Three Roles of Empirical Information in Philosophy: Intuitions on Mathematics do Not Come for Free

1.1 Sources of Empirical Information

One well-known categorization by Prinz (2008) was further developed by Löwe (2016): it evokes a nice metaphor with tailoring. According to this idea, philosophers have three different options how to get empirical information:

1. Ready to wear-style: by looking into the literature for empirical results that are relevant to their endeavour/analogous to going into the mall and taking a suit off the rack.

2. Customization: by requesting custom-made results/analogous to going to a tailor who would tailor them a custom-made suit.

3. Tailoring themselves: by doing the empirical work themselves/analogous to tailoring the suit themselves.

The first option is close to what is within acceptable boundaries to a lot of people in the armchair community. As the upcoming section on philosophy of mind shows, philosophers are often willing to use empirical information from psychology and neuroscience. We can look at existing scientific literature and decide which results are important to us. If we cannot find something that fits, we can do the work ourselves (third option), which is excluded from the methodology of armchair philosophy. But this makes it necessary to get trained in adequate methodology that may be foreign to philosophers. A more resourceful option would be to divide labour. But here (second option), it remains to find problems and questions that are important for both communities, for an empirically working scientist would need a reason to collect the data that philosophers would be interested in. While their non-philosophically motivated data may be useful to address philosophical questions, the “suit” may not fit the philosopher’s size. This idea is very influential in the philosophy of mathematical practice, especially in what Carter (2019) calls in her recent overview of the discipline “empirical philosophy of mathematics”. It is a tradition which substantially profits from a series of conferences in Brussels, see f.i. (François and van Bendegem 2007). As Carter observes, this branch of philosophy is closely related to mathematics education, where the processes by which students learn mathematics are studied. Recently, one of the most prestigious journals in that field included an introduction to philosophy of mathematics for mathematics educators, see Hamami and Morris (2020).

1.2 Uses of Empirical Information: A New Categorization

Löwe’s distinction addresses the different sources from which philosophers may get empirical information that is relevant to their work. While this opens the door to important practical debates,^[1] we focus on the use that philosophy makes of empirical information, because this is more relevant for the current dichotomy between narrow empiricism and wider empiricism (see below). For this aim, we propose two distinctions that capture different ways in which philosophers can use empirical information.

The first distinction discriminates between empirically informed philosophy that answers classical philosophical questions on the one side and empirically informed philosophy that aims to understand the practice of a field on the other. The subject matter is different. A related, partially overlapping distinction can be made between empirically informed philosophy that presupposes an onto-epistemic frame (or features broad philosophical intuitions) and empirically informed philosophy that does not (the frame arises a posteriori for the philosopher; it is developed during the empirical scrutiny of practices).

• Distinction 1: solving philosophical issues or describing and understanding a field

• Distinction 2: presupposing/anticipating an onto-epistemic frame (which includes the objects of analysis) or accepting the onto-epistemic frame of what is empirically investigated (e.g. information as is provided by scientists or other actors, or scientific practices)

Together, these distinctions allow us to discriminate between three types of empirically informed philosophy. Empirical philosophy that sets out to solve already formulated philosophical dilemmas necessarily features onto-epistemic frames in advance: the terms of the debate are set, the ontologies are defined, and what would count as evidence for or against a given point is largely predetermined. Therefore, the first side of Distinction 1 (solving philosophical issues) is not compatible with the second side of Distinction 2 (finding the entity of analysis in the empirical investigation), and we obtain the following three types of empirically informed philosophy:

1. Apostate speculator: Addressing philosophical questions via empirical information framed by pre-established epistemologies and ontologies/discursive entities

2. Informed analyst: Describing, studying or analytically investigating practices via a philosophical epistemic lens that includes pre-established ontologies/discursive entities

3. Freeway explorer: Describing, studying or analytically investigating practices mainly drawing from epistemologies and ontologies/discursive entities established in the empirical research

Since any analysis depends on previous theoretical baggage however minimal it is (see f.i. Chang et al. 2011; Pitt and Pitt 2011; Rittberg and van Kerkhove 2019), the freeway explorer will feature onto-epistemic frames as well. But there is a difference of degree between the informed analyst and the freeway explorer: the latter works with minimal notions and is easily willing to accept the notions and objects of the practitioners that become apparent or emerge in the empirical analysis, while the former usually works with more philosophically sophisticated notions and ontologies, which they tend to cherish and therefore favour over alternatives. The notion of ontologies is understood broadly as including for example genes and sets, but also abstract concepts such as justice or the maximize principle put forward by Maddy (2007). There is an important difference between ontologies in the philosophy of mathematics and in the philosophy of science. The scientific intended domain often is found in the real world, and therefore more readily available than the mathematical intended domain. This is of course not clear cut, because for instance parts of physics are more rooted in the mathematical domain and parts of mathematics can be quite embodied in the real world, as for example in basic arithmetics. This difference is crucial regarding the philosopher’s access to the respective intended domains.

