The central idea of Lakatos’ quasi-empiricism view of the philosophy of mathematics is that truth values are transmitted bottom-up, but only falsity can be transmitted from basic statements. As it is falsity but not truth that flows bottom-up, Lakatos emphasizes that observation and induction play no role in both conjecturing and proving phases in mathematics. In this paper, I argue that Lakatos’ view that one cannot obtain primitive conjectures by induction contradicts the history of mathematics, and therefore undermines his quasi-empiricism theory. I argue that his misconception of induction causes this view of Lakatos. Finally, I propose that Wittgenstein’s view that “mathematics does have a grammatical nature, but it is also rooted in empirical regularities” suggests the possibility to improve Lakatos’ view by maintaining his quasi-empiricism while accepting the role induction plays in the conjecturing phase.
Lakatos propounds a view of the philosophy of mathematics that embraces quasi-empiricism and fallibilism. He distinguishes his approach from three other views: the Euclidean view, Inductivism, and Empiricism. Lakatos’ rejection of the Euclidean approach is based on the idea that its “top-down” determination of truth-values conflicts with the history of mathematics, which is characterized by disagreement and lack of progress. Regarding the empiricist view, Lakatos criticizes the problematic, bottom-up supposition that infallible knowledge can be obtained only by factual propositions and their inductive consequences. In particular, he objects to the Carnapian neoclassical empiricist position which holds that induction from observation data can confirm knowledge to some degree, though it cannot guarantee infallible knowledge. For Lakatos, both classical and neoclassical empiricism argues that truth can be transmitted in a bottom-up manner. In contrast, the central tenet of Lakatos’ quasi-empiricism is that bottom-up transmission is only possible for truth-values that yield falsity. This tenet preserves a degree of fallibilism in Lakatos’ quasi-empiricist view: the focus on the transmission of false values admits the very real possibility that we will find counterexamples to scientific claims in the future. Lakatos’ focus on the falsity of basic statements also provides a basis according to which we can reject existing proofs and, thereby, help mathematicians improve the primitive conjecture and contribute to progress in mathematics. In this paper, by primitive conjecture, I mean the conjectures that mathematicians first propose, which they find worth proving. These primitive conjectures are also referred to by Lakatos as naïve conjectures or bold guesses, which might be improved by proofs and rejections.
Rather than focusing on “how a primitive mathematical conjecture is discovered”, Lakatos develops his quasi-empiricism with attention to the way conjectures are justified. In order to understand Lakatos’ quasi-empiricism, it is also necessary to examine his view on how mathematicians generate primitive hypotheses in the first place. Lakatos’ philosophy of mathematics has been interpreted in myriad ways. Among others, there are interpretations that focus on the dialectical approach (Larvor 1998), fallibilism (Kadvany 2001; Larvor 1998), mathematical skepticism (Kadvany 2001; Koetsier 2002), and the evolution of mathematical concepts (Mormann 2002). In this paper, I consider Lakatos’ quasi-empiricism with an attention to his thinking about the discovery of a conjecture. I note that Lakatos denies that primitive conjectures can be generated based on induction from observational data, a denial which contradicts historical thinking about mathematics. The argument of this paper is twofold. First, I argue that Lakatos’ anti-induction thinking is a consequence of his Popperian misconception of induction. This is his belief that what we can obtain from induction are merely atheoretical patterns with high probability. Second, I argue that Wittgenstein’s view that mathematical propositions are grammatical rules based on hardened empirical regularities suggests a possible improvement to Lakatos’ theory. For Wittgenstein, induction from observations is not subjective or arbitrary; it is something objective insofar as it has a sociological and empirical basis. From these two arguments, I conclude that Lakatos’ worries about using induction when generating conjectures is unfounded, as induction does not necessarily contradict the rationality and objectivity of mathematics. Accepting the role induction plays in generating conjectures could make Lakatos’ quasi-empiricism more plausible.
2 How Do We Obtain Primitive Conjectures?
Some might argue that concern about how mathematicians generate primitive conjectures is primarily a sociological or psychological problem. However, though the problem can be considered a sociological or psychological issue, it does have philosophical implications. One way of thinking about these implications is by using the practice-centered approach that has been thriving in the philosophy of mathematics for decades. This approach investigates the practices of mathematicians and aims to understand the philosophical implications of these practices. Given that the generation of primitive conjectures is the starting point of the “proving process,” understanding how mathematicians generate these primitive conjectures helps us to understand the nature of mathematical knowledge.
I will begin by examining Lakatos’ most influential work, Proofs and Refutations (henceforth PR). In PR, Lakatos illustrates his quasi-empiricism and fallibilism by reference to the development of Euler’s conjecture for polyhedra. Euler’s conjecture proposes that ‘V + F − E = 2’ is valid for regular polyhedra, where V represents vertices, F represents faces, and E represents edges. Lakatos’ intention in PR is to demonstrate how such conjectures might have been developed in dialogue, for instance, in discussions involving a teacher and their students. The discussions in PR represent an approach referred to as the ‘Method of Proofs and Refutations.’ The approach begins with a primitive conjecture (or, in Lakatos’s term, “bold guess”), then proceeds with a proof of that conjecture. The proof is then reconsidered as counterexamples (i.e., refutations) are found, and, in light of counterexamples, attempts are made to add corresponding hidden lemmas to the conjecture as conditions. As counterexamples are found and hidden lemmas added, the primitive conjecture must be replaced by an improved conjecture, which takes previously identified counterexamples as support. The main purpose of PR is to emphasize that it is the repetitive process of offering proofs and finding refutations that drives the development of mathematics. Moreover, Lakatos posits that the repetitive process of proofs and refutations is never complete, as it is always possible that counterexamples for any “proven” claim might be found. In this sense, our mathematical knowledge is not static, but ever-improving.
