Metamathematics

Mathematical foundations: what are mathematical entities?

Position: Ante rem mathematical structuralism

We’ll immediately distinguish two themes:

The ontological question: do mathematical entities exist?
The foundational question: is there an ultimate ground for mathematical knowledge?

Positions in the philosophy of mathematics often get hopelessly muddled for lack of a clear distinction between these two. Ontological positions get split, as usual, between realist and anti-realist views. Foundational positions often relate to one of the "big three": logicism, intuitionism and formalism.

Ontological positions

Mathematical realism:

I will adopt a realist position about mathematics. That may not be a shock announcement in light of my defense of logical realism and my endorsement of abstract object existence, but there’s no strict entailment between the two. Various forms of nominalism (fictionalism, constructivism, etc.) aim to avoid the ontologization of mathematics at all costs: in other words, to argue that numbers, and the likes, do not exist. Let me re-state the main issues plaguing these views.

First, nominalism struggles to explain the staggeringly successful use of mathematics in the natural sciences. Quine couched his support for realism in an influential argument from indispensability: since (1) we need to have ontological commitment to entities that are indispensable to our best scientific theories, and (2) some mathematical entities are indispensable to those theories, then we need to accept the existence of at least some mathematical entities. The first premise is debatable, but is really hard to reject for any "mainstream" nominalist who would accept quarks or black holes in their ontology, despite having no direct evidence of their existence. Rejecting it would seemingly push one towards solipsism or at least some form of Berkeleyan idealism. The second premise has been rejected, most notably by Field, who then attempted to prove that physical theories can be "nominalized". Given the complexity of the enterprise and the limited success in the simple case of Newtonian gravitation, I will seriously doubt the prospects of ever nominalizing our "best theories" of quantum fields or spacetime curvature.

Another problem for nominalists lies in their semantics. As regards the entities it is committed to, realism is completely transparent: that $\pi$ is transcendental is true, because $\pi$ satisfies that property, in the same way that an atom satisfies the property of having mass. But nominalists can’t accept that, since to them, numbers don’t refer. Although they have to recognize the usefulness, maybe even the methodological necessity of mathematics in its standard form, they need to re-write mathematical theories, changing the syntax and semantics to preserve verbal agreement with the truth-values from mathematical works. Such a re-writing should also impose a uniform semantics: science is rife with mixed statements involving mathematical and physical propositions. Inference from such mixed statements to others is key to scientific work, thus it is key that physical truth and the truth of mathematical statements be part of compatible semantics.

For all these reasons, and having no inclination towards conceptual parsimony in the first place, I am a staunch realist.

Mathematical platonism:

As we’ve seen when exploring the abstract realm, there are many ways to be a realist.

One would be to recognize that mathematical entities exist, but are grounded in mental states. For instance, psychologism à la Mill or Locke has it that numbers or algebraic structures are just ideas in our heads. I had already shown psychologism in general to be a highly unstable position: by most accounts, minds haven’t always existed; certainly, minds capable of thinking about mathematics haven’t always existed. So, committing to mathematical entities that are ontologically dependent on mental phenomena doesn’t help much: either one doesn’t accept that the physical world exists independently of mathematically capable minds, in which case this is another slide towards subjective idealism, or one will need to on-board all the resources (and all the problems) of nominalism anyway.

Alternately, we could adopt a form of realism that’s independent from minds, but dependent on the physical world. Quine’s style of thin realism seems to fit the bill. As we’ve seen above, the argument from indispensability marked Quine’s breakup with nominalist views. The way we’ve presented it is that we should commit to at least some indispensable objects. To be consistent with his naturalist view of philosophy, Quine added that we should commit to only the indispensable ones. As we gain more and more insights about the world, a hyper-general core forms at the center of our collective web of beliefs (Quine’s concept) of which mathematical knowledge is a part. This is compatible with the idea of grounding mathematical facts in physical facts. However this makes Quine’s account at odds with mathematical practice, which is methodologically distinct from scientific inquiry. The incredible success of modern physics in taking decades-old, unapplied swathes of speculative mathematics (Riemannian geometry, Lie groups, Hilbert spaces...) and to make them indispensable parts of cutting-edge formalism should give pause. What should have been the ontological status of these objects in the intervening years? Besides, it can be highlighted that physical theories can be multiply realized by mathematical frameworks: there are several equivalent formulations of quantum mechanics and quantum field theory. A good case could be made there is an infinity. Quineans tend to put lots of stock in the idea of "best theories" to get out of this quagmire, but that vague phrase is a definite chink in their armor.

Lots of these critiques of Quinean doxa have been made by Maddy, who expounds her own version of thin realism. It postulates that the objects of set theory are part of the physical realm and can be perceived. Maddy sits between Quine’s limited realism and Gödel’s expansive platonism, which makes this interesting position quite unstable, albeit promising.

The version of realism I will thus defend is platonism: not only do mathematical entities exist, they are not grounded in either the mental or the physical realm. Since I’ve already endorsed a theory of abstract objects, that is not earth-shattering and we can fully expect mathematical objects to find a place there.

