Philosophy of science

Summary

The geometry over a Euclidean field (every positive number has a square root)may have no straight line segment of length π. For example, the model containing only the constructible real numbers does not contain π. We want to find a theory which proves the circumference and area formulas for circles. Our approach is to extend the theory EG so as to guarantee that there is a point in everymodel which behaves as π does. In this chapter we will show that in this extended theory there is a mapping assigning a straight line segment to the circumference of each circle.We first introduce π to the Euclidean scheme by forming a theory EGπ. In a second direction (Chapter 10.2), we note Tarski's axiomatization of ‘Cartesian’ plane geometry, E2. Then we combine the two in a theory E2π to give the theory of real closed fields that include π. Given that the entire project is modern, we give the arguments entirely in the style of modern model theory.

For Archimedes and Euclid, sequences constructed in the study of magnitudes in the Elements are of geometric objects, not of numbers. In a modern account, as we saw already while discussing areas of polygons in Chapter 9.5, we must identify the proportionality constant and verify that it represents a point in any model of the theory.1 Thus this goal diverges from a ‘Greek’ data set and indeed is orthogonal to the axiomatization of Cartesian geometry in Theorem 10.2.1.

This shift in interpretation drives the rest of this chapter. We search for the solution of a specific problem: is π in the underlying field?

π in Euclidean and Archimedean Geometry

We now describe the rationale for placing various facts in the Archimedean data set2 in Notation 9.3.2. Three propositions encapsulate the issue: Euclid VI.1 (area of rectangle), Euclid XII.2 (area of a circle is proportional to the square of the diameter), and Archimedes’ proof that the circumference of a circle is proportional to the diameter. Hilbert showed (Theorem 9.3.4) that VI.1 is provable already in HP5. While Euclid implicitly relies on the Archimedean axiom, Archimedes makes it explicit in a recognizably modern form. Euclid does not discuss the circumference of a circle.

Summary

Recall (page 15) Detlefsen's first question:

(IA)Which view is the more plausible – that theories are the better the more nearly they are categorical, or that theories are the better the more they give rise to significant non-isomorphic interpretations?

Button and Walsh distinguish two issues for philosophers investigating categoricity ([Button & Walsh 2016], 283): ‘determinacy of reference of mathematical language and the determinacy of truth values of mathematical statements.’ As in their paper we focus on the question of determining reference for theories that are intended to describe a single isomorphism type of a structure. With the normal background of ZFC, one can define the vocabulary τ, the domain A, and the interpretation of the τ -relations on A. This description unambiguously refers to a particular structure. The isomorphism type of A is the class of all τ -structures isomorphic to A. The number theorist Barry Mazur wrote,

the objects that we truly want enter the scene only defined as equivalence classes of explicitly presented objects. That is, as specifically presented objects with the specific presentation ignored, in the spirit of ‘ham and eggs, but hold the ham.’ [Mazur 2008]

Importantly, one can also ‘take the ham’. For example, use linear algebra to study a matrix group and apply the result to any isomorphic group (Chapter 4.7).

There are several ways that a serious issue arises. Attempting to reify the notion of isomorphism type strikes the obstacle that the collection of structures isomorphic to A is not a set. This can be resolved by bounding the set theoretic rank of the structures or considering the isomorphism class as definable in ZFC; either of these is adequate from a model theoretic perspective. Secondly, one can demand that the description be formulated in the original vocabulary and then the problem becomes categoricity. In this context, the study of categoricity begins with the choice of logic. We argue, working from our pragmatic notion of virtue, that categoricity is interesting for a few second 68 order sentences describing particular structures.

But this interest arises from the importance of those axiomatizations of those structures and not from any intrinsic consequence of second order categoricity for arbitrary theories.

Summary

The announcement1 for a conference on Philosophy and Model Theory in 2010 began:

Model theory seems to have reached its zenith in the sixties and the seventies, when it was seen by many as virtually identical to mathematical logic. The works of Gödel and Cohen on the continuum hypothesis, though falling only indirectly within the domain of model theory, did bring to it some reflected glory. The works of Montague or Putnam bear witness to the profound impact of model theory, both on analytical philosophy and on the foundations of scientific linguistics.

