The most important thing: Wittgenstein, engineering, and the foundations of mathematics

ABSTRACT

This paper revisits Wittgenstein’s heavily criticized claims about the admissibility of inconsistencies in mathematics. It argues from the perspective of mathematics as a tool and combines material from the history and practice of engineering that makes Wittgenstein’s claims about contradiction and inconsistency look much more plausible. Against this background, the paper interprets passages from Wittgenstein, including his exchange with Alan Turing where he highlights that basic laws of thought are at issue and that reflecting on them would be “the most important thing” he has talked about.

KEYWORDS:

• Anti-maths movement
• engineering
• inconsistency
• practice

1. Introduction

Ludwig Wittgenstein’s (1889–1951) philosophy of mathematics presents a conundrum of extraordinary charm. This paper focuses on the passages on inconsistency.¹ Apparently, Wittgenstein set out to argue that contradictions might happen in mathematics without harm to the foundations and without harm to using mathematics. This baffled even his most ardent followers. Michael Dummett (Wittgenstein’s Philosophy of Mathematics), for instance, deemed these passages of “poor quality” (166). Crispin Wright, who made a thorough attempt to come to grips with Wittgenstein’s views on the foundation of mathematics, excluded the discussion of inconsistency: “Here the impression is not so much that of ordinary attitudes or assumptions questioned, as of good sense outraged” (Wright, Wittgenstein Foundations Mathematics, 295). In short, Wittgenstein proposed a liberal view toward inconsistency in mathematics that most readers, from mathematics as well as philosophy, take as simply untenable.²

Recent work in both history and philosophy reassesses the case and develops a friendly reading of Wittgenstein.³ The present paper ties in with this work and brings to the case a new perspective, combining material from the history: and practice of engineering that makes Wittgenstein’s claims about contradiction and inconsistency look much more plausible.

When Wittgenstein set out to study engineering at TU Charlottenburg (Berlin) it was one of the leading technical universities of its time and also a place where engineers debated a new conception of mathematization. Section 2 reconstructs the debate between Alois Riedler (1850–1936), then the leading engineer at Berlin, and Felix Klein (1849–1925). Over their controversy, Riedler made key contributions to the ‘anti-maths movement’ in German engineering that shook the academic landscape in the outgoing nineteenth century. Contrary to what one would expect from the name of the controversy, Riedler and fellow engineers did not oppose mathematization, but promoted a different conception of mathematization for engineering that sees mathematics as a tool. This conception was firmly established when Wittgenstein studied at Berlin.

Section 3 fosters the point and discusses two examples where engineering works with models (or calculus) of questionable consistency, if not outright inconsistency. One is contemporary to Wittgenstein, namely Oliver Heaviside’s ‘operational calculus’ that brought great progress in electrical engineering, but was met with hostility in the Royal Society because of missing rigour in mathematics. The second example is a recent one from engineering thermodynamics, in particular from the field of fluid mixtures. The common model shows inconsistencies whereas the rules for how to use the model make sure that the inconsistencies do not prohibit working with the model.

On the basis of Sections 2 and 3, the paper offers a friendly reading of Wittgenstein on contradiction and inconsistency in mathematics, or rather in mathematical calculus.⁴ Section 4 discusses seminal passages in Wittgenstein’s work, including the geographical swamp metaphor according to which users of a mathematical model do not need a proof of inconsistency but can avoid existing inconsistencies like hikers can avoid a swamp when they know a (reliable) bypass. According to Wittgenstein, operating in a model must be reliable to such high degree that miscalculation, that means not following the rules, can be detected without ambiguity. This criterion does not require consistency, let alone proof of consistency. Viewing the foundations of mathematics from this perspective is in stark contrast to the common philosophical discussion, but in good agreement with the perspective from engineering.

Section 5 consolidates the interpretation by focusing on Wittgenstein’s exchange with Alan Turing (1912–1954) which happened during Wittgenstein’s 1939 lectures on the foundation of mathematics. At a crucial point in Wittgenstein’s examination of inconsistency, Turing famously remarks that the danger of inconsistency will become obvious when a bridge constructed with mathematical methods will fall down. Wittgenstein concedes the relevance of Turing’s remark but insists that the question of whether the bridge stands or not is not a question of proven mathematical consistency. He highlights that basic laws of thought are at issue and that reflecting on them would be “the most important thing” (Diamond, Wittgenstein's Lectures, 235). In fact, based on what was introduced in Chapter 3, I argue that Wittgenstein brought home much more of his claims than the extant literature acknowledges.⁵

Finally, the concluding Section 6 reflects on what the examination of mathematics and inconsistency can tell philosophy. Much of philosophy of mathematics defends a conception of mathematics that leans toward logic.⁶ Wittgenstein, in contrast to his earlier position in the Tractatus, approaches the foundations of mathematics in a different way, from the perspective of mathematics as a tool. This angle illuminates the conception of rationality and what we think of as ‘laws of thought’. Arguably for this reason, Wittgenstein remarked that – if he succeeds – this will be “the most important thing” he has talked about (Diamond, Wittgenstein’s Lectures, 235).

2. The anti-mathematicians movement in German engineering around 1900

Wittgenstein studied mechanical engineering from 1906 to 1908 at TU Charlottenburg, Berlin, before he moved on to Manchester, UK, where he followed his interests in aeronautics and worked at the Glossop Station that developed and used kites for meteorological research. It is widely acknowledged that Wittgenstein had a deep and life-long interest in mechanical engineering.⁷ Furthermore, engineering played an important role in Wittgenstein’s philosophy. Alfred Nordmann, for instance, argues that Wittgenstein always remained an engineer, not because he philosophized about problems in engineering, but because the engineering way of dealing with difficulties and obscurities shaped his way of dealing with philosophical problems (Nordmann, “Another New Wittgenstein”, 358). Typically, the late Wittgenstein deconstructs a philosophy that is oriented at truth, first causes, certainty, and the ‘real’ representation of the world. In contrast, he cast his language games more in terms of the workings or failures of a machine (Kroß, “Engineering Phenomena”).⁸ The question is, whether Wittgenstein’s background in engineering could inform him about the foundations of mathematics. According to Brian McGuinness, problems dealing with the foundations were “ … unconnected with his technical concerns as an engineer; … ” (McGuinness, Wittgenstein: A Life, 76). Others, like Mark Wilson, are more open, but when they reason about the relation between mathematics and engineering, they all look at Franz Reuleaux’s work as the major influence on Wittgenstein (see, e.g. Wilson, “Wittgenstein: Physica Sunt”, 290). Indeed, Reuleaux was among the most prominent German engineers in the 1870s and 1880s. He held a professorship at Berlin and elaborated a rational stance in his famous “Kinematics of Machinery” (1876, original German title: Theorie der Kinematik, 1875). Reuleaux intended “to make the science of machinery deductive” (Kinematics, 22) by taking mathematical axiomatics as a methodological paradigm. In this way, engineering should develop into a mathematized science.

