The Quasi-Empirical Epistemology of Mathematics

2.1 Euclidean Theories and Quasi-Empirical Theories

In one of his under-discussed papers^[4] A Renaissance of Empiricism in the Recent Philosophy of Mathematics, Imre Lakatos makes the claim that mathematics is quasi-empirical. But what does it mean for a theory to be quasi-empirical? In Renaissance, the definition of quasi-empirical theories is presented in contrast to Euclidean theories. According to Lakatos, both Euclidean theories and quasi-empirical theories are deductive systems. To draw the distinction between them, we follow Lakatos in calling the sentences in the deductive systems whose truth-values are initially injected basic statements. Then a deductive system is a Euclidean theory if it is the deductive closure of the true basic statements, otherwise it is quasi-empirical (Lakatos 1976, p. 206). Lakatos states in Renaissance that the ideal of a theory (e.g. Euclidean geometry) was once thought to be Euclidean, which, in greater detail, is:

a deductive system with an indubitable truth-injection at the top (a finite conjunction of axioms) – so that truth, flowing down from the top through the safe truth-preserving channels of valid inferences, inundates the whole system. (Lakatos 1976, p. 205)

However, it turned out that scientific theories could not be modelled as a Euclidean theory. Instead, they are quasi-empirical theories:

Scientific theories turned out to be organised in deductive systems where the crucial truth-value injection was at the bottom – at a special set of theorems. But truth does not flow upwards. (Lakatos 1976, p. 205)

It is important to note that, in Lakatos’ characterisation of quasi-empirical theories, there is no requirement on the nature of a theory.^[5] That is to say, a quasi-empirical theory doesn’t have to be an empirical theory. Whether a theory is empirical depends on whether its basic statements are concerned with spatio-temporal entities. While whether a theory is quasi-empirical is decided by the direction of the truth-value^[6] flow of the theory, i.e. the upward flow of falsity.

Further comparisons between Euclidean theories and quasi-empirical theories are made in Renaissance (Lakatos 1976, p. 206). A Euclidean theory may be claimed to be true; while a quasi-empirical theory is either conjectural or false. The characteristic logical flow for a Euclidean theory is the transmission of truth from axioms to the rest of the theory; while the characteristic flow for a quasi-empirical theory is the re-transmission of falsity from false basic statements to the hypothesis. In a Euclidean theory, the true basic statements prove the rest of the system; while in a quasi-empirical theory, the true basic statements are explained by the rest of the system.

However, the distinctions between Euclidean theories and quasi-empirical theories made above can seem somewhat confusing: surely in any deductive system truth flows downwards and falsity flows upwards, how do these two truth-value flows help to differentiate two kinds of theories? In addition, why does Lakatos note that in a quasi-empirical theory the true basic statements are explained by the rest of the system instead of just saying that they are proved by the rest of the system? To clear up the confusion, I believe that it is crucial to make clear Lakatos’ motivation for suggesting the classification of theories into Euclidean ones and quasi-empirical ones in the first place, which can be found in another one of his papers Infinite Regress and Foundations of Mathematics. In the opening section of Regress, Lakatos reports a sceptical argument which concluded there can be no foundations of any knowledge:

One can try to pin down the meaning of a term either by defining it in other terms – this leads to infinite regress – or by defining it in “perfectly well known terms”. But what does one mean by “perfectly well known terms”? The abyss of infinite regress opens up again …… But even allowing for “exact” concepts, how can we prove that a proposition is true? How can we avoid the infinite regress in proofs, even if we could avoid the infinite regress in definitions? Meaning and truth can only be transferred, but not established. But if so, how can we know? (Lakatos 1962, pp. 156–157)