Discriminating between these three types is critical because they deserve differential treatment regarding the narrow empiricism or wider empiricism dichotomy. Furthermore, it helps make visible the particularities of the third type, to which the philosophy of mathematical practice counts, as we will argue. The first two types constitute the bulk of empirically informed philosophy where the onto-epistemic frame is presupposed or anticipated by the philosophers. Hence, we have only few examples of philosophical investigations where the onto-epistemic frames are found in empirical research. Our case study in set theory will provide an example of such a case.

While these three categories are not exhaustive and may overlap, we will next justify that they are useful to understand and characterize empirical philosophy, and to address the distinction between narrow empiricism and wider empiricism.

1.3 The Effect of Empirical Information: Narrow and Wider Empiricism

Apart from the use that philosophy can find for empirical information, there is a debate on the effect that empirical information should have on philosophical knowledge: what kind of information overrides the other in situations of conflict? There is a distinction between First Philosophy and Second Philosophy established by Maddy (2007). Because Maddy proposes a very specific approach under the label of ‘Second Philosophy’ that is applied to philosophy of mathematics (especially set theory), we instead embrace the labels “narrow empiricism” and “wider empiricism”, due to Cornelius Benjamin (1939). We use them in the following way: Narrow empiricism, closely related to armchair philosophy, refers to the position that philosophical intuitions should have primacy over empirical information, while wider empiricism defends that, in case of discrepancy between the two, empirical information overrides previous philosophical intuitions. Cornelius Benjamin (1939), while presenting an early notion of empirical philosophy (more encompassing than current conceptions), uses these terms, acknowledges the pros and cons of each position and situates them in a continuum:

The positivist ends by being simple-minded; the metaphysician, by being muddle-headed. Narrow empiricism, while not positivistic in the extreme form, prefers simple-mindedness to muddle-headedness; wider empiricism, while not allying itself with speculative metaphysics, prefers muddle-headedness to simple-mindedness. We have, therefore, a continuum of positions: strict positivism, narrow empiricism, wider empiricism, speculative metaphysics. The problem of empiricism, it seems to me, is to specify more precisely the nature of this serial arrangement of attitudes. Apart from such considerations, to say that one is an empiricist is like saying that a stick is long or an object is heavy; specifications of how much are imperative (Cornelius Benjamin 1939, p. 520).

We believe that the kind of use made from empirical information is crucial to determine whether a narrow empiricism or wider empiricism approach is pertinent. As we will see in the upcoming section, the apostate speculator or informed analyst types admit both approaches in interesting ways. Therefore, deciding between the two might be a task with no straightforward rules. Both extremes seem to have crucial shortcomings: dismissing strong philosophical intuitions in favour of dubious empirical results naively errs on the side of narrow empiricism, towards strict positivism, while ignoring strong and systematic empirical evidence in favour of armchair theorizing is succumbing to blind speculation. Ultimately, this may depend on case-by-case local judgement. We agree with Benjamin that there is a continuum which the “strict positivism” versus “speculative metaphysics” (and perhaps First Philosophy vs Second Philosophy) dichotomy cannot capture. Below we provide examples on how to judge specific cases in the philosophy of mind, science and in ethics.

The freeway explorer type, on the other hand, leans towards wider empiricism. The freeway explorer, after all, does not start from invasive philosophical intuitions or ontological frameworks in their analysis of practices, and instead takes for granted what the practitioners claim about their subject matter. Without tension between preceding philosophical baggage and empirical content, the freeway explorer does not resort to philosophical speculation but collects what they find. For instance, philosophical analysis on an ontological entity cannot precede the entity itself, and the freeway explorer works with ontologies elucidated during the empirical investigation. The case study in set theory in Section 3.2.1 will support this claim.