I argue that the reference to Reuleaux is misleading because Wittgenstein, the student at TU Charlottenburg, experienced a different mindset concerning the relationship between engineering and mathematics. This was because Alois Riedler had become the leading engineer in Berlin. He had frozen out Reuleaux from teaching and had established a concept of mathematization that almost reversed Reuleaux's.⁹ Thus, Reuleaux is the wrong lens when evaluating how Wittgenstein’s engineering background may have influenced his claims on the foundation of mathematics. The appropriate lens is Riedler’s viewpoint.

Riedler was a politically well-connected campaigner for the academic emancipation of engineering and technical institutes. Two landmark achievements on the side of engineering were the introduction of engineering laboratories, which opened engineering methodology to systematic experimentation, and, second, the right to award doctorates that the German Emperor granted in 1899. For both, Riedler and Charlottenburg played a role as trailblazer. Riedler is of particular significance in the context of the present paper because of his stance on mathematics. This stance elucidates best from the ‘Anti-Math Movement’ (Anti-Mathematikerbewegung der Technikwissenschaften) that erupted when the 1895 General Assembly of German Engineers debated whether mathematicians should be thrown out of technical institutes because they had a view on mathematics inadequate for engineers. The movement sent shock waves through the academic system. After all, it touched upon questions of academic hierarchy. While Riedler was a ringleader in the Movement of the 1890s, he did not oppose mathematics per se, rather called for a new conception of mathematization.¹⁰ According to this conception, the goal of mathematization was “not theoretical consistency, but practical usability” (König, “Technische Hochschule Berlin”, 21).¹¹ This viewpoint, elaborated in stark opposition to the contemporary mathematics education at technical universities, resonates with Wittgenstein’s claims on inconsistency.

A good way to introduce Riedler’s stance on mathematics is by looking at his controversy with Felix Klein. Both were leaders in their respective disciplines; both were convinced that new technologies would call for a new conception of mathematization; both were pushing new initiatives of teaching and research, but they came to opposite conclusions. The role of mathematics in engineering science marks the core of their disagreement. Klein defended the viewpoint that mathematics is a foundational science and, consequently, engineering is scientific to the degree it is subsumed under mathematized applied science, much in line with Reuleaux. Riedler granted the opposite: mathematization is a good and useful thing only as much as it fosters the autonomy of engineering, founded in using mathematics as a tool.

Felix Klein was a geometer who studied in Bonn and Berlin and held professorships in Erlangen, TH München, Leipzig, and (from 1886 onward) Göttingen. In mathematics he is still known for his 1872 “Erlangen program” to characterize spaces based on geometry and group theory. Later in his career he organized the rise of Göttingen to a world-leading centre of mathematics and mathematical physics (in the early twentieth century). Klein was deeply involved in teaching and in reasoning about mathematical education. The nineteenth century had seen the rise of pure mathematics to the forefront in nearly all quarters of math research. Klein fully appreciated this fact that opened many new avenues for the discipline (not least pursued in Göttingen), but he also acknowledged the relevance of applied mathematics, since the prominence of mathematics (as a discipline in the university) rested on their significance for the sciences. Klein realized that technology was posing a new challenge: it demanded accurate predictions in increasingly complex circumstances where traditional rational mechanics turned out to be rather helpless.

Klein perceived this as an opportunity to increase the relevance of mathematics. Applied mathematics should be developed to serve as backbone for applied science, including mechanical engineering. According to Klein, pure and applied mathematics share the basic rationality – natural laws are described as mathematical functions – but additional conditions of technology have to be considered in the applied case. How one should do this, Klein held, called for a mathematical research programme. Klein’s strategy was to embed applied mathematics into a broader appreciation of pure science and mathematics. When proposing his vision, Klein usually addressed an audience of mathematicians and physicists where he faced opposition against his appreciation of applied science (see also Manegold, Universität). For many engineers, however, such embedding looked like a colonization of their discipline through mathematics.

Klein wrote a memorandum on a new institute for ‘technical physics’ in spring 1895 which attracted immediate attention and negative reaction from the side of engineers. In a series of further talks, both to mathematical and engineering audiences, Klein tried to show his plans were in the good interest of every party, including the engineers. These talks were met with criticism, often expressed in direct commentaries to the publication. Klein’s talk to engineers on “Demand of engineers and education of mathematics teachers”, for example, appeared 1896 in ZVDI (the publication journal of German engineers) with harsh comments by Riedler. Where Klein argued for integration into modern mathematics, Riedler called for separation.

Klein did not only design a new institution, but also argued about mathematics itself, in particular about the mathematization of approximation techniques. Throughout his work, he was “always underlining the relationship between the exactness of the idealized concepts and the approximations to be considered in application” (Menghini, “Precision and Approximation”, 181). Overall, Klein stressed the unity of science and a sort of top-down mathematization. In his view, this step would elevate engineering mechanics to the rational mode of science. Klein did not find approval from the side of engineers. Riedler, Klein’s most outspoken counterpart on the side of the engineers, reversed emphasis and advocated the autonomy of (engineering) science together with what can be called bottom-up mathematization.

