## 1. Introduction



Deductivism is a philosophy of mathematics that was popular in the
late nineteenth and early twentieth century. It understands a
mathematical sentence \(s\) as expressing the claim that \(s\)
deductively follows from appropriate axioms. For instance,
deductivists might construe "\(2 + 2 = 4\)" as



> 
> 
> the sentence "\(2 + 2 = 4\)" deductively follows from the
> axioms of arithmetic,
> 
> 
> 



or "2 is prime" as



> 
> 
> the sentence "2 is prime" deductively follows from the
> axioms of arithmetic,
> 
> 
> 



or, for a non-arithmetical example, "every group \(G\) has
exactly one identity element" as



> 
> 
> the sentence "every group \(G\) has exactly one identity
> element" deductively follows from the axioms of group
> theory.
> 
> 
> 



Deductivists thus reject the surface syntax of mathematics. They do
not take "2 is prime" to attribute a property (primeness)
to an entity (the number 2); for them, it states that the sentence
"2 is prime" follows from some axioms.



This entry will assess deductivism's strengths and weaknesses,
discuss its turn-of-the-twentieth-century heyday, and compare it to
contemporary versions of structuralism.


## 2. Motivation



Deductivism is interesting and important for a host of reasons. The
first is that it gives proof, indisputably of paramount importance in
mathematics, pride of place. Mathematicians take a result to be
established only if it has been proved. It is natural to explore what
happens if we take this one step further and take mathematical truth
to be, not just *established*, but *constituted*, by the
existence of a (possibly not yet known) proof.



Second, many mathematicians profess to believe in deductivism or
something like it (Maddy 2022: 9). Ask mathematicians what their
subject is about and at least some will reply that it is about
"what can be deduced from the axioms". Naturally, a slogan
like this is not a worked-out philosophy, and is more often
encountered in conversation than in print. Still, if the slogan and
the sentiment behind it point to any specific philosophy, surely it is
to deductivism. Moreover, most parts of mathematics are explicitly
understood as exploring consequences of stipulated
 axioms,[1]
 which deductivism spells out in its own way and generalises to all
mathematics. Not only that, but as
 SS3
 will briefly document, many influential philosophers of mathematics
from the late nineteenth to the early twentieth century were, or could
be interpreted as, deductivists.



Third, and this might go some way towards explaining the second point,
deductivism is seen as philosophically "clean" (Rheinwald
1984: 49) or a way to "avoid philosophical quicksand"
(Maddy 2022: 9). On the epistemological side, the deductivist's
main task is to account for the epistemology of deductive
logic--there is no further specific problem of mathematical
epistemology. On the metaphysical side, the deductivist construal
appears not to presuppose any abstract mathematical objects, often
seen as problematic. A further point in deductivism's favour is
that it respects the presumed objectivity of mathematics, because what
follows from the axioms is an objective matter, independent of us.
Moreover, deductivism seems to offer a promising account of
applications: whenever you find a physical system that satisfies the
axioms of a particular branch of mathematics, the branch's
theorems must apply to the system. (We shall return to these alleged
attractions below.)



The fourth and final reason is that deductivism and structuralism are
related. Indeed, deductivism is seen by some (e.g., Bostock 2009: 183)
as a syntactic version of eliminative structuralism. And
structuralism, in its various guises, and its various problems
notwithstanding, is one of the most attractive metaphysics of
mathematics on the market
 today.[2]



Deductivism and eliminative structuralism share some general features.
Roughly speaking, both understand a mathematical claim \(p\) as
"the axioms imply that \(p\)" (where "the
axioms" may be understood as those of the germane branch of
mathematics or otherwise--for different versions of deductivism,
see
 SS4).
 Deductivism understands the relevant implication deductively,
eliminative structuralism semantically. For deductivists, the relevant
notion of implication or consequence is the syntactic one which in
logic is denoted by "\(\vdash\)"; to say that \(\Gamma
\vdash \delta\) is to say that the sentence \(\delta\) is derivable
from \(\Gamma\) in some specified deductive system. For eliminative
structuralists, the relevant notion of implication or consequence is
instead the semantic one which in logic is denoted by
"\(\vDash\)". Eliminative structuralists, for example,
construe "\(2 + 2 = 4\)" as a claim about what holds in
any structure in which the axioms of arithmetic are
 satisfied.[3]



When we dig down into the details, however, some clear differences
emerge. Deductivists take mathematical claims to be about sentences,
whereas eliminative structuralists take them to be generalisations
over structures. Unlike deductivists, eliminative structuralists
therefore owe us an account of the metaphysics of structures.
Eliminative structuralists also usually couple their account with the
existential assertion that a structure of the right kind exists (or
could exist), whereas deductivists often omit the corresponding claim
that the axioms are consistent. One well-known version of eliminative
structuralism, first articulated by Putnam (1967b) and developed by
Hellman (1989), is modal, whereas deductivism is usually not modal.
(Although see
 SS5
 for how to spell out deductivism in terms of a primitive modal
operator.) Finally, eliminative structuralism is but one of the many
forms of structuralism around today. Some, such as Hellman's
version, have much in common with the eliminative variety closest to
deductivism; others, such as Shapiro's ante rem structuralism
(Shapiro 1997), less so.



Still, despite these differences, there are some clear historical and
philosophical continuities between the two, so that getting clear
about deductivism's flaws can help us better appreciate the
virtues of a broadly structuralist approach. What is certainly true is
that prior to Godel's 1929 Completeness Theorem for
first-order logic (published as a paper the following year), the
differences between deductivism and eliminative structuralism were not
clearly understood. This point will be illustrated in the historical
 SS3.
 Also relevant to the story is Godel's 1931 First
Incompleteness Theorem, which will form the basis of a key objection
to deductivism in
 SS9.[4]



A final point of connection is that, when the background logic is
axiomatizable, eliminative structuralism and deductivism offer an
extensionally equivalent account of mathematical
 truth.[5]
 The simple reason is that, for an axiomatizable logic, such as
first-order logic, semantic implication is extensionally equivalent to
deductive inference. So suppose you are tempted by eliminative
structuralism in the philosophy of mathematics and use first-order
logic as your default
 logic.[6]
 You will then take \(p\) to be a mathematical truth exactly when a
(first-order) deductivist does. Avoiding a "lapse into a
semantic equivalent of deductivism" was precisely one of the
reasons Hellman chose a second-order framework for his modal version
of eliminative structuralism (Hellman 1989: 18-19). As
explained, however, eliminative structuralism and deductivism are not
equivalent in any stronger sense than this extensional one, even when
their shared background logic is axiomatisable.