My astonished reply to the organizers began:

It seems that I have a very different notion of the history of model theory. As the paper at [Baldwin 2010] points out, I would say that modern model theory begins around 1970 and the most profound mathematical results including applications in many other areas of mathematics have occurred since then, using various aspects of Shelah's paradigm shift. I must agree that, while in my view there are significant philosophical implications of the new paradigm, they have not been conveyed to philosophers.

This book is an extended version of that reply to what I will call the provocation. I hope to convince the reader that the more technically sophisticated model theory of the last half century introduces new philosophical insights about mathematical practice that reveal how this recent model theory resonates philosophically, impacting in particular such basic notions as syntax and semantics, structure, completeness, categoricity, and axiomatization. Thus, large parts of the book are devoted to introducing and describing, for those not familiar with model theory, such topics as the stability theoretic classification of first order theories, its applications across mathematics, and that its interaction with classical algebra is inevitable (Chapter 5.6). Much of this exposition will be in the context of discussing the paradigm shift. In short, the paradigm around 1950 concerned the study of logics; the principal results were completeness, compactness, interpolation, and joint consistency theorems. Various semantic properties of theories were given syntactic characterizations but there was no notion of partitioning all theories by a family of properties.

Summary

Bourbaki wrote ‘We emphasize that it [logical formalism] is but one aspect of this [the axiomatic] method, indeed the least interesting one.’ In this chapter, we begin the argument that Bourbaki have it backwards. Formal methods are not a pedantic puddle on the path to find the methods and principles that underlie a mathematical result but rather a stepping stone to understanding.

We explore the epistemological purposes of formalization along the lines of Manders (Chapter 2.1) and contrast Bourbaki's views with the contemporary mathematician, Kazhdan (Chapter 2.2). In Chapter 2.3 we take up Detlefsen's questions (page 15), and provide a criterion for the virtue of a property of a theory. In Chapter 2.4 we begin the explanation of how virtuous formal properties can provide a more fruitful way of organizing mathematics than that proposed by Bourbaki.

The Process of Formalization

In Chapter 1.1 we gave a static description of a formalization because such full formalizations of various topics are taken as the data for the organization of mathematics via classification theory. Here, we take a somewhat broader view. Lakatos writes:

For more than two thousand years there has been an argument between dogmatists and skeptics … The core of this case-study will challenge mathematical formalism but will not challenge directly the ultimate positions of mathematical dogmatism. Itsmodest aim is to elaborate the point that informal, quasi-empirical,mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. ([Lakatos 1976], 4–5)

In contrast, I take formalization as part of that ‘incessant improvement’ and one that aims toward ‘indubitably established theorems’. Our goal here is to move beyond case studies, to show how formalization can be a tool for systematizing the changing definitions of mathematical notions. In this section, I expand more on how this tool will be developed in the book and where it is applied.

This kind of systematic analysis is developed byMarkWilson for physical science. My approach is less ambitious than Wilson's; I deal only with mathematics, not science writ large.

Summary

We reported in the introduction (page 15) several questions raised by Mic Detlefsen. In less precise fashion, they were: Is it better for a theory to be categorical or not? How do we justify whether categoricity is a virtue for a theory? Is completeness a good approximation to categoricity? This part has several themes that together respond to these questions.

Summary

In Chapters 3.3 and 4.2, we discussed the gradual development of notions of completeness in the first third of the twentieth century. We recounted confusions amongst deductive and semantic completeness (for theories and/or logics) and categoricity. In this chapter we look from another angle on the distinction between descriptive and Gödel completeness and another sense of completeness identified by Kreisel. Today the difference between the topological and the logical notions of completeness is clear. But, since the topological notions are deeply entangled with categoricity concerns about the real numbers, the ideas intermingled at the beginning of the twentieth century.We discuss mathematical and logical attempts to clarify the notion from Dedekind to contemporary logicians.

We consider first Kreisel's ‘Logical foundations, a lingering malaise.’ A quick reading of the excerpt below might suggest Kreisel refers to the necessity of the Dedekind axiom for semantic completeness of Hilbert's geometry. But we argue he is actually referring to a deeper uneasiness on the part of Poincaré. Kreisel remarks of Hilbert's rules for geometry:

True, [they] provided a genuine surprise at the time, and it does not seem to be well-known that Poincaré still doubted the completeness of Hilbert's rules. Presumably everybody else did too who had taken Kant's view on the role of visualization (Anschauung) in mathematical reasoning literally (that is, some kind of logical need) and not in the sense of effective use – a distinction already stressed earlier. [Kreisel 1984]

In the first paragraph of the piece, Kreisel had identified one element of the malaise: ‘a preoccupation with a universal framework (a universal language, for example) and thus with logical possibilities. This preoccupation is at the heart of the malaise; it concerns a potential conflict between pursuing these logical ideas and effective knowledge.’