Mathematical structuralism:

the classic version of platonism, however, is not without major challenges itself. In that framework, all mathematical objects we may encounter in mathematical discourse exist. That includes $6$, the $\textrm{log}$ function, the unit circle, etc.

The first objection is common to all forms of realism about abstract objects: humans refer to mathematical objects and state mathematical facts, yet platonism is characterized by the idea that those are metaphysically remote. How do we have epistemic access to them? The second objection is tied to the foundations programme in metamathematics that’s engulfed the start of the twentieth century. Indeed, one may just endorse platonism over an undistinguished blob of objects and properties, but such profligacy and lack of structure doesn’t sit well with most views of mathematics. Hence the drive to reduce all mathematical theories to a more fundamental one.

The canonical example is the attempt to reduce mathematics to set theory. To the regular platonist, this is supposed to mean that only set-theoretic objects and concepts are ontological primitives, which ground all others. This means that conversely, objects like numbers are not primitives and have a set-theoretic internal structure. The problem lies with the identification of those structures: case in point, natural numbers have multiple equivalent set-theoretic representations. For instance, in terms of Von Neumann ordinals, $2=\{\emptyset,\{\emptyset\}\}$, but in terms of Zemerlo ordinals, $2=\{\{\emptyset\}\}$. This problem was central in promoting structuralism: the idea that mathematics is the study of structures.

For structuralists, what we think of as mathematical objects are only places in a mathematical structure. So $2$ is just the name we give to the third place in a natural number structure. That structure can be instantiated by many systems, for example the Zemerlo ordinal system. Since objects in that system are set-theoretic, they refer to places in set theory structure, and so on. This seemingly dispels our earlier problem: by reconceiving mathematical objects as places in a structure, we defuse questions about their identification. This is also consistent with mathematical practice, where the primacy of algebra and, later, the emergence of category theory, have put the idea of isomorphic representations of a given structure at the forefront of mathematical reasoning.

Structuralism is not prima facie a platonist doctrine. In re structuralists do not ontologize structures. In line with my professed platonism, I will instead adopt ante rem structuralism. The corresponding position is that structures exist, as do places in those. The independence component of platonism implies that there could be structures not instantiated by any system (and a fortiori not grounded in the physical or mental realms).

Mathematical structures in the formal ontology

I agree with Zalta and Nodelman’s recent contention that formal ontology in the fashion of their Abstract Object Theory offers a hospitable environment to this kind of mathematical structure. Despite significant nuances compared to Shapiro’s sui generis structuralism and to Hellman’s modal version, it’s been linked to that tradition.

Zalta starts defining a theory as such: \[\mathfrak{t} \text{ is a theory} \hat{\longleftrightarrow} \mathbf{A}\mathfrak{t} \wedge (\forall P(\mathfrak{t}P \equiv \exists \phi (P=[\lambda x \phi])))\] In plain English: a theory is an abstract object that only encodes properties that are propositional (that is how we call the likes of $[\lambda x \phi]$: "the property that $\phi$"). From there we can define quite naturally the concept of truth in a theory: \[\mathfrak{t}\vDash \phi \text{ ('$\phi$ is true in the theory $\mathfrak{t}$')} \hat{\longleftrightarrow} \mathfrak{t} [\lambda x \phi]\] A proposition is true in a theory when the abstracted property is encoded by that theory. To ensure this definition is consistent with the mathematical meaning of a theory (a class of theorems derived from an axiom schema), we will also need the following rule of closure: \[(\mathfrak{t} \vDash \phi) \wedge (\phi \vdash \psi )\rightarrow \mathfrak{t} \vDash \psi\] So, if a formula is true in a theory, then any logical consequence of that formula will also be true in the theory.

Let’s illustrate with a simple case, for instance the Peano Number Theory (henceforth ’PNT’), which is entirely defined by a set of axioms: \[\begin{aligned} &N0 \\ &\forall n (Nn \rightarrow\exists p (Np \wedge Snp)) \\ &\neg\exists n(Nn \wedge Sn0) \\ &\text{etc.} \end{aligned} \quad \begin{aligned} &\text{'$0$ is a number'}\\ &\text{'Every number $n$ has a successor $p$'}\\ &\text{($0$ is not the successor of any number'}\\ & \end{aligned}\] As with any axiomatic system in mathematics, the concepts of the theory (number, successor, $0$...) don’t have any meaning a priori. They’re just symbols, carvings on a page, and anything else we’ll know of them shall be deduced from the axioms themselves.

However, there is something pre-mathematical in the formulas above and that is the language in use: having read the previous sections, no reader will be surprised. $0$ exemplifies the property $N$ "being a number". We quantify over a domain of discourse covering all numbers. $S$ is a binary relation for "being the successor of". In other words: this mathematical theory is couched in the syntax of logic. It may seem like a truism but is an important assumption we can call language logicism: that the language of mathematics consists of logical expressions.