Lakatos opens PR by introducing the historically relevant fictional dialogue between a teacher and their students. The dialogue begins with the following passage:
After much trial and error, they noticed that for all regular polyhedral V − E + F = 2. Somebody guesses that this may apply for any polyhedron whatsoever. Others try to falsify this conjecture, try to test it in many different ways-it holds good. (Italics Lakatos’)
Here, ‘V − E + F = 2’ represents a starting point, a conjecture that requires proof and offers the basis according to which knowledge can develop. Lakatos does not clarify what kind of “trial and error” mathematicians had before noticing this pattern, which, for him, seems to be less significant than what happens after the primitive conjecture is discovered. In footnote that attends to Euler’s thinking about this conjecture, Lakatos observers that “Euler tested the conjecture quite thoroughly for consequences. He checked it for prims, pyramids and so on …”. Lakatos also notes the similarity of Euler’s examination of the “conjecturing and testing” phase to the examination found in Polya (1954). It is natural to take Lakatos’ remark in this footnote as agreeing with Polya’s thoughts about the “conjecturing and testing” phase, which explains why Lakatos proceeds directly to the proving phase. For Polya (1954):
Both the common man and the scientist are led to conjectures by a few observations, and they are both paying attention to later cases which could be in agreement or not with the conjecture.
Polya (1954) presents one way in which Euler might have discovered this conjecture, ‘V − E + F = 2’. Development of the conjecture begins with the consideration about whether F always increases with V. Examining the examples of polyhedra in Table 1, we find that this is not the case. We can also conclude from the table that neither V, nor F, necessarily increase with E. Considering the table in further detail, mathematicians find that F + V seems to increase steadily with E, which leads to the primitive conjecture that F + V = E + 2.
Table of polyhedra.
For Polya, primitive conjectures are discovered by observation of examples at our disposal, and they represent a tentative pattern that we arrive at by induction from observable data. Though Lakatos purports to agree with Polya’s reconstruction of the “conjecturing and testing” phase, there is a dialogue in PR that seems to show that Lakatos has a different view about the role of induction in discovering a primitive conjecture.
In the discussion regarding the origin of primitive conjectures, student Beta argues that despite failing many times, he finally found the regularity ‘V − E + F = 2’ in the table of polyhedron by observation (see Table 1):
All that we have to do now is to complete our table with the data for non-Eulerian polyhedra and look for a new formula: with patient, diligent observation, and some luck, we shall hit on the right one; then we can improve it again by applying the method of proofs and refutations!
Student Sigma argues in response that Beta’s induction is merely contingent; including certain data about anomalies like picture-frames or urchins in this table would undermine the integrity of Beta’s induction. The point here is that induction will not be successful if the table is characterized by chaotic data. Beta continues by asking, “where do you want to start if not with an inductive low-level generalization as a naive conjecture?” To answer, student Zeta suggests that the naive conjecture ‘V − E + F = 2’ might be generated from a process of deductive reasoning. Zeta demonstrates this by reference to the way that ‘V − E + F = 2’ can be deduced from the fact that V = E holds true for polygons. Zeta’s argument is that it is not necessary to use induction with the data of Table 1 to arrive at this naive conjecture.
Objecting to Zeta’s argument, Beta points out that induction provides the basis according to which famous conjectures in the history of mathematics have been established, including the four-color conjecture and Goldbach’s conjecture. Beta also emphasizes that deductive guessing only happens accidentally, and that induction is more frequently used in cases wherein deductive guessing is not possible. In other words, mathematicians do start with naïve inductive conjectures in most cases. The teacher interrupts at this point to offer a criticism:
Teacher: We certainly have to learn both heuristic patterns: deductive guessing is best, but naïve guessing is better than no guessing at all. But naïve guessing is not induction: there are no such things as inductive conjectures!
Beta: But we found the naïve conjecture by induction! ‘That is, it was suggested by observation, indicated by particular instances … And among the particular cases that we have examined, we could distinguish two groups: those which preceded the formulation of the conjecture and those which came afterward. The former suggested the conjecture, the latter supported it”. Both kinds of cases provide some sort of contact between the conjecture and “facts” …’ This double contact is the heart of induction: the first makes inductive heuristic, the second makes inductive justification or inductive logic.
Teacher: No! Facts do not suggest conjectures and do not support them either!…
Teacher: Naive conjectures are not inductive conjectures: we arrive at them by trial and error, through conjectures and refutations.
As Corfield (1997) and other Lakatos scholars indicate, it is highly probable that the teacher’s arguments in PR represent Lakatos’ own position. If this reading is correct, it is clear that Lakatos’ view is that observations and induction do not play any role in discovering primitive conjectures or in justifying a conjecture. We can glean from the teacher’s argument that, for Lakatos, Beta’s work of finding a pattern in the data of Table 1 is a process of “trial and error”, not induction. In a footnote, Lakatos suggests that it is this process of “trial and error” that is reconstructed by Polya.