In his review of the 1899 edition of the Grundlagen, Poincaré wrote,

But this geometry, strange to say, is not quite the same as ours, his space is not our space, or at least is only a part of it. In the space of Professor Hilbert we do not have all the points which there are in our space, but only those which we can construct by ruler and compass, starting from two given points. In this space, for example, there would not exist, in general, an angle which would be the third part of a given angle.

Summary

Suppose we want to clarify the fundamental notions and methods of an area of mathematics and choose to formalize the topic. Our notion of a formalization of a mathematical topic involves not only the usual components of a formal system, specification of ground vocabulary, wellformed formulas, and proof but also a semantics. From a model theoretic standpoint the semantic aspect has priority over the proof aspect. The topic could be all mathematics via e.g. a set theoretic formalization. But our interest is more in the local foundations of, say, plane geometry or differential fields.We set the stage for developing Thesis 1, by focusing on a specific vocabulary, designed for the topic, rather than a global framework. In any case, a mathematical topic is a collection of concepts and the relations between them. There are course other less restrictive notions of axiomatization, and often such a ‘formalism’ deliberately omits the semantic aspect. But we want the wider notion here as it reflects the model theoretic perspective.

It is not accidental that ‘formalization’ rather than ‘formal system’ is being defined. The relation between intuitive conceptions about some area of mathematics (geometry, arithmetic, Diophantine equations, set theory) and a formal system describing this area is central to our concerns. The first step in a formalization is to list the intuitive concepts which are the subject of the formalization. The second is to list the key relations the investigator finds among them. In stipulating this view of formalization, we are not claiming to fix the only meaning of the term but only the meaning most suitable for the discussion here.

Definition 1.0.1A full formalization involves the following components.

(1) Vocabulary: specification of primitive notions.

(2) Logic:

Summary

On page 12, we transcribed Maddy's Second Philosopher's questions about the role of set theory to questions about model theory – What sort of activity is model theory? How does model theoretic language function? What are models and how do we come to know about them? We see in this chapter that as the paradigm changes there is increasing emphasis on the study of theories and the role of types1 as tools in assigning invariants to models and eventually these developments lead to an entanglement with classical mathematics.

As background for the key concepts of the stability hierarchy in Chapter 5.3, we start with another founding paper of modern model theory, Vaught's ‘Denumerable models of complete theories’ [Vaught 1961], delivered inWarsaw in 1959.

Analogy to Theorem toMethod

MacLane ([MacLane 1986], 37) describes analogy as ‘finding a common structure … underlying different but similar phenomena.’ In his example the concept of vector space is the common notion underlying geometry, linear equations, and linear differential equations. A main line of this book is the discovery of ‘well-defined notion of dimension’ as the common structure underlying a flock of mathematical areas. The result is not merely analogy as inspiration but analogy manifested as actual mathematical theorems. A fundamental tool for this work is itself a product of an analogy turned into a theorem and then into a model theoretic method. As Schlimm [Schlimm 1985] explains, successive analogies of propositional logic with algebra by Boole, of Boolean algebras with rings by Stone, and of deductive systems with Boolean algebra by Tarski led to a remarkable unification of topology and logic.2 Even more remarkable is that Tarski's topological description of syntactic objects was transformed in the 1950s into a powerful method for studying semantics. First, we explore the Boole–Stone– Tarski axis; then we pass to the more modern incarnation in model theory.

Theorem 5.1.1There is a functorial 1–1 correspondence between Boolean algebras and totally disconnected Hausdorff spaces.

The exact nature of this correspondence is crucial. To describe it we introduce the notion of the Stone space of a Boolean algebra and relate it to logic. Consider propositional logic with a set V of k propositional variables.

Summary

In a 1967 letter to Hao Wang, Gödel explained why others had missed his proof of the completeness theorem,

This blindness (or prejudice, or whatever you may call it) of logicians is indeed surprising. But I think the explanation is not hard to find. It lies in a widespread lack, at that time, of the required epistemological attitude towardmetamathematics and toward nonfinitary reasoning. …

I may add that my objectivist conception of mathematics and metamathematics in general, and of transfinite reasoning in particular, was fundamental also to my other work in logic.