A terminological note: deductivism is sometimes called
"if-thenism". However, "if-thenism" sounds
more like the genus whose syntactic and semantic species are
deductivism and eliminative structuralism respectively. Rheinwald
(1984: 49) uses the name "implicationism" for the genus,
reserving "syntactic implicationism" for deductivism and
"semantic implicationism" for structuralism. This
terminology manages to cover both genus and species and also has the
great merit of transparency. The present entry sticks to
"deductivism" and "structuralism" for the two
species, because these names are the more usual ones today (and are
shorter).


## 3. Historical Accounts



This section covers four historical accounts of deductivism, advocated
by Russell
 (SS3.1),
 Hilbert
 (SS3.2),
 Pasch
 (SS3.3),
 and Curry
 (SS3.4)
 respectively. The first three are pre-Godelian; the fourth is a
post-Godelian attempt to defend deductivism.



Attributing deductivism to older authors is harder than it may
initially appear, and the existing literature offers no attempts to do
so in a comparative way. The difficulty is owed to several conflating
factors. The main one is that, before Godel, there was no clear
distinction between semantic and syntactic consequence. And
terminological confusion does not help: Curry, for example, refers to
his deductivism as "formalism", a label usually reserved
for a different position. This article thus focuses on classifying
authors' substantive claims. A comparison with Brouwer
 (SS3.5),
 who emphasised proof in mathematics but was no deductivist, will also
be relevant.


### 3.1 Russell



Russell's *The Principles of Mathematics* was published
in 1903. The book's very first sentence contains Russell's
credo about what mathematics is:



> 
> 
> Pure mathematics is the class of all propositions of the form
> "\(p\) implies \(q\)", where \(p\) and \(q\) are
> propositions containing one or more variables, the same in the two
> propositions, and neither \(p\) nor \(q\) contains any constants
> except logical constants (Russell 1903 [2009: 3]).
> 
> 
> 



It may seem odd that the propositions \(p\) and \(q\) are supposed to
contain at least one variable. Yet, according to Russell, even
"\(1+1=2\)" contains variables once made fully
explicit--namely "If \(x\) is one and \(y\) is one, and
\(x\) differs from \(y\), then \(x\) and \(y\) are two" (Russell
1903 [2009: 6]).



Is Russell's credo a form of deductivism or structuralism, or
some admixture of the two? At the turn of the twentieth century, no
one clearly distinguished semantics and syntax, and Russell was no
exception. This led Putnam to claim that what Russell really had in
mind was, in our terminology, eliminative structuralism. (Putnam
christened the structuralism he attributed to Russell
"if-thenism"; as explained in
 SS2,
 this entry avoids the term.) He urged:



> 
> 
> What he meant was not, of course, that all well formed formulas in
> mathematics have a horseshoe as the main connective! but that
> mathematicians are in the business of showing that *if* there
> is any structure which satisfies such-and-such axioms (e.g., the
> axioms of group theory), *then* that structure satisfies
> such-and-such further statements (some theorems of group theory or
> other) (Putnam 1967a [1975: 20]).
> 
> 
> 



However, Putnam's structuralist interpretation of Russell is
controversial, not least because a structuralist reading appears to
explicitly contradict the deductivist wording of several passages
(Kraal 2014: 1498). For example, with pure mathematics defined in the
above way, Russell went on to assert that



> 
> 
> [b]y the help of ten principles of deduction and ten other premisses
> of a general logical nature (e.g., "implication is a
> relation"), all mathematics can be strictly and formally
> deduced; and all the entities that occur in mathematics can be defined
> in terms of those that occur in the above twenty premisses (Russell
> 1903 [2009: 3]).
> 
> 
> 



The emphasis here is clearly on deduction.



To prove the above deducibility claim is the stated objective of
*The Principles of Mathematics* (Russell 1903 [2009: xiii]).
And like everyone else at the time, Russell tended to assume that
logical implication is a deductive relation. Thus, even if it was not
his intention to differentiate semantics and syntax sharply, as he
historically was not in a position to do, his credo combined with the
equation of logical with deductive implication committed him to
deductivism. Hence *The Principles*' key doctrine is more
plausibly construed as a version of deductivism than eliminative
 structuralism.[7]


### 3.2 Hilbert



Other contemporaries of Russell's also displayed deductivist
leanings. Resnik classified Hilbert's work of around 1900 as a
"deductivist period" (1980: 105). However, Hilbert at the
time endorsed what can now be separated out as deductivism on the one
hand and some form of proto-structuralism on the other. He made no
distinction between the two, as we would today, because he assumed
complete axiomatisations and proof systems were available.



Consider Hilbert's 1900 "About the Concept of
Number". In the introduction, Hilbert declares that he intends
to apply the axiomatic method, known from geometry, to the concept of
number (1900a: 181). He contrasts the axiomatic method with the *ad
hoc* approach usually found in the literature of the time. The
purpose of his axiomatic approach is to achieve the "final
presentation and logical certainty of the content of our
understanding" (ibid). He begins by setting out eighteen axioms
for the real numbers. To give a flavour, the first axiom is



> 
> 
> From the number \(a\) and the number \(b\) results through
> "addition" a specific number \(c\), formally: \(a + b =
> c\) or \(c = a + b\). (1900a: 18)
> 
> 
> 



Hilbert then asserts that propositions about the real numbers are only
valid (*gultig*) if they can be proven from the set of
axioms in a finite number of steps (1900a: 184).



The reason for this insistence is explained in his famous lecture to
the International Congress of Mathematicians in Paris in the same year
(published as "Mathematical Problems"). Here Hilbert
expresses the "absolute conviction" that any mathematical
problem, no matter how difficult, must be solvable "through a
finite number of purely logical steps" (1900b: 261). He
maintains that this conviction is shared by all mathematicians and is
a major source of motivation; but he also notes that we have no proof
that the finite-derivability claim is true (1900b: 262). Despite the
absence of such a proof, he takes this conviction to be the natural
*modus operandi* of the working mathematician. In
"Mathematical Problems", Hilbert also maintains that, to
the extent that new concepts are introduced in mathematics--say,
due to their usefulness in scientific explanations--those
concepts need to be axiomatized to make them useful for deduction:



> 
> 
> [Their introduction] gives birth to the task for mathematics to
> research the principles associated with the new concept and then to
> fix them in a simple and complete axiom system in such a way that the
> new concepts and their usability *for deduction* is in no way
> inferior to the old arithmetical concepts. (1900b: 259, our
> translation and emphasis)
> 
> 
> 