Alois Riedler studied machine construction in Graz and Brünn before he held professorships at TH München, TH Aachen, and (since 1888) TH Berlin Charlottenburg where he also acted as president (‘Rektor’) 1899–1900. He was a specialist in pumps and motors, making a fortune with patents on high-performance pumps. Riedler was also a prolific publisher on engineering education, including his (“Zur Frage der Ingenieurerziehung”) which tailored for the Aachen assembly of engineers. Apparently, Riedler rarely avoided controversy. He made Reuleaux retire and then threw kinematics (Reuleaux’s pride) out of the curriculum. At the same time, Riedler hammered home in every publication that mathematics “is an indispensable basic tool, but not itself foundation” (“unerlaessliches Grundwerkzeug, aber nicht Grundlage selbst”) (Riedler, “Die Ziele”, 305). Conversely, according to Riedler, engineering science is not defined by mathematics (as Klein suggested), rather by the way it uses mathematical tools for solving engineering problems.

For Riedler, connecting to some rational scientific structure was irrelevant when it meant following the contemporary trends of arithmetization and rigour; instead he underlined the flexibility of mathematics, in construction (graphical methods) as well as in prediction (parameterization). One could use coefficients and parameters whose determination was left open and filled in later by measurement. To Riedler, such empirical ingredients were no threat to rationality – on the contrary, rationality lied in efficient determination of coefficients. The engineer would formulate a problem whose efficient solution requires mathematical and scientific means. Therefore, technical knowledge is on a higher level than mathematical (scientific) knowledge (Riedler, “Technische Hochschulen”, 5).

Thus, for Riedler, mathematization amounts to seeing engineering and science from a methodological point of view, i.e. as a tool for tackling engineering problems – not as an entry point into the rational structure of nature. The most important goal for engineering is designing a device or organizing a system of them so that they are “efficient in industrial practice” (betriebsbrauchbar) (Riedler, Unsere Hochschulen, 115). Riedler saw that the complex conditions of modern technology required experimentation (in engineering laboratories) as part of finding an adequate formulation of problems.

In sum, Riedler proposed mathematization, but along a very different line from Klein. Not as a link to the broader theoretical structures of science, but as a flexible tool that follows the needs of the problems at hand. If one puts practical problems at the bottom, this kind of mathematization can be called bottom-up. Since mathematization then is guided by engineering problems, while the question of unity or overarching theoretical framework remains open, this perspective on mathematization goes well with the autonomy of the technical (or engineering) sciences.

During Wittgenstein’s student years in Berlin, Riedler was at the height of his influence.¹² Riedler’s point of view escaped interpreters of Wittgenstein’s. However, this perspective sheds some new light on Wittgenstein’s take on the foundations of mathematics. Although it is unknown how much the experience with engineering and the bottom-up conception of mathematization actually influenced Wittgenstein’s standpoint, it is a starting point quite different from the logic endeavours that Frege and Russell pioneered.

3. Two examples in inconsistency

Admittedly, in the context of the present paper, the perspective on the use of mathematics is relevant only to the extent this use involves inconsistency.¹³ This section examines two examples of mathematical tools whose success is not plagued by the inconsistencies they entail.

3.1. Operational calculus

The first example is from an electrical engineering contemporary to Wittgenstein. Oliver Heaviside (1850–1925) had an outsider career in England; he started to work as a telegraphist and electrician while he taught himself mathematics, physics, and electrical engineering.¹⁴ James Clerk Maxwell is famous for his theory of electrodynamics, often referred to as ‘Maxwell’s equations’. However, it was Heaviside who formulated these four equations (Hunt, The Mawellians, 245–7), attempting to cast Maxwell’s theory into a practically useful form, i.e. the vector notation invented by Heaviside. His main field of work was long-distance telegraphy, exemplified by the then spectacularly new and challenging Atlantic cable. Even though the Maxwell equations are relevant and valid (an insight that Heaviside pioneered), actually predicting the very exact form of how a signal propagates in an imperfectly insulated and very long cable poses formidably complex mathematical problems.

Heaviside developed his ‘operational calculus’ as the right tool for solving these problems. In principle, the concept of an operator was already well established. A seminal example is the Laplace transform that translates analytical equations into algebraic problems (and vice versa). However, Heaviside’s calculus is based on a fractional expansion of a divergent series – something that mathematical doctrine does not accept as a well-defined object. Working with it therefore does not guarantee staying consistent with the assumptions one started with, i.e. the equations derived from Maxwell’s theory.

Importantly, Heaviside was not concerned with this demand for rigour-based consistency. Instead, he saw mathematics as an experimental tool. In a way, Heaviside treated his calculus like a rule with some adjustable parameters. After adjusting parameters to observed cases, his calculus would be able to make predictions. And the very fact that useful predictions come out of it made further concerns about consistency dispensable.

That procedure combines mathematization with experimentation and is akin to the bottom-up conception discussed in Section 2. This brought Heaviside fierce opposition from mathematicians and physicists who took rigorous definitions as indispensable parts of mathematization.¹⁵ Although Heaviside had become a fellow of the Royal Society, he struggled for recognition by the scientific establishment. His 1892/3 paper series “On operators in physical mathematics” that included years of experimentation with operators abruptly ended after Part II because the Royal Society refused to publish more. Although papers by fellows were normally not reviewed at all, the mathematicians objected against Heaviside’s operational calculus and commissioned a review of Part III from the mathematician William Burnside. He did not take into account whether the results of the calculus worked, rather stated: “Detailed criticism of results obtained in this way seems out of place. They may or may not be true, but the way in which they are arrived at makes them absolutely valueless” (cited according to Nahin, Oliver Heaviside, 224). In other words, missing rigour puts consistency into doubt and results whose consistency is not certain in advance are automatically valueless. Heaviside, in contrast, held that

Mathematics is an experimental science, and definitions do not come first, but later on. They make themselves when the nature of the subject has developed itself. It would be absurd to lay down the law beforehand.

(Nahin Oliver Heaviside, 222/223)

In short, Heaviside invested his energy in developing a calculus that was useful in practice but he did not much care about consistency.