However, Polya himself has no intention to exclude induction from the process of developing conjectures. On the contrary, he emphasizes the role of induction in conjecturing: “The scientist’s procedure to deal with experience is usually called induction. Particularly clear examples of the inductive procedure can be found in mathematical research … Induction often begins with observations”.
There is an apparent tension here between Lakatos’ rejection of induction and Polya’s emphasis on the role of induction in conjecturing. Both Polya and Lakatos admit the process of “trial and error” in finding the primitive conjecture. For Polya, observations and inductions are involved in the “trial and error” process; whereas, for Lakatos, “trial and error” is essentially different from induction. The tension between their views, therefore, is based on the way they differ in their understanding of the process of “trial and error”.
Polya (1954) distinguishes two different kinds of reasoning used in mathematics: demonstrative reasoning, and plausible reasoning. For Polya the process of generating conjectures is an instance of plausible reasoning, whereas the process of proving a conjecture represents demonstrative reasoning. Demonstrative reasoning is rigid, and it cannot provide us with new knowledge; plausible reasoning is creative and can bring us new knowledge. Polya notes that plausible reasoning is ‘hazardous, controversial, and provisional’, but it helps us learn new things about the world. For Polya, both demonstrative reasoning and plausible reasoning play significant roles in mathematical practice. A formal proof is a kind of demonstrative reasoning. However, before we engage the process of formal proof, we must generate a conjecture and reinforce our belief that it is mathematically interesting and worth proving. Polya suggests that inductive reasoning should be used to this end. Polya emphasizes that inductive reasoning should be seen as a kind of plausible reasoning, and in mathematics inductive reasoning plays a similar role as it plays in empirical sciences. He refers to the Goldbach conjecture to illustrate how induction based on observable data allows mathematicians to generate a conjecture, and to demonstrate how more data works to reinforce the credibility of conjectures.
Induction thereby plays an essential role in Polya’s conjecturing phase, while deductive methods are reserved for building formal proofs. The distinction between plausible reasoning and demonstrative reasoning corresponds to the distinction between the “informal proof” stage and the “formal proof” stage, a distinction which Lakatos (1978) also emphasizes. For Lakatos, the “informal proof” stage includes inspiring thought experiments that yield new ideas about how proofs of a conjecture might be demonstrated. It is imperative, then, that mathematicians should spend more time at the “informal proof” stage, where counterexamples might be found and conjectures could be revised, before proceeding to offer a formal proof. It is important to note, though, that Lakatos does not include observations and induction as essential components of the “informal proof” stage. In order to understand Lakatos’ thinking about the “informal proof” stage, it is necessary to determine the motivation for Lakatos’ insistent anti-induction position, a position starkly opposed to Polya’s.
3 Lakatos’ Misconceptions of Induction
Some modern philosophers of mathematics (Cellucci 2013; Ippoliti 2019) argue that the practices of working mathematicians demonstrate that most discoveries of primitive conjectures are generated from induction and analogy. Ippoliti (2019) refers to the way Hamilton invented quaternions by analogy to the geometric and algebraic representations of complex numbers as a basis for the claim that mathematical knowledge is developed from “the search for small regularities and similarities”. Similarly, Cellucci (2013) considers the non-deductive rules employed in the search for hypotheses. He argues that induction and analogy are the “two oldest and better-known rules of discovery”, though he admits that the set of rules is not static. This general agreement about the import of induction for the discovery of conjectures is in stark contrast to Lakatos’ practice-centered rejection of induction. Cellucci (2013) suggests four typical misconceptions about induction that might be taken to explain Lakatos’ position:
There is no such thing as induction and there is no need for it.
Induction is closely linked to probability.
Induction is a means of justification.
Induction ultimately depends on intuition.
In what follows, I will argue that the first and second of these misconceptions characterize Lakatos’ thinking about induction in PR and elsewhere.
4 Misconception 1: “No Such Thing as Induction and No Need for It”
Cellucci notes that the first misconception, that “there is no such thing as induction and there is no need for it”, is represented by Popper’s view. Cellucci points out that for Popper, induction is redundant in the process of discovering conjectures and in justifying them. Popper posits that creative intuition is used to generate discoveries, and that falsification contributes to our justification of those discoveries. It follows that there is no way for induction to make a novel contribution to the sciences. This position is echoed by the teacher in Lakatos’s PR:
We certainly have to learn both heuristic patterns: deductive guessing is best, but naïve guessing is better than no guessing at all. But naïve guessing is not induction: there are no such things as inductive conjectures!
Here, it is worth noting the significant influence of Popper and Polya on Lakatos’ philosophy of mathematics. Regarding the discovery of primitive conjectures, though, Popper and Polya have opposing views. Writing as the teacher in PR, Lakatos posits that naive guessing is not induction, and that conjectures cannot be generated from the method of induction. Lakatos’ clear rejection of induction in this passage is evidence that his view is more consistent with Popper’s thinking about induction than it is with Polya’s.
Lakatos’ intention is to apply Popper’s anti-induction position in the philosophy of science to his philosophy of mathematics, that is, arguing that induction cannot contribute to mathematics either. Lakatos says: “The Inductivist Programme was a desperate effort to build a channel through which truth flows upwards from the basic statements”. Lakatos also rejects the weak version of Inductivism, Probabilistic Inductivism, which holds that partial truth (i.e., degree of confirmation) can flow bottom-up because “there is nothing conclusive about the truth-value of basic statements”, which is to say that a theory can either be conjectural or false. I will consider Lakatos’ objection to Probabilistic Inductivism later. Now, it is necessary to look at the historical practice of working mathematicians to glean the role played by induction.