According to this sceptical argument, it seems impossible to pin down the exact meaning of a term. And even if we have managed to pin down the exact concepts, how can we establish the truth of a proposition through a proof while there can be an infinite regress in finding more and more fundamental premises for the proof?^[7] The sceptics thus provided a negative answer to the question “how do we know?” by denying that we can have any knowledge at all. Consequently, Lakatos remarks that “it has become a vital problem for rationalism to stop these exasperating twin infinite regresses^[8] and to provide knowledge with a firm bedrock” (Lakatos 1962, p. 157). Lakatos then introduces three enterprises which tried to achieve this: the Euclidean Programme, the Empiricist Programme and the Inductivist Programme.^[9] The Euclidean Programme, roughly speaking, models a theory into a Euclidean theory; and the Empiricist Programme organises a theory into what he called an empiricist theory – the term was later replaced by “quasi-empirical theory” (Shaffer 2015, p. 5). Therefore, we can see that the classification of a theory as a Euclidean or a quasi-empirical theory is essentially an attempt to explain the epistemology of that theory. The epistemic bedrock of a theory is thus its basic statements, which for a Euclidean theory is the set of axioms and for a quasi-empirical theory is some special theorems.^[10] It now becomes clear that the truth-value transmissions possess an epistemic significance for Lakatos. In a Euclidean theory, the axioms receive initially injected truth values True, which establish the truth of the rest of the system. The initial truth-value injection should be non-arbitrary: the axioms need to be indubitably true to be able to justify our beliefs in them and their consequences. A Euclidean theory is thus infallible. While in a quasi-empirical theory, some statements at the bottom of the theory receive initially injected truth values. If the truth value injected is True, it doesn’t flow upwards – it can’t establish the truths of the rest of the system. Only the falsity of the basic statement can establish the falsity of (at least one of) the statements at the above. A quasi-empirical theory is thus fallible. Therefore in a quasi-empirical theory true basic statements are not proved by the rest of the system simply because the rest of the system is not true^[11] so they are unable to transfer truth to the basic statements.^[12]

To mark the epistemic motivation behind Lakatos’ claim that mathematics is quasi-empirical, I shall call this view of mathematics the quasi-empirical epistemology of mathematics.

2.2 “Mathematics is Quasi-Empirical”

In the first section of Renaissance, Lakatos lists a number of quotations from philosophers and mathematicians suggesting that the search for a set of infallible self-evident axioms for mathematics had failed; some axioms are justified by the consequences which could not be derived without them. The quotations, Lakatos believes, showed that “mathematical empiricism and inductivism (not only as regards the origin or method, but also regards the justification, of mathematics) is more alive and widespread than many seem to think” (Lakatos 1976, p. 205). In other words, there was a renaissance of empiricism in the philosophy of mathematics. Philosophers of mathematics – for example, Russell, Carnap, Curry and Quine, from whom Lakatos quotes – were having second thoughts on the infallibility of mathematics. Also, a similarity between the justifications for mathematics and natural sciences was observed.

Additionally, in Regress and Renaissance, Lakatos extensively discusses the turn in Bertrand Russell’s views on mathematics. As reported by Lakatos, in the early years of his career, Russell ambitiously took up Frege’s unaccomplished task of founding mathematics on logic (Irvine 1989, p. 306). However, later Russell changed his mind and advocated for a regressive methodology of mathematics (Russell 1907). Russell’s main argument was that some axioms^[13] that serve as the foundations of mathematics are not analytic truths. Moreover, Russell claimed that it’s often not the case that people who know the basic arithmetical propositions are able to write out the proofs of them from the logical principles – they may even not know these logical principles (Russell 1907, p. 572). People don’t tend to think that it’s the lengthy proofs from the logical principles to the arithmetical propositions that establish the truths of them: if the logical principles turn out to negate the arithmetical propositions, we are more likely to amend the logical premises rather than the arithmetical propositions (Irvine 1989, p. 308). Therefore, Russell argued that our beliefs in the arithmetical propositions are not justified by the axioms.^[14] Russell also observed there is a close analogy between the methodology of mathematics and that of natural sciences. Just like Maxwell’s equations are believed because of the observed truth of their logical consequences, axioms in mathematics are believed on account of their consequences (Lakatos 1962, p. 175).

