Two Challenges for Fictionalism in Mathematics
Ahti-Veikko Pietarinen
University of Helsinki
Abstract: Fictionalism in philosophy of mathematics faces two major objections: the
philosophy of mathematical practices and Brouwer‟s intuitionism. The philosophy of
mathematical practices is in this paper traced back to the key tenets of Peirce‟s pragmaticism.
The core argument concerns the modalisation of mathematical entities at the heart of
substantive fictionalism: they are shown to assume a metaphysical notion of modality that is
unviable from the point of view of pragmaticism and its scholastic realism and fallibility of
mathematical practices. Another challenge comes from Brouwer‟s original, languageless
conception of mathematics. It is pointed out that fictionalism and its offshoot of figuralism
fail to analyse the semantics of the specific tropes on which they nevertheless rest their
arguments.
Key
words:
Fictionalism,
figuralism,
philosophy
of
mathematics,
pragmaticism,
mathematical practices, Peirce, Brouwer, intuitionism, language, realism, fallibilism.
1. Introduction
It is perhaps needless to label the arguments in this paper as challenges, as they are calculated
to refute fictionalism as a defensible philosophy of mathematics. They are thus not challenges
in the usual sense of the term: they are not to be settled from a shared epistemological vantage
point. Hence my arguments are likely not to convert a committed fictionalist trenched in
cartesian epistemology. But even the most committed fictionalist must grant that a discourse
on the fundamental nature of mathematics and its preferred entities is possible.
Mathematical fictionalism has arisen as one of the major themes in contemporary
philosophical debates about the fundamental notions of mathematics (Balaguer 2008; Field
1980, 1989; Yablo 2005). The underlying arguments in its favour are to a notable degree
derivative from parental fictionalisms in general philosophy of science, ethics or religion.
1
Yet it is not at all obvious what the scientific appeal of a fictionalist deconstruction of
mathematical theories is supposed to be. Minimally, it is a sceptical claim about the feasibility
of classical platonism. In its semantic outfit, it is both an acceptance of a platonist attitude „as
if‟ we were referring to subjects of meaningful discourse, followed by a denial that such
reference or denotation actually succeeds. What is, then, can be its own positive contribution?
A malicious commentator might be tempted to say that, no matter how appealing fictionalism
may to a sceptically minded philosopher be, if it does not answer questions relevant to actual
mathematical or scientific inquiry, it ain‟t real at all (and so fictionalism risks liquefying into
relativist impasse). The status of such theories, the thought continues, is to be classified
among the „easiest way out‟ types of explanations, which not uncommonly have been
proffered by philosophers whenever interpreting the hard dilemmas faced in the exact
sciences. When all else fails, take refuge in the fictitious, non-platonic haven of the
instrumental utility of theories.
According to mathematical fictionalism, statements of mathematics are not about
mathematical things (Field 1980, 1989). Statements build up narratives, and just as narrative
stories of fiction give rise to fictional entities. Fictionalism serves to deny the existence of
abstract objects of mathematics. Since there are no such objects, mathematical statements are
“in fact” and “strictly speaking” false. They may be taken to be true in some other sense, such
as true in the domains of these fictional narratives. In that sense, they pertain to that
Vaihingerian class of “as if” statements. What accounts for the importance and relevance of
human mathematical activity among the vast array of other kinds of activities – and if we get
really lucky, could even explain the possibility of progress in terms of that human activity – is
the instrumental expedience and value human beings are able to ascribe to these creative
activities. This position nevertheless implies a hard-to-swallow contention that mathematics is
just one kind of cultural phenomenon among a wealth of others, with no privileged role to
play in human culture in comparison with myths, rituals, chess boxing or cheese rolling.
As a mere denial of the existence of abstract objects, fictionalism is, I will maintain,
an empty theory that presupposes, perhaps unconsciously, a sceptical epistemology.
Therefore, what I will be concerned with in this paper are versions of substantive fictionalism
that rely on the possibility of succeeding with passable and testable analogies between
mathematics and some articulated conceptions of fiction and narratives (see the helpful
comparison of the two sides in Thomas 2007).