Let’s go back to the axioms. I’ve expressed them "naively" in the formalism of PNT, the way it would be done in mathematical practice. But here, we are trying to define a mathematical theory as an abstract object we’ll call $\mathfrak{pnt}$. We’ll follow a principle of axiom importation, hereby applied to the first axiom: \[\vdash_{PNT} N0 \rightarrow \mathfrak{pnt}\vDash N_{\mathfrak{pnt}}0_{\mathfrak{pnt}}\] We have just imported the mathematical axiom as a true proposition in the mathematical theory, expressed within our formal language. To avoid ambiguity, we’ve added a subscript to each term from that expression.

Let’s now call $\alpha_1...\alpha_n$ the axioms of a general mathematical theory. Then we can define the abstract object representing the theory in our model as: \[\mathfrak{t} = \imath x ( \mathbf{A}x \wedge \forall P ( xP \equiv \exists \phi ( P=[\lambda y \phi] \wedge \alpha_1,...\alpha_n \vdash_T \phi_T ) ))\] Here’s what we’ve done here, step by step:

We start from the axioms $\alpha_i$ in the mathematical theory $T$
Any formula that can be derived from these axioms, we import in our formalism and abstract into a propositional property
The theory is the abstract object that encodes all those properties.

We can verify that this object behaves exactly as expected: it complies with the rule of closure, and guarantees that all theorems are true in the theory.

What about the objects that have popped up when we’ve imported the axioms? They’re part of our formal language and thus need to have a specific type. Any mathematical theory has a ground level of basic objects that don’t act as predicates. In the case of number theory, these basic elements are numbers. Therefore $0_{\mathfrak{pnt}}$ is taken to be a type $i$ individual term. Since $N_{\mathfrak{pnt}}$ is acting on numbers, it is a property: a type $\langle i \rangle$ predicate.

However, in our theory $N_{\mathfrak{pnt}}0_{\mathfrak{pnt}}$ is false! That’s where it differs from classic Platonism. $0$ or $1$ are not numbers in the same way that I am human, because of the dual predication. They exemplify extra-mathematical properties like "being abstract", but not mathematical ones like "being a number". All the theory says is that \[\mathfrak{pnt}\vDash N_{\mathfrak{pnt}}0_{\mathfrak{pnt}}\] which means that \[\mathfrak{pnt}[\lambda x N_{\mathfrak{pnt}}0_{\mathfrak{pnt}}]\]

We will identify the elements of a theory as: \[x_\mathfrak{t}=\imath y ( \kappa y \wedge \forall P ( yP \equiv \mathfrak{t} \vDash P x_\mathfrak{t} ) )\] Mathematical objects are then represented as codes that encode predicates which, in the theory, they exemplify. It is important to realize that this formula applies to any type $t$. Let’s go back to our PNT number theory example. Applying this definition to our two main objects, we get: \[\begin{aligned} 0_{\mathfrak{pnt}} &=\imath y ( \mathbf{A}y \wedge \forall P ( yP \equiv \mathfrak{t} \vDash P 0_{\mathfrak{pnt}} )\\ N_{\mathfrak{pnt}}&=\imath P ( \kappa P \wedge \forall \mathcal{F} ( P\mathcal{F} \equiv \mathfrak{t} \vDash \mathcal{F} N_{\mathfrak{pnt}} ) \end{aligned}\] Each of these objects encodes exactly the properties that it exemplifies in the theory. Objects, properties, higher-level predicates, etc. can thus be represented in our ontology in terms of codes of ascending types. But let’s remember that every code can be abstracted down to the realm of abstract objects, via comprehension. This is consistent with our ontological understanding of the abstract realm as the locus of objects that could not possibly have mental or physical existence, and is fully in line with our professed mathematical Platonism.

We can then confirm the principle that every mathematical theory $T$ and every concept used therein can be represented by objects from the abstract realm. Furthermore, that representation is a natural fit for ante rem mathematical structuralism. The main tenet of structuralism is that mathematical objects don’t have intrinsic properties, only relational ones in virtue of the roles they play in a structure. We can completely assimilate the structuralists’ structures to theories as we’ve defined them here. That seems justifiable: an abstract object like $2_{\mathfrak{pnt}}$ is akin to a role, being entirely defined by properties it is bound to exemplify in the structure.

Set-theoretic definitions in the von Neumann and Zemerlo pictures tell us that both $\{\{\emptyset_\mathfrak{st}\},\emptyset_\mathfrak{st}\}$ and $\{\{\emptyset_\mathfrak{st}\}\}$ (with $ST$ some set theory) can play the "PNT $2$ role" with the right bridging function. This was taken as a paradigmatic motive for structuralism by Benacerraf, who noted that reducing number theory to set theory could lead to awkward questions ("how many elements does the set $2$ have?"). This is dealt with handily in Zalta’s object theory since there’s never any question of equating those objects. Two mathematical term cannot be equal if they stem from different theories.