Similarly, it was observed by Lakatos that the set-theoretical axioms of mathematics that were devised to avoid Russell’s paradoxes are not indubitably true or even indubitably consistent (Lakatos 1976, p. 208). Therefore Lakatos argues that one shouldn’t organise mathematics into a Euclidean theory – instead, he claims that mathematics is quasi-empirical. That is to say, our beliefs in axioms of mathematics are justified by their consequences, which play the role of the basic statements. Axioms are not indubitably true – they are always conjectural. Mathematics is therefore fallible. But this, Lakatos claims, doesn’t commit us to mathematical scepticism, as we only need to admit the fallibility of daring speculations (Lakatos 1962, p. 183), by which he means axioms.

In Renaissance and Regress Lakatos also mentions Hilbert’s program. But it is a bit unclear what Lakatos actually thinks of Hilbert’s program. He classifies it as a Euclidean Programme (Lakatos 1976, p. 209), but he then remarks that “analysis is a quasi-empirical theory but the Euclidean consistency proof will see to it that it should have no falsifiers” (Lakatos 1976, p. 209). Also, in Regress, he characterises Hilbert’s meta-mathematics as an attempt to help mathematics pass the test for consistency as an empiricist theory (Lakatos 1962, p. 179). So it seems that Lakatos doesn’t think that Hilbert’s program can organise mathematics into a Euclidean theory. But then why is it Euclidean itself? An editorial note on (Lakatos 1976, p. 209) sheds some light on the question:

Lakatos is, perhaps, wrong to think that Hilbert’s philosophy, at least as here presented, can be subsumed easily under Euclideanism. Metamathematics is an informal unaxiomatised theory and such theories do not have the required deductive structure to be candidates for Euclidean status.

Therefore, Lakatos may mistakenly think that Hilbert’s program was intended to be a Euclideanisation of meta-mathematics which, if successful, would establish the infallibility of the consistency of mathematics. Unfortunately, its goal was proven to be impossible by Gödel’s incompleteness theorems (at least in Lakatos’ eyes). As a result, the impossibility of establishing the consistency of mathematics provides another source for the fallibility of mathematics.

3.1 Logical Falsifiers

Rejecting an axiom which derives a logical contradiction is a commonly accepted practice in mathematics. Most famously, the axiom (schema) of naive comprehension (which states that every property determines a set) was “abandoned” as it leads to Russell’s Paradox. Lakatos also deems consistency an essential property for a theory. Hence, for him, the impossibility to establish the consistency for mathematics (Section 2.2) opens up a source for the fallibility of axioms, for it could be the case that some day mathematicians find that current axioms derive a logical contradiction. Thus he characterised logical contradictions as logical falsifiers which are able to falsify axioms.

Nevertheless, the presence of a logical contradiction in a system doesn’t have to be “fatal”. The above example doesn’t tell the full story of naive comprehension. It wasn’t put to an end by Russell’s Paradox. Although the majority of mathematical community no longer work with the axiom of naive comprehension, there are still mathematicians sticking to it, as Mortensen described, “to retain the full power of the natural comprehension principle (every predicate determines a set), and tolerate a degree of inconsistency in set theory” (Mortensen 2017). After all, apart from Russell’s Paradox, there seems to be no obvious reason why one should reject naive set theory (Mortensen 2017). In order to retain naive comprehension, and to avoid the theory being trivial,^[16] these mathematicians give up the logical principle ex contradictione quodlibet (ECQ)^[17] which allows anything to be derived from a logical contradiction in their system. A logic which drops (ECQ) from its rules of inference is called a paraconsistent logic. Mathematical theories with paraconsistent logics comprise inconsistent mathematics. Naive comprehension thus can be retained in the inconsistent set theory. It has been proved that there is a classical recapture in the naive set theory formulated with a paraconsistent logic: the main theorems of ordinal and arithmetic can be proved (Weber 2010) and a theory of cardinal numbers can be developed as Cantor’s Theorem is retained (Weber 2012). Besides, the proof of the well-ordering principle and thus the axiom of choice can also be provided (Weber 2010). These results suggest that inconsistent set theory is not only interesting in its own right, but is also a promising theory that is able to provide new insights on foundations of mathematics. Therefore, the axiom of naive comprehension, though leading to a logical “falsifier”, is not sentenced to death as the quasi-empirical epistemology of mathematics would expect. Quite the opposite, it is vibrantly alive somewhere in mathematics, albeit not be in the mainstream.