2
Burgess (2004: 19) has argued that the successful modal recasting of structuralist
entities of mathematics compels us to appreciate a fictionalist reinterpretation of mathematics.
This remark is part of his argument against fictionalism, I should emphasise, but at the same
time also an expression of a presumed ontological innocence of the modalities evoked in the
argument. The modalisation of mathematics which, given the rich history of pure mathematics
is as such no novel suggestion, has since Putnam (1967) become standard in characterising
the hypothetical nature of mathematical propositions. The question of exactly how the
modalities are to be conceived in that role has not yet been explored in full, however.
2. Pragmaticism as non-Platonic non-fictionalism
And not all else fails. Modalities are not ontologically innocent. From the perspective of the
philosophy of mathematical practices, what Burgess proposes to be happening turns out to be
an oversimplification. Given Charles Peirce‟s pioneering work on these topics (Peirce 1980-;
Moore 2010a, 2010b), I will term the attempts to get at the philosophy of mathematics by
focussing on the actual practices of mathematicians mathematical pragmaticism. According to
mathematical pragmaticism (and Peirce‟s pragmaticism in general), possibilities are just as
real as actualities. This falls from the tenet of scholastic realism at the back of his
pragmatistic, modal theory of meaning. Modalities in scholastic realism have very little to do
with fiction: what is real, and what is a real possibility, are from this perspective contrary and
opposing notions to that of a fiction.
By pragmaticism, it needs to be clarified, I do not mean what probably many are led to
believe is at issue here, namely a philosophy of pragmatism in the sense of William James
and that tradition in American philosophy. That „practical‟ notion is not what I have in mind.
James indeed once thought that scientific theories are associated with something like their
expediency, satisfaction, or utility values, and came to designate that idea with the unfortunate
label of pragmatism. Though there is a seed of truth in James‟s adaptation, his line of thought
created more controversy than satisfaction. The reasons can be traced back to ultimate
presuppositions on which one‟s system of thought hangs. James associated the
presuppositions of utility values of theories with what may be termed the „one-world‟
assumption. The one-world assumption states that what those assignments of expediency or
utility measures are can be exhaustively and definitely resolved by us, individual
investigators, within this one, actual world of under our scrutiny. No recourse to the apparatus
of possible worlds or modalities is needed. (For details of this argument, see citation omitted.)
3
John Dewey largely followed suit and took the instrumentalist reinterpretation of the utility
values of theories to be a sole matter of “concrete and actual situations” (citation omitted). If
either James‟s or Dewey‟s instrumentally reinterpreted pragmatism was to be taken as the
guiding theory in the philosophy of mathematics, then fictionalism would largely be
compatible with it.
However, this tradition of pragmatism is not what the historically faithful account of
the philosophy of mathematical practices means. Peirce‟s original formulation of
pragmaticism is a theory of meaning in its broad semantic and pragmatic senses of meaning.
It interprets possible worlds realistically. I will not rehearse the main tenets of pragmaticism
as a philosophy of mathematics in this paper (see citation omitted, citation omitted for more
details): briefly, its rendering of scholastic realism holds that what might, could, or would
follow from assumptions made along the course of studying different kinds of mathematical
structures are just as real as any structure readily construed by a mathematician who performs
mathematical reasoning. The reason is that the former can exert counterfactual force to the
structures singled out by a creative mathematical thought, say by experimenting on a set of
mathematical axioms. But these counterfactually effective structures do not exist and they
need not be in any causal or law-like connection with sui generis structures. Possible worlds
are real though non-extant. Precisely this differentiates pragmaticism from individualist and
instrumentalist appropriations. There is a world of difference (or rather „many-worlds‟ of
difference) as to the meaning of counterfactual statements whether the semantics of modal
constructions is evaluated in terms of real yet non-extant unactualised alternative worlds or
not.
From this denial of the actual existence of mathematical things fictionalism will
follow only if one is to make much stronger epistemological assumptions that pragmaticism
does not make. According to pragmaticism, all cognitions rest on previous cognitions, yet
there is no and need not be any first cognition. Thinking, knowing and intention are forms of
action, and action is not preceded by anything else. Knowledge follows action, and action
follows general habits of acting in certain ways in certain kinds of situations. There is no
atomicity, intentionality, or stimulus-response model of action involved and hence there need
not be any absolute certainty or foundation for human knowledge characteristic of cartesian
scepticism.