At the same time as emphasising the importance of a deductive
framework, Hilbert endorsed proto-structuralist ideas. He mentions
"thinking of a system of objects", saying we "call
those objects numbers" and "denote them with *a*,
*b*, *c*" (1900a: 181). In his correspondence with
Frege one year earlier, for example, Hilbert noted that



> 
> 
> E.g., instead of points, think of a system of love, law, chimney-sweep
> ... which satisfies all axioms; then Pythagoras' theorem
> also applies to these things. Any theory can always be applied to
> infinitely many systems of basic elements. (Letter from 29 Dec 1899,
> Frege 1976 [1980: 42])
> 
> 
> 



Hilbert moved fluidly between both deductivism and proto-structuralism
by relying on the completeness assumption. For example, in his letter
to Frege dated 29 December 1899, he stated that if the axioms in an
axiom system do not contradict one another,



> 
> 
> then they are true, and the things defined by the axioms exist. This
> for me is the criterion of truth and existence. (Letter from 29 Dec
> 1899, Frege 1976 [1980: 42])
> 
> 
> 



Note that Hilbert does not maintain that the axioms are true if they
have a model--consistency and model-existence are equivalent only
if the axiom system is complete.



Hilbert at the time, then, was firmly committed to two views. Both
understand mathematical claims as true in virtue of implication from
axioms. One is that mathematical validity amounts to mathematical
provability with a finite number of steps from a finite number of
axioms. The second is that mathematics is not about specific objects
but that mathematical truth (in a particular domain) is defined by the
axioms.



We can also observe differences between Russell's and
Hilbert's varieties of deductivism. By asserting that



> 
> 
> all the entities that occur in mathematics can be defined in terms of
> those that occur in the above twenty premisses,
> 
> 
> 



Russell closely links deductivism to the claim that any non-logical
vocabulary in axioms and theorems is ultimately eliminable (1903
[2009: xliii]). Thus Russell combines deductivism with a core tenet of
logicism, namely that non-logical terms be definable by logical
ones.



For the turn-of-the-century Hilbert by contrast, numerals and other
non-logical terms (as we could call them) in our deductions remain
primitive. But although not eliminable, they are generic. Hilbert asks
us to imagine "a system of things, which we will call
numbers" (Hilbert 1900a: 181). The axioms then provide a
"precise and complete description of numbers"
"mutual relations" (Hilbert 1900a: 181). For as he
memorably put it, "one must be able to say at all
times--instead of points, straight lines, and
planes--tables, chairs and beer mugs" (Kennedy 1972: 133).
Here Hilbert's stance effectively derives from Pasch. Pasch had
called his method "a formalism" and maintained that
mathematical proofs must remain valid if the basic concepts are
replaced throughout "by any concepts or by meaningless
signs" (Schlimm 2010: 103).


### 3.3 Pasch



This brings us to Moritz Pasch himself, another contemporary of
Russell and Hilbert's who defended a flavour of deductivism.
Pasch is best known for his axiomatisation of geometry in his 1882
*Vorlesungen uber neuere Geometrie*. A key passage found
in the 1882 first edition and, with minimal changes, also in the 1926
second edition, reads as follows:



> 
> 
> Indeed, if *geometry is to be truly deductive [*deduktiv*],
> deduction [*Deduktion*] must be independent everywhere from the
> meaning of the geometric terms*, just as it must be independent of
> diagrams; only the relations between the geometric terms specified in
> the utilised theorems and definitions may be considered. *During
> the deduction [*Deduktion*] it is permitted and useful, but in
> no way necessary, to think of the meanings of the geometric
> terms*; in fact, if it is necessary to do so, this makes plain the
> gaps in the *deduction*, and under certain circumstances even
> the inadequacy of the theorems invoked in the proof (1882/1926, our
> translation and
>  emphasis).[8]
> 
> 
> 



This passage does two things: it maintains that the meaning of
geometrical terms has no role to play in geometry at its most
rigorous; and it emphasises proof. So it can be seen as advocating
deductivism or, possibly, formalism. "Formalism" here
means what is sometimes called "game formalism", i.e., the
thesis that mathematics is a mere play with symbols according to
defined rules. However, other passages in Pasch's corpus favour
a deductivist interpretation:



> 
> 
> *Mathematics* is a system with two parts that must be clearly
> distinguished. The first, the properly mathematical part, *is
> devoted entirely to deduction*. The second makes deduction
> possible by introducing and elucidating a *series of insights*
> that are to serve as material for deduction, preparing new material
> whenever the deductive part requires it. This second part, dwarfed by
> the properly mathematical one, can be called the pre-mathematical
> part, while I might refer to the other as the *rigid part*,
> since it forms a structure *characterized by the absolute rigor of
> its inference rules*. Though it is tempting to ignore how rigorous
> these rules are, that does not make them any less rigorous. *The
> rigid part is, as I said, mathematics proper. It proceeds by
> mathematical proof and mathematical definition.* (Pasch 1918
> [2010: 51], our emphasis)
> 
> 
> 



Pasch thus distinguished between mathematics and what we may call
"pre-mathematics". Mathematics is entirely deductive,
whereas pre-mathematics serves to motivate the "material for
deduction". Mathematics "dwarfs" pre-mathematics and
is characterised by "absolute rigor of its inference
rules", which is why Pasch calls it "rigid"
(*starr*). In light of the above passages, and others, Pasch
may reasonably be classified as a deductivist.



Pasch takes a notably different view from Hilbert as to what, among
all the possible sets of axioms a deductivist could adopt,
characterises a "good" set of axioms. Hilbert's
response to this question foreshadows his future formalism. In
"About the Concept of Number", he maintains that, once the
set of axioms has been proven to be consistent, any further doubt
about the existence of the concept number "loses all
justification" (Hilbert 1900a: 184). Classically, consistency is
a necessary condition for a good axiom set, so all forms of
classical-logic-based deductivism will adopt it. But whether or not
consistency is also *sufficient* is a matter over which various
deductivisms may differ. From a modern perspective, we can think about
the sufficiency claim as follows: absent a completeness proof, the
consistency of a set of axioms does not in general entail that they
have a model. To spell out the familiar argument, let PA2 be the set
of second-order Peano Axioms, and Con(PA2) a consistency statement for
PA2. Then \(\textrm{PA2} \cup \{\neg \textrm{Con}(\textrm{PA2})\}\) is
consistent provided PA2 is, since \(\textrm{PA2} \not\vdash
\textrm{Con}(\textrm{PA2})\) by Godel's second
incompleteness theorem. However, \(\textrm{PA2} \cup
\{\neg\textrm{Con}(\textrm{PA}2)\}\) lacks a model, and so, under a
traditional language-world picture, the language of the axiom system
does not pick out anything in the world. But for a deductivist who
asserts that only consistency matters, the alleged absence of
successful reference is of no significance.