1.4 Our Argument in a Nutshell

There is an ongoing debate about the right methodology for Philosophy of Mathematical Practice(s), see f.i. (Rittberg 2019) or the many programs mentioned by Carter (2019). One can see a parallelism between the development of PMP and certain events in Philosophy of Science. Kuhn (1962) is credited with being a pioneer of such events or at least with popularizing ideas that we can find f.i. as well in Fleck (1935) with his The Structure of Scientific Revolutions, endorsing a perspective on the sciences that stress the irrational component of scientific practice. Theory choice or Paradigm Shifts seemed not to fit purely rational proceedings. These developments yielded many important pieces of Philosophy of Science, some with an institutional backbone like the Society of Philosophy of Science in Practice (SPSP). One anonymous reviewer mentioned a non-exhaustive list of authors which belong in this tradition: Bruno Latour, Nancy Nercessian, Joe Rouse, Karen Barad, Hans-Jörg Greiffenhagen, among others. While we are very sympathetic for these more anthropogenic, ethnomethodological, or in our words freeway explorer-ish endeavours, one can investigate whether there is a point where the parallelism between PMP and Philosophy of Science does not apply. Philosophers could argue that whenever they want to talk about the real world the exact practice of those researching the real world is not relevant for them. Social-constructivists could not argue that, but realism is more mainstream. Philosophy of mathematics presents a very different scenario. Here even the realist must see that mathematical entities (again without a strict metaphysical meaning of entities) are not accessible in the sense that the real world is: Mathematical entities remain abstract. Thus the main point of the paper – besides the categorization – is: Philosophers of science, philosophers of mind etc. can go for all three approaches, especially for the apostate speculator and informed analyst types. But philosophers of mathematics need to go for a freeway explorer type approach as much as possible.

2.1 Empirically Informed Philosophy of Mind

We first look at empirical philosophy of mind that we identify mainly with an apostate speculator type.

Due to the abstract conceptualization of the mind, its philosophical study has traditionally relied heavily on speculative methods rather than what today would be considered appropriate empirical investigation by modern scientific tenets.^[2] A classic example is Descartes’ rationalist approach to the mind-brain problem (insofar as the proposition is not founded on empirical findings), and plenty of modern examples can be found in the form of thought experiments (the p-zombie argument (Campbell 1970; David 1996; Kirk 1974a, 1974b), Mary’s room (Jackson 1982, 1986), and so forth, which if even some consider empirical in the sense that one’s thought is observed, they consist on thought experiments rather than empirical observations of the world). And in fact, some philosophers defend that this is all that philosophy of mind can ever hope to achieve with some degree of reliability (Brandt 1967; Kim 1966). Tibbetts (1973) claims that, rather than an empirical issue, the mind-brain problem is a methodological issue. Bermúdez (2014) acknowledges that empirical information is critical in philosophy of psychology but irrelevant in philosophy of mind. Irvine et al. (2014) sustains that when a thesis in philosophy of mind is vague, general and/or tries to fit rather than predict empirical data, chances are that it cannot be tested empirically. Carreno and Pérez-Escobar (2019) argue that empirical information is useful to tackle practical issues and make clinical decisions based on preliminary notions on the character of the (abnormal) mind and its brain bases but does not settle metaphysical issues.

However, as psychology, cognitive science, and neuroscience have matured as disciplines, more and more philosophers of mind have drawn from empirical information to avail their premises. An extreme supporter of these proceedings is Foss (1987), who argues that the mind-brain problem is merely an empirical issue. As such, positions that entail metaphysical stances regarding the mind-brain problem (like materialism,^[3] dualism,^[4] emergentism,^[5] functionalism,^[6] idealism^[7] and epiphenomenalism^[8]), are open and must be subjected to empirical testing. In this line or in more moderate grounds, some philosophers have appealed to and interpreted empirical findings in the sciences to underpin their arguments on the mind-brain problem.

For instance, let us consider eliminative materialism, the position that denies the existence of (most) folk-psychological mental states, such as beliefs and desires. Its proponents insist on paying attention to neuroscientific empirical findings and claim that no neuronal basis of folk-psychological mental states has been or will be found. Therefore, they hold that folk-psychological mental states do not exist in an ontologically strong sense.^[9] For instance, Paul and Patricia Churchland dismiss propositional attitudes on these grounds (Churchland 1981, 1989).