4
Accordingly, from the vantage point of this understanding of mathematical
pragmaticism, Burgess‟s view that a modalisation of mathematics is liable to lapse into the
fictionalist camp is not defensible. If it does so lapse, extra assumptions must have crept in.
And even if that was to happen, fictionalism would nonetheless run into problems. First, there
are the generic questions such as: How can fictionalism account for progress in mathematics?
(One is reminded here of the „no miracles‟ argument in philosophy of science which Peirce
discusses in his 1905 What Pragmatism Is paper.) What is the ontology of fictional entities in
the first place?
I do not think adequate answers have yet been given, nor do I intend to grant any of
these extra assumptions. Proposals include the „consistency of stories‟ explanation as well as
the ontological maxims of parsimony and economy. Consistency is not sufficient, since there
would be uncountably many „stories‟ consistent with some previous instalments of the alleged
mathematical narrative in question. Parsimony and economy are dubious for much of the
same reason as they are regarding the ontology of non-fictional entities: a closer look at how
mathematics is practiced disconfirms that mathematicians ascribe to maxims stating that what
they do is simple, parsimonious, elegant, inexhaustible, or striving for the economy of
theories. Hardly any mathematician or physicist would hold mathematics to be “an easy way
of saying what we want to say about the physical world”, contrary to what a proponent of
fictionalism wrote in order to recapitulate a point in favour of dispensing with the truth of
mathematical propositions in mathematical theories (Balaguaer 2008: 4). No, it is not at all
easy, and there are no non-mathematical but difficult ways of communicating what we would
want to say about the worlds of physics.
Instead, what interests us here is not so much the dwelling on any in-depth evaluation
of the attempted analyses made of these qualitative facets of mathematics, but the role of
mathematical practice in relation to certain substantive fictionalist assumptions. Second, then,
let us take up the following thesis: If the ontology of mathematics corresponded to fictional
entities as novels or narratives correspond to fictional entities, mathematics could not occupy
that key position in the hierarchy of the sciences upon which much of other sciences depend.
The argument proceeds by reductio: the contrary case would force taking the ontological
status of the objects of all sciences as fictional. Mathematical fictionalism would imply
scientific fictionalism and would be vulnerable to the same arguments against it as
fictionalism in science in general.
5
A subsidiary corollary from this state of affairs is worth drawing before proceeding
with my argument. If you are keen on denying the feasibility of scientific fictionalism in the
manner that the defenders of instrumentalist access to observables are willing to, then you
cannot have your mathematical fictionalism as a getaway from the looming reality of abstract
objects, either. In other words, constructive empiricism does not imply commitment to
fictionalism about the objects of mathematical theories. Far from being a defence of such
empiricism, however, this incidental observation merely points out that the apparent reality of
abstract objects remains a problematic feature in instrumentalism that takes science to be a
rational form of inquiry. However, pragmaticism steers clear from accepting any
unsophisticated interpretation of instrumentalist theories of science. Dewey serves as a fine
example of an early pragmatist who recognised the value of the specifically pragmatic notion
of instrumentalism in steering the middle course. His pragmatic instrumentalism involves the
denials of most of our obstinate dichotomies (e.g., theory-practice, material-empirical, reasonexperience, fact-value) that emerge from our empiricist cum rationalist conception about the
aims of scientific inquiry. Peirce‟s pragmaticism is essentially alike Dewey‟s on this count.
The proposition my thesis asserts is that mathematics occupies the key position in the
hierarchy of sciences. I do not think that its truth needs to hinge on an indispensability
argument, though its essential bearings might well be guaranteed by an accurately specified
version of indispensability. I take the proposition to be an expression of a historical truth that
has been verified in the course of the history of science and mathematics time and again. It
was only the post-1950s conceptions of naturalisms that muddied the waters. But naturalism
and pragmatism are not in cahoots. If the early notions of naturalism, not altogether unlike
Dewey‟s pragmatic instrumentalism and even better Peirce‟s pragmaticism rather than the
modern ahistorical version, is correct, then we are compelled to draw the conclusion that the
ontology of mathematics cannot correspond to objects of fiction in the same way in which
novels or narratives are on the whole taken to correspond to them.