5 Induction in Mathematical Practice
Bendegem (2003) suggests that thought experiments play a significant role in mathematics. He proposes four types of constructive mathematical thought experiments: informal proofs; induction; mathematical experiments; and probabilistic arguments. To illustrate the function of induction in mathematics, Bendegem considers mathematicians’ work on Fermat’s Last Theorem, which holds that there is no solution in positive integers for the equation + = when n is greater than 2. Bendegem points out that before a general proof is given, Wagstaff attempted to test Fermat’s Last Theorem using values of n up to n = 12,500. Bendegem argues that in order to prove a conjecture over an infinite domain, mathematicians have taken time to test a number of specific cases, as Wagstaff did. For Bendegem, mathematicians conduct this kind of inductive work not only because counterexamples might be discovered in the process, but also because inductive work provides valuable information indicating the possibility of counterexamples and insights that might lead to more plausible proofs.
Lakatos and Popper would claim that the inductive work that Wagstaff did for Fermat’s Last Theorem does not make any mathematically significant contribution, as the work is not a proper proof for the universal statement. However, as Bendegem indicates above, mathematical practice throughout history demonstrates that working mathematicians practice with the opposite view in mind, that induction is an indispensable tool not only in the conjecture phase, but also when testing a conjecture before a rigid proof is discovered. Bendegem’s reference to Wagstaff is a demonstration of how inductive methods can be helpful when finding a proof is extremely difficult.
Working mathematicians might have very little sense of what inspires their conjectures – whether it is “a creative intuition” or “a pattern they find from small regularities”, or a combination of both. However, it is evident that working mathematicians do not tend to rule out the inductive method in their work, as Lakatos insists they should. Accordingly, we find that Lakatos’ argument about eliminating induction in both proving and pre-proof processes is not supported by the real practice of mathematicians. As Polya (1954) and Bendegem (2003) illustrate, both (i) diligent observation and the induction based on these observations (as in Beta’s table), and (ii) testing a number of instances of a conjecture that haven’t been formally proved, are normal tactics for working mathematicians. In the next section, I will consider another reason for Lakatos’ strong opinion about induction found in his criticism of Carnap’s logic of discovery.
6 Misconception 2: “Induction is Closely Linked to Probability”
Lakatos’ criticism of Carnap’s logic of discovery is a demonstration of the second misconception about induction that Cellucci mentions, that is, that “induction is closely linked to probability”. In a long footnote in his paper “Changes in the problem of inductive logic”, Lakatos notes that although both Popper and Carnap emphasize “improvement of a guess” and “improving our general methods for making guesses” in the development of our knowledge, these statements have different meanings for the two philosophers. Lakatos argues that, for Popper, “improvement of a guess” means substituting one theory with a new theory that has “more empirical content”, which is “a critical, creative and purely theoretical affair”. For Carnap, “improvement of a guess” means “a mechanical (or almost mechanical) and essentially pragmatic affair”. Lakatos defends Popper’s position by arguing that Carnap’s approach is problematic because it merely focuses on comparing the probabilities of different hypotheses and selecting one with higher probability “according to the purpose of one’s action”. More importantly, Lakatos also attacks Carnap’s idea that ‘inductive logic not only improves single guesses, but also helps us improve the general method we guess.’
For Lakatos, what Carnap claims to be the ‘improvement of our general method of making guesses’ is not an essential improvement of method. Rather, it is merely a process of selecting a “better” function, that is, a function that leads to a higher probability. The pragmatic improvement here is not mathematically significant. Lakatos uses Kemeny’s three stages of hypothesis selection to illustrate the Carnapian logic of discovery. Kemeny’s three stages of hypothesis selection can be summarized as (1) a language to express the hypothesis is decided, (2) the statement to be used as a hypothesis is selected, (3) decide whether the hypothesis can be considered as justified based on given evidence”. For Lakatos, Kemeny’s methodology represents ‘Bayesian conditionalization’, which is both “atheoretical” and “acritical”. Lakatos emphasizes that if what we do during the second stage (when a statement is selected to be a candidate hypothesis) is programming the calculation of probability on an inductive machine, then the third stage becomes trivial. Lakatos raises two concerns about this approach. First, important explanations and criticism cannot be made in this stage as the discussion is limited to the language selected in stage one. Second, the explanation in this stage focuses on statistics, whereas meaningful and inspiring refutations are ruled out. Lakatos posits that this inductive approach features “guesses, with different and changing degrees of probability, but without criticism.” It follows that with this approach, we lose both explanation and testability.
Lakatos’s main concern here seems to be that if probability is all that matters, then any patterns the empirical data show with high probability should be considered as a conjecture worth proving, even if we cannot find any explanatory power or rationality in the conjecture. Consequently, we might end up proving a hypothesis that turned out to be a pattern that contingently suits the empirical data, rather than a conjecture with more concrete valuable knowledge. Therefore, for Lakatos, probability is not convincing enough to be considered as a good reason to accept a hypothesis. Explanation and possibility of criticism should be regard as much more essential than probability.