Of course we can develop a model-theoretic arsenal to define useful concepts such as reduction or embedding (unlike traditional model theory however, it will need to do without set theory, which it precedes in the construction). For instance, $PNT$ could be said to be embedded within $ST$ with the right bridging laws, which would allow us to say something like \[2_{\mathfrak{pnt}}\hookrightarrow_{vN}\{\{\emptyset_\mathfrak{st}\},\emptyset_\mathfrak{st}\}\] where $vN$ represents the von Neumann embedding of natural numbers into set theory. If we use the notation that for any element $x$ of $PNT$ (be it an individual, predicate or formula), our bridging laws say that \[x \hookrightarrow_{vN} x^*\], then the embedding could simply be stated as \[\forall \phi (\mathfrak{pnt} \vDash \phi \rightarrow \mathfrak{st} \vDash \phi^*)\] $PNT$ is embedded in $ST$ because the "translation" of any true proposition in $PNT$ into the $ST$ language is also true in $ST$.

Crucially, embedding does not only allow mathematical objects from other theories to fulfil roles, but any object. That includes abstract objects. For example, if logicism is true, there is no purely mathematical object and the basic elements of mathematics, be they numbers, sets or categories, should be built from purely logical grounds. In that instance, we should exercise care not to mix up the "PNT-numbers" (abstract objects which are elements from that theory) with "logicist-natural-numbers" (logically constructed abstract objects which play the PNT-number role).

That also applies to concrete objects: we are used to saying that concrete objects, together with their properties, can be modelled by a mathematical theory. Yet, Frege’s "Julius Caesar problem" is thoroughly defused. We know that Julius Caesar cannot be a number, because he is a concrete object. Even though he exemplified properties that made it appear he fulfilled a PNT-number role, that would be entirely distinct from being a "PNT-number".

Another benefit of assimilating mathematical structures to Zalta’s theories is to avoid lots of vexing puzzles thanks to the principle of dual predication.

Some philosophers have argued that numbers can’t only stand in relation to other mathematical concepts; for instance $4$ is not just the successor of $3$, it’s also the number of Dr. March’s daughters. In our model, that doesn’t raise any issue. $4$ exemplifies being the number of the good doctor’s daughers, but it encodes being the number that follows $3$. Abstracts objects like $4$ are entirely characterized by properties they encode, hence in a sense it is correct to say that numbers only "have" relational properties if that ambiguous term is associated to the principal predication for abstract objects: encoding.

In a similar vein, pesky paradoxes can be dispelled. Let’s say some mathematical object "has" exactly four properties, according to the relevant theory. Then it also has the property of "having four properties", which means it’s got a fifth property, hence a paradox. But although the constitutive qualities of an abstract object are expressed via encoding, the property of encoding $n$ of them is exemplified, hence the paradox cannot rear its ugly head.

Foundational options

These developments help us represent mathematical theories in our system. But we are still confronted with a plenitudinous, flat ontology: every mathematical structure, either known or undiscovered, exists, necessarily.

That problem is reminiscent of the crisis of foundations that befell mathematics at the turn of the twentieth century. Kant’s idea had briefly become a consensual ground, whereby mathematical truths were both synthetic and a priori, knowable as they were through mysterious "intuitions" that were characteristic of our cognition. In that view, the Euclidean axiom of parallels, for instance, is somehow "self-evident" (the axiom states that for any line $L$, and any point off that line $p$, there is a unique parallel line running through $p$ that never meets $L$). According to Kant, it cannot be proved and does not need to be (which is convenient, since every attempt to derive it from other axioms had failed). Instead we have some non-empirical knowledge of it from our intutive representation of space.

Such views were severely compromised in the nineteenth century, when mathematicians’ work led to spectacular results. One cannot under-estimate the impact of non-Euclidean geometries. Gauss and Lobashevsky, among others, developed an alternate geometry that is broadly similar to the familiar Euclidean one but either denies or amends the axiom of parallels. What could be the status of the resulting theorems? Kant thought we cannot prove the axiom of parallels because it is intuitive. As it turns out we cannot prove it because it’s not a necessary consequence of the other axioms at all. It still seemed possible to salvage Kantian intuitions about space and time as a core, with non-intuitive concepts being written off as notational conventions or analogies (the way a 2D curved space can be interpreted as an object in a flat 3D space). Yet continuing developments rendered that position more and more untenable. First, the ancient priority between geometry and arithmetic was turned on its head by modern algebra. Calculus had relied for centuries on the obscure usage of "infinitesimals", but rigorous derivation of the real numbers via Dedekind cuts or Cauchy sequences gave it arithmetic roots instead. Riemann formalized non-Euclidean geometries by giving them a basis in differential algebra. And Cantor’s studies of sets and infinities only cemented the advent of a "pure mathematics" divorced from empirical experience. Besides, the development of physics at the turn of the century, with relativistic and quantum concepts supplanting Newton’s as the most fruitful representations of the world, made the mere possibility of an "intuitive core" quickly recede into the horizon.

Letting go of the Kantian anchor was welcomed by those who embraced the total freedom mathematicians afforded themselves during that era. Didn’t Cantor say that the very essence of mathematics "lies in its freedom"? But this anything-goes approach was jeopardized by the ever-growing number of paradoxes, courtesy of Cantor himself, Hilbert, Russell and others. With set theory thus undermined, right when it developed into a possible common standard for all of mathematics, the question of foundations took center stage for ensuing decades.