The axiom of naive comprehension is not falsified by Russell’s Paradox. The inconsistent theories are not falsified by the logical contradictions within them. Therefore, we shouldn’t describe logical contradictions as falsifiers – a role that logical contradictions don’t have. Certainly we can choose not to work in an inconsistent theory as probably most mathematicians do. But such choices shouldn’t be pictured as falsifications. To say that inconsistent theories are falsified ignores the values behind inconsistent theories and disregards the actual mathematical research.

Two possible objections can be raised at this point. First, Lakatos notes that refutations don’t happen every day for quasi-empirical theories. There are occasional long stagnating periods – the periods when some theory is dominant. During the stagnating periods people tend to forget the fallibility of the axioms (Lakatos 1976, p. 219). One might argue that logical contradictions do impose falsifications to inconsistent theories but the falsifications are currently being ignored because we are in a stagnating period. This objection can be refuted straightaway. Inconsistent theories are by no means dominant in current mathematical research. Nor are the paraconsistent logics. There isn’t any authority of inconsistent theories which prevents people from falsifying them. Therefore it cannot be the case that it is the stagnant periods which keep inconsistent theories from the falsifications by logical contradictions.

The second objection is based on the idea that a falsification doesn’t always lead to a complete abandonment. For example, Newtonian mechanics is falsified but it is in use, since it is convenient to explain many phenomena using Newtonian mechanics. One might argue that for inconsistent theories, similarly, that they are alive because they offer conveniences in some ways. But in essence they are falsified, and they are merely handy substitutions for some other not-yet-falsified consistent theories, or for better consistent theories to be proposed in the future. That is to say, eventually mathematicians strive for consistency; an inconsistent theory is only present when there is no better alternative consistent theory. Although it is hardly the case for the inconsistent set theory, this objection does raise an interesting question concerning the motivations and the values of studying inconsistent mathematics in general. To properly address the objection, I briefly discuss inconsistent mathematics.

There are in fact many motivations for studying inconsistent theories. For example, Priest believes that the existence of semantic paradoxes such as the Liar Paradox suggests that our linguistic principles are inconsistent. Also, it is natural to have the phrases like “… is true” in human’s languages, i.e. “inconsistency is the natural outcome of spontaneity” (Priest 2006, p. 5, p. 6). Thus, instead of striving for consistency by amending our linguistic principles, it may be more favourable to resolve the paradoxes with paraconsistent logics. Besides, the idea of relevance that a correct natural entailment should rest on the conceptual connections between the antecedent and the consequence casts suspicion on (ECQ) and provides a reason in favour of paraconsistent logics (Mortensen 1995). Moreover, Mortensen suggests that the epistemic limit of finite humans and computers and the time limit for a machine to familiarise itself with the environment give rise to a possibly inconsistent database to reason with.^[18] In such inconsistent data environments, the rule (ECQ) should be abandoned for it allows us to derive everything. Mortensen believes that “the study of inconsistent theories is indicated as part of the long-term goal of artificial intelligence” (Mortensen 1995, p. 9).