Lakatos is right that a higher probability does not necessarily make a conjecture interesting and worth proving. However, Lakatos ignores one important fact, that although it is theoretically possible that some conjectures with high probability turn out to have a lack of explanatory power, in real mathematical practice, this rarely happens. Therefore, it is reasonable for working mathematicians to consider probability as an indicator of the plausibility of a conjecture. This follows because high probability is normally a sign of mathematically meaningful conjectures; in most cases, explanatory power and rationality co-occur with high probability. With this in mind, mathematicians are happy to use the inductive method when generating and testing conjectures that have high probability. Mathematicians use the inductive method in the conjecturing phrase, not because they think probability has mathematical significance, but because probability is a useful indicator that helps them become more convinced a conjecture is worth proving. Certainly, plausibility merely suggests a conjecture is worth proving; a rigorous proof is always needed to preserve the integrity of a conjecture. Now, it is clear that, for Lakatos, mathematicians do not discover primitive conjectures by diligent observation and induction, but by naïve guess. However, it is not so clear how mathematicians can make trustworthy naïve guesses that generate primitive conjectures. I will try to answer this in what follows by considering Lakatos’ heuristic view on mathematics.
7 Heuristics and Objectivity
Lakatos’ rejection of the inductive method is influenced by his heuristic view of mathematics. Lakatos (1978a) defines ‘heuristic’ as “a powerful problem-solving machinery, which, with the help of sophisticated mathematical techniques, digests anomalies and even turns them into positive evidence”. Larvor suggests that Lakatos’s conception of heuristics is similar to Polya, who believes “heuristic exploits objective features of mathematics, rather than subjective quirks of human psychology”. Kadvany (2001) points out that Lakatos suggests a “heuristic pedagogy” and highlights the essential difference between the Euclidean approach to mathematics and a Lakatosian heuristic approach. For Lakatos’ approach, “one actively contributes to the historical development of mathematics and culture through a pedagogically inspired philosophy that explains the mathematical present through its, or our past”; for the Euclidean approach, it is “intuitive grasp” that matters, and growth or progress in mathematics are not expected. Similarly, Koetsier (2002) emphasizes that for Lakatos, “Mathematics is fallible but its development is not arbitrary. Its development is in general in accordance with certain norms of rationality.”
With the perspectives of Larvor, Kadvany and Koetsier in mind, we might characterize Lakatos’ “heuristic” view as something like “mathematical intuition”, or as a set of “mathematical skills” that one obtains by learning mathematical methods and the history of mathematics. It is not purely intuitive, but pedagogical. On Lakatos’ view, the heuristic thinking of mathematicians enables them to discover primitive conjectures that are worth proving without relying on “diligent observation and induction from empirical data”.
Although Lakatos rejects the Euclidean approach, for which rationality guarantees true mathematical knowledge, he does not intend to diminish the significance of rationality and objectivity in mathematics. On the contrary, for Lakatos, there exists objectivity and rationality in the history of mathematics despite the fact that mathematics is fallible and continues to develop. The challenge for Lakatos, then, is to defend the rationality and objectivity of fallible mathematics. Wittgenstein offers one such defense of the objective but not a priori nature of mathematics. However, in contrast to Lakatos, Wittgenstein does not deny the empirical basis of primitive mathematics. I will consider Wittgenstein’s position on the relation between human experience and mathematics, and then Bloor and Steiner’s interpretation of this position.
8 Wittgenstein—Mathematics as Rules Built from “Hardened Empirical Regularities”
In Remarks on the Foundation of Mathematics (henceforth RFM), Wittgenstein focuses on the relation that mathematics has to the world:
III.15. It is a fact that different methods of counting practically always agree.
When I count the squares on a chess-board I practically always reach ‘64’ …
III. 66. … What if we said that mathematical propositions were prophecies in this sense: they predict what result members of a society who have learnt this technique will get in agreement with other members of the society? ‘25 × 25 = 625’ would thus mean that men, if we judge them to obey the rules of multiplication, will reach the result 625 when they multiply 25 × 25. —That this is a correct prediction s beyond doubt; and also that calculating is in essence founded on such predictions. That is to say, we should not call something ‘calculating’ if we could not make such a prophecy with certainty. This really means: calculating is a technique. And what we have said pertains to the essence of a technique.
III. 69. … Experience teaches that we all find this calculation correct.
VI, 21. … It is of the greatest importance that a dispute hardly ever arises between people about whether the colour of this object is the same as the colour of that, the length of this rod the same as the length of that, etc. This peaceful agreement is the characteristic surrounding of the use of the word “same”.
Steiner (2009) cites the above remarks to support his reading that for Wittgenstein’s later philosophy of mathematics, the idea that mathematical propositions are rules established by “hardening empirical regularity” is central. Fogelin (1987) also emphasizes that Wittgenstein “carefully considers an empiricist view”. The idea of the passage can be summarized as follows. When we learn to calculate, we try to find the answer to questions like ‘what is 2 + 3?’ by counting physical objects, for instance by adding three apples to two apples. Although we might count in different ways, it is quite likely we always get five apples. When we find that three added to two always makes five, we take the regularity of ‘2 + 3 = 5’ as convincing enough that we can make it into a rule. In future cases, we will only consider five as the correct calculation when adding 2 to 3. For a more complicated calculation, such as ‘25 × 25’, it is possible that 90% of a group calculating will answer 625, and 10% of the group will answer 624 or 626. Regarding this possible difference in calculation, Wittgenstein argues this:
Well, suppose that 90 per cent do it all one way. I say, “This is now going to be the right result.” The experiment was to show what the most natural way is—which way most of them go. Now everybody is taught to do it—and now there is a right and wrong. Before there was not.