At this point, the historiography typically introduces three schools: logicism intuitionism and formalism.

Logicism

originates, as so many things, in Gottlob Frege’s work. Frege rejected the implicit tenet of the anything-goes era, whereby any entity we may need can be conjured into existence (for instance, solutions to a previously unsolvable equation). Since contradictions are sometimes far from evident, it is not sufficient to just use trial and error. Frege assumed that the only way to ensure consistency was to find a consistent ground from which to derive mathematics. Logicism strives to prove that all mathematical truths can be derived from logical truths.

This constitutes a break both from the freewheeling practice of nineteenth-century set theory, and from Kantian views: mathematics cannot be a form of synthetic knowledge if it is grounded in logic, with logical inference a paradigmatic example of analytic thinking (if one accepts such distinctions). The mathematician’s job is then clarified: it is not to create new pieces for some mental games, but to discover which theories and entities can be deduced from logical laws. Frege describes his goal as "preventing anything intuitive from penetrating [his theory] unnoticed"...

One advantage of logicism that’s often touted versus "plain-vanilla" Platonism is its epistemology: the lack of reliance on synthetic knowledge arguably removes the need for a transmission mechanism between mathematical abstracta and the mind. On the metaphysical side, logicism is often associated with a realist stance, and indeed Frege and Russell’s position on the matter casts a long shadow. Yet there are attempts to expound a nominalist logicism out there that are convincing enough.

Intuitionism

holds, according to its founder, L.E.J. Brouwer, that mathematical truths are grounded in the human mind. On its face, this may seem like a simple retread of Kant’s conceptualism. Intuitionism certainly draws inspiration from Kantian views about mathematical objects: where Kant evokes the a priori intuitions of space and time that ground our mathematical concepts, Brouwers describes mathematics as a "languageless activity of the mind" that relies on our perception of time. This temporality establishes both a core of purely intuitive mathematics: the finite natural numbers (elements in a temporal sequence), and the key criterion for mathematical reasoning: that it be constructive. In that view, there is no abstract truth nor existence: a mathematical object can be said to exist when it has been constructed, i.e. when we can exhibit an example of it.

Like Frege, Brouwer thus proposes a core of "self-evident" truths and a standard for valid reasoning, and hopes this will naturally enforce consistency. Unlike Frege’s however, the intuitionist principles entail a massive pruning of mathematics, to the point where classical logic is invalid. Brouwer completely turns logicism on its head with his insistence that language has nothing to do with mathematics. To him, the reliance on formal systems is what ails modern mathematics and symbolic notations are an illegitimate (and content-less) surrogate for the intuitive content of mathematical concepts. This philosophical conceit is supported by a historical narrative: the laws of classical logic would have emerged from ancient mathematics as handy properties of derivations that were indeed constructively valid for finitary arithmetic or geometry, however, these shorthands are completely invalid for the newfangled objects of higher mathematics, in particular infinities. Mathematicians mistakenly kept using the same logical rules, but are unable to cash them out in the form of actual constructions. For Brouwer, the justification is in the construction, not the logical derivation.

It is not easy to describe "the" ontology implied by intuitionism. Brouwer himself was as clear as it gets about his anti-realism, but intuitionism at large proved itself disjointed to the point of mild schizophrenia. Other prominent intuitionists eschew metaphysics to focus on developing an intuitionist logic (Heyting) or to re-set the battlefied into semantics (Dummett).

Formalism

is indelibly linked to its foremost supporter, David Hilbert, and the eponymous programme. Hilbert’s reaction to the paradoxes he contributed to unearth ran, in many respects, completely opposite the two other "schools". Both logicism and intuitionism, as we’ve seen, provide an epistemically accessible core (respectively, a set of logical axioms and an intuitive notion of finite integers) and a method of proof (respectively, rules of inference and principles of construction). These foundations are meant to define what mathematics is (and to exclude all that cannot be built upon them). The hoped-for byproducts are the consistency of thus-built theories, and the quelling of an excessive creativity that drove some areas of mathematics into paradoxical dead-ends.

Hilbert faces the conundrum from the other end. Rather than discovering a foundation to govern how mathematicians do their job, we should embrace their unlimited creativity and just endeavour to do mathematics better. Specifically, he takes that to entail (1) working with fully axiomatizable theories, and (2) proving that those theories are consistent (i.e. that we cannot infer a contradiction from the axioms). The existence of objects and the truth of sentences is then simply admitted once we’re working in a safe environment. As Hilbert wrote to Frege, we shouldn’t search for "obviously" true axioms and hope that consistent theories spring out; we should search for consistent axioms, and consider that all the resulting theorems are true by virtue of that consistency.

In a sense, Hilbert answers the foundational question by the negative. Although he is no simplistic deflationist and like Brouwer, admits that natural number arithmetic forms a contentual core, the formalist programme does tend to de-ontologize existence questions. Frege thought that mathematicians have discovered laws and objects that exist in another realm. Brouwer, that they are perpetually constructing them through concrete mental processes. To Hilbert, anything beyond the contentual core of finitist mathematics is like a game played with simple rules by way of symbols (scratches on papers, characters on a screen...). Finitary statements should be provable by way of a detour into this non-finitary ("ideal", "contentless") mathematics, thus making the viewpoint less vulnerable to Fregean onslaughts about applicability.