As we'll see this ‘objectivist conception’, at least in the sense of envisioning models, is central to model theory. Part II traces the historical roots of the paradigmshift and then its effect on doing and organizing mathematics.We begin in Chapter 4 by seeing how the influence of Tarski and Malcev in the 1930s along with Robinson and Henkin almost 20 years later distinguished model theoretic concerns fromthose that prompted Gödel's work. Then we see the development of the tools of quantifier elimination,model completeness, indiscernibility, and interpretation in the 1950s. In Chapter 5, with Morley and Vaught, properties of theories such as Stone spaces, saturated models, and categoricity in power come to the forefront. We then discuss the key ingredient: Shelah's syntactic hierarchy with its dividing lines culminating in the main gap theorem, specifying which theories are classifiable. Zilber's trichotomy for combinatorial geometry highlights the interaction of model theory with other areas of mathematics. Chapter 6 demonstrates the role of formalization in sharpening the notion of tame and the consequent deep interaction of model theory and algebra. First order analysis moves the impact beyond algebra. An interlude in Chapter 7 considers some reasons for generalizing infinitary logic and the interaction of first order and infinitary logic including Vaught's conjecture. Chapter 8 recounts the role of the paradigm shift in the separation of set theory from first order model theory. We discuss such issues as the ‘identity of indiscernibles’ and Voevodsky's univalent type theory. We examine the relationship of model theory with both axiomatic and combinatorial set theory, seeing a greater entanglement of infinitary logic with axiomatic set theory.

Summary

We chronicle here the divorce of first order model theory from set theory. Recalling Maddy's [Maddy 2007] injunction (page 13) to ‘adjudicate the methodological questions of mathematics – what makes for a good definition,’ we explain this major aspect of the paradigm shift as the decision to choose definitions of model theoretic concepts that reduce the set theoretic overhead. To explore this development we follow the prescription of Thesis 2: connect various local formalizations. That is, we look at formalized set theory ZFC and analyze its use in model theory.

These investigations expand on recent work studying the entanglement of mathematics with set theory [Kennedy 2015, Kennedy 2013, Parsons 2013, Väänänen 2012]. These works, as on page 46, find that the amount of entanglement depends crucially on the logic chosen for formalizing mathematics. Paraphrasing Väänänen [Väänänen 2012], it is very difficult to tell the difference between the second order view (mathematics is the study of higher order properties of structures) and the set theory view (formalizing mathematics in set theory); see Chapters 7.3 and 11. The high entanglement of second order logic with set theory is expressed more precisely by it being symbiotic with the power set [Kennedy 2015]. Parsons’ general theme in [Parsons 2013] is closer to the approach in this chapter. He considers, a bit more generally, logic and mathematics, and argues that the ontological commitments of first order logic are relatively weak. But his argument is not a general metaphysical one. Similarly to Maddy's Second Philosopher he precisely analyzes the commitments within second order arithmetic for specific results in the basic model theory. For example, he points to the fact [Simpson 2009] that the completeness theorem is equivalent to weak König's lemma over RCA0. Analogously we investigate the necessity in contemporary first order and infinitary model theory of specific ZFC techniques or specific axioms. To what extent should, could, or must a model theorist go beyond ZFC?

What are the specific commitments for doing model theory (as opposed to mathematics in general)? We consider six aspects of the interaction of set theory and model theory. In the first section, we describe an apparent close relation between model theory and set theory that was dissolved by the paradigm shift.

Summary

Pillay writes,

The notion of truth in a structure is at the centre of model theory. This is often credited to Tarski under the name Tarski's theory of truth. But this relative, rather than absolute, notion of truth was, as I understand it, already something known, used, and discussed. In any case, faced with the expression truth in a structure there are two elements to be grasped. Truth of what? And what precisely is a structure? [Pillay 2010]

We defined the notions Pillay refers to in Chapter 1. In this chapter we consider his questions more closely and consider some key episodes in the development of the notion of a first order theory as a focal point.

