Fallibilism and Formality

Let us now turn to another aspect of Lakatos’s philosophy: fallibilism about mathematical knowledge. In the introduction, Lakatos presents his work as a new step in the battle for certainty:

In this great debate [between sceptics and dogmatists], in which arguments are time and again brought up to date, mathematics has been the proud fortress of dogmatism. Whenever mathemat- ical dogmatism of the day got into a ‘crisis’, a new version once

again provided genuine rigour and foundations, thereby restor- ing the image of authoritative, infallible, irrefutable mathematics [...] Most sceptics resigned themselves to the impregnability of this stronghold of dogmatist epistemology. A challenge is now overdue. (Lakatos 1976, p. 5)

The dogmatist position in mathematics which is Lakatos’s (indirect) target is that of theformalist, or in keeping with our terminology, theFormalist- Reductionist. As always, the Formalist-Reductionist takes pieces of knowl- edge of mathematical propositions to be certain and infallible as a result of formal, logical proofs. The advent of modern logic brings with it the ‘fortress’ of dogmatism, settling what it is for a proof to be immune to doubt, error and leaps of reasoning. Closely related to the Formalist-Reductionist position is the deductivist approach, discussed in the second appendix if

Proofs & Refutations.

In deductivist style, all propositions are true and all inferences are valid. Mathematics is presented as an ever-increasing set of eternal, immutable truths. Counterexamples, refutations, criti- cism cannot possibly enter. An authoritarian air is secured for the subject with disguised monster-barring and proof-generated definitions and with the fully-fledged theorem, and by suppress- ing the primitive conjecture, the refutations, and the criticism of the proof. Deductivist style hides the struggle, hides the adven- ture. (Lakatos 1976, p. 142)

The deductivist style thus also encompasses a presentational style of defin- ing concepts, setting out theorems and providing proofs, which separates these from the dialectical environment in which they were created. Indeed, Lakatos suggests that mathematics would be greatly improved if we pre- sented mathematics in the heuristic style which makes explicit the growth of the theorem, proof and concepts involved.

The challenges to the Formalist-Reductionist, the deductivist and the dogmatist, then, come from presenting an alternative picture of mathemat- ics and its development. By making the dialectical logic of mathematical discovery clear, it demonstrates that mathematics cannot be equated with its formal shadow, that mathematical concepts change and emerge from mathe- matical practices, and that the status of mathematical theorems will change

as they and the concepts they feature change. In a way, this deflates some of the radicalism that talk of fallibilism—and indeed scepticism— might sug- gest, for the sense in which mathematical statements are fallible is merely as the by-product of the idea that concepts and theories are changeable through dialectical logic. Meanwhile, this doesn’t cast doubt in a more radical sense, as Larvor puts it:

[I]t has been and will remain the case that an apple taken to- gether with two oranges makes three pieces of fruit. (Larvor 1998, p. 36)

The point is more subtle than the radical sense of fallibilism. Larvor con- tinues:

It is also the case that an apple released in mid-air will fall to earth. Nevertheless, in both cases the theoretical apparatus we use to describe and to account for the phenomenon is highly complex and open to criticism. (Larvor 1998, p. 36)

Ultimately, then, the sense of fallibilism which we find in Lakatos is about the fact that we don’t find certainty in having established some mathemat- ical theory because that theory will always been open to potential change, revision and development in light of new counterexamples, new ideas and new mathematics. One might even see the major part of the project as being the attempt to show that this does not collapse into pure subjectivity and that there are good mathematical criteria for these kinds of dialectical changes.

Moving on for the moment, what is of particular interest to consider for our purposes is the distinction between formal and informal proofs in Lakatos’s picture. In arguing against ‘formalism’ and ‘deductivism’ so widely, there is a point of view which is natural to Lakatos: that informal proofs are the central method of mathematical demonstration, with their own associ- ated notion of rigour operative in differentiating correct and incorrect proofs. While at time controversies do arise, these are beneficial for mathematics in that they drive the development of proofs and concepts, as discussed above. Conversely, formal proofs and the purely deductive picture of mathematics cannot underlie the rigorousness of proofs as they appear in mathematics nor can they account for mathematical discovery. In fact, in the second

chapter ofProofs & Refutations Lakatos has the student Epsilon go through a separate proof of the Euler conjecture in the ‘Euclidean’ style, where the theorem and concepts are translated into the language of vector algebra:

I analysed our concepts of polyhedra and showed that they are

reallyvector algebraic concepts. I translated the circle of ideas of the Euler-phenomenon into vector algebra, thus displaying their essence. Now I am certainly proving a theorem in vector algebra, which is a clear and distinct theory with perfectly known terms, neat and indubitable axioms, and with neat, indubitable proofs. —Epsilon (Lakatos 1976, p. 118)

The class raise several problems for this line of thinking. Firstly there are the standard worries of whether the translation fully captures the informal concepts they are formalising and whether the ‘certain’ system is guaranteed to be consistent.15 This, of course, relates closely to the discussions of chapters 1 and 2. Moreover, though, the Lakatosian view seems to be that while such a translation does successfully limit the counterexamples that may appear within the theory, it also loses out on much which we had before the translation:

[Y]ou may push out the original problem into the limbo of the history of thought— which in fact you do not want to do. (foot- note: This process is very characteristic of twentieth-century for- malism.) —Alpha (Lakatos 1976, p. 122)

and

Epsilon wanted, “in virtue of a series of startling definitions to save mathematics from the sceptics”, but what he saved was at best some crumbs. —Gamma (Lakatos 1976, p. 123)

The standard idea in this direction, then, is that we can formalise any math- ematics we choose, but formality is balanced against meaning and so fully formalising leads to theories devoid of meaning, pushing content into the meta-mathematical interpretation.