What distinguishes Pasch's deductivism from Hilbert's is
that Pasch simultaneously maintained an empiricism about the axioms
(Schlimm 2010: 95-96). He held that the axioms are to have
independent empirical grounding ("the pre-mathematical
part" above). For example, Pasch grants the possibility of other
consistent axiom sets for geometry, but maintains that without
empirical grounding such a system constitutes merely
"hypothetical geometry" (1917 [2010: 46]). In particular,
Pasch maintains, the hypothetical sentences derived in such a geometry
may lack applicability (1917 [2010: 46]; Schlimm 2010: 101).


### 3.4 Curry



A more recent version of deductivism was defended by Haskell Curry in
his 1951 *Outlines of a Formalist Philosophy of Mathematics*.
As he sees it, "mathematics is the science of formal
systems" (1951: 56). For Curry, formal systems are to be
understood deductively--they are comprised of a formal language,
axioms and inference rules called "rules of procedure"
(1951: 11-13). Writing after the discovery of incompleteness,
Curry has this to say about its implications:



> 
> 
> The incompleteness theorems [...] show that it is hopeless to
> find a single formal system which will include all of mathematics as
> ordinarily understood. Moreover the arbitrary nature of the
> definitions which can constitute the primitive frame of a formal
> system shows that, in principle at least, all formal systems stand on
> a par. The essence of mathematics lies, therefore, not in any
> particular kind of formal system, but in formal structure as such.
> (Curry 1951: 56)
> 
> 
> 



Curry calls his conception *formalist.* But by formalism, Curry
just means the science of what can be deduced in formal systems.
Formalism and deductivism (as understood in this entry) are similar in
their focus on syntax over semantics. However, deductivism holds that
mathematical claims have truth-conditions, unlike games; and it
regards only logical inferences as permissible, whereas a formalist
can in principle condone any symbol-derivation practice. In line with
the deductivist rather than the formalist interpretation, Curry uses
the notion of truth extensively in his writings and rejects the view
that formal systems are to be conceived of as a game:



> 
> 
> There must be an *objective criterion of truth*, and the first
> task of the mathematical logician is to find it. The first thesis of
> this book is that such a definition can be found. (1951: 3)
> 
> 
> 
> Before we proceed to the discussion of the formalist definition of
> mathematics, it will be necessary to consider the definition of its
> fundamental notion, that of a formal system. [...] The conception
> here expounded differs superficially from the usual one in that
> *such a system is conceived not as a game but as a body of
> propositions.* [...] The first point is that the words
> "proposition", "true",
> "predicate", etc., used in the following, are intended in
> the sense of ordinary discourse. [...] [A proposition] indicates
> certain conditions for verification; when the proposition is
> asserted--taken as true --, that means that these conditions
> are fulfilled, so that processes of verification are what determine,
> essentially, the meaning of the proposition. (1951: 8)
> 
> 
> 
> For the purposes of mathematics it is sufficient to have criteria for
> recognising a formal system as such, and for *deciding which
> propositions are true*. (1951: 31)
> 
> 
> 



One of the main advantages of this perspective, according to Curry, is
the absence of metaphysical speculation--a key motivation for
deductivism (see
 SS2):



> 
> 
> The formalist conception of mathematics is thus *free from
> metaphysical bias*, and is therefore compatible with practically
> any sort of philosophy. It is the only conception so far proposed
> which has that character. (1951: 58; our emphasis)
> 
> 
> 


### 3.5 Intuitionism and Brouwer



Intuitionism is sometimes spoken of in the same breath as deductivism.
Curry, for example, maintains that



> 
> 
> Indeed, if we subtract from intuitionism its metaphysics, the
> differences between it and formalism as to the definition of
> mathematical truth are superficial. The emphasis on constructive
> methods--Hilbert's "finite
> Einstellung"--on recursive definitions and mathematical
> induction, are made by formalist and intuitionist alike. (1951:
> 58)
> 
> 
> 



Brouwer's intuitionism is related to deductivism but differs
from it in an interesting way. What, briefly, are the similarities and
differences?



In his Cambridge lectures, Brouwer rejects "mathematical
creation by means of logical axioms" (1981: 7). Indeed, for him,
"all axioms became illusory" (1981: 92). Brouwer's
target here is what he calls the "Old Formalist School",
which for him includes Dedekind, Peano, Russell and Hilbert, and from
which he goes to lengths to distance himself. For Brouwer, "the
criterion of truth or falsehood of a mathematical assertion" is
not a deduction but "mathematical activity itself"
(Brouwer 1981: 92). The relevant activity is the process of providing
a construction. Thus, for the intuitionist, truth and falsehood are
replaced with, or cashed out as, the existence or absence of an
intuitionistically acceptable construction (Brouwer 1981: 92).



Constructions are a series of mental acts; they are thus distinct from
proofs which are sequences of appropriately linked sentences. For
Brouwer, the *first act of intuitionism* is to completely
separate "mathematics from mathematical language and hence from
the phenomena of language described by theoretical logic" (1981:
4). For Brouwer, no linguistic inscription or utterance can convey the
true nature of a construction in intuition. The severance of
mathematics from language and logic, a cornerstone of Brouwer's
intuitionism, sharply distinguishes it from deductivism.



That said, intuitionism does have some parallels with deductivism. A
notable one is the replacement of the usual referential account of
truth in mathematics with a verification-based approach: deductivism
does this in terms of proofs, intuitionism in terms of constructions.
The development of intuitionistic logic brings out an important fact
about deductivism: deductivism is not tied to any particular formal
system. It could be based on classical logic, or Heyting's
intuitionist formal system, or some third
 alternative.[9]


## 4. Meaning Matters



We outlined some of the historical support for deductivism in
 SS3.
 In
 SSSS5-9,
 we consider some of the challenges it faces. In the present section,
we aim to achieve greater clarity on what semantic claim the
deductivist makes.



The first point is that we have characterised deductivism as
"understanding" or "construing" or
"rendering" a mathematical sentence \(p\) as "\(Ax
\vdash p\)" This characterisation is deliberately neutral
between two more precise readings. One is that deductivists take their
construal to capture the meanings of mathematical statements as
ordinarily
 spoken.[10]
 The other is that they aim to do no such thing but wish instead to
provide a reconstruction of mathematics answering to philosophical
criteria. One such criterion *might* be proximity to
mathematical discourse as ordinarily understood--but typically
this will not be an overriding factor. Other criteria will include
giving an attractive metaphysics, epistemology and account of the
applications of mathematics. This entry will stay neutral on this
point, though it should be noted that the first approach faces a
considerable burden of proof: there are no obvious grounds to think
that we *actually* speak in deductivist fashion.