I am here progressing towards the key argument. According to Peirce‟s fallibilism
(and inter alia corrigibilism), not all scientific statements can be false in one go, although any
one of them may turn out to be subject to revision or rejection in the long run. Fallibilism lies
at the heart of pragmaticism and is nowadays subscribed to by the majority of scientists. It
needs to be evoked by philosophical investigations of mathematical practices. For Peirce,
fallibilism was is an outcome of the synechist model of continuity: metaphysics co-evolves
with human epistemology. The model of synechism cannot be set-theoretic or well-founded,
6
however; its purpose is to “objectify” (Peirce 1931-58, 1.171) fallibilism in taking all and any
possibility, including any possible and future mathematics, their objects, relations, structures,
propositions and theorems as real ingredients of the “true continuum” (Ehrlich 2010; Peirce
1931-58, 6.170). Real modalities trigger non-absolute knowledge, including mathematical
knowledge, to “swim, as it were, in a continuum of uncertainty and of indeterminacy” (Peirce
1931-58, 1.171). No scepticism can follow from this, as sceptical arguments aim at
establishing precisely the contrary case, namely that no scientific statement can resist being
subject to a possible revision or rejection in this actual world of our current investigation.
And the reason given in support of such actualism is the presumed lack of our epistemic
access to objects abstracted from ordinary space-time coordinates.
The fallibilist nucleus of mathematical pragmaticism contradicts the consequences
fictionalists are compelled to draw from the situation that mathematical statements, though
used referentially, are “in fact”, “actually”, or “strictly speaking” false. It is a factual error to
call any mathematical statement “in fact” false. It is the fact of synechism that any one of the
mathematical propositions entertained by any mathematician may turn out to be in the need of
revision in the long run, and it is an equal fact that any of the entities of mathematics may in
the course of future mathematical inquiry turn out to be non-existent though perfectly real.
But mathematical statements that concern synechist reality of the continuum cannot mean
their all-round actual falsification, just as any perpetual unactualisation of mathematical
entities in the future would indicate nothing about their unreality.
A couple of implications are now worth drawing. For one thing, it follows that
fictionalism cannot be upheld as a plausible reconstruction of structuralist philosophies of
mathematics. And it does not need to, for it is perfectly conceivable to have a realist
reinterpretation of modal category-theoretic structuralism. Hellman (2004) proposes that
mathematics should be understood and rewritten as subjunctive claims, such as: „If a suitable
structure of natural numbers were to be at hand, then there would be infinitely many prime
numbers.‟ According to Hellman‟s proposal, however, the interpretation of modalities
involved in such subjunctive conditionals is not compatible with pragmaticism and its
scholastic realism (citation omitted).
The option available to a pragmaticist, however, is to develop a proper semantics for
an axiomatisation of second-order modal logic. Hellman‟s axioms are S5, but he provides no
interpretation for them. In contrast, the pragmaticist‟s possible worlds are entities of reality.
7
Since model theory for second-order logic is not a set-theoretic one in the normal set-theoretic
sense of axiomatic systems, a new avenue for non-nominalistic semantics for second-order
modal logic is opened up.
The fruits of such semantics for second-order modal logic are many. For one thing, the
semantics is free from standard foundationalist disputes concerning the nature of sets. Second,
we can have a realist quantification of higher-order notions. Second-order quantification
presupposes a cross-identification of higher-order notions on the semantic level of possible
worlds. It can keep track of which of the higher-order notions actually exist and which pertain
to possibilia. Such cross-identification cannot be achieved in Hellman‟s theory, whose
possible-worlds talk is “heuristic only” and in which there are “literally no such things” that
merely might have existed (Hellman 2004: 147). The impossibility of cross-identification
follows from the fact that his second-order comprehension schema is not a cross-modal notion
and applies only “within a world”: all worlds are isolated creators of their own mathematics
(Hellman 2004: 147).