3.2. Mixtures of liquids

The second example is from contemporary engineering thermodynamics and discusses a common way for modelling mixtures of liquids. Knowing the properties of substances is crucial for designing chemical processes and operating the machinery. There are large databases that record known properties, like viscosity at certain temperatures and pressures. However, the full space of values is nearly infinite, especially when one takes into account that substances can be mixed and mixtures separated. Thus, mathematical models are essential for the prediction of relevant properties. The following example introduces a common but inconsistent model.¹⁶ Modelling liquid mixtures is a demanding task, as the liquid state is neither unstructured such as the ideal gas nor fully structured such as an ideal crystal. For describing the structure in a liquid mixture, local concentrations are used.

What follows is a text that introduces a standard model in direct language as is usual for a student of thermodynamics. Assume that N₁ is the amount of particles of the component 1 and N₂ that of component 2 in a mixture. The overall concentrations of components 1 and 2 are: (1) $x_1=N_1/(N_1+N_2)$(1) and (2) $x_2=N_2/(N_1+N_2).$(2)

We are now interested what a particle of component 1 ‘experiences’ in the mixture. Think of sitting on a particle of component 1 and observing your vicinity. How many particles of component 1 or 2 will you find in the vicinity? This leads to the concept of local concentrations. If the central particle is of component 1, there is a local concentration of 2 around 1 (x₂₁) and a local concentration of 1 around 1 (x₁₁). As there are only two components, we have: (3) $x_{11}+x_{21}=1.$(3)

Similarly, if the central particle is of component 2, we have the local concentrations of 1 around 2 (x₁₂) and that of 2 around 2 (x₂₂), that are connected by (4) $x_{12}+x_{22}=1.$(4)

This is illustrated for a simple lattice model of a liquid in Figure 1.

Figure 1. Local concentrations in a lattice model where each particle has four next neighbours (left, right, up, down) for a binary mixture of components 1 and 2. Left: the central particle is of component 1, the local concentrations are x₂₁ = 3/4, i.e. three neighbours of type 2 among 4 neighbours overall, and x₁₁ = 1/4. Right: the central particle is of component 2, the local concentrations are x₁₂ = 2/4 and x₂₂ = 2/4. The overall concentrations are x₁ = 5/9 and x₂ = 4/9 in the respective cases.

In an unstructured fluid, the local concentrations are identical with the overall concentrations, i.e.: (5) $x_{11}=x_{12}=x_1$(5) and (6) $x_{12}=x_{22}=x_2.$(6)

In a lattice model this would be the case if the particles of component 1 and 2 were placed randomly on a large lattice and the average for all particles were taken. However, in real mixtures, there are preferences. Particles of component 1 may prefer being surrounded by their kin and the same might hold for the particles of component 2. This would lead to high values of x₁₁ and x₂₂ and low values of x₂₁ and x₁₂. On the other hand, particles of component 1 might find particles of component 2 highly attractive. This would lead to high values of x₁₂ and x₁₂.

The concept of local concentrations can be used for counting contacts. The number of 1 - 1 contacts is proportional to x₁ · x₁₁, the number of 2 - 2 contacts among particles is proportional to x₂ · x₂₂, and the number of 1 - 2 contacts is proportional to x₁ · x₂₁ or x₂ · x₁₂, which must be the same as you can look at the 1 - 2 contact from the two sides. Hence, the four local concentrations x₁₁, x₂₂, x₁₂, x₂₁ are not only connected by Equations (3) and (4) but also by a symmetry relation (7) $x_1\cdot x_{21}=x_2\cdot x_{12}.$(7)

Now, why do I bother you with all this? The probably most famous model for describing properties of liquid mixtures is the so-called NRTL model by Renon and Prausnitz (“Liquid Mixtures”). It is the workhorse for describing liquid mixtures in industry and still often used in academia.¹⁷ NRTL stands for Non-random two-liquid and it is a lattice model of the type described above. ‘Non-random’ refers to the fact that preferences are accounted for and ‘two-liquid’ refers to the fact that in a two-component mixture either particles of component 1 or component 2 can be considered as central particle. The NRTL-model describes the so-called Gibbs excess energy from which various thermodynamic properties can be calculated, such as the vapor–liquid equilibrium. This is used for designing separation processes such as distillation. As NRTL is the workhorse for such calculations, it can be assumed that a substantial share of the tens of thousands of distillation columns that are operative now has been designed using the NRTL model.

For describing a given mixture 1 + 2, NRTL must be trained using experimental data. But after having been trained using only a few data points, NRTL can then be used to make predictions. This includes making predictions for multicomponent systems based on a training on binary systems only. The wide acceptance that NRTL has found is owed to its success in such predictions. How is the NRTL model trained? The model has three parameters, usually labelled as α, τ₁₂ and τ ₂₁. The numbers of these parameters are adjusted to data for the binary system 1 + 2. In the derivation of the model τ₁₂ and τ₂₁ are related to the interaction energies of the three types of contacts 1 - 1, 2 - 2, and 1 - 2. Hence, preferences for certain types of contacts show up in the numbers that are found for the parameters τ₁₂ and τ₂₁. A central element in the NRTL model are the local concentrations, and the paper of Renon and Prausnitz gives the equations for x₁₂ and x₂₁ (Renon and Prausnitz, “One Phenomenon, Many Models”, Equations 13 and 14), from which x₁₁ and x₂₂ can easily be found from Equations (3) and (4). The results for the local concentrations depend on the values of the three NRTL parameters. However, it is found that the symmetry relation Equation (7) is fulfilled only for certain choices of the parameters. In other words, using the NRTL means following the rule not to look back at Equation (7). Using the parameters found from the adjustment to experimental data, leads to a violation of these requirements in all cases except for some lucky incidents. As a consequence, from the NRTL model, even though its derivation is based on local concentrations, no meaningful information on the local concentrations in liquid mixtures can be obtained. This has been known for a long time, and there is no reason to believe that Renon and Prausnitz were unaware of this fact. But they have found that the model was successful in describing data and, more importantly, it was successful in predicting data that was not used for the training. All this works well, despite the inconsistency in the derivation of the NRTL equation.