The mathematician’s true role is then not in finding proofs, which in this formalized, axiomatic mindset appears as a kind of mechanical operation. Instead, it is to ensure the safety of the playing field by making sure that (1) cutting-edge mathematics are expressed in the formal system of a safe, well-know theory, typically arithmetic or set theory and (2) that we can prove those safe havens, in their simplest axiomatized guises, to be complete and consistent. This moves the real work towards metamathematics: the use of mathematical methods to prove results about mathematical theories themselves: whether a problem is solvable, whether adding a certain axiom jeopardizes a theory’s consistency, etc.

Crisis redux?

Which of those three ended up gaining prominence? Well, none.

The proof by Gödel of his incompleteness theorems, in many ways the defining event in metamathematics, dampened the enthusiasm of each school’s proponents. It wasn’t just a new act in the unfolding crisis that Dedekind and Cantor inaugurated; in a sense, it opened a whole new crisis of its own. Still, I shall avoid emphatic statements about the "end" of this and that... None of these programmes came out unscathed, yet work has continued. Still, they’re far from being the be-all-end-all some hoped they would be one century ago. Then why put so much thought into them? Simply put, the ideas at the center of each one keep informing mathematical work, even as practitioners may be unaware. It seems important to me to understand the original, if failed, version of each position to adequately judge the foundational approaches that are compatible with our formalized ontology.

But let’s come back to Gödel. His two theorems make use of three concepts in metamathematics:

a theory is called recursively axiomatizable (RA) if an algorithm could enumerate its axioms. In particular, a theory with a finite set of axioms is RA. A countable set of axioms also defines a RA theory (for instance, in first-order Peano arithmetic, saying that every number has a successor is actually making an infinity of statements, but a countable one).
a consistent theory generates no contradiction: one cannot derive both $A$ and $\neg A$ from its axioms.
in a complete theory, one can always derive either $A$ or $\neg A$ from the axioms. We can say that every statement is decidable. (This is a notion of syntactic completeness, unlike the completeness of a logical system, which is a semantic notion: the equivalence between truth and provability).

The incompleteness theorems can be stated this way:

If $\mathfrak{t}$ is a theory that is
- consistent,
- RA,
- able to express elementary arithmetic,
then
1. $\mathfrak{t}$ is incomplete.
2. $\mathfrak{t}$’s consistency cannot be derived within $\mathfrak{t}$.

Sketching the proof of the first theorem is quite illuminating, as it is essentially an update of the old riddle about the Cretean who says "All Creteans are liars" (is he lying?).

Qualitatively, Gödel arithmetizes syntax in such a way he can create a similar, self-referential sentence in the language of arithmetic, say: $G="\nvdash G"$ (a sentence $G$ that says "$G$ cannot be proved"). If $G$ was provable, then the theory would not be consistent, so we know that $G$ is not provable. Similarly, $\neg G$ cannot be proved, because it would entail that $G$ can be proved, in contradiction with what precedes. We’ve just created a Gödel sentence that shows arithmetic to be incomplete!

The importance of these results goes well beyond metamathematics and cannot be overstated. Their reach is near-universal, towards the philosophy of mind, the philosophy of language, ethics, and much more. Let’s focus here on the fallout regarding the foundations of mathematics.

Frege’s logicism had already been through quite a shock, as did Frege himself, when Russell showed his system to be inconsistent. Basic Law V, an axiom of universal comprehension famously led to a contradiction that can be exemplified by $E=\{x,x\notin x\}$, the set of elements that are not in themselves. If $E \in E$, then by definition of that set, $E \notin E$, and vice versa. That’s a direct contradiction. Frege got discouraged, but not Russell, who teamed up with Whitehead and set out to continue the logicist mission on better grounds this time. This took the form of Principia Mathematica, an ambitious, if unfinished, piece of foundational work. It relied on type theory to tame the paradoxes in Frege’s model, building a ramified structure that ensure that sets, sets of sets, and so on, could not be treated on equal footing, each level replicating the axioms and structures of the one before it. Since entities on each level can be quantified upon, this is an example of higher-order logic, and for the first time, the same doubt we’ve evoked before were levied against the authors: is it really logic at all? To make their system consistent, Russell and Whitehead had to add an axiom of infinity, an axiom of choice and an axiom of reducibility, which appeared suspiciously as mathematical rather than logical statements.