A parallel objection can be levelled against David Lewis‟s modal-realist interpretation
of possible worlds (Lewis 1969). It is important to note how different scholastic realism is
from modal realism. Lewis‟s interpretation implies that any matter concerning the meaning of
quantifying higher-order notions must be decided within the one, actual world of our
mathematical investigation. Problems concerning higher-order quantification cannot
according to his interpretation be decided in terms of alternative yet unrealised and
unactualised possibilities. Lewis‟s realism, also known as „extreme realism‟, is a misnomer: it
turns out to be a much less extreme case of nominalism in realist‟s clothing. For note how allembracing the indexical reference is: what there is in Lewisian possible worlds according to
their observers is just as like assuming that all observers in any one of such worlds are using
their own comprehension schemas in generating their mathematics.
3. Linguistic facts in intuitionism and in fictionalism
The second argument against fictionalism emerges from L. E. J. Brouwer‟s intuitionism.
Brouwer believed that mathematics has nothing to do with language (Brouwer 1975;
Heijerman & Schmitz 1986). Mathematicians‟ activities are prior to those to do with
language. Nothing expressed or described using language is relevant in solipsist mathematical
investigation. By the same token, the activities of mathematics are prior to logic, too,
8
including theories axiomatised by logical axioms. Since substantive fictionalism presupposes
the possibility of the existence of narratives – whatever they in the context of mathematics
may be – and since any reasonable account of the possibility of narratives must see them as
exemplifications of linguistic phenomena, according to the Brouwerian tenet of languageless
mathematics, fictionalism is unattainable.
The reason for any substantial account of mathematical fictionalism having to
presuppose a theory of narratives follows from the fact that a non-narrative fiction can at best
accomplish some „story-math‟: imaginary cases and thought experiments to test the reasoner‟s
mathematical skills. But that of course is not what fictionalism is after. Story-math
experiments never suggest that the mathematical elements referred to in non-narrative stories
are themselves fictional or created by the stories. On the contrary, the success of story-math
derives precisely from a realist understanding of real possibilities.
To put the same point in other terms, it is a commonplace feature of theories of
narration that narrative structures are built up from some elementary propositions or
constituents describing the events, motifs, and the principal agents of those structures. In the
story-math account, these constituents are supposed to act so that the outcome is an adequate
performance of some imaginary mathematical tasks. From the fact that these events, motifs,
or agents are fictional in the story-math account, no stretch of imagination or reasoning can
make the mathematics that these events and characters describe to refer to fictional
constructions, structures or entities.
Therefore, fictionalists‟ proposal that the claims of mathematics are not to be
interpreted literally is from the intuitionistic point of view on mathematical activities not a
false, unprovable, or unconstructible assertion. It is simply a meaningless utterance of words,
as in Brouwer‟s thinking there is no room for the distinction literal–non-literal.
Similar conclusion is inevitable with respect to the figuralist reinterpretation of
fictionalism (Yablo 2005). It even holds in a stronger sense. Figuralism is an offshoot of
broadly the same genre as fictionalism, and presupposes that mathematical language, or the
formal prose that gives rise to mathematics, must have some special, sophisticated devices of
generating non-standard meaning tropes. The kinds of tropes that Yablo adduces are
metaphors. However, not all figurative meanings are metaphors, and that there are subtle
logical, linguistic and cognitive issues to do with the meaning of metaphors (citation omitted).
Metaphor is not just one trope among many others or something that could be explained by a
9
plain reference to the nature of language being able to carry and communicate non-standard
meanings. It needs to be explained how language is capable of performing that sort of
trickery. Metaphors have complex, overlapping pictorial, diagrammatic, visual and schematic
characteristics that make use of several alternative, parallel media in order to represent the
essential characters of these special kinds of linguistic meaning. Yet none of these topics and
complications to do with the fundamentals of metaphors has been taken up in the literature on
mathematical fictionalism, including Yablo‟s figuralism.