Eliminative materialists dismiss introspection as a valid method to gather reliable empirical information, since it is influenced by folk-psychological notions themselves (Churchland 2013). Thus, empirical support of folk-psychological notions derived from introspection is eschewed by eliminative materialists. To defend this attitude, they may appeal to empirical information suggesting that introspection fails to provide appropriate empirical information (see Nisbett and Wilson 1977 for a critique of introspective methods). This is another pathway by which eliminative materialism is informed by empirical results.

However, such empirical support does not straightforwardly lead to the veracity of eliminative materialism, for several reasons. First, there is controversy regarding whether stances on the mind-brain issue are empirical matters at all, as we mentioned before. For example, it could be contended that it contains unfalsifiable premises. Second, as Foss (1985, 1987 notes, eliminative materialism not only appeals to current empirical information, but also to future empirical findings: neither current nor future neuroscience does/will support the existence of desires, beliefs and so forth. Therefore, this position is not only based on empirical information but also on aggressive induction. And third, empirical information can receive heterogeneous treatment and interpretation by different communities: some may consider introspection to be a better source of information than the measurement of neurophysiological parameters, some consider folk-psychological concepts warrant enough predictive power to deserve a more serious treatment (Lahav 1992), etc. But this issue does not undermine the structure we want to stress here. What is relevant is that some philosophers propose solutions to philosophical problems claiming that there is empirical evidence that backs up their accounts. There is a philosophical issue (the character of the interface mind-brain), empirical information that may shed some light on the issue (results from psychology, cognitive science, and neuroscience in the mind-brain interface), and a philosophical stance on the issue informed by such empirical information (eliminative materialism). This use of empirical information in philosophy is a clear example of the apostate speculator style.

2.2 Empirically Informed Philosophy of Science

Another area of philosophy that draws from empirical information is philosophy of science. There are three ways in which philosophers of science can make use of empirical information:

1. Philosophers intending to ascertain the character of scientific theories, explanations, concepts, reasoning and so forth based on empirical data about scientific practices (apostate speculator)

2. Philosophers intending to ascertain how scientists conceive of and work with scientific theories, explanations, concepts and so forth to check whether they adhere to “good scientific practices” (informed analyst, normative)

3. Philosophers intending to ascertain how scientists conceive of and work with scientific theories, explanations, concepts and so forth to better understand the field, work out why and how certain theories succeed or fail in their own terms, contribute with clarifications, and so forth (informed analyst, descriptive)

Here, we present a concise example to illustrate this distinction. Machery et al. (2016) discusses work in experimental philosophy of science where the uses of concepts by philosophers, scientists and lay people are compared. He discusses empirical work on the scientific notion of “gene” (Stotz and Griffiths 2004; Stotz, Griffiths, and Knight 2004) and concludes that it is quite variable in meaning across scientific communities and different epistemological situations. He also points to empirical research on the notion of “innateness” (Griffiths, Machery, and Linquist 2009). In this case, a formal characterization of innateness in terms of sufficient and necessary features (fixity, typicality, and teleology) is put to the test by contrasting it to the actual conception of innateness by lay people.

In the case of “innateness”, the authors propose to debunk previous philosophical views on the concept on the grounds that it lacks empirical validity. For this reason, it qualifies as the apostate speculator empirical philosophy. In the case of “gene”, it is descriptive work of the informed analyst style.

However, it does not qualify as freeway explorer. Although there was not a philosophically rigorous notion of the object “gene” to be tested, more minimal or primitive intuitions were displayed (the word itself for example, and phenotypic effects as essential properties of genes) to at least be able to conduct the analysis of practices. Therefore, the object preceded and conditioned the analysis of practices. For instance, a philosopher may study biological practices in situ with the aim to test their conception of “gene” and how it is relevant in their knowledge and techniques. However, if the philosopher found that the scientist call a cup of tea “gene”, and spilled tea “gene expression”, the philosopher is not likely to take the practices seriously, but instead they would dismiss them: “they are out of their minds”, “they are pulling my leg so that I leave their lab”.

2.3 Empirically Informed Ethics

The last empirically informed philosophical subdiscipline that we will discuss in this section is empirical ethics. Similar to the aforementioned case of empirically informed philosophy of science, empirical ethics may set up to solve philosophical questions (support or argue against consequentialism, intuitionism, pluralism, deontology …) and, with regard to practices, may be not only descriptive but also normative.