For Wittgenstein, mathematics should be distinguished from experiment. In experiments, one can get different results without claiming which is right and which is wrong. In mathematics, there are rules dictating what is a right calculation and what is a wrong one. We can see from this line of thinking that Wittgenstein, like Lakatos, rejects the Euclidean position that mathematical knowledge is a priori and there can be absolute mathematical truths. For Wittgenstein, mathematical objectivity is a natural result of our behavioral regularity. Although there can be exceptions and counterexamples to rules, significant disagreements about rules seldom happen in mathematical practice. This is to say that although it is theoretically possible to have an alternative way of doing mathematics, it remains practically difficult to conduct or think about a sort of mathematics that is different from the one we practice.
Bloor (1976, 1983 interprets Wittgenstein’s position as a sociological view of mathematical knowledge. Bloor (1983) offers the following characterization of the behavioral regularities involved in mathematical practice:
Of all the countless games that can be played with pebbles, only some of the patterns that can be made with them achieve the special status of becoming ‘characteristic ways’ of ordering and sorting them. In exactly the same way, all the countless possible patterns that may be woven into a rug are not all equally significant for a group of traditional weavers. There are norms for those who would weave carpets just as there are norms for those who would learn mathematics (p. 99).
Bloor’s interpretation of Wittgenstein holds that although it is possible for multiple patterns to be generated from the same empirical data, this possibility is merely theoretical. In practice, we accept some patterns, and we reject others. It might be that we naturally follow these patterns, despite our awareness of them. The fact that we seldom disagree about acceptable patterns suggests that patterns are not selected arbitrarily; there must a good reason to explain our general agreement. The central thesis of Bloor (1983) is that for Wittgenstein, the behavioral regularities, and thereby the patterns we accept, are based on the common social norms we share. Bloor emphasizes that considering the sociological nature of mathematical knowledge is necessary to understand Wittgenstein’s philosophy of mathematics. On Bloor’s reading, Wittgenstein’s view is that mathematical knowledge is objective because it is sociological, not because it is rational a priori. In the passage cited above, we can see that Wittgenstein does emphasize that our agreements on calculating are related to the common training and conventions. Accordingly, it is plausible that this sociological reading of Wittgenstein is correct.
However, Steiner offers a different reading of Wittgenstein’s position, which does not rule out Bloor’s reading, but aims deeper in its analysis. In contrast to Bloor’s reading, which emphasizes the sociological nature of mathematical knowledge, Steiner stresses that for Wittgenstein, the behavioral regularity that guides mathematical knowledge has a deeper root, namely, certain “empirical regularities”. Steiner refers to the following passages to support his reading of Wittgenstein.
In RFM, Wittgenstein states that
It is as if we had hardened the empirical proposition into a rule. And now we have, not a hypothesis that gets tested by experience, but a paradigm with which experience is compared and judged. And so a new kind of judgment.
In his 1939 Lectures on Foundation of Mathematics (henceforth LFM), Wittgenstein argues as follows:
We have invented multiplication up to 100; that is, we’ve written down things like 81 × 63 but have never yet written down things like 123 × 489. I say to [the trainee], “You know what you’ve done so far. Now do the same sort of thing for these two numbers.” —I assume he does what we usually do. This is an experiment—and one which we may later adopt as a calculation.
What does that mean? Well, suppose that 90 per cent do it all one way. I say, “This is now going to be the right result.” The experiment was to show what the most natural way is—which way most of them go. Now everybody is taught to do it—and now there is a right and wrong. Before there was not.
It is like finding the best place to build a road across the moors. We may first send people across, and see which is the most natural way for them to go, and then build the road that way. (LFM, X, p. 95)
Moreover, in Philosophical Investigations, Wittgenstein writes about the possibility of understanding the language of a foreign country. He suggests that in order to call the sound and orders that people make in this unknown country “language”, we need to find sufficient regularity in it. Steiner posits that, for Wittgenstein, mathematics is like language in that it is based on behavioral regularities. Although Steiner does not deny Bloor’s thesis that behavioral regularities have a sociological nature, Steiner stresses that these behavioral regularities are not merely sociological; rather, and more importantly, they are rooted in the empirical world. This is to say that mathematics has both empirical and normative nature, but also that empirical regularities are more fundamental than our behavioral regularities. On this reading, Wittgenstein’s position can be considered empiricist insofar as he holds that at least primitive mathematics is developed from our contact with the empirical world.
If Steiner’s reading is correct, Wittgenstein strives to emphasize the fact that without the stableness of the physical world, there cannot be behavioral regularities: behavioral regularities in the counting of apples are possible because apples do not disappear or increase. It follows that in order to develop mathematical rules, we need more than common training which helps us follow rules in similar ways. Importantly, we require the regularities that define patterns that we learn in our experience of the physical world. It should be noted here that Wittgenstein is not strictly an empiricist. He does not suggest that the mathematical system itself is empirical a priori, but rather that empirical regularities are a starting point for building a mathematical system. When we have decided the rules which we will follow, we do not need the help of empirical regularities anymore; and from this moment mathematics becomes grammatical. Pérez-Escobar (2022) recently provides a reading that is similar to Steiner’s, which clearly distinguishes Wittgenstein’s view from both formalism and empiricism. He argues that for the later Wittgenstein, mathematics is not a “description of the world” but “rules of descriptions of the world”. By “rules of descriptions”, Pérez-Escobar characterizes both the grammatical nature and the empirical root of mathematics, i.e., mathematics describes the world with patterns made by humans.
