics is an axiomatic system because it is grounded in sets of axioms (structural axioms for each subset of mathematics, and general axioms fundamental to all of mathematics). Given the stunning complexity of modern mathematics, it is a remarkable fact that the entire mathematical universe can be reduced to a simple set of axioms.Footnote 18 But moreover, mathematics is perhaps the system of thought with the highest credibility of all – the (established) claims of mathematics are accepted beyond any doubt, both by mathematicians and non-mathematicians; its foundational axioms thus have a very strong epistemic status. Setting aside the possibility of erroneous chains of reasoning, which would render our mathematical knowledge unjustified even if true, accepting established mathematics is widely considered to be perfectly rational.
With few exceptions, the large majority of mathematicians consider the axioms of Peano-arithmetic (PA) to implicitly define the concept of number – arguably one of the most fundamental concepts of mathematics.Footnote 19 In modern notation (and starting with the number zero rather than the number one, as Peano did in his initial formulation), the axioms can be rendered as follows:
• 0 ∈ N (‘Zero is a natural number’)
• n ∈ N ⇒ n’ ∈ N (‘Every natural number n has a successor n’ which is a natural number’)
• n ∈ N ⇒ n’ ≠ 0 (‘Zero is not the successor of a natural number’)
• m, n ∈ N ⇒ (m’ = n’ ⇒ m = n) (‘Natural numbers that have the same successor are identical’)
• 0 ∈ X ∧ ∀ n ∈ N : (n ∈ X ⇒ n’ ∈ X) ⇒ N ⊆ X (‘If X contains the number zero, and with every natural number n also its successor n’, then the natural numbers are a subset of X’)
These axioms are pretty straightforward and understandable even to a non- mathematician, and their simplicity befits the simplicity of their definiendum, i.e. the natural numbers. If we take PA for example, the immediate acceptability of their axioms can be understood as self-evidence in the sense of ‘immediately obvious', and it is this sense of self-evidence that lends high credence to belief systems that are built upon them.
However, the large majority of fundamental mathematical axioms are far from immediately obvious. Even after thorough reflection, their truth seems to require further explanation. Let's take for example the axioms of the standard set theoretic system ZFC (‘Zermelo-Fraenkel-Choice’), to which all of contemporary mathematics can be reduced.Footnote 20
The first axiom of ZFC I will mention is also the only one which can be argued to be self-evident in the sense of ‘immediately obvious', or at least obvious upon reflection. This is the axiom of Extensionality:
Extensionality: If X and Y have the same elements, then X = Y.
This axiom implicitly introduces the idea of set-hood and membership (being an element of something). Employing an entirely intuitive understanding of these ideas, Extensionality is obviously true. It doesn't require much reflection to get to that conclusion, much less mathematical training. Hence, it seems reasonable to accept belief in this axiom as rational.
The next three axioms already require much more substantial intuitions about the nature of sets and membership:
Pairs: For any a and b there exists a set {a,b} that contains exactly a and b.
Union: For any X there exists a set Y = ∪ X of all elements of X.
Power Set: For any X there exists a set Y = P(X), the set of all subsets of X.
It is not clear from where we take the intuitions required to accept Pairs, Unions, and Power Set, but let's be generous and assume that the way in which we use the concepts of ‘set’ and ‘membership’ in ordinary natural language contexts suffices to explain our intuitive grasp of the truth of these axioms. Together with Extensionality, we can thus say that the first four of the ZFC axioms are arguably rationally acceptable because their truth is obvious to any competent natural language user, based on their intuitive grasp of the concepts ‘set’ and ‘membership’.
So far, so good. However, we now face two axioms whose claims are diametrically opposed to our intuitive understanding of what a set could be, and what it could mean for a set to have members. Here they are:
Infinity: There exists an infinite set.
Empty Set: There exists an empty set.
There is no way we can make sense of these two axioms using our ordinary language intuitions, according to which sets are made up of a definite number of members, rather than an infinite row. Our intuition insists ‘But where is the border of the set?’, whereas the axiom tells us that there isn't one.
Even harder to understand is the claim that there is a set that has no members. Intuition says: no members, no set. The axiom, however, tells us that the opposite is true. Wouldn't it be reasonable to assume that it is here, at the latest, that our intuitions stop to confirm the alleged truth of the ZFC axioms? And consequently, wouldn't it be entirely appropriate to stop calling the acceptance of those axioms rational?
However, ZFC doesn't stop here. We still have another three axioms whose intuitive appeal is almost zero:
Foundation: Every nonempty set has an ∈-minimal element.
Replacement: If a class F is a function, then for any X there exists a set Y = F(X) = {F(x): x ∈ X}.
Separation: If P is a property (with parameter p), then for any X and p there exists a set Y = {u ∈ X : P(u,p)} that contains all those u ∈ X that have property P.