In the case of ethics, the latter point may sound less intuitive: how can information on the values, choices and behaviour of people lead to any insight about what values people ought to have, what choices they ought to make and how they ought to behave? This seems to violate the is-ought dichotomy. Nevertheless, new developments of empirical ethics challenge the strength of this distinction and propose further study of the relations between facts and norms (see for example the introduction of Christen et al. 2014). Such facts are not limited to those of descriptive ethics but also include empirical results from psychology, cognitive science, neuroscience, anthropology, and more (see Greene 2015) for an example. Kauppinen et al. (2014) therefore distinguishes two positions in ethics: “armchair traditionalism” – the view that empirical information is not relevant for the justification of normative theories – and “ethical empiricism” – the view that empirical information can be insightful for ethics.

We will briefly discuss a typical experimental approach in ethics: subjecting people to situations where they must make choices that reveal their moral character. Experimentally studying ethical intuitions of people is useful, first, because it helps to determine who holds the burden of proof (Levy 2009). That is, given two opposing ethical accounts, the one which must find arguments in its favour is the one which is further away from lay intuitions. Second, empirical information may provide direct support for a given ethical account. The usual procedure consists in presenting the subjects with moral dilemmas and predefined options or paths of action. Then, the experimentalist draws conclusions from the choices made by the subjects: such values are present, such strategies are followed, such imperatives apply, such circumstances are relevant, and so forth. The experimental philosopher abstracts from the choices made by the subjects. Furthermore, the subjects may not have any interpretation or apparent intuition on their moral choices, or, if they have an interpretation, it may be in tension with that of the experimenter. If the subjects are asked to give reasons about their choices, they tend not to reflect the “real reasons” according to the experimenters, or fail to give reasons at all, being “morally dumbfounded”^[10] (Haidt 2001). It is argued that the choices made by the subjects are based on a more socially distributed process of reason-giving (Haidt et al. 2008), but for some reason they are still unable to access them. This contrasts with the case study in philosophy of mathematical practice that will be described next, where the intuitions of the mathematician are the ones that matter the most, and therefore they are not questioned.

There are more examples from these and other philosophical disciplines – philosophy of language, aesthetics, philosophy of technology … – which cannot be mentioned here due to space limitations. In the light of the examples discussed, our view is that the apostate speculator and the (descriptive or normative) informed analyst types are common and arguably fruitful modalities of empirically informed philosophy, perhaps especially in those areas featuring ontologies accessible and/or familiar to philosophers to a certain degree. These modalities feature a broad philosophical (especially ontological) framing of the issue empirically investigated.

Furthermore, we can see how it is not quite clear whether these types of empirical philosophy should adhere to a narrow or wider empiricism approach. In the case of eliminative materialism, some empirical information is considered to support the position, while other information is either disregarded or presupposed before it comes. So, there is an important degree of speculation. In the example of empirical philosophy of science, we see how (some) philosophers are willing to revise their previous notions, be they minimal (gene) or slightly more elaborated (innateness), relegating them to the results they obtain. In the case of empirical ethics, some philosophers are willing to incorporate the results of the studies into their accounts even if they are strange to them, but on the other hand they are not willing to let the interpretation of the subjects of their own behaviour (which is also empirical information) trump theirs.

It must be acknowledged that some ethnographic approaches (Latour 2005; Latour and Woolgar 1979 is a prominent example) may lean more clearly towards the freeway explorer type of empirical philosophy. For instance, Latour (2005) puts forward a practical metaphysics which consists in accepting the metaphysical perspectives of actors and the ontologies they use as “real”, without the social scientist trying to reformulate them (in other words, actors are not “dumbfounded”). Whether Latour’s later stances deviate from this approach, for instance regarding his taking sides in the issue of climate change, and whether his approach is ultimately feasible in science, is a matter of debate (Stamenkovic 2020), and outside the scope of this work.

In any case, these are a few examples among many, and the aim of this paper is not to offer a comprehensive analysis of empirical philosophy in general. The takeaway message is that, in the areas of empirical philosophy discussed here, there does not seem to be an overall rule of thumb to go one route or the other: which route is more fruitful may depend on local, case-by-case judgement on how solid the preceding philosophical intuitions are, and how reliable the empirical data gathered is. We now proceed to argue that the philosophy of mathematical practice is different. We will illustrate this via a case study.