The second point is that the deductivist has to offer more than the
simple claim that a mathematical sentence \(p\) should be construed as
"\(Ax \vdash p\)". For one thing, her (meaning-capturing
or reconstructive) semantics for mathematics cannot be confined to the
sentence level but must also apply to words. The deductivist must, for
instance, give a semantic account of the numeral "2" and
the predicate "is prime" compatible with her construal of
"2 is prime" as "\(Ax \vdash 2\) is
prime".



For another, she must give an account of speech acts involving
mathematical statements. It is natural to suppose that she construes
an assertion of "569 is prime", say, as an assertion of
"\(Ax \vdash 569\) is prime". Likewise, presumably, for
many propositional attitudes: for the deductivist, what we might
ordinarily describe as wondering whether 569 is prime should be
understood as wondering whether the axioms deductively imply that 569
is prime; what we might ordinarily describe as doubting that 569 is
prime should be understood as doubting that the axioms deductively
imply that 569 is prime; and so on. A tentative suggestion might then
be that the deductivist should take "\(Ax \vdash\)" to
have narrow scope over force operators (asserting, wondering,
doubting, etc.).



Neither of these tasks--providing a word-level semantics for
mathematical language and an account of speech acts involving
mathematical sentences--seems entirely straightforward. As for
the first, deductivists' account of mathematical language must
allow for its use in empirical contexts to make straightforwardly
literal claims. For example, "there were 569 people in the
audience" should presumably not be understood as a covert
implication from axioms but rather as a literal claim about a physical
situation. For the deductivist, then, this is quite a different use of
"569" from a purely mathematical one.
 SS8
 discusses the problem of applications for the deductivist (though at
the sentence- rather than word-level).



As for the second, it's not clear that the deductivist will
always wish to apply the "\(Ax \vdash\)
"-takes-narrow-scope strategy in the same mechanical way. Take
for example proof by assumption, in which we first assume \(p\_1\) and
deduce \(p\_n\) via \(p\_2 , \ldots , p\_{n-1}\). Our initial supposition
\(p\_1\) might be that \(m\) is an odd perfect number and the final
line \(p\_n\) might be some interesting fact about any such \(m\).
It's not at all obvious that "assume" here takes
wide scope and "\(Ax \vdash\) " takes narrow scope. For if
it did, "Assume \(p\_1\)" would be construed as
"Assume \(Ax \vdash p\_1\)", which intuitively is a
different claim. A more technical way of seeing this is that because
\(m\) does not occur free in \(Ax\), it seems that we can generalise
to obtain "\(Ax \vdash \forall m p\_1\)". If so,
"Assume that \(Ax\) proves that \(m\) is an odd perfect
number" implies "Assume that \(Ax\) proves that for every
\(m, m\) is an odd perfect number". But that latter assumption
is an outright false claim. A more promising approach is to take the
first step in a supposition as allowing us to add \(p\_1\) to the axiom
set, and to then construe the second line as "\(Ax + p\_1 \vdash
p\_2\)" (or its equivalent via the Deduction Theorem "\(Ax
\vdash p\_1 \rightarrow p\_2\)"), the third line as "\(Ax +
p\_1 \vdash p\_3\)" (or "\(Ax \vdash p\_1 \rightarrow
p\_3\)"), and so on. All in all, there is work for the
deductivist to do on this front to spell out these ideas into a
coherent theory.



A third point is based on Parsons's (2008: 55 n. 35) distinction
between what might be called *reflexive* and
*unreflexive* deductivism. Deductivism as this entry defined it
is reflexive, because it interprets \(p\) as "\(Ax \vdash
p\)". Unreflexive deductivism, by contrast, interprets \(p\) as
"if \(\wedge Ax\) then \(p\)" (\(\wedge\) is an operator
that conjoins all the elements of the axiom set) and asserts that this
conditional is a deductive logical truth. Rheinwald (1984: 50) opts
for the former, reflexive form, as we have. Commenting on Rheinwald,
Parsons advocates adopting the latter, unreflexive, form instead. For,
as he sees it, the reflexive form introduces an inessential element of
reflection--explicit mention of the axioms--into the content
of mathematical statements, which he thinks they lack. Parsons has a
point, especially if deductivism is intended as meaning-capturing.
Even so, this entry takes deductivism to be reflexive to avoid the
problem faced by unreflexive deductivism when the set of axioms is
infinite, as it is for example in the case of the first-order Peano
Axioms. For in that case, the sentence "if \(\wedge Ax\) then
\(p\)" is infinitely long, so not an ordinary-language one. In
contrast, "\(Ax \vdash p\)" is an ordinary sentence of
(mathematical) English, where "\(Ax\)" is a name denoting
the relevant axiom set. That said, it may be that unreflexive
deductivism is better equipped to handle some of the problems
deductivism faces.



A fourth and final point. So far, this entry remained neutral on what
the axioms "\(Ax\)" in the statement of deductivism are. A
"local" deductivist, as illustrated in
 SS1,
 deductively conditionalises the theorems of a branch of mathematics
on the branch's axioms, e.g., theorems of arithmetic on the
axioms of Peano Arithmetic. Another sort of deductivism might
conditionalise on the axioms of a particular alleged foundation for
mathematics, say ZFC set theory. Thus a ZFC-deductivist understands a
mathematical claim \(p\) as "\(p\) follows deductively from the
ZFC axioms". The choice of ZFC is not essential: if the
deductivist's preferred foundation is not ZFC, but some other
set theory, or a topos theory, or something else altogether, they
could base their deductivism on that instead. A third, similar but
looser form of deductivism than this might be called
"whatever-it-takes deductivism". On this view, the axioms
provability from which constitutes mathematical truth are not a
specific collection, such as the ZFC axioms, but are more vaguely
described: they are whatever axioms it takes to prove the theorems of
mathematics. This last version has the advantage over the second of
not tying provability, perhaps artificially, to a single foundational
system. However, it achieves this at the cost of considerable
 imprecision.[11]
 In short, different forms of deductivism result from different
understandings of "\(Ax\)" in deductivism's
canonical statement.


## 5. Provability: to reify or not to reify?



The rest of this entry examines five challenges to deductivism, one
per section in
 SSSS5-9.
 The first turns on the notion of provability. Since deductivism
claims that mathematics is about what is provable in certain axiom
systems, providing an account of provability is central to it. The
issue is whether to cash out provability in terms of the existence of
proofs or not. In short, to reify or not to reify?