Practice demonstrates that students can write exams about the NRTL model and teaching assistants can reliably determine whether there is a mistake or not. This is in line with Wittgenstein’s idea that the very notion of calculation excludes confusion and implies repeatability. Furthermore, the examples show how a calculus (or model) might be used without being consistent. It must be clear when someone has calculated correctly and when not. The results of using a calculus have to be reliably the same – else the process does not count as calculation.¹⁸

4. A fresh look at Wittgenstein on inconsistency

The perspective from engineering, as laid out in Sections 2 and 3, is indeed a helpful preparation for reading Wittgenstein. He wrote down his controversial stance on contradiction and the foundations of mathematics in manuscript 177, in entries with chronological order that date over the first quarter of 1940 and that he often repeats in changed wording, showing the tentative character. These entries have been substantially shortened, edited and translated into English by von Wright, Rhees, and Anscombe and now build the final paragraphs 77–90 of part three in the Remarks on the Foundation of Mathematics (RFM), subtitled “contradiction” by the editors.¹⁹

My claim is that the confluence of two factors lends plausibility to Wittgenstein’s (in)famous passages. First, he had developed a stance on language games in the Philosophical Investigations. Importantly, he wanted to concentrate on describing practices, rather than explaining them by some rational reasoning. Wittgenstein’s verdict in RFM: “What we want is to describe, not to explain” (RFM, III, 78) indicates the guiding role of this first factor. Second, from this perspective, the mathematical practices in engineering (examined in Sections 2 and 3) become relevant for the foundations of mathematics. If these practices would entail inconsistencies, they would support a position in stark contrast to the mainstream viewpoint on the foundations of mathematics where consistency is a precondition, the main pillar without which the whole endeavour would not make sense. Wittgenstein is fully aware how provocative the approach from practice to the foundations is:

We shall see contradiction in a quite different light if we look at its occurrence and its consequences as it were anthropologically—and when we look at it with a mathematician’s exasperation. That is to say, we shall look at it differently, if we try merely to describe how the contradiction influences language-games, and if we look at it from the point of view of the mathematical law-giver.

(RFM, III, 87)²⁰

Wittgenstein knows that mathematical calculation might yield useful results even if the calculus is reliable but not consistent – as long as the user knows how to handle the calculus in a way that prevents the inconsistencies from becoming effective.

I should like to ask something like: ‘Is it usefulness you are out for in your calculus?—In that case you do not get any contradiction. And if you aren’t out for usefulness—then it doesn’t matter if you do get one.’

(RFM, III, 80)

Putting things this way sounds nonsensical if one does exclude the possibility that some calculus that actually is inconsistent might still be useful.

However, the examples from engineering discussed above suggest that one should not exclude this possibility; they present positive instances. Wittgenstein argues along a different pathway. He is committed to analysing the use of language related to a calculus, or rather to examine how inconsistency might play out. The first impulse of course is to see inconsistency as perilous. Since a contradiction implies anything, an inconsistent calculus must be useless – or so ex falso quodlibet has it. Wittgenstein certainly feels the challenge: How should one describe the situation in which inconsistency arguably is not harmful? Wittgenstein makes repeated attempts to employ the medical metaphor that a contradiction is like a local inflammation that can be contained and does not imply the whole body is ill.²¹ He also brings in a geographical metaphor according to which using a math calculus is like walking through a landscape. There might be swamps but when one knows how to avoid them, one can safely navigate. Therefore, finding a pathway does not require a proof that there is no swamp.

And if they [the mathematicians, jl] now demand a proof of consistency, because otherwise they would be in danger of falling into the bog at every step—what are they demanding? Well, they are demanding a kind of order. But was there no order before?

(RFM, III, 78)

Arguably, when Wittgenstein addresses “the mathematicians”, he also includes logicists like Russell.²² His point is if you can avoid the swamp, everything is in order, even if a swamp exists. Indeed, Wittgenstein’s metaphor fits to our example from Section 3. There are rules for how to extract useful predictions from the mathematical model, but not all manipulations are allowed (see footnote 17). In the same section from RFM, Wittgenstein escalates his reasoning:

Can we be certain that there are not abysses now that we do not see? But suppose I were to say: The abysses in calculus are not there if I don’t see them!

Is no demon deceiving us at present? Well, if he is, it doesn’t matter. What the eye doesn’t see the heart doesn’t grieve over.

(RFM, III, Sect78, 205)

In this stunning passage, he seems to stress the following point: When we use a calculus according to certain rules (no predictions about local concentrations), it is irrelevant what happens if we modify the rules (and allow any predictions in the model). We do not have to worry whether some Cartesian demon only makes us believe there is no inconsistency. These passages were met with disbelief, because interpreters could not fathom how an inconsistent model could possibly be useful. However, these passages make good sense if we suppose that Wittgenstein aimed at dismantling the foundational meta-project à la Frege and Russell and instead sided with the engineer who uses the calculus (model) in certain ways in certain contexts.²³ The crucial point is that these certain ways deliver the extraordinarily reliable results that turn a procedure into a calculus. Of course, the usability of the calculus is never guaranteed once and for all.

But is it wrong to say: ‘Well, I shall go on. If I see a contradiction, then will be the time to do something about it.’?—Is that not really doing mathematics? Why should that not be calculating? I travel this road untroubled; if I should come to a precipice I shall try to turn around. Is that not ‘travelling’?

(RFM, III, 81)

In other words, Wittgenstein entertains a pragmatic perspective: The question whether mathematics is well-founded has to be addressed by examining the usability and reliability of a calculus, which is quite independent from a consistency proof. Such proof relies itself on a calculus, a specific practice of using language, and therefore cannot decide on usability. From here, the debated final statement of RFM’s Part III makes good sense: “I have not yet made the role of miscalculation clear. The role of the proposition: ‘I must have miscalculated’. It is really the key to an understanding of the ‘foundations’ of mathematics” (RFM, III, 90). Indeed, Wittgenstein argues that the extraordinary reliability of calculation does not hinge on a proof of consistency. He puts ‘foundations’ into quotation marks, arguably to indicate the difference between his approach and the Frege-Russell project.