But Gödel’s first theorem posed a more fundamental challenge. The kind of theory that logicists wanted to develop was naturally RA (they wanted any set of axioms to be "simple enough" and any proof to be obtained mechanically) and consistent. Since the goal was to derive all useful mathematics, it is quite clear that simple arithmetic should be part of it: numbers are often used in mathematics, to say the least. Per the first incompleteness theorem, the logicist theory is (syntactically) incomplete: there is at least a statement $\phi$ such that neither $\phi$ nor its negation can be derived from the axioms using the rules of inference: \[\nvdash \phi \wedge \nvdash \neg \phi\] For a theory expressed in first-order logic, this entails (since first-order logic is semantically complete) that neither $\phi$ nor its negation are true in every model: \[\nVdash \phi \wedge \nVdash \neg \phi\] In a given model, $\phi$ needs to be true or false, so the conclusion is that there are necessarily some models were it is true, others where it is false. So the hope for a first-order theory to pin down a unique set of objects like the natural numbers is dashed. As a matter of fact, it is possible to define plenty of non-standard semantics over first-order arithmetic. Gödel’s theorem tells us there is no way to amend the axiom system so that it’s not the case anymore.

Things are hardly better in higher-order logic, for converse reasons. It is demonstrably possible, with full semantics, to define higher-order axiom systems with a single model (up to isomorphism). Case in point: second-order arithmetic admits only $\mathbb{N}$ (and its isomorphs) as models. The consequence is that higher-order logic is incomplete. In that case the undecidable statement $\phi$ is either universally true or false (since there’s effectively only one model), but cannot be proved or disproved. Which means the mechanical application of rules of inference fails to demonstrate the whole truth, and there is actual synthetic knowledge. It appears that logicists have to abandon either their goal of deriving exactly the desired objects from first principles (that is, an axiom system that practically admits one semantics), or the idea that every truth is analytic. This is a very severe blow to the original programme, but not the end of the line. Neo-logicism has been conceived as a way to sidestep these problems, in various fashions, such as expanding the scope of what "logic" means, or changing the way we understand "reduction" or "analyticity".

Intuitionism, in its purest form, seems immune to Gödel’s theorems. Brouwer’s allergy to the formalization of mathematical language could suggest that such results would be of little import to his ideas. Gödel doesn’t have much to say about intuitions indeed and saw intuitionism’s lack of precision as a serious drawback. In principle, Brouwer had expressed sympathy for the idea of unprovable statements: after all, the rigors of his constructive tenets made many classic theorems constructively unprovable (and of course their negation wasn’t provable either). However, intuitionism also equates provability with truth, which is deeply contradictory with Gödel’s insights. Ultimately the cursory manner in which intuitionists have developed their philosophy and their metamathematics have doomed its pure, Brouwerian version to retreat from mathematics into a niche of phenomenology. A contrario, some intuitionist logicians did bring great precision to hitherto vague concepts, but did so by adopting Brouwer’s rivals’ conceits, like the primacy of logic or the separation between syntax and semantics.

Yet it is for formalism that Gödel’s theorems were truly devastating. The first theorem denies Hilbert’s deep-seated belief that every problem is solvable. More precisely, the principle that any truth from infinitist, ideal mathematics should be expressible and provable in a safe theory like RA arithmetic is irremediably false. The second theorem is the final nail in the coffin: Hilbert’s programme had aimed to demonstrate the consistency of safe theories "from the inside", by virtue of their own axiomatic system. That is out of the question now: we can prove that arithmetic is consistent using set theory. But we’re just kicking the can, and of course we cannot prove the consistency of a set theory from that very same set theory, etc. In the end, formalism may be more damaged than logicism and intuitionism in its lofty goals, but in its apparent failure, it has also been the most fertile ground for later mathematicians. Foundational questions in mathematics today are less ambitious, but are couched in terms of metamathematical (proof-theoretic and model-theoretic) problematics that are a direct inheritance from formalist investigations.

A way forward

Let’s recapitulate where we stand post-Gödel.

We will never be able to prove that mathematics is consistent. We think it is, or at least hope it is. But a clever future mathematician may well prove the axioms we use to be inconsistent. At which point, we may change them, or add new axioms to remove that specific inconsistency... But we can never prove the outcome to be consistent.
There is no "simple" set of axioms from which we can deduce analytically all mathematical truths or solve all problems.

That is certainly demoralizing to anyone who’s felt bursts of enthusiasm for the rigor, simplicity and beauty that a formal reduction brings to any field. If even mathematics falls victim to this very twentieth-century type of malaise, what hope is there left? But no, all is not lost. Not only did humans keep doing mathematics after Gödel (as they had before him), the foundational topics still form a vibrant field of mathematics.

Nowadays, foundational systems are associated with either set theory or category theory.

The prominence of set theory as a mean to formalize and unify all of mathematics (a remarkable fact which only seems banal out of sheer habit) has endured for a century and a half now. The most popular axiomatic system is ZFC (Zemerlo-Fraenkel with the axiom of choice), extended by large cardinals. From these axioms plus definitions, most known theorems have been derived. This partial fulfilment of Hilbert’s formalist plans, which the Bourbaki group, among others, aimed for, is now greatly helped by computerized proof generators.

More recently, category theory emerged from algebraic topology, but soon entered the foundational arena. In a nutshell, category theory considers structures and morphisms between them (structure-preserving functions) as fundamental (rather than the membership of elements within sets). It cannot be described as a rival, let alone a replacement for set theory, as the two are deeply interconnected and can be handily derived from one another. But the choice of one as a foundation is often made pragmatically, based on what it helps us achieve for a given theory.