Take the reifying approach first. It reads "\(p\) is provable
from \(Ax\)" as "there is a proof of \(p\) from axioms
\(Ax\)" (in a particular deductive system understood). This is
the most natural and most standard understanding of provability. On
this approach, to say that a system \(T\) is consistent is just to say
that there is no formula \(\phi\) in the language of \(T\) such that
there is a proof of \(\phi\) from \(T\) and a proof of \(\neg \phi\)
from \(T\). However, some formulas and proofs are far too long to have
concrete physical instantiations; a simple example would be the proof
of a numerical equation involving extremely large numbers. Even a
formula with concrete instantiations should not be identified with a
particular one of these, or with the class of its instantiations. In
other words, proofs as understood here are types rather than tokens:
abstract strings conforming to the rules set by a deductive system,
which exist irrespectively of any physical instantiations in speech,
or on whiteboards, paper or screens. Indeed, if the deductivist wishes
to retain the ability to refer to theories, as she will surely want
to, she will have to accept linguistic types anyway. (Compare
Field-like nominalists, who must also refer to scientific and
mathematical theories, even as they try to avoid reference to
mathematical objects.)



The moral is that the deductivist will have to admit linguistic types.
Her motivation for being a deductivist therefore cannot be to avoid
*all* abstract objects (i.e., nominalism), since linguistic
types are abstract. She must consequently motivate why admitting some
abstract objects such as types is acceptable, whereas admitting
mathematical objects is not. She might point to considerations of
economy: linguistic types are but a small part of the abstract realm.
Or she might argue on the epistemological front that linguistic types
are more easily knowable, since they have concrete instances, unlike
mathematical objects, which do not. However, the literature is not
short of arguments that those who reject abstract mathematical objects
should also reject abstract linguistic types; see for example
Wetzel's 2009 book *Types and Tokens*.



Of course, one could be a deductivist for other reasons than an
aversion to abstract objects. As we saw in
 SS4,
 one sort of deductivist might hold that a semantic analysis reveals
the *meaning* of mathematical statements to be of the form
"\(p\) deductively follows from the axioms". It is
perfectly consistent to combine this view with an acceptance of
abstract objects and the standard account of deductive consequence in
terms of the existence of proofs. Yet, although perfectly coherent,
the view appears to lack independent motivation.



The alternative to reifying, evidently, is not to reify, and the most
obvious way of doing so is to go down a modal route. Instead of
cashing out provability by means of proof-existence, deductivists
could take provability to be a modal primitive. Logical possibility
and necessity are these days typically understood model-theoretically:
a statement is logically possible or consistent just when it is true
in some model; and necessary just when it is true in all models. The
deductivist must abjure this account of consistency (which she equates
with logical possibility), on pain of re-introducing mathematics in
the metatheory. Against the prevailing model-theoretic account, she
must take "it is logically possible that" as a primitive
notion, and forgo its analysis in terms of models (Putnam 1967b). In
so doing, she is in the same boat as all anti-platonists who cannot
help themselves to model theory or set theory in the metatheory. This
particular boat is a crowded one: it holds not just eliminative
structuralists, but also formalists of various stripes and nominalists
as well. Nominalists, for example, do not believe in abstract objects.
Hence they face the same predicament--they cannot appeal to the
existence of models in the metatheory. Hartry Field (1991) develops a
nominalist logic that takes the notion of logical consistency as
primitive, which we may symbolise as \(\diamond\), with the associated
notion of logical necessity being \(\square\). In particular,
\(\diamond T\) is the claim that the theory \(T\) is logically
consistent, and claims of logical implication are conditionals
prefaced by
 \(\square\).[12]
 The non-reifying deductivist can avail themself of these resources
and thereby construe \(G \vdash p\) as \(\square(\wedge G \rightarrow
p)\).



Such "modalist" accounts of logical consequence a
la Field have had a fairly low take-up among logicians. Many see
little reason to prefer them to the model-theoretic account of
consequence. Modalist accounts suffer from problems of expressive
adequacy, because they cannot quantify over possibilities; and they
are theoretically less virtuous than the usual model-theoretic one,
positing as they do primitive links between the existence of set
models and primitive consistency facts. Chapter 4 of Melia (2003) is a
lucid overview of the problems faced by modalism about
*metaphysical* modality, which have counterparts in the logical
case. If the deductivist takes the non-reifying route, they will have
to come up with a satisfactory response to these sorts of
challenges.


## 6. Unaxiomatic Mathematics



A second objection to deductivism is that deductivism cannot account
for unaxiomatic
 mathematics.[13]
 This includes pre-axiomatic mathematics such as, say, "Old
Babylonian" mathematics of the early second millennium BC, in
which first principles are never stated explicitly (Robson 2008:
 90).[14]
 But it also includes non-deductive mathematics; for example, we might
wish to credit the eighteenth-century Swiss mathematician Euler with
knowledge of results obtained through analogical reasoning, extensive
verification of cases, and the like.



It seems right to attribute mathematical knowledge to Babylonian
mathematicians who, among other things, were able to solve specific
quadratic equations. Still, the challenge may not be that worrying.
That first principles were not explicit does not mean that they were
not implicit. Presumably, any mathematical principle that Babylonian
mathematicians were willing to accept without argument should count as
a first principle or axiom. Independently of this point, deductivists
should also point out that unaxiomatic methods can yield knowledge if
they track what a proof would yield. Babylonian mathematicians used
methods that reliably produced solutions to specific quadratic
equations, even if they could not formulate, still less solve, the
general quadratic equation. Euler's methods reliably tracked the
existence of proofs for his results; if they had not, he would not
have been right as often as he was. This deals with the vast majority
of such cases. Moreover--thinking about the residual
cases--there is some reason to think that non-deductive reasoning
can yield knowledge of mathematical truth even in the absence of proof
(Paseau 2015).


## 7. A Regress?



We turn now to the third challenge. To introduce it, consider whether
deductivism should apply to a sentence such as



> 
> 
> the sentence "\(2 + 2 = 4\)" deductively follows from
> \(Ax\)
> 
> 
> 



(As usual, we write \(Ax\) to preserve neutrality on what the axioms
are exactly; see
 SS4
 for the options here.) Since proof theory appears to be a branch of
mathematics like any other, you might suppose that a consistent
deductivist should construe this sentence like any other mathematical
one. In which case, the sentence "\(2 + 2 = 4\)" should be
construed not as



> 
> 
> the sentence "\(2 + 2 = 4\)" deductively follows from
> \(Ax\),
> 
> 
> 



but instead as



> 
> 
> the sentence "the sentence '\(2 + 2 = 4\)'
> deductively follows from \(Ax\)" deductively follows from
> \(Ax'\).
> 
> 
> 



(Where \(Ax'\) may or may not be identical to \(Ax\)--this will
not affect the main point.) Now if a deductivist is to go down to this
route, she will also have to construe this last sentence like any
other mathematical one. So it should be glossed in the same way. The
same applies to the resulting gloss, and so on without end. Clearly,
we are off on an infinite regress.