5. Strengthening the interpretation: Turing and Wittgenstein

In 1939, Wittgenstein held thirty-one weekly lectures on the Foundations of Mathematics (LFM). Diamond (Wittgenstein’s Lectures) has synthesized the lectures from notes that four people in the audience took.²⁴ Many issues overlap with Wittgenstein’s manuscripts, but Wittgenstein at times interacted significantly with his audience. His disputes with Turing on inconsistency are of particular value to this paper. By that time, Turing had already published his seminal work on computable numbers, introducing the symbol manipulating machine later called the Turing machine. Hence Turing was not a regular student, but a logician and mathematician of first. When Wittgenstein asked him what harm should come from an inconsistency in a calculus, Turing replied that this harm might materialize later, when a bridge falls down that had been constructed with the help of this calculus. In the lecture, Wittgenstein did not give in. He recognized that Turing had formulated an important objection that would require special effort to refute. Wittgenstein remarked that this effort would have to touch upon the fundamental conception of scientific rationality²⁵ and would be “the most important thing” he lectured about.

I disagree with commentators like Chihara (Wittgenstein's Analysis) who take Turing’s point to be obvious and devastating.²⁶ Furthermore, I argue that Wittgenstein’s effort to save his position was successful and that he was on to an important insight regarding the relationship between scientific rationality and mathematization. Wittgenstein does not grant a special role for logic when the foundations of mathematics are at stake, rather he looks at how mathematics fares in applications.

I should like to show that one tends to have an altogether wrong idea of logic and the role it plays; and a wrong idea of the truth of logic. If I can show this, it will be easier to understand why logic doesn’t give mathematics any particular firmness.

(181)

Wittgenstein questions that logic would add a particular rigidity to mathematics. Both are about using symbols. “I am speaking against the idea of a “logical machinery”. I want to say there is no such thing. The idea of a logical machinery would suppose that there was something behind our symbols” (194). In a typical move, Wittgenstein equates the idea that something is ‘behind’ some x with the idea that there is some ‘super-x’ – and then moves on to show that such thing does not exist. In particular, Wittgenstein thinks there are no super-rules that exclude contradictions and guarantee usefulness.

However, ruling out logic as a foundation for mathematics does not tell us why contradictions might be admissible. Not surprisingly, for Wittgenstein, the law of contradiction emanates from language use. There is no natural way to act on a contradiction, hence we have learned how to react to it. We can take a contradiction as indicating a mistake, but we can also introduce rules for how to manage contradictions so that the calculus remains useful. Hence, Wittgenstein proposes, contradictions need not be harmful.

Turing resolutely disagrees: “The real harm will not come in unless there is an application, in which case a bridge may fall down or something of that sort” (211). Wittgenstein replies: “Ah, now this idea of a bridge falling down if there is a contradiction is of immense importance. But I am too stupid to begin it now; so I will go into it next time” (211). And from lecture 22 onwards, they discuss the merits and limitations of the bridge-objection. Wittgenstein argues that bridges do not falter because of contradictions. Turing accepts this point and specifies:

The sort of case I had in mind was the case where you have a logical system, a system of calculations, which you use in order to build bridges. You give this system to your clerks and they build a bridge with it and the bridge falls down. You then find a contradiction in the system.

(212)

Wittgenstein, in turn, concedes that “[w]e as a matter of fact avoid contradictions and are even inclined to call it illogical thinking if there are contradictions” (213). But avoiding contradictions for him is merely a matter of fact, not a matter of a super-rule. If one has not noticed a contradiction and the bridge does not fall down, everything is all right. Wittgenstein’s stance sounds like irresponsible ignorance from the side of engineers using mathematics. However, what sounds stubborn or even non-sensical to most readers makes good sense with the engineering examples in mind. Wittgenstein rightly points out that “if something does go wrong – if the bridge breaks down – then your mistake was of the kind of using a wrong natural law” (217).

Two questions are on the table: (1) Does a mathematical model enable the calculation of relevant consequences? (2) Do these consequences adequately support the intended applications? If the second question has a negative answer, the bridge might fall down. However, the first question might still be answered in the positive, even if the model (calculus) is not free of contradictions. One just needs reliable procedures to deal with them. In other words: Wittgenstein maintains that having reliable procedures does not presuppose having a calculus free of contradictions. The engineering examples from Section 3 support this option.

Turing insists: “You cannot be confident about applying your calculus until you know that there is no hidden contradiction in it” (217). Wittgenstein apparently sees the discussion at the spot he wanted to have it: “There seems to me to be an enormous mistake there” (218). Any calculation, irrespective of contradictions, can lead into collapsing bridges. And if contradictions occur, one can still use a calculus to derive predictions if one can circumvent these contradictions, like circumventing a swamp. Whether the rules work in terms of bridge stability, however, is a question of physics. Turing does not back down: “But that would not be enough. For if one made that rule [draw no conclusion from a contradiction], one could get round it and get any conclusion which one liked without actually going through the contradiction” (220). Only a consistent calculus would exclude this possibility. In a way, Turing does not buy the swamp metaphor, i.e. does not accept the possibility of finding viable pathways without falling into the swamp. A contradiction at one spot, so Turing, leads to all kinds of mistakes, thus no safe pathway exists.

A bit like in a Socratic dialogue, Wittgenstein starts anew. Logic cannot provide foundations to mathematics. Frege’s logic can be used as a calculus containing “and” etc. “But this has nothing to do with the idea of giving foundations for mathematics” (227). When Turing interjects that contradictions are detrimental to Frege’s calculus, Wittgenstein agrees: “I should say the same. – The point I’m driving at is that Frege and Russell’s logic is not the basis for arithmetic anyway – contradiction or no contradiction” (228). This statement shall convey, I believe, that arithmetic is a rule-based calculus of great value in many contexts of mathematization. It helps to extract what some mathematical model has to say about a case under investigation. The usefulness in these contexts is what counts, not the relationship to some logic.