Without explanation the intuitive appeal of those three axioms is negligible, but even with additional explanation (Foundation prevents cyclical and infinite sequences of sets, so that a set cannot be a member of itself; Replacement guarantees the set-hood of every image of a set under any definable mapping, which is a precondition for the construction of certain infinite sets; Separation says that we can construct a subset for any set S and any predicate P which is made up of all the elements which satisfy P) they don't seem to be what we would typically call ‘immediately acceptable’ or ‘self-evident’.
The final axiom of ZFC is the most mysterious one. One the one hand, what it says seems pretty straightforward:
Choice: Every family of nonempty sets has a choice function.Footnote 21
The axiom is widely used by mathematicians because it has been proven to be useful in many contexts (it implies a well-ordering of the Reals, for example), even though it has counter-intuitive implications in other cases (such as the Banach-Tarski paradoxFootnote 22). On the other hand, Kurt Gödel and Paul Cohen proved that the axiom is logically independent of ZF, i.e. that both the axiom and its negation are logically consistent with the rest of the set-theoretic axioms. This result leaves no immediate reason whatsoever to assume or to reject the Axiom of Choice, other than its usefulness. And most certainly, it doesn't give us any reason to consider it immediately acceptable.Footnote 23
We usually consider it rational to accept a claim as an axiom if it is immediately obvious. But as we have seen, this is not the case for most of the foundational axioms of mathematics. It also won't help to claim that axioms are definitions rather than assertions: the foundational axioms of mathematics don't define their central notions of ‘set’ or ‘membership’ at all, for example, precisely because they are taken to be understood a priori. So it is not at all clear that the axioms of set theory are immediately acceptable.
Let's now turn back to religious belief, and the question why religious belief has such a poor epistemic reputation. Why is it that religion, at least in the Western hemisphere, has become such a rare phenomenon? Because modern enlightened man considers the acceptance of a belief system irrational if it is unclear which objects the key concepts of the system refer to, and how we acquire knowledge about them – precisely what holds true of mathematics. However, the absence of an ontological and epistemological backing seems unproblematic for mathematics, whereas it is a reason to reject other abstract systems of thought as irrational. I propose that this double-standard is unjustified and that mathematical and religious belief should be considered either equally rational or equally irrational.
4. Religion
How can I propose such an outrageous claim? I will show that the way in which religion fails and succeeds to fulfil the four conditions for the rational acceptability of an abstract system of thought parallels the way in which mathematics fails and succeeds to do so. Let's go through the four conditions once again, this time for the case of religion. But before that, a few things need to be noted.
First, I will not go into the intricate discussion of what exactly constitutes a religion. Suffice it to say that I take the three monotheistic religions to fall under the concept, as well as Hinduism, Shintoism, Zoroastrianism, etc. There are certainly many more established religions but I want to concentrate on those examples whose status as religious systems of thought is uncontroversial enough for the purpose of this argument.Footnote 24
Second, one needs to keep in mind that belief in religion has become increasingly unpopular in the Western hemisphere since the Age of Enlightenment, while faith in the all-encompassing explanatory power of the natural sciences today is a matter of course: lightning conductors are more effective defences against the forces of nature than prayers.Footnote 25 On the other hand, religion often fares better than science with regards to providing explanations of the meaning of life, morality, etc. However, distinctions of this kind are often ignored; mathematics is an abstract system of thought with universally high credence, religion is an abstract system of thought with increasingly low credence.Footnote 26 As a result, arguments against the rationality of religious belief don't need to break steadfast convictions anymore (this is often sadly reflected in the simplistic argumentations of anti-theists). Canonical arguments for the irrationality of religion include that religion is not based on evidence (Richard Dawkins, Sam Harris); that religion can be explained naturalistically (Daniel Dennett); that religion causes more harm than good (Christopher Hitchens), etc. The most often cited argument, however, is that religion is not evidence-based.
My view is that the absence of evidence is a characteristic of abstract systems of thought in general, and as such not a sufficient indicator of irrationality. Rather, in order to judge the rationality of a system of thought, we have to examine whether or not it fulfils the four conditions given above. We already saw that mathematics passes the test only insufficiently. So let's see how religion fares with respect to the four conditions, starting with condition one, immediately acceptable axioms.
First of all we need to ask: are there any abstract axiomatic systems of thought besides mathematics and logic at all? In a narrow way of speaking, the term ‘axiomatic system’ is applied only to the formal sciences (logic and mathematics) because only there we find agreed upon sets of axioms that are commonly accepted as the foundation of the whole system of thought. However, the fundamental assertions of the natural sciences (physics, astronomy, geoscience, chemistry, biology), more often referred to as ‘laws' than ‘axioms', serve as the foundational cornerstones of the respective system of thought in an equivalent way: Einstein's general theory of relativity and quantum mechanics, the two main pillars of contemporary theoretical physics, are an example here, as well as the law of conservation of mass in chemistry, and perhaps some biological laws such as Mendelian Inheritance.