It seems to follow that for the deductivist there is no finite
formulation of the sentence "\(2 + 2 = 4\)". Any such
formulation would have to be infinite, and, if it ends at all, must
end in a sentence \(s\) that is a "fixed-point" of this
iteration. In other words, \(s\) must be identical (not just logically
equivalent) to "the sentence \(s\) deductively follows from
\(Ax\)" (for some \(Ax)\) since \(s\) is construed by the
deductivist as



> 
> 
> the sentence \(s\) deductively follows from \(Ax\)
> 
> 
> 



Whether such a sentence exists is highly dubious. Even if it does, an
infinitely long sentence, with infinitely many embeddings, is an
absurdly complicated way to interpret "\(2+ 2 = 4\)",
which after all is one of the simplest sentences of mathematics,
indeed of the English language.



The objection just raised is premised on the idea that provability
claims are mathematical. To support this premise, the
objection's proponent might point out that results in proof
theory are typically infused with mathematics. Our conviction that
some statement is provable in a deductive system is not usually a
result of having carried out the proof ourselves, starting from the
axioms and applying the rules of inference to derive the theorem.
Rather, it often follows from more abstract mathematical
reasoning.



How should the deductivist reply to this challenge? She can agree that
we use mathematics, sometimes even quite sophisticated mathematics, to
determine what deductively follows from some axioms. That is an
*epistemological* point. But it does not affect her thesis
about what mathematics *is*. For the deductivist, the truth of
a mathematical claim \(p\) consists in its being a theorem of an
appropriate deductive system \(D\). How we determine what
\(D\)'s theorems are may require mathematics--this is to
apply mathematics to proof systems--and perhaps other machinery,
but that, to repeat, is an epistemological point. The fact that \(p\)
is a theorem of \(D\) is a fact about a deductive system, namely that
there is a sequence of formulas starting from axioms and proceeding by
rules of the system concluding in \(p\). As the deductivist sees
it--as she *has* to see it, to avoid the
regress--*this* fact is not mathematical, even if we must
use mathematics to determine whether it holds or not. (Compare the
claim that a large meteor will land on Earth before 2050: a physical
fact, but one we verify or falsify using sophisticated
mathematics.)



In short, to avoid the regress, deductivists should not apply their
interpretation strategy to claims of deductive consequence. For any
sentence \(s\) and well-specified deductive system \(D\), whether
\(D\) proves \(s\) or not should be taken to be a determinate fact.
Such facts are the cornerstone of her philosophy and should not be
"deductivised". Rather than have the deductivist recipe
applied to them, they must be taken at face
 value.[15]


## 8. Applications



The fourth challenge concerns applications.
 SS2
 mentioned that deductivism appears to provide a straightforward
account of the applicability of mathematics: provided a physical
system satisfies the axioms of a particular branch of mathematics, the
branch's theorems must apply to the physical system. But what
exactly does it mean for a system to satisfy an axiom? The simplest
account of applications is the mapping or isomorphism account. Given a
physical set-up \(P\), stipulate an isomorphism from \(P\) to some
mathematical structure \(M\), deduce some facts about \(M\), and then,
applying the isomorphism in the reverse direction, read off the
corresponding facts about \(P\). Resnik (1980, 125), following Russell
and Putnam, worries that this account is knee-deep in
mathematics--for what is an isomorphism if not a mathematical
object? More generally, a satisfactory account of mathematical
applications will have to employ mathematics to explain how
mathematics relates to the physical
 world.[16]



This is a major concern that applies to a host of other philosophies.
All anti-platonists must set out how to understand the mathematics
itself used to explain the applicability of mathematics. Eliminative
structuralists must give an account of the mathematics relating
structures to the physical world. Nominalists must likewise explain
how scientific theories laden with commitment to abstract objects can
correctly depict the physical facts despite there being no abstract
objects. Each specific anti-platonism seems to face some version of
this general problem. Allowing the use of mathematics in the
metatheory enables a philosophy to relate mathematics and the natural
world; disallowing it appears to hamstring a philosophy--or so
this challenge maintains.



So can the deductivist explain mathematics' applicability?
Perhaps her best hope of doing so is by means of a combined
mathematical-scientific "total" theory, which we may call
*TT*. *TT* incorporates all the axioms of mathematics,
as well as established scientific principles and data. Through its
combination of mathematics and science, *TT* is able to posit
all sorts of relations between mathematical "structures"
and physical situations and thereby explain mathematical applications
(at least as well as anyone can). The truth-conditions of physical
claims derivable from *TT* should be understood in the usual
referential manner, as relating to the physical world; whereas the
truth-conditions of mathematical claims should, for the deductivist,
be understood as above, namely as claims of the form "\(p\) is
deducible from *TT*". Importantly, for the deductivist,
"structures" and "isomorphisms" are to be
understood deductively within the theory *TT*. Effectively, the
deductivist internalises model theory into the deductive
framework.



This response looks like it dispatches the challenge from
applications. Mathematics and science can be discussed and compared
within the single overarching theory, *TT*. However, the
response has an Achilles heel: mixed statements. The deductivist
construes talk of numbers in deductivist fashion and talk of chairs
(say) in straightforwardly literal fashion. So what is she to make of
a claim such as "My favourite things are the number 7 and this
chair"? The claim cannot be read literally, as it involves
reference to a number; nor should it be read in deductivist fashion,
since it involves reference to a concrete entity, a particular chair.
That a chair is one of my favourite things is a claim with empirical
content, to be taken literally. The sentence "my favourite
things are the number 7 and this chair" is of course a toy
example. The point applies much more widely, since mathematics is
seamlessly interwoven with ordinary and (especially) scientific
language.



The deductivist therefore owes us a recipe for cleaving the
mathematical from the non-mathematical in all mixed statements. This
is a problem faced by other anti-platonist approaches, for example by
fictionalists about mathematics such as Mary
 Leng.[17]
 They take applied mathematics to be an "extended fiction"
that combines the (alleged) fiction of mathematics with science, which
they construe literally (as it is about the physical world). But which
is which, where do we draw the line, and how do we deal with
statements mixing fiction and the literal, apparently inextricably?
Finding good answers to the corresponding questions is a significant
challenge for deductivists just as much as it is for
fictionalists.