This ought to be an imminent worry in the figuralist account, however. If
mathematical meaning is essentially metaphorical in its nature, it will necessitate the existence
of a language-like system that has at least equal expressive capacities and essentially the same
underlying workings as natural languages have. It is quite plausible, however, that the
negation of the consequent in the last sentence is likely to be the case. Natural languages
differ from mathematical ones precisely because they have evolved to accommodate complex
mechanisms creating a range of tropes and non-standard meanings, first and foremost
metaphors and metonyms. Even the simplest metaphor, for instance, typically has very
specific way of meaning transfer at its disposal, enabling subtle comparisons between
representational domains.
Yablo‟s appeal to metaphoric language of mathematics fails in other respects, too. He
assumes that likeness is the primary mover of the particular linguistic trope that explains the
nature of mathematical objects. First, the quality of likeness does not yet distinguish
metaphors from metonyms, however. Second, what the likeness means is left open. It is
precisely the question of how to understand the Aristotelian idea of comparison that makes it
possible to understand the phenomenon of metaphors in the first place – or any other
figurative meanings of language for that matter.
Third, figuralism is falsified by evidence: wealth of metaphors are better or worse, and
many are even clearly true or false. Metaphors have semantic content that is grounded in our
domains of discourse. They are deemed better or worse or even to have truth values by virtue
of their domains of the states of affairs. The properties of metaphors follow the semantic and
pragmatic rules and conventions of meaning. Yet the reality of such rules is something that a
consistent figuralist must go on to deny.
The pragmatic meaning of metaphors is the fourth problem. An important
phenomenon related to the use of metaphors, and quite commonplace in scientific prose, is
10
that metaphors tend to explain essential features of the world and hence strive to make the
complex underlying structures more understandable. Metaphors are routinely used to model
these structures in terms of the specific mechanisms shared by linguistic tropes, for instance in
view of simplifying these structures for explanatory purposes. And explanations are not
limited to metaphors in scientific prose: poetic metaphors can serve an analogous role in
explaining the general features of the underlying ideas of literary texts. But figuralism is not
about explaining the hidden facets of some underlying domains: it makes a stronger claim that
those domains themselves consist solely of metaphoric entities. Keeping in mind that this
claim assumes a tight parallelism between mathematical and natural languages, all meaning in
language must be constituted by some such metaphorical entities. Yet it is absurd to think that
any non-mathematical language would in reality be of this sort, notwithstanding the truism of
metaphors being far and wide in all communication.
It appears, then, that fictionalism and figuralism err in taking the appeal in nonstandard meanings and metaphorical entities to rest on the notion that fiction is a relatively
straightforward and commonsensical issue, more or less equal to make-believe, posturing or
imagination. However, this would be a conceptual error. We can imagine and make-believe
lots of things that are not fictional in the least. In theories of fiction, and in particular in those
that emphasise that fiction is distinguished from both narratives and literature in fundamental
ways, the character of make-belief or imagination is typically not sufficient for some textual
object or a story to emerge as fiction. Much more is needed, such as semantic or intentional
domains related to the domain of reality (Ryan 1991). And in order to build up an adequate
notion of fiction, semantic domains need to be constructed by way of author‟s or artist‟s
purposeful intentions, such as semantic gestures or plans by them for something to be
interpreted as fictional. But when we observe what is happening in mathematicians‟ actual
mathematical practices, there is nothing remotely resembling such intentional gestures.
Whatever the gestures may be, they are not to create the semantic domains to play the role of
fictional entities, quite the contrary. A mathematician routinely intends things and
propositions to be hypothetical, or to concern hypothetical constructions and assumptions that
provide the basis for the study of consequences from those constructions and assumptions.
But what a hypothesis is and what a fictional entity or character is have next to nothing in
common. Hence it is a mistake to think of hypotheses, no matter what their truth-values may
be, in fictional terms.