I will turn now to consider whether Wittgenstein’s idea, with the help of the above readings, can provide any insights for Lakatos’ thesis. In PR, student Sigma objects Beta’s idea that we can arrive at a pattern from induction of observations:
Sigma: … Impossible! Look at the vast crowd of counterexamples. Polyhedra with cravities, polyhedral with ringshaped faces, with tunnels, joined together at edges, vertices … V-E + F can take any value whatsoever! You cannot possibly recognize any order in the chaos! …
From the discussion above, we find that Sigma’s argument seems to represent Lakatos’ view, that it is possible for Beta to begin their induction from polyhedral like ring-shaped faces and tunnels rather than convex polyhedral (see Table 1). In that case, it is not possible to generalize the pattern ‘V−E + F = 2’ from what is observed. Certainly, this statement is theoretically possible. However, to reply to this concern about observation and induction, Wittgenstein might respond that this situation will never occur in reality, as there is a high possibility that most people would start to observe from similar instances. To illustrate: if we conduct a survey either among mathematicians or among laypeople by asking them to draw 10 different polyhedra, it is highly probable that most of them would come up with polyhedra included in Table 1; very few people will answer with anomalies like ring-shaped faces or tunnels. If this assumption is correct, one must admit that polyhedra in Beta’s table are not randomly chosen. The fact that most people naturally associate the polyhedra included in Table 1 with the word “polyhedron” is based on our behavioral regularities, the common understanding of the concept of “polyhedron” that most people share. Therefore, although the pattern Beta obtains from induction turns out to have a high probability, this high probability is not contingent or “atheoretical”, in Lakatos’ sense. Instead, the pattern can be explained by appeal to the similarity of our behaviors and our understanding of normal polyhedra.
Lakatos might object that the problem of Euler’s formula is not as primitive as the problem of ‘what is 3 + 2?’, and thus there might not be a behavioral regularity in discovering a pattern from the observation of 10 polyhedra that is convincing enough. Fortunately, this doubt can be resolved by Steiner’s emphasis on “empirical regularities”. If Steiner’s reading is right, Wittgenstein would argue that most people probably will conduct the same induction as Beta does because we share a common understanding of “regular polyhedron”, and thereby selecting the same polyhedra to begin the observation. More importantly, the reason we start our observation with similar polyhedra (ex. those in Table 1) is that they are easily found in our daily life and geometric practices. Very few people would deny that in our everyday life, we experience more physical objects like cubes and triangular prisms than objects with tunnel-like shapes. The frequent appearance of basic polyhedra in the empirical world leads to the fact that these are the shapes that mathematicians first observe when searching for a universal pattern for polyhedra. In other words, the ultimate explanation of our behavioral regularities, is not the social norms that Bloor emphasizes, but the empirical regularities that characterize our experiences. Careful examination of Wittgenstein’s writing shows that Bloor’s reading is not wrong, but that Steiner and Pérez-Escobar’s interpretations more accurately represent Wittgenstein’s view. In RFM, Wittgenstein repeatedly emphasizes the importance of distinguishing proof from experiment. Wittgenstein points out that a mathematical proof is not an experiment but the “picture of experiment” by using an instance of laying out marbles. Wittgenstein states that when we arrange 100 marbles in the 10 × 10 grid pattern, the process of arranging is an experiment. But if we “film” this process, the picture of this experiment is not itself an experiment; rather, it is a proof that the possibility of a 10 × 10 arrangement is a property of 100 marbles. Wittgenstein’s remarks here supports Steiner’s reading. In this marble example, and other examples involving observations of objects, Wittgenstein stresses that primitive mathematics is established on the way humans encounter and observe the empirical world. Therefore, neither human behavior nor the empirical world alone should be seen as the reason why we have mathematics as such rather than an alternative mathematics. Certainly, once some basic mathematical rules are proved to be effective and trustworthy, advanced mathematical knowledge can be built on these rules without being tested by empirical experiments. In this sense, Wittgenstein’s position on how humans obtain primitive mathematical conjectures and proofs has an empiricism flavor.