Inconsistent theories in mathematics other than the inconsistent set theory are also of research value. Here I briefly present one example of them: inconsistent arithmetics. The relevant arithmetic R#, which is basically Peano arithmetic formulated with the relevance logic, can be shown to have a non-trivial model with only finistic methods in Hilbert’s sense (Meyer and Mortensen 1984, p. 919). Meyer believes that this result provides a way to revive Hilbert’s program: the relevant arithmetic distinguishes negation inconsistency^[19] from triviality while Peano arithmetic doesn’t, though R# cannot prove its negation consistency as a result of Gödel’s second incompleteness theorem, its non-triviality can nevertheless be established in a finistic way. Besides, there are a series of non-trivial models of R^#. These models are called RM3 ⁱ where three represents that there are three truth-values, and the domains of RM3 ⁱ are modulo arithmetics with modulo i (i is a natural number). RM3 ⁱ was proven to have be inconsistent for every i (Meyer and Mortensen 1984, p. 919). They provide inconsistent models for R^#, which comprise inconsistent arithmetics. They are interesting in their own rights (their properties were explored in Meyer and Mortensen 1984) and they have triggered some philosophical discussions. For instance, Priest^[20] raises the question of which arithmetic – Peano classical arithmetic or one of the inconsistent arithmetics – is the correct arithmetic? Priest points out that anything that the classical Peano arithmetic endorses, inconsistent arithmetics endorse. So advocates of consistent arithmetic must provide reasons why consistency is good, which he suggests are lacking. The question becomes more interesting as inconsistent arithmetics can be desirable in some respect. For instance, every RM3 ⁱ is decidable (Meyer and Mortensen 1984, p. 920); and inconsistent arithmetics can contain their own truth predicates without running into any problem (Priest 2006, p. 236).

Besides inconsistent arithmetics, there are inconsistent theories in various other fields of mathematics: calculus, topology, category theory, etc.^[21] But I think the example of inconsistent arithmetics itself suffices to show the point that inconsistent theories have intrinsic value in their inconsistency. It is not the case that inconsistent theories are only present since there is no better alternative consistent theory, nor is it the case that consistency always outweighs inconsistency. The second objection raised above is hence refuted. Therefore, to re-state the interim conclusion, logical contradictions don’t always falsify. The quasi-empirical epistemology of mathematics makes a too strong claim that logical contradictions are falsifiers.

In addition, the “birth processes” of the two inconsistent theories discussed above don’t seem to match the quasi-empirical epistemology of mathematics. The truth-value flows are top-down, as in the Euclidean theories. But what is different is that mathematicians don’t seem to be troubled with dubitable truth-value injections at the top. The naive comprehension is in some theory true (inconsistent set theory) and in some theory not true (for example, ZFC). But that doesn’t stop some mathematicians from using it to transmit truth-values downwards. Remember that the indubitability of axioms is required for the epistemological account of mathematics (Section 2.1). The examples in this subsection indicate that the truth-value injection and transmission in mathematics don’t necessarily carry epistemic weight, which I will argue in details in the later part of this paper.

3.2 Heuristic Falsifiers

Although Lakatos doesn’t offer a definite account of which mathematical propositions are heuristic falsifiers in Renaissance, our discussion is not hampered, since the quasi-empirical epistemology of mathematics can be refuted without considering what heuristic falsifiers for mathematics exactly are, if they exist at all. This is because, as I will show, no matter what they are, the epistemic priorities assigned to them by the quasi-empirical epistemology of mathematics do not conform to mathematical practices.

Heuristic falsifiers of mathematics are essentially mathematical statements which play the role of empirical facts in natural sciences. Suppose that they do exist for a mathematical theory, then, first of all, axioms are tested against heuristic falsifiers. Secondly, axioms are supposed to explain them. Thirdly, the quasi-empirical epistemology requires an epistemic source for the truth-value injections to the heuristic falsifiers. This is because, as remarked earlier, the heuristic falsifiers are the epistemic bedrock for mathematics according to the quasi-empirical epistemology of mathematics (Section 2.1). Hence their truth-value injections must be epistemically justified. I hereby call the source where the epistemic justifications for the truth-value injections to heuristic falsifiers come from the epistemic source for heuristic falsifiers. In Renaissance, Lakatos points out that one question for the quasi-empirical epistemology to settle is that: “what is the ‘nature’ of mathematics, that is, on what basis are truth-values injected into its potential falsifiers?” (Lakatos 1976, p. 218). The question concerns exactly the epistemic sources for the potential falsifiers. Finally, as when testing axioms against heuristic falsifiers, we test axioms indirectly against the epistemic sources for heuristic falsifiers, the epistemic sources for them should also act just like heuristic falsifiers for mathematics in this respect. That is to say, epistemic sources for heuristic falsifiers should also be able to provide falsifications.