The Downward Löwenheim–Skolem–Tarski Theorem

The meaning of ‘contradictory’ underwent a vast change in the early decades of the twentieth century. It is important to read the famous letter from Hilbert ([Frege & Hilbert 1980], 39) in the 1899–1900 Frege–Hilbert correspondence as written by the turn-of-the-century Hilbert not the later Hilbert:

if the arbitrarily given axioms do not contradict one another with all their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence. The proposition ‘Every equation has a root’ is true, and the existence of a root is proven, as soon as the axiom ‘Every equation has a root’ can be added to the other arithmetical axioms, without raising the possibility of contradiction, no matter what conclusions are drawn. ([Frege & Hilbert 1980], 39)

As we observed on page 33, the Hilbert writing this passage has not yet made the distinction between formal and informal language. Today we might replace ‘do not contradict one another’ by ‘do not imply 0 = 1 in an ambient formal system’ and the word ‘true’ by ‘satisfiable’ and read the first sentence as an instance of Gödel's completeness theorem for first order logic. Such replacements are anachronistic in several respects; not only has a formal sentence entered the discussion but it assumes the first/second order distinction.

Summary

Martin Davis wrote,

Gödel showed us that the wild infinite could not really be separated from the tame mathematical world where most mathematicians may prefer to pitch their tents.

We will now see howmodernmodel theory avoids the Gödel phenomena; the key is to formalize topics locally by axioms which catch the relevant data but avoid accidentally encoding arithmetic and, more generally, pairing functions. We do not attempt a general definition of tame but provide a number of examples of sufficient model theoretic conditions. For more details, see page 160 and [Teissier 1997].

The most basic examples are when mathematicians are already studying definable relations on a class of structures and the natural axiomatization of the area yields a tame theory. In studying real or complex algebraic geometry, the formalization is automatic; Steinitz (ACF) and Artin-Schreier (RCF) defined concepts that happen to be first order; these theories provide the framework for much of the development of the geometries. The theories are ℵ1-categorical (Chapter 3.3; indeed, interpretable in a strongly minimal structure, Example 4.3.1) and o-minimal (Chapter 6.3), respectively.

One of the earliest algebraic discoveries linking algebraic structure with stability properties echoes the Bourbaki assertion of the importance of groups. An ω-stable group cannot have a descending chain of ‘definable subgroups’. This condition extends to stable groups (for uniformly definable chains) and the distinction between the stability classes is signaled by the size of the allowed quotient groups. This principle is now seen to apply to different algebraic structures and gives a unified explanation for finding various kinds of radicals.5 (See [Baldwin 1979] for a very early account of this phenomenon and [Altinel & Baginski 2014], [Aldama 2013], and [Freitag 2015] for recent updates.)

Groups of FiniteMorley Rank

In this section we give a short case study of one research area that serves both as a tool for unifying studies in several areas of mathematics and for isolating the role of basic concepts. In particular, it is seen that finite plays a dual role in the study of finite groups: both as the size of structure and as a dimension on which one can do inductions. In the case at hand, the study is extended to infinite groups by introducing a broader definition of ‘dimension’, Morley rank (Chapter 5.3), and requiring it to be finite.

Summary

We discussed ‘formalization’ at length in Chapter 1. What do we mean by ‘formalism-freeness’? Kennedy begins with

That mathematics is practiced in what one might call a formalism-free manner has always been the case – and remains the case. Of course, no one would have thought to put it this way prior to the emergence of the foundational formal systems in the late nineteenth and early twentieth centuries. [Kennedy 2013]

Kennedy goes on to explore what Gödel [Gödel 1946] calls the search for absolute notions of computability, provability, and definability. We try to distinguish such practice from the use of logical methods (either for metamathematical studies or as tools for proving mathematical results in the normal sense (in number theory or geometry or graph theory or …)). Formalism-freeness does not argue against formal methods per se. Here we point to a number of instances where model theorists have studied informal equivalents of fully formalized frameworks (Tarski) or found such frameworks inadequate for certain goals (Makkai–Reyes, Shelah, Hrushovski– Zilber). Our general theme differs from Kennedy's by focusing specifically on the development of formal theories for parts of mathematics.

Formalism-free is a matter of degree. It involves dropping some of the components of Definition 1.0.1. Clause 1 of Definition 1.0.1 is sacrosanct. We deal only with structures of a fixed vocabulary.

To clarify the situation, we distinguish between ‘formalism-free’ and ‘formalism-independent’. We call a definition formalism-free if it is given semantically without any formal distinction between syntax and a semantics. For example ‘a group is a set and a binary function satisfying …’ We call an investigation formalism-free if it is a semantic study. Thus there are two aspects: definition and investigation.