Wittgenstein reformulates his question in a critical way:

How do we get convinced of a law of thought? (…) Turing said that a proof in mathematics may play either of two different roles, (1) of convincing someone; (2) of proving the truth: really prove it, to make it indubitable—he used some phrase like that.

Let’s talk about that. If I can talk about it, it will be the most important thing I have talked about. I want to show that there is some sort of muddle here.

(235)

At this point, Wittgenstein invokes his engineering past.

There are cases where we actually give a proof simply to quieten the conscience of the person: so that he can say he’s had a proof. For instance, I used to learn engineering formulae; and in one case—torsion—there were three different ways of calculating which up to a certain point gave closely agreeing results. In that case, one might just say that a formula agrees with the results obtained experimentally; (…) One can give a person a proof of some formula in physics to make him feel comfortable; but this will be no good unless the formula agrees with experience.

(236)

There are different mathematical formulae for dealing with torsion. The proof of one particular formula (from physical modelling assumptions) counts as a way to establish the rationality of using this formula. The point, however, is the success in predictions, not (in)consistency in a mathematical sense.

On the other hand, if thinking according to another law of thought does not mean we conflict with any experience or that we get into difficulties, but just that we use language differently—then there can be different kinds of mathematics. (…) then there is nothing wrong in your adopting this technique. If I then gave you a proof, I would not be giving you a foundation of it.

(236)

In other words, giving a proof is relative to some mathematical system or model. Thus, a proof cannot count as a foundation – unless foundation is also used in a relative, pragmatic sense. If mathematics is conceived as a tool, then only application can say whether the rules/calculus work successfully. In general, there might be different, alternative ways of ‘techniques’ – illustrated by three approaches to torsion. Thus, the development of mathematics does not rest on logic. “I want to say it in no way rests on logic. And the fact that you can make logical formulae agree with it, in no way shows that it rests on logic” (260). For Wittgenstein, the reliability of arithmetical procedures is not less basic than that of logical procedures.

We can say that arithmetical propositions are laws of thought in the same sense in which logical propositions are. (…) But it is misleading to think this an explanation: to think that when we get down to predicates and predicative functions, we see what mathematics is really about.

(271)

Overall, Wittgenstein argues for examining the foundations of mathematics through the use of mathematics.

6. Contradiction, prediction, and progress

I would like to close this paper with two remarks. One is about scientific practice. What matters is not so much, whether there is an inconsistency (a swamp) or not, but rather whether there are means to avoid it safely. The NRTL model (Section 3), for instance, must not be used for calculating local concentrations. Interestingly, as most people do not follow Wittgenstein and find inconsistency abhorrent, even known inconsistencies are deliberately hidden. Tellingly, the inconsistency of the NRTL model is not mentioned in the original paper nor in almost any of the plethora of subsequent papers on the subject. This is a grave mistake, as the argument of Turing shows. The model-specific rules for how to circumvent inconsistencies and, accordingly, for when to refrain from certain types of predictions remain hidden as long as inconsistency is seen as something to cover up.

The second remark stresses how much philosophy of mathematics and science can learn from the perspective of engineering. In his controversial discussion of contradiction and the foundations of mathematics, Wittgenstein enters a confrontation that is not unlike the one Riedler faced in the equally controversial discussion with Klein. Much like Klein’s attempts to provide foundations for applied mathematics, the Frege-Russell project of giving mathematics a logical foundation approaches the matter top-down. According to them, logic is a foundation, because it expresses laws of thought and therefore provides a sort of super-rules. Wittgenstein sides with the bottom-up standpoint. In this view, mathematization aims at providing useful tools (predictions based on models and calculations). Importantly, practice does not strictly require consistency. Formal structures and their exploration have (rightly) attracted much attention from philosophy. While laws of thought correspond to mathematical practices, they do not dictate them. Rather, exploring uses of mathematical models in the world, like for torsion or the mixture of fluids, promises new practical pathways into the world’s complexities. If Wittgenstein was after the insight that the laws of thought are not fixed, but are constantly being worked out anew in exploratory activity, then I fully join his opinion that this is one of the most important things.

Acknowledgements

I profited from the enlightening discussions with Matthias Brandl, expert on Wittgenstein, and with Hans Hasse, expert on thermodynamics, on mixtures of fluids, and mixtures of philosophy and engineering. I am grateful to two anonymous referees for helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was funded by Deutsche Forschungsgemeinschaft, DFG LE 1401/9-1.

Notes

1 These passages are the closing §§77–90 of Wittgenstein’s Remarks on the Foundations of Mathematics (RFM), part III. The editors of RFM have extracted the text from manuscript 177 that is now openly available at the Bergen website.

2 For more examples of the negative reception, see (Kreisel, “Wittgenstein’s Remarks”, Bernays, “Comments on Wittgenstein’s Remarks”, Chihara, “Wittgenstein’s Analysis”, or Gödel, Works).

3 There is a remarkable number of recent papers that deal with Wittgenstein’s views on inconsistency and their position in the philosophy of mathematics (Berto, “Wittgenstein’s Reasons”, Marion and Okada, “Wittgenstein on Contradiction”, Persichetti, “Wittgenstein’s Guide to Contradictions”, Schroeder, Wittgenstein on Mathematics, Chapter 11; Matthíasson, “Wittgenstein’s Reply”), especially concerning the perceived need of foundations (Wagner, “Does Mathematics Need Foundations?”, Pérez-Escobar, “Wittgenstein as a Forerunner”, Wheeler, “Wittgenstein on Miscalculation”).

4 What Wittgenstein calls calculus shows great affinities with what today is discussed under the notion of model. This paper does not depend on a precise terminological fixation.

5 I agree with Juliet Floyd’s finding that the disputes between Wittgenstein and Turing do not so much arise from contrasting philosophical standpoints, but rather display their joint interest in “common sense” (Floyd, “Turing on Common Sense”, 103). Importantly, Ásgeir Matthíasson (“Wittgenstein’s Reply”) has recently contributed an analysis of Wittgenstein’s discussion with Turing on the bridge example. I very much agree with his general point that contradictions in a math calculus are not necessarily fatal to scientific practice (that uses this calculus). See also Marion and Okada (“Wittgenstein on Contradiction”) and Pérez-Escobar and Sarikaya (“Purifying Applied Mathematics”).