Foundations are therefore as important as ever, albeit in a different way from what was anticipated.

With the climb-down from earlier foundational ambitions, it seems we cannot ever fully "solve" mathematics epistemically (we can’t derive all knowledge from knowing a few axioms) or metaphysically (foundations can give us a good model of how to represent numbers with sets, but no one expects they’ll tell us that numbers are sets). Yet foundations play an incredibly important role in illuminating notions by making them more precise, more rigorous, and safer to use. So far at least, the set-theoretic formulation of arithmetic doesn’t provide much help to the practicing mathematician: the Peano axioms or other purely arithmetical systems are more than enough, and nobody "does arithmetic" in ZFC. In contrast, the set-theoretic foundation of real numbers by Dedekind is extremely illuminating: it’s not just a surrogate that reflects known features of the reals, it’s an improvement that allows mathematicians to rigorously prove theorems which henceforth had stemmed from age-old geometric reasoning. The hope for such illumination is a powerful motive to connect theories to a foundational system.
Foundations also provide a fertile ground for metamathematics. Gödel’s theorems are often seen as weakening the idea of foundations, but a foundational set theory (Principia or ZFC) is needed to prove it! Only it can act as a "final court of appeal" to determine the validity of a proof: these days validity typically means that a proof can be formalized in the language of ZFC. And even if we can’t prove the consistency of our foundations using only their own resources, we can prove relative consistency in another framework (for instance, ZFC is provably consistent in ZF), thus creating some approximation of Hilbert’s "safe environment". The field of reverse mathematics seeks to work back from known theorems towards foundations to ascertain which minimal group of axioms is needed to support the corresponding theory.

As I’ve mentioned above, this Brave New World of mathematics can leave those who appreciate simplicity and rigor slightly traumatized. But it does not invalidate in the least the construction of our formalized ontology. That is why I clearly separated the ontological from the foundational questions. If classical logicism was true, there wouldn’t be any theory object since every truth could be derived from the laws of logic through simple inference. If formalism was true, there could ultimately be a single theory object that would contain all mathematical truths (including its own consistency).

None of that came to pass; where does that leave us?

Remember we’ve defined a theory as \[\mathfrak{t} = \imath x ( \mathbf{A}x \wedge \forall P ( xP \equiv \exists \phi ( P=[\lambda y \phi] \wedge \alpha_1,...\alpha_n \vdash_T \phi_T ) ))\] That is the abstract object that encodes every theorem derived from the axiom schema $(\alpha_i)$. From the comprehension principle we know there is mathematical plenitude: for any class of well-formed formulae $(\alpha_i)$ we can build the corresponding theory object. Of course, the consistency of the resulting theories is in no way guaranteed. Since any proposition can be derived from inconsistent axioms, such systems simply point to a theory that encodes every well-formed formula of the language.

One outcome from Gödel’s theorem is that we cannot ever distinguish one axiom schema from which to derive all useful mathematics. In the language of our formalized ontology: there is no single abstract object representing mathematics. There is an abstract object $\mathfrak{pnt}$ that encodes the Peano axioms and all resulting theorems, but not every truth from arithmetic. There is another abstract object that encodes every truth from arithmetic, but it’s axiom schema is infinite and not recursively enumerable.

Using bridging laws, embedding, and in general, re-constructing model-theoretic notions, we can simplify the picture with a network of relationships, for instance the fact that a model of Peano arithmetic can be derived from the ZFC axioms by taking \[\begin{aligned} 0_{\mathfrak{pnt}} &\hookrightarrow_{vN}\emptyset_\mathfrak{zfc}\\ S_{\mathfrak{pnt}} &\hookrightarrow_{vN}[\lambda x \{ x \}]_\mathfrak{zfc} \end{aligned}\] However there is no ontological compression going on: PNT numbers are distinct from ZFC sets, as expounded when we argued in favor of structuralism: they just represent a role in their original structure.

Most Hilbertian ambitions are gone, courtesy of Gödel, but this general outlook of mathematics can be seen as distinctly neo-formalist: a set of syntactically correct sentences can be used to generate a theory, and it is for metamathematics to teach us about the outcome. The universe is only playing a "formal game" in generating these abstract objects.

It could thus seem like there’s no attempt to distinguish the theory of mathematics and we must stay completely agnostic in that regard, perhaps taking a Quinean cue and using the physical world as a litmus test. That’s not necessarily the case though, as this formalized ontology is also fully conceivable as a neo-logicist enterprise. Zalta and Linsky have proposed ways to derive Peano arithmetic from the bare structure of the ontology. Armed with quasi-logical principles, one could derive basic theories and define "real" mathematics as the class of abstract theory objects that’s compatible with (per some model-theoretic notion to be defined) this neo-logicist basis. That development is as vulnerable to the usual critiques of neo-logicism as its brethren though: we may be accused of smuggling set theory in through our use of higher-order logic.

This is not the place to settle such questions — I am content with having shown, hopefully, that the formalized approach herein advanced is sufficiently flexible to not close any doors in the current mathematical and philosophical debates.