I argue that if Lakatos adopts Wittgenstein’s viewpoints discussed above, his concern that induction contradicts the rationality and objectivity of mathematics would be assuaged, and his quasi-empiricism would become more plausible. From the analysis above, we find that Wittgenstein’s stance is neither Euclidean nor empiricist. In this sense, Wittgenstein’s and Lakatos’ positions are similar. We can see that Lakatos’ rejection of induction in the conjecturing phase is not supported by mathematical practice, which undermines his quasi-empirical thesis. Wittgenstein’s idea yields a way for Lakatos to treat the role of induction in the conjecturing phase and the proving-phase differently, that is, using induction to find primitive conjectures and using deductive methods to prove them. Although Polya’s thesis distinguishing demonstrative reasoning from plausible reasoning also illustrates this solution, he does not offer a theory to ease Lakatos’ concern that induction can be conducted arbitrarily. However, Wittgenstein’s concept of “empirical regularities” might work to convince Lakatos that there is a low possibility that induction will yield meaningless patterns with high probability. The fact that mathematicians use induction even in solving very profound mathematical problems demonstrates the integrity of induction. If Lakatos’ view is that mathematics is fallible, then he should accept that the objectivity in mathematics is not absolute, but something constantly improving. In general, as long as we use the proof-and-refutation approach to conjectures, Lakatos has no reason to insist on his rejection of induction. Similarly, Pérez-Escobar (2022) suggests that Wittgenstein’s philosophy of mathematics can solve another anxiety of Lakatos’, namely, the existence of counterexamples. He argues that if we apply Wittgenstein’s view that “mathematics as rules of descriptions of the world” to Lakatos’ presentation of Euler’s conjecture, we could agree that hardened empirical regularities based on observation lead to the “relative resilience” of Euler’s conjecture. This “relative resilience” in mathematical concepts makes mathematics immune to counterexamples.
9 Final Remarks
In this paper, I have examined Lakatos’ quasi-empiricism with a focus on how mathematicians generate meaningful primitive conjectures. I argued that, for Lakatos, induction plays no role in the conjecturing phase, and I showed that this position contradicts mathematical practices. Influenced by Popper, Lakatos holds that empirical data in mathematics can only function in rejections of a conjecture, not in the construction of one. Lakatos’ hostility to the idea that a primitive conjecture can be obtained by induction from observation is caused by his concern that observation and induction can be arbitrary, that focusing on probability might lead to patterns with little mathematical significance. However, ruling out induction in the discovery of conjectures ignores mathematical practice, which contradicts Lakatos’ practice-focused approach to the philosophy of mathematics, and therefore undermines his quasi-empiricism.
I considered Wittgenstein’s suggestion that our observations and inductive behaviors are objective, not arbitrary. Bloor interprets Wittgenstein’s objectivity of our mathematical knowledge as sociological. Steiner’s more plausible interpretation develops Bloor’s by pointing out that for Wittgenstein, the behavioral regularities in mathematics are rooted in physical regularities, and so not merely sociological. In other words, although mathematical rules are grammatical, these rules do have empirical roots which should not be ignored. In this sense, Wittgenstein’s view provides a solution for Lakatos’ concern about induction: in order to maintain rationality and objectivity when generating conjectures, Lakatos does not need to reject induction.
Bloor, D. 1976. Knowledge and Social Imagery. Chicago: The University of Chicago Press.Search in Google Scholar
Corfield, D. 1997. “Assaying Lakatos’s Philosophy of Mathematics.” Studies in History and Philosophy of Science, Part A 28 (1): 99–121, https://doi.org/10.1016/s0039-3681(96)00002-7.Search in Google Scholar
Fogelin, R. J. 1987. Wittgenstein, 2nd ed. (The First edition was published in 1976). London: Routledge & Kegan Paul.Search in Google Scholar
Koetsier, T. 1991. Lakatos’ Philosophy of Mathematics: A Historical Approach. North-Holland: Elsevier Science Publishers BV.Search in Google Scholar
Lakatos, I. 1976. Proofs and Refutations: The Logic of Mathematical Discovery, edited by J. Worrall, and G. Currie, 6–89. Cambridge: Cambridge University Press.10.1017/CBO9781139171472.003Search in Google Scholar
Lakatos, I. 1978a. “Introduction.” In The Methodology of Scientific Research Programmes (Philosophical Papers Vol. I), edited by J. Worrall, and G. Currie, 1–7. Cambridge: Cambridge University Press.10.1007/978-3-030-97095-6_1Search in Google Scholar
Lakatos, I. 1978b. In Mathematics, Science and Epistemology (Philosophical Papers Vol. II), edited by J. Worrall, and G. Currie, 1–7. Cambridge: Cambridge University Press.Search in Google Scholar
Larvor, B. 1998. Lakatos: An Introduction. London: Routledge.Search in Google Scholar
Mormann, T. 2002. “Towards an Evolutionary Account of Conceptual Change in Mathematics: Proofs and Refutations and the Axiomatic Variation of Concepts.” In Appraising Lakatos: Mathematics, Methodology and the Man, edited by G. Kampis, L. Kvasz, and M. Stöltzner, 139–56. Hungary: Kluwer Academic Publishers.10.1007/978-94-017-0769-5_9Search in Google Scholar
Pérez-Escobar, J. A. 2022. “Showing Mathematical Flies the Way Out of Foundational Bottles: The Later Wittgenstein as a Forerunner of Lakatos and the Philosophy of Mathematical Practice.” KRITERION–Journal of Philosophy 2022, https://doi.org/10.1515/krt-2021-0041.Search in Google Scholar
Wittgenstein, L. 1956. “III, IV, VI.” In Remarks on the Foundation of Mathematics, edited by G. H. von Wright, R. Rhees, and GEM Anscombe, 1st ed. 143–254, 303–53. Oxford: Basil Blackwell.Search in Google Scholar
Wittgenstein, L. 1958. Philosophical Investigation, 2nd ed. (The first edition was published in 1953). Basil Blackwell Ltd.Search in Google Scholar
Wittgenstein, L. 1976. Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge, 1939, The Lecture X., edited by C. Diamond, 92–102. Cornell University Press.Search in Google Scholar
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