Here are three illustrative examples of formalism-free definition; we study some such investigations in a few pages. We say a notion, such as ‘computability’, is formalism-independent (formalism-free in Kennedy's terminology) because of the known equivalence among the Turing,Markov, ƛ-calculus, and Gödel characterizations. Only the Gödel characterization as _01 -definable functions in number theory is completely formal in the sense defined here; Turing machines are formalism-free.

Secondly, as Kennedy has pointed out, if L is constructed by adding relatively definable sets then the usual L is obtained by several different logics, e.g. by iterating either first order definitions or weak second order logic definitions.

Summary

But attempts to find compelling extralogical principles that can provide us with perfect knowledge of abstractmathematical objects have largely stalled…. It seems likely that if one is aiming to justify one's axioms and basic concepts on broader grounds, one will ultimately have to attend to the roles they play in organizing our mathematical knowledge, and the role that knowledge plays in organizing and structuring our scientific experiences. [Avigad 2010]

We renounced on page 11 Tarski's goal of creating a methodology of the deductive sciences. More modestly, we have argued directly for the significance in mathematics of a particular methodology – formalization. In this last part, we heed Avigad's admonition and ‘attend to the role our concepts’ play in organizing mathematical knowledge. Having adopted a stance of local foundations, the choice of those axioms will be a part of the organization.

In Chapter 12 we explore the role of formalization in evaluating the purity of proofs, focusing on Hilbert's analysis of Desargues’ theorem, and conclude such a method is not sufficient. In Chapter 13, we consider Shelah's method of finding dividing lines as a general technique in developing an area of mathematics. We distinguish definition, classification, and taxonomy as different modes of organizing information and identify Shelah's dividing as creating a taxonomy.While Shelah originally applied his strategy only to the classification of first order formal theories, the technique readily extends to the formalism-free notion of abstract elementary class. Chapter 14 acts as a foil to the rest of the book. As opposed to our emphasis on the fruitfulness of formalizing mathematics, in this chapter we elaborate Kennedy's notion [Kennedy 2013] of formalism-freeness. It might seem that formalism-freeness is incompatible with the syntax–semantics distinction fundamental to model theory. We argue otherwise by considering several formalism-free approaches to specific areas of model theory that are formulated entirely semantically. Finally, we return to the relation between set theory and model theory by outlining connections of work in abstract elementary classes with large cardinal hypotheses. Chapter 15 rehearses the main points of the book to show how they support the argument.

Summary

As promised in the introduction, we now consider the following goal of Maddy, in the light of classification theory.

In sum, then, the Second Philosopher sees fit to adjudicate the methodological questions of mathematics – what makes for a good definition, an acceptable axiom, a dependable proof technique – by assessing the effectiveness of the method at issue as means towards the goal of the particular stretch of mathematics involved. ([Maddy 2007], 359)

We focus on how to make useful definitions. That is, we study, not the role of formal definition, but the strategy for making informal definitions in mathematics and particulary in model theory. We analyze the motivation for the dividing line strategy.

For context, we consider some earlier work on the justification of definitions. In [Tappendon 2008a, Tappendon 2008b], Tappenden argues for the necessity of studying the notion of definition in context, asserting ‘the math matters.’ In explaining how the actual mathematics matters, he proposes that ‘fruitfulness’ and ‘naturality’ (‘possesses a certain kind of appropriateness or correctness’) are markers of ‘good’ definitions. We expand below on the meaning of ‘appropriate and correct’ in our context. But first we emphasize the point of one of Tappenden's examples: the role of the Legendre symbol. With this symbol, the law of quadratic reciprocity, which is complicated to state because of many cases, becomes a single equation. But, Tappenden argues, this ease of statement is not the reason for its significance; rather the symbol simplified proofs and was a tool for the eventual solution of Hilbert's 9th problem. The crucial point for us is that while a specific technical definition (Legendre symbol) might be a focus at one stage, the significance of the definition is investigated in the context of its role in a wider theory (algebraic number theory).

Similarly, Lakatos emphasizes variations of proof-centered definition. His guiding illustration is the development of the concept of a polyhedron. Lakatos expounds, in his Socratic exploration, the attempt to verify Euler's assertion that for any ‘polyhedron’ with E edges, V vertices, and F faces, E=V +F−2. He shows how the interpretation of these variables changed in the 2,000-year passage from the Greek picture of a polyhedron to chains of simplices and the notion of Euler characteristic in topology. The test of a definition is how it performs in proving the result.