3.1 What is Empirically Informed Philosophy of Mathematics?

There is a group of areas in philosophy with the form “philosophy of …”. When we insert “physics” | “social sciences” etc. we get a subject where, since the Practice Turn, a sizable part of the community is focusing on the practice of the respective academic field. Philosophy of mathematics used to be an exception; it had to wait longer to receive the same treatment as Philosophy of Science. The main focus was on ontological and epistemological questions. A subfield stressed the need to work empirically informed and to focus on the actual practice. From this the discipline “philosophy of mathematical practice” emerged.

Prominent names in this field stressed the need to incorporate the work of mathematicians via case studies.^[14] Or in the words of a founding member of the 2009 created Association for Philosophy of Mathematical Practice, Paolo Mancosu:

[A]nyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice (Mancosu 2008, Preface).

This community is constantly growing. Philosophers of mathematics often ignored mathematical practice, and philosophers and sociologists of science/humanities often ignored mathematics. Bettina Heintz for instance wrote about her profession that “Sociology meets mathematics with an idiosyncratic mixture of devotion and disinterest”.^[15]

The shift to philosophy of mathematical practice leads not necessarily to the abandonment of old questions of philosophy of mathematics. It rather extends the scope of philosophical questions and mainly brings in a new methodology. It is a diverse branch of research. To get a broad overview, based on distinctions made by van Bendegem (2014, p. 221), Jullien et al. (2014), and Mancosu (2008, pp. 3–7), here are some traditions subsumed under the label philosophy of mathematical practice:

1. The Lakatonean tradition (sometimes labelled as the maverick tradition): Here we mainly focus on a historically informed approach. A key view in this tradition is anti-foundational, which stands especially against the importance of mathematical logic and set theory.

2. A naturalistic (or Second Philosophy) position states that the authority lies with the researchers of the analysed field and not with the philosopher. This may include ontological, methodological, or epistemological dimensions, see especially (Maddy 2007).

3. A normative (or First Philosophy) approach, which wants to revise mathematics based on philosophy.^[16]

These approaches are accompanied by non-philosophical dimensions. Those are:

1. A sociologically informed approach and empirical studies of the mathematical community (Greiffenhagen 2008; Heintz 2000; MacKenzie 2001).

2. A mathematics educational approach, including both philosophical aspects of mathematics education and aspects of the process of learning mathematics (François and van Bendegem 2007; Paul 1991; van Kerkhove 2007).

3. Ethnomathematical ideas, taking cultural aspects into account (François and van Kerkhove 2010).

4. Approaches that include cognitive and biological factors. See f.i. Giaquinto (2007), Mancosu, Jørgensen, and Pedersen (2005), Pease, Guhe, and Smaill (2013), and Schlimm (2008, 2018).

Each of these modalities can be realized via different methodological strategies. For instance, G could be conducted via experiments or by literature review. Thus, this might be armchair philosophy or experimental philosophy.

We think that all of these approaches introduced fruitful investigations into the philosophy of mathematics. Important topics addressed in the philosophy of mathematical practice are the epistemological roles of diagrams (Manders and Mancosu 2008), visualization (Giaquinto 2007), and terminology (Schlimm 2018), the evaluation of ‘good’ definitions (Tappenden and Mancosu 2008), and the dimensions of proof appraisal (Inglis and Andrew 2015, 2016). Recent work looks into social aspects such as the background of mathematics represented in web blogs (Pease, Aberdein, and Martin 2019) and peer reviewing practices (Andersen 2017). All of them have in common that they reveal aspects of mathematics that are not accessible to the philosopher who does not know about the views of the practitioners.

We will look at one debate that is of interest both to classical philosophers of mathematics and for philosophers of mathematical practice. We will look at the study of so called large cardinal axioms and argue that we need to adapt to a wider empiricism approach (understood as described in Section 1) and empirical input is essential, as the preceding philosophical baggage is simply insufficient to onto-epistemically frame the relevant content.

3.2 The Case of Large Cardinals

The questions of independence are well known in philosophical and mathematical literature. On one side there are the Gödelian incompleteness theorems. In 1931, Gödel (1931) showed that every consistent, effectively axiomatized theory containing ‘some arithmetic’ is incomplete.^[17] This means that there is a formula in the language of the theory which cannot be proven and whose negation cannot be proven either. His second incompleteness theorem gives even an example for such a statement, namely consistency statements of the theory in question (again with the above-mentioned restrictions).

There are approaches which aim to find independent statements that are relevant to mainstream mathematics.^[18] A branch of mathematics that is particularly engaged in questions that are independent from its current axiomatic setting is set theory. There are several results independent from ZFC.