## 9. Truth-Completeness



We move on to the fifth and final challenge to deductivism. One
version of it, owed to Michael Resnik, runs as
 follows.[18]
 Consider the deductivist's set theory \(T\) and its Godel
sentence \(G\_T\). By construction, \(G\_T\) is not derivable from
\(T\). So the deductivist cannot explain the truth of \(G\_T\).
Resnik's response: the deductivist can simply challenge the
antideductivist to explain how she "sees" the truth of
\(G\_T\). Whatever mathematical argument the antideductivist offers,
the deductivist can simply extract from it an axiom set \(T^{+}\) and
point out that \(G\_T\) follows from \(T^{+}\). Objection parried.



Godel's First Incompleteness Theorem does create a
significant problem for deductivism; but it is not the one just
formulated and responded to. The above version focuses on how, given a
particular theory \((T)\), we are supposed to appreciate the truth of
its Godel sentence \((G\_T)\). A stronger form of the objection
casts it in more global terms and brackets the issue of the Godel
sentence's truth. What deductivism appears to get wrong is that
it disrespects the entrenched understanding of most parts of
mathematics as *truth-complete*: for any precisely formulated
sentence \(p\) from a truth-complete area of mathematics, exactly one
of \(p\) and not-*p* is true. Analysis and arithmetic are
examples of areas standardly understood to be truth-complete. It is
standardly thought, for example, that a statement such as
Goldbach's Conjecture is true or false, even if it turns out to
be unprovable from any reasonable axioms, not just present
 ones.[19]
 But on the deductivist view, undecidable sentences in any
formalisation of arithmetic and analysis are false and their negations
are also false, since neither deductively follows from the axioms.
That there are undecidable sentences follows from Godel's
First Incompleteness Theorem together with standard constraints on
deductive
 systems.[20]
 Hence deductivism misdescribes our understanding of truth-complete
branches of mathematics such as arithmetic, analysis and geometry.



This objection seems to affect two of the three variants of
deductivism we identified at the end of
 SS4:
 local deductivism and ZFC-deductivism. A form of deductivism the
Godelian objection does not affect is constructivist deductivism,
because constructivists do not assume the law of excluded middle in
mathematics, so do not assume truth-completeness. Though, as we
glimpsed in the discussion of Brouwer
 (SS3.5),
 constructivists do not tend to be deductivists.



Sticking to classical-logic-using deductivism, its proponent could
respond to the objection by idealising the axiom system or the proof
system or both. The first approach is to construe "\(2 + 2 =
4\)" as:



> 
> 
> the sentence "\(2 + 2 = 4\)" deductively follows from the
> *ideal* axioms of arithmetic
> 
> 
> 



The resulting position is essentially whatever-it-takes deductivism,
mentioned in
 SS4.
 It adds a little more detail to the position articulated in
 SS4
 in that the ideal axioms of arithmetic by stipulation give rise to a
complete system, in order to overcome the present objection. (Of
course, an informal version of whatever-it-takes deductivism would
evade the objection altogether, because it is not based on a formal
system. But critics will retort that it does so only because its
account of mathematical truth is inherently vague.) The second sort of
approach might, for example, take the proof system to include
infinitary rules of reasoning, such as the omega-rule, which, as is
well-known, restores completeness if added to first-order Peano
 Arithmetic.[21]



Idealising deductivism in either of these ways, however, is not to
rescue but to abandon it. Any combined axiom-and-proof system that is
consistent, deductively complete and strong enough to capture all the
truths of mathematics lacks an effective proof procedure, by
Godel's First Incompleteness Theorem. To see this, consider
say the deductive "system" whose "theorems"
consist of all the true sentences of
 arithmetic.[22]
 Assuming the Church-Turing thesis, this list of
"theorems" cannot be generated by any mechanical proof
procedure (i.e., algorithm). Idealised deductivism is thus eliminative
structuralism in thin disguise, since it advocates taking as the
combined axiom-and-proof system whatever it takes to capture the
structural content of the axioms, be it by beefing up the axioms or
the "proof" system or both. If the idea is that
mathematicians take a sentence to be true if and only if it is a
logical consequence of the semantic content of
structure-characterising axioms, one should come clean and accept
deductivism's semantic counterpart: eliminative structuralism.
The clear moral is that what lands deductivism in trouble is its focus
on syntax. Instead of taking a syntactic turn as it did in the late
nineteenth and early twentieth centuries, the "mathematical
statements are conditional on axioms" idea must instead take a
semantic form.



Deductivists might try another defence. We quoted Curry in
 SS3.4
 as holding that "all formal systems stand on a par". And
this is surely a thought many deductivists will find attractive: in
principle, any formal system is acceptable, the differences between
one formal system and another being owed to the fact that one of them
is better suited to applications, or to the fact that one of the
systems happens to be the one \(we\) use, and so on. So, for the
deductivist, if the axioms of arithmetic \(Ax\) we explicitly or
implicitly accept are incomplete, then any arithmetical sentence \(p\)
not decided by them is neither true nor false, precisely because its
truth-value is relativised to deducibility from \(Ax\). Of course, we
can strengthen \(Ax\) in one way to derive \(p\) and in another way to
derive not-\(p\), and in those respective strengthenings \(p\) will be
respectively true and false. This, it appears, is Curry's
suggestion about how deductivists should react to Godel's
result.



The resulting pluralist position rejects truth-completeness for a
principled reason. But it rejects it nonetheless, and that must count
as a significant cost. For the truth-completeness of arithmetic is a
fundamental commitment of mathematics, and permeates most of our
thinking about the natural numbers. (A similar point applies to many
other areas.) Some would say that respecting it is a non-negotiable
constraint on a philosophy of mathematics. Of course, the
truth-completeness objection challenges deductivism only in certain
areas of mathematics--specifically those typically held to be
truth-complete but for which no deductively complete, consistent
axiomatisation exists. A complete theory-axiomatisation may be
provided even in first-order logic not just for finite structures but
for some infinite ones as
 well.[23]
 Thus one could still try to defend a deductivism about all other
areas. But there is great value in having a uniform philosophy of
mathematics. More importantly, given deductivism's motivations,
as outlined in
 SS2,
 it is hard to see what the point of this sort of eclectic deductivism
might be.



Truth-completeness presents the deductivist with a formidable
challenge. To meet it, she cannot privilege any system of arithmetic
over any other by maintaining that "PA is consistent" is
true in one system and false in another. The post-Godelian
realisation that eliminative structuralism fares better
here--because it allows for the possibility of truth-completeness
if we move to an incompletable semantic consequence relation--led
to structuralism's gain in popularity at the expense of
deductivism in the second half of the twentieth century.