In this subsection, I show that there can be no satisfactory epistemic source for any mathematical statements that supports them to be heuristic falsifiers. I do so by discussing two kinds of epistemic sources: intuition and empirical experience. This is because these are the two primary epistemic sources. A piece of knowledge comes either from experience or not. If it doesn’t come from experience, nor is it an analytic truth,^[22] then we must know it by some kind of intuition. I argue in the rest of this subsection that neither intuition nor experience is able to support heuristic falsifiers in fulfilling the above expectations that the quasi-empirical epistemology of mathematics has for them.

3.3 Mathematics is not Quasi-Empirical

Given that (1) logical contradictions don’t always falsify, and (2) there can’t be any heuristic falsifier in mathematics, then the quasi-empirical epistemology of mathematics is not correct. As the above analysis shows, the problem for the quasi-empirical epistemology – or, more generally, for Lakatos’ classification of mathematics into Euclidean theories and quasi-empirical theories in the first place – is that the epistemic requirement on the truth-value injections and truth-value flows in mathematics is too strict. Lakatos requires indubitable initially-injected truth-values for axioms in a Euclidean theory since he wants to stop the infinite regress in proofs and to account for the epistemology of the theory (Section 2.1). And since set-theoretical axioms are not indubitably true and they don’t seem to have initially-injected truth-values, Lakatos rejects the Euclidean picture of mathematics and argues for the quasi-empirical epistemology of mathematics. He argues that the statements at the bottom of mathematics are epistemically prior to axioms – they receive initial truth-value injections. Then the epistemological question for mathematical axioms – i.e. how do we come to know mathematical axioms? – is transferred to basic statements at the bottom. But as Section 3.2 shows, the question “how do we know mathematical basic statements?” is by no means easier to answer, since there isn’t a satisfactory epistemic source which can support any basic statement founding all of mathematics.

The above arguments indicate that the truth-value injections in mathematics don’t always come together with adequate epistemic justifications since there may not be any. Hence the truth-value flows don’t have to start with epistemically justified propositions to transmit truth. A mathematical theory can start with non-indubitable (fallible) axioms at the top (such as the inconsistent set theory in Section 3.1) or with non-falsifying statements at the bottom. The quasi-empirical epistemology of mathematics does capture the idea of extrinsic justifications for axioms in mathematics. But attaching epistemic values to these justifications and viewing them as the only kind of justifications in mathematics cause problems.

Lakatos himself also notices the problem of the quasi-empirical epistemology as he writes in Renaissance:

A heuristic falsifier after all is a falsifier only in a Pickwickian sense: it does not falsify the hypothesis, it only suggests a falsification – and suggestions can be ignored. It is only a rival hypothesis …… The crucial role of heuristic refutations is to shift problems to more important ones, to stimulate the development of theoretical frameworks with more content. One can show of most classical refutations in the history of science and mathematics that they are heuristic falsifications. The battle between rival mathematical theories is most frequently decided also by their relative explanatory power.

Lakatos observes that, even if there is a heuristic falsifier at all for a theory, it doesn’t have to falsify. Since a falsification is essentially a rival hypothesis, people usually choose between the rival between hypotheses on the basis of their “relative explanatory power”. A similar remark of Lakatos can be found in Lakatos (1978, p. 155). It seems that Lakatos might have questioned his quasi-empirical epistemology after he proposed it, so he later shifted to a different view of mathematics^[27] where mathematical theories are compared and selected based on their explanatory power. But we don’t really need to choose a theory at the cost of another as the co-existence of inconsistent set theory and ZFC and the co-existence of Euclidean geometry and non-Euclidean geometries shows that there doesn’t have to be. The evidence in this section suggests a more liberal picture of mathematics, which I explain in the next section.