A prominent example is the continuum hypothesis (CH): If we denote the cardinality of the natural numbers by ℵ 0 and of its power set (i.e., the set of all subsets of the natural numbers, or equally of the continuum) with 2 ℵ 0 , then the continuum hypothesis states that 2 ℵ 0 is the second smallest infinite cardinality, or in formula:

This was the first question on Hilbert’s list of the most important questions at the beginning of the 20th century. It happens to hold that one can neither prove CH nor its negation within ZFC. So, one might think that ZFC is not a good basis to do set theory and we should rather look for new axioms. Gödel held this position and wanted to justify new axioms (partly) by inductive arguments. He said that

probably there exist other […] axioms of infinity even disregarding the intrinsic necessity of some new axiom, and even in the case it has no intrinsic necessity at all, a decision about its truth is possible also in another way, namely, inductively, by studying its ‘success’.^[19]

So, he already introduced a quasi-empirical component, the study of such axioms. But this can be done by respecting the literature and seeing which results were proved.

Large cardinal axioms are interesting in that respect since they are the most acceptable ones in set-theoretic research. They are much used and set theorists have substantial knowledge about them.^[20]

A large cardinal axiom states the existence of a specific large cardinal, whose existence is not provable in ZFC. Although their consistency cannot be proven, just one turned out to be inconsistent, namely the “Reinhardt cardinal”, whose existence was shown to be inconsistent with the Axiom of Choice (Kunen 1971).

We saw above that Gödel had some hope that large cardinal axioms may settle the continuum problem. However, due to the theorem of Levy and Solovay (1967), it is known that large cardinal axioms are compatible with both the continuum hypothesis and its negation.^[21]

When it comes to the question of how large cardinal axioms can be justified, there are two different claims to differentiate: we either try to argue in favour of the consistency of large cardinal axioms, or in favour of the actual existence of large cardinals. Both claims are important in set-theoretic research. The consistency of large cardinals is used to prove the consistency of other new axioms. The actual existence of large cardinals is often used in the form of their consequences since they settle some independent questions, for example, many questions about the projective subsets of the reals.

One important strategy for the justification of large cardinal axioms proceeds through reflection principles (justification of the existence of large cardinals not only the consistency of the axioms). There is a reflection principle provable in ZFC (which says for each formula that it is equivalent to the existence of a bounded part of the set-theoretic hierarchy satisfying the formula). The strategy is to strengthen this principle and use it to prove the existence of specific large cardinals. Koellner investigated this line of thought in detail relating technical results (which reflection principles imply the existence of which large cardinals?) to philosophical justification (how can a specific reflection principle be justified on the iterative conception of set?). He concludes that

the Erdös cardinal κ(ω) appears to be an impassable barrier as far as reflection is concerned. This is not a precise statement. But it leads to the following challenge: Formulate a strong reflection principle which is intrinsically justified on the iterative conception of set and which breaks the κ(ω) barrier (Koellner 2009, p. 217).

So, he claims that it seems very hard to justify the existence of larger cardinals than κ(ω) following the strategy of reflection principles (κ(ω) is not very large). Recently, Welch et al. (2019) tried to give a stronger case by means of his ‘Global Reflection Principle’, which implies the existence of Woodin cardinals (which are much larger than κ(ω)).

Another strategy is pursued by Maddy, who collects first arguments in favour of large cardinals in her two articles Believing the Axioms I, II (Maddy 1988a, 1988b). In the first article, she considers measurable cardinals and concludes that

[here] we have, as Gödel predicted, an axiom so rich in extrinsic supports that ‘… whether or not [it is] intrinsically necessary, [it is] accepted at least in the same sense as any well-established physical theory’ (1947/64, p. 477) (Maddy 1988a, p. 508).

Her reasoning works by identifying specific rules of thumb that guide the set theorists’ judgments whether some axiom (or consequence of an axiom) is appealing or not. Extrinsic support means that the consequences of the axiom are judged appealing.

In view of the distinction between the consistency of large cardinal axioms and actual existence of large cardinals, Woodin developed an interesting argument, in which he sees the existence of large cardinals as the best explanation for the consistency of large cardinal axioms, which is, in turn, an observation in the set-theoretic community: so far, set theorists did not find an inconsistency in any theory of the form ZFC + Large Cardinal Axiom, see below. Rittberg elaborated on this argument (Rittberg 2020).