6 In the literature on Wittgenstein, there is another line of argument that lends support to his claims about inconsistency. The point of paraconsistent logic is that one can allow contradiction without making logic impossible. That means one can meaningfully restrict ex falso quodlibet – from a contradiction does not follow anything (in this logic). Paraconsistent logic might save some of Wittgenstein’s most criticized statements, since this logic drops consistency as a strict requirement, see, for instance, (Priest, Beall, and Armour-Garb, The Law of Non-Contradiction). Although paraconsistent logic is of systematic importance to Wittgenstein’s claims (Marconi, “Wittgenstein on Contradiction”, Goldstein, “Wittgenstein's Development”, Bromand, “Wittgenstein über Paradoxien”), Wittgenstein did not elaborate on paraconsistent logic.

7 Here is an illustration: “That he chose to study engineering was a consequence of his early interests and talents, rather than of his father’s influence. Throughout his life he was extremely interested in machinery. While a small boy he constructed a sewing machine that aroused much admiration. Even in his last years, he could spend a whole day with his beloved steam-engines in the South Kensington Museum. There are several anecdotes of his serving as a mechanic when some machinery got out of order” (von Wright, “Biographical Sketch”, 3).

8 The significance of Wittgenstein’s experience in engineering for his philosophical reasoning remains a debated issue. Sterrett (Wittgenstein Flies a Kite) argues that the conception of model in the Tractatus is directly influenced, whereas David Hyder holds that Wittgenstein’s logical theory is unrelated to mechanical engineering (Hyder, Mechanics of Meaning, 157).

9 König (Der Gelehrte) wrote an instructive double-biography of Reuleaux and Riedler.

10 This movement is only rarely addressed in the history of technology, with contributions like Gispen (“Theory to Practice”) who does not cover mathematics, other than the contributions of Hensel, Ihmig, and Otte (Mathematik und Technik).

11 For a more comprehensive treatment, including original literature by protagonists, see (Johnson and Lenhard, Cultures of Prediction, Chapter 3).

12 Hamilton (“The Mind’s Eye”, 58) documents the courses Wittgenstein attended. Furthermore, Wittgenstein lived in the household of Stanislaus Jolles, a mathematician teaching at Charlottenburg from 1893 onward, who became full professor in 1907, during Wittgenstein’s stay – when Jolles gave courses on descriptive geometry. It is not unlikely that Jolles was still deeply impressed by the anti-maths movement and the changes it provoked.

13 Additionally, studying the foundations of mathematics by studying actual practices (and not by studying logic or metamathematics) is in good agreement with Wittgenstein’s general pragmatic position expressed in the Philosophical Investigations.

14 The books authored by Hunt (The Maxwellians), Nahin (Oliver Heaviside), and Yavetz (From Obscurity to Enigma) together give an outstanding coverage of historical, philosophical, and engineering-scientific aspects.

15 Nahin refers to the controversy as “the battle” (Oliver Heaviside, 196). Over the course of the nineteenth century, rigour had become a defining feature of mathematics. A typical case is the ϵ-δ-definition of a continuous function that expresses continuity via an arithmetical relationship.

16 There is philosophical literature on inconsistency between different models that deal with the same target system from different perspectives, e.g. atomic nucleus via quark models, or shell models, see Morrison (“One Phenomenon, Many Models”). The present case is different since the (one) model itself is mathematically inconsistent.

17 The paper has been cited more than 6,000 times in the academic literature. This number is much lower than the number of papers in which the model has been used, as it is so well-known that it is deemed sufficient to simply say that NRTL was used without referring to the original paper.

18 This paper is (deliberately) based on evidence from engineering history and practice. One can explore further what standpoint in logic is compatible with accepting inconsistency (in a certain way). While actually attempting such analysis is beyond the scope of this paper, I would like to point out that there is relevant and controversial literature, partly in line with Wittgenstein, partly not. Chihara, for instance, argued in his analysis of the liar paradox that the detrimental effect comes from questions that do not stop and suggests a strategy that avoids asking certain questions (Chihara, “Semantic Paradoxes”, 228). Recently, (Persichetti, “Wittgenstein’s Guide to Contradictions”) argued along similar lines.

19 Wittgenstein’s notes on the foundations of mathematics are voluminous. Felix Mühlhölzer’s careful and sympathetic analysis, a commentary on part III of Wittgenstein’s RFM, also consulting the original manuscripts, takes a book of more than 500 pages (Mühlhölzer, Kommentar). Although the manuscript is the original and often better source, this paper cites from RFM because the selection of thoughts and quotes from Wittgenstein can be found there, too.

20 There is considerable debate about whether Wittgenstein considered language-games as actual descriptions or rather hypothetical tools. Since the quoted passage mentions ‘anthropologically’, the former interpretation seems to be more suitable here.

21 This is nicely illustrated by our second example from Chapter 3: if the goal of the NRTL-model were to calculate local concentrations, it would be useless, as that part is infected by the inconsistency. But it is used for calculating other properties of mixtures, for which the inconsistency does not prevent usefulness.

22 Whether he also includes his own earlier position is unclear. Recent literature has made the point that even the early Wittgenstein of the TLP was critical of a logicist standpoint, see, e.g. Floyd’s review (“Mathieu Marion”) review of Marion (“Wittgenstein, Finitism”).

23 Purpose-dependency is an important facet in the newer philosophical debate about models.

24 She used the notes of R. G. Bonsanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. About a dozen hearers attended. Quotations from Diamond’s compilation are just by page number.

25 Wittgenstein speaks about how we get convinced of a law of thought. This is what I have earlier called rationality.

26 One of the authors whom I join is Matthíasson (“Wittgenstein’s Reply”) in his assessment that consistency is one virtue among other theoretical virtues and has no special status. Although many commentators highlight how strong Turing’s point is, opinion is not unanimous. Disagreeing voices have gained ground recently, see also footnote 3.