Yet there are also other axiomatic systems of thought besides mathematics and the natural sciences. The humanities are full of systems of thought – religious studies and theology, aesthetics, ethics, etc. – which arguably invoke fundamental claims that are taken for granted, i.e. they are assumed as the starting point for subsequent chains of reasoning, as fundamental assertions on which the whole body of discourse rests: ‘God created the world’ is arguably a religious axiom, ‘Beauty is a metaphysically substantial property’ might be an axiom of aesthetics, ‘Killing is wrong’ an ethical one.
Of course it is not obvious which sets of claims exactly could be considered the definitive sets of religious, aesthetic, or ethical axioms (i.e. those axioms from which all theorems of ethics, aesthetics, etc. can be derived). For one thing, this is because scholars working in those fields simply haven't put much work in identifying definitive sets of axioms yet. Rather, they are interested in examining questions of existence arising for specific, singular axioms. For example, much work in philosophical aesthetics has been invested into clarifying how we are to understand the property of beauty metaphysically, i.e. what kind of property it could be, where in our world the property is located, etc. Similarly, philosophers of religion have put much effort into the question how to understand the claim that God created the world in six days in light of contemporary geoscience. And for centuries already, moral philosophers have tried to give an answer to the question what kind of truth an ethical truth such as ‘Killing is wrong’ is, and how it relates to other truths about this world, such as ‘The earth revolves around the sun’ or ‘All beings with hearts also have kidneys'. In short, scholars working in those fields have focussed on examining the ontological commitments of the most central claims of their fields, rather than searching for sound and complete foundational sets of axioms. But: There is no principled reason to believe that basic sets of axioms cannot be given for some specific areas of discourse in religion (theology, for example), aesthetics (for instance discourse revolving around the nature of beauty), or ethics (for example questions concerning the semantics of moral truths); rather, it is simply that nobody has made the effort yet.Footnote 27
Another reason for the absence of agreed upon sets of axioms for those systems of thought is that most philosophers would feel reluctant to call basic religious, aesthetic, or ethical claims ‘axioms'. This is because that term seems to be somehow ‘reserved’ for mathematics and maybe the natural sciences: many feel a vague intuition that the latter areas of thought are somehow ‘more rational’ than the former, and that axioms belong to the more rational areas of thought.
However, taking the above definition of axioms as our guideline, there is no reason why we shouldn't be able to find claims of exactly this kind – i.e. the axiomatic kind – in religious discourse. In fact, if we look at things from a certain perspective, mathematics as a system of thought appears to be more closely related to those areas of thought than to the natural sciences: what mathematics has in common with religious, aesthetic, or ethical discourse is that it is a purely abstract enterprise, i.e. the objects forming the central elements of discourse are abstract, not concrete: numbers, God, beauty, wrongness are all highly abstract concepts whose ontological status is obscure and which cannot be falsified by empirical data. The central elements of discourse in the natural sciences, on the other hand, tend to be easier to understand: physics, for example, is about physical objects, their structure, properties, relations, and the laws governing the spatial and temporal changes they undergo. However, highly obscure objects appear even in the natural sciences: electrons, spins, gravitons cannot be understood in the way ‘ordinary’ objects like tables, chairs, and beer mugs can be understood. Even so, that neither prevents them from being useful or even necessary for physical discourse, nor does it prevent physicists from assuming their existence despite their deviant nature. Axiomatic systems of thought could thus not only be found in mathematics but arguably also in religious discourse.
The question whether or not religious axioms can be considered immediately acceptable in the sense of ‘obvious' will naturally receive different answers from theists than from atheists. A theist way of supporting the acceptance of a claim like ‘There is a God’ is to draw on religious experience; an atheist would insist on the absence of such experiences in order to support his belief that ‘There is no God’. However, as we have seen in the case of mathematics, that an axiom could seem counterintuitive to some (like the existence of an empty set) is not a sufficient reason to abandon it, at least not from the point of view of the system's proponent.
Regarding consistency, religious systems of thought definitely strive for internal consistency (a large part of theology and biblical exegesis, for example, is concerned with rendering certain claims in the religious scriptures consistent). Whether or not a religious system of thought is internally consistent depends on the axioms one assumes for the system: for example, if one takes as the axioms of Judaism the Thirteen Fundamental Principles extracted from the Torah by Maimonides (belief in God as the sole creator of the world who is unified, non-corporeal, eternal, omniscient, who communicates through prophecy, gave the Jews the unchangeable Torah, rewards and retributes good and evil action, resurrects the dead, and will eventually send a messiah), then these are certainly internally consistent. The difficulty lies more with external consistency, i.e. with the question of the (in)compatibility of a religious system with other, commonly accepted systems of thought, such as the natural sciences. As long as a religious system of belief doesn't contradict scientifically established facts, there is no reason why we should consider it externally inconsistent. The existence of God has so far not been falsified by science and can therefore not be said to stand in contradiction to science.