Summary

Infinitary logic occupies a peculiar place between first and second order logic. Since infinite conjunctions are manifest in the very definition it seems less secure than second order logic which after all has formation rulesmuch like first order. Only after a little thought does one see that second order logic is immensely more powerful and has much stronger metatheoretic commitments, although incomparable in definability strength.

Recall that Lω1,ω allows countable Boolean combinations of formulas but only finite strings of first order quantifiers; L∞,ω strengthens the logic by allowing conjunctions of any cardinality (Chapter 1.3.5). Moore [Moore 1997] describes the ‘continuous development of infinitary logic’ as beginning with the work of Henkin, Karp, Scott, and Tarski in the middle 1950s. Moore also reports a substantial prehistory including Boole, Peirce, Schroeder, Léwenheim, Skolem, Hilbert, Zermelo, and others. The key contentual distinction seems to be that the work up to Zermelo is entirely semantic. Only whatMoore calls the ‘continuous development’ culminating in Karp's completeness theoremand the active school of the 1960s–70s fully incorporates the Tarskian syntax–semantics distinction.

We first consider the role of infinitary logic in formalizing mainstream mathematics by reviewing attempts to formalize the Lefschetz principle. This analysis relies on the completeness of the theory of the algebraically closed field of a fixed characteristic. In Chapter 7.1 we describe the role of categoricity in power for the logic Lω1,ω and investigations of the theory of complex exponentiation. Chapter 7.2 surveys the connections among many areas of logic arising from Vaught's famous question, ‘Does the continuum hypothesis hold for the countable models of a first order sentence?’ In Chapter 7.3 we return to second order logic with a slight twist.We consider second order logic with infinite conjunctions of various lengths.

Why Infinitary Logic? The Lefschetz Principle: We discuss an example of logicians carefully choosing an appropriate logic to formalize a well-known heuristic principle. These local foundations are tested against a goal set by AndréWeil. The Lefschetz principle was long known informally by algebraic geometers and appeared in Lefschetz's 1953 text, Algebraic Geometry. Barwise and Eklof describe the issues around formalizing the Lefschetz principle as follows.

Summary

In this chapter, we draw on our earlier analysis of the epistemological role of vocabulary (Chapter 1.2) as a tool for considering the question of whether/how formalization can be used to capture the notion of ‘purity of method’. Our attention was drawn to this topic by the papers of [Hallett 2008] and [Arana & Mancosu 2012]. We first use our perspective of the role of vocabulary to set out the mathematical issues involved in the study of Desargues’ theorem in a somewhat different way than theirs and to use the notion of explicit definition to investigate purity concerns. However, careful analysis (Chapter 12.5) of Hilbert's embedding theorem shows that even first order explicit definition can introduce impurities. This emphasizes the impotence of purely formal methods (such as logical and strong logical purity discussed in Chapter 12.3) to characterize purity. We see that to diagnose the impurity of Hilbert's embedding theorem one must consider the content of what is in fact a ‘formalism-free’ (Chapter 14) proof. The appendix to [Baldwin 2013a] provides what I argue is a pure proof.

We discuss the relation between the choice of vocabulary and basic axioms and the ‘content’ of a subject in Chapter 12.1. In Chapter 12.2 we lay out some of the mathematical background in terms of formal axiomatic systems to clarify the relation between the Desargues’ proposition in affine and projective geometry. This analysis is based rather directly on a first order axiomatization of geometry. Consideration in terms of Manders’ [Manders 2008] ‘diagram proofs’ would raise other pertinent issues. Chapter 12.3 analyzes several attempts to define the notion of purity and concludes that formalization can highlight impure methods but does not give a criteria for ‘purity’. In Chapter 12.5, we come to grips with the essence of Hilbert's work on Desargues’ theorem and assert the impurity of his argument that the three-dimensional proof is impure.

The discussion of purity below will make clear that the context of a proposition is crucial for the issues we are discussing. But, as discussed in Chapter 9, the notions of what geometry actually is have changed radically over the centuries. The distinction between the coordinate geometry of Descartes and synthetic geometry is crucial for our purposes. As [Arana & Mancosu 2012] makes clear the study of geometry in the late nineteenth century involved multiple dimensions, surfaces of various curvatures, and coordinate and coordinate-free approaches.

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