15_{Indeed, Epsilon admits that they must}

[...] forget about the old meaning. I create freely the meaning of my terms while scrapping old vague terms. —Epsilon(Lakatos 1976, p. 122)

There is some dispute over where Lakatos stands on this exactly, though. On the one hand, there are Davis & Hersh (Davis & Hersh 1981, pp. 345– 359) and Larvor in (Larvor 1998, pp. 33-34), offering a reading much like the above, but on the other hand there are Worrall & Zahar, the editors of Proofs and Refutations, who insert several substantial footnoted com- ments into the book offering an alternative reading, backed up by Corfield in (Corfield 1997) who defends their interpretation of Lakatos. In the in- serted footnotes, Worrall & Zahar suggest that Lakatos is mistaken about the need to reject the infallibilist idea that “deductive, inferential intuition is infallible” (Lakatos 1976, p. 138), writing in their footnote that:

This passage seems to us mistaken and we have no doubt that Lakatos, who came to have the highest regard for formal deduc- tive logic, would himself have changed it. First order logic has arrived at a characterisation of the validity of an inference which [...] does make valid inference essentially infallible. (Lakatos 1976, fn. 4, p. 138)

Both Davis & Hersh and Larvor see Worrall & Zahar as making mistakes of the precise sort that Lakatos is arguing strongly against. Firstly, Davis & Hersh argue that the mistake is one of conflating mathematical proof with its formal representation as a derivation in some fixed formal system. In essence, they accuse Worrall & Zahar of Formalist-Reductionist thinking, and reject it for several of the classical reasons. Secondly, Larvor argues that their mistake is to focus on language-statics rather than language-dynamics. That is, the fallibility is not of the logical validity in some given system (say, first-order logic) but rather that the counterexamples will be heuristic in a way that might lead to change or abandonment of the system altogether.

Corfield sets out a defence of Worrall & Zahar against Davis & Hersh’s criticism, trying to show that Lakatos was more conservative than they appreciated. The claim is that while they are correct that Lakatos focuses on informal mathematics, and that most mathematics found in practice is informal, in fact Lakatos made a distinction between different levels of informality. The difference lies between informal proofs proper and ‘quasi- formal’ proofs which really are formal proofs with some of the interim steps left out or supressed. Corfield then says:

informal mathematics extends to present day proofs in estab- lished branches of mathematics, surely the majority of the es- timated 200,000 produced each year, Lakatos himself wishes to count them as ‘almost formal’ or ‘quasi-formal’. (Corfield 1997, p. 115)

Furthermore, the point is that formal systems, logic and axiomatics do play a major role in modern mathematics and that even in these cases the de- velopment of new concepts does not cease, but instead works alongside the axioms. With respect to axiomatics, their role in mathematical discovery is brought out particularly clearly by Schlimm in his case study of lattice theory in (Schlimm 2011).

In general, I think that Corfield’s reading, along with that of Worrall & Zahar, cannot be right. The claim that most modern proofs are quasi- formal rather than informal, and as such are not subject to the Lakatosian arguments, strikes me as entirely wrong. Indeed, the central argument of my first chapter, that of the over-generation of formalisations relative to some given informal proof, stands strongly against such a view. The Formalist- Reductionist line seems to hold that the modern work in formal mathematics supports their view of proofs, as is made explicit by Corfield:

Given the stabilization that has occurred in the idea of what constitutes a rigorous proof, this gives them [Davis & Hersh] little room for manoeuvre against their formalist adversaries. (Corfield 1997, p. 117)

I think the advantage of my argument against such a view is precisely that it draws so directly on work in formalisation, showing that the idea that modern proofs are no longer informal in a strong sense is false, but also delivering the fact that formal mathematics projects are no help in defending against the criticisms of the informalist camp. In addition to this argument, it seems wholly unlikely that the Formalist-Reductionist interpretation of Lakatos is correct when read in the context of the rest of the text. For instance, the opening passage quoted above on the constant retreat of the dogmatist in the face of sceptical challenges, and it being time to finally storm the stronghold of dogmatism in mathematics, makes for a poor fit with the acceptance that actually most modern mathematics is suitably infallible by virtue of being quasi-formal.

Despite believing that Corfield is totally mistaken in following the Wor- rall & Zahar interpretation, I do think there is a point to his argument that cannot be ignored. That point is that the Lakatosian framework does not fit well with the methodology of modern mathematics. Axiomatisations, formalisms and metamathematical results are used alongside traditional, in- formal proving; the relationship between syntactic proofs and associated model theory is highly complex and fruitful; reverse mathematics offers gen- uine insights into the provability strength of mathematical statements; and computational mathematics is developing at a phenomenal rate. The point is that while I have been stressing the importance of taking informal math- ematics and proof seriously, this has to be taken alongside formal methods rather than instead of them. I think Corfield is right to stress that the strict dichotomy found in the Lakatosian picture between fruitful and con- tentful informal mathematics and the sterile and static formal theories does not fully do justice to modern mathematical practice. I will come back to this final point shortly as a motivation for investigating whether other work towards conceptual development, conceptual change and conceptual engi- neering might be fruitfully applied to the case of mathematics. Before then, let us consider another dialectical proposal for mathematics: that of G. T. Kneebone.