In terms of applicability, we have noted above that the larger part of (pure) mathematics has definitely no application in the concrete world. The rest of mathematics can be used by natural sciences as a tool for the calculation and formulation of scientific theories, yet it is an open question whether mathematics as a tool is indispensable to the natural sciences. What matters for our purposes is that mathematics is nothing but a tool; mathematical expressions don't refer to concrete objects and as such, don't add content to a scientific theory.
Let's try to look at religion in this way. Let's say that certain core terms of religious systems of thought (‘God’, ‘angel’, etc.) and the central assertions containing them (‘God created the world’, ‘Ezra argues with the angel Uriel’, etc.) do not refer to concrete objects whose existence could be scientifically measured or verified. I suppose a theist and an atheist would agree on that. Religious statements definitely have content, but on the assumption that their core terms don't refer to concrete objects, religious content does not add substantial content to a concrete, scientific theory. In this sense, religious belief is externally consistent with science, and other non-abstract theories of thought.Footnote 28 Let's further say that religion is a tool used by religious people in order to make sense of their lives, structure decision-making processes, ground their ethics, etc. The dispute between the theist and the atheist is then over the question whether or not religion, given the stunning explanatory success of science, is dispensable to an overall project of, say, making sense of one's life - an open question, like the question of the (in)dispensability of mathematics. And just like the mathematician can happily continue his work while others agonize about the ontological and epistemological issues arising for mathematics, the believer can continue believing in and living according to her religion without worrying about ontological and epistemological issues arising for religion.
Regarding predictability, one can say that religious texts definitely make predictions. The problem is that believers and non-believers disagree on whether the predictions became true. For example, Christians argue that the arrival of the messiah was predicted in the Old Testament, and they believe that the prediction was fulfilled when Jesus appeared on earth. Jews, on the other hand, are still waiting for the arrival of the messiah. What is more important than arguing about past predictions, though, is whether religions make any predictions today which can be verified or falsified scientifically. And I think the answer is no. Religions tend to refrain from making scientifically verifiable or falsifiable claims. Rather, their predictions focus on the afterworld and on the palpable benefits of leading an observant life. A typically Jewish way of arguing in favour of being religiously observant, for example, is to say that it provides meaning to people's lives, increases one's appreciation of family, etc. Such claims are hard to verify or falsify by statistical means because of the difficulty of determining to which degree a person genuinely believes in a given religious system of thought, to which degree she leads an observant life for other reasons, such as habit or societal pressure, to what extent other factors influence her happiness or unhappiness, etc. So in terms of predictability, religion certainly doesn't fare well. But then again, the same holds of mathematics.
5. Conclusion
The ability of a system of thought to fulfil the four conditions of rationality I stated above depends crucially on its degree of abstractness, i.e. whether it is a purely abstract system, such as mathematics or chess, or whether its abstract concepts are used to explain concrete phenomena, such as religion, physics, chemistry, or biology. Systems of thought form a continuum of rational acceptability according to the level of abstraction involved: purely abstract systems are less likely to fulfil all four conditions of rationality than the natural sciences (which come up with theories which can be verified or falsified by repeatable experiments) as well as other disciplines working mainly with statistical evidence, such as sociology, economics, etc.
Mathematics may play a significant role in the sciences and religion may not be able to answer certain questions the sciences can answer, but in the same way religion provides answers for questions which science doesn't even have the proper vocabulary to address (e.g. what is the meaning of life, why is there something rather than nothing, why is life worth living, etc.).
I have argued in this article that mathematics and religion fail and succeed in similar ways to fulfil the four conditions of rational acceptability for abstract systems of belief. Mathematics is an abstract axiomatic system of thought which may or may not be consistent (though generally assumed to be by mathematicians), which may or may not be an essential tool for the natural sciences, and which makes no scientifically falsifiable claims. Religion is an abstract system of thought that arguably relies on fundamental premises which can be considered its axioms (such as ‘God exists'), which may or may not be consistent (though generally assumed to be by believers), which may or may not be essential in explaining the existence of the world and the meaning of life, and which makes no scientifically falsifiable claims.Footnote 29
The crucial question is now: what do we do with this result? As the saying goes, ‘one person's modus ponens is another person's modus tollens’, so I leave it to the reader to decide whether to become an ‘atheist about mathematics’ or to reconsider her opinions about the irrationality of religious belief. Note that my account does not oblige anyone to take on religious beliefs. All it shows is that it is epistemically permissible to hold religious beliefs, just as it is permissible to hold scientific and mathematical beliefs.