11
We can approach fiction from broadly two perspectives: as referential fiction, namely
as a mode of being that opposes the mode of being non-fiction, and as intentional fiction,
which entertains specific modes of speech, language or narrative that gives rise to fictioncreating communicative acts. Two points are to be noted. The referential account of fiction is
dubious, since non-factuality (falsities, errors, lies, embellishments) is not sufficient for
something to emerge out as fiction. Second, referential fiction does not imply intentional
fiction. However, mathematical fictionalism assumes the referential account and rests content
with the fallacious presumption that to fictionalise and to imagine something to be non-factual
amount to the same linguistic and ontological class of things.
The troublesome implications of this confusion cannot fail showing up. Fictionalism
has sometimes been classified among the „error theories‟ (Eklund 2005): the ceteris paribus
propositions and theorems of mathematics do seem true for us as well as for those stating such
propositions and theorems; however, we all are in fact mistaken about such attributions. There
are two problems with taking fictionalism to be an error theory. As noted, first, it is widely
accepted in theories of fiction that non-factuality, such as errors and mistakes, is not a
sufficient property to yield fiction. To classify fictionalism among the error theories is in fact
to commit a double error, since it means that fiction is taken to be a referential phenomenon,
yet one that could be characterised in terms of something going wrong with the reference.
What really goes wrong in this line of thought, however, is the ascription of mathematical
fictionalism to referential accounts of fiction.
Second, fallibilism does not permit that “we all” could “in actual fact” be “entirely
mistaken” about the status of mathematical truths. It only permits that at any given time, any
single such attribution may turn out to be in need of revision. This is virtually orthogonal to
the scepticism of error theories.
To return to the main argument of this section, I noted that fictionalism‟s appeal to the
use of linguistic information has problematic consequences. Even more worrying is that these
appeals to semantic and pragmatic aspects of language involve not only what the Gricean
theory of distinguishing between the literal and speaker‟s meaning is intended to capture but
also the specific understanding of speaker‟s meaning as the mathematician‟s meaning and as
interpreted by fictionalism. Assertions and intentions to assert play an equally important role
in those interpretations. Fictionalism eventually needs to claim that a proposition that a
mathematician utters is not an assertion of that proposition. However, if so, two points are
12
worth making. First, such notions are altogether irrelevant to Brouwer‟s intuitionism. Second,
even if we may not accept the intuitionistic standpoint, as most would probably not, to be
meaningful the fundamental linguistic facts will nevertheless have to be realistically
interpreted.
There is quite another sense in which figurative meanings are important to
mathematical practices, however. In reasoning hypothetically mathematicians frequently
experiment on diagrammatic constructions and then go on to observe the results of the
experiments (Hadamar 1949). By doing so they can gain important insights into what may,
might, or could be the case, something which they would not be able to accomplish without
such imaginative and iconic constructions. But this does not make mathematics any more
fictional than an architect‟s sketch of a building is fiction. Peirce‟s pragmaticism took
diagrammatic representations to be icons that have objects by virtue of some structural,
abstract or intellectual similarity or resemblance. According to pragmaticism, it is by virtue of
acting on and using such diagrammatic icons that real mathematical discovery is possible
(Stjernfelt 2007).
4. Conclusions
The moral of fictionalist philosophies of mathematics is a warning against a common fallacy:
mistaking the view that mathematical reasoning concerns mental, imaginary, hypothetical,
diagrammatic, iconic or phenomenal schemas and structures created in the mind in the course
of mathematical investigations for the view that takes the objects of observation of these
mental, imaginary, or phenomenal schemas and structures to be on a par with fiction. The
former view is both approved by actual mathematical practices and a common denominator of
such diverse, positive theories about mathematics as pragmaticism and intuitionism, whereas
fictionalism is antithetical to either of them.
On the other hand, Brouwer‟s languageless mathematics puts forth very different ideas
compared to the pragmatistic philosophy of mathematics. Yet the two have a common
motivation: the power of entities at issue in creative activities of mathematicians cannot be
undervalued or downplayed without losing the essence of mathematical inquiry. Both
entertain a notion of mathematical reasoning that concerns hypothetical creations of the mind.
Both agree that these creations concern mathematical objects, including structures. The
underlying conception of the mind is very different in the two, however.
13
On the whole, fictionalist attempts echo foundationalist projects in philosophy of
mathematics. In opposing the platonist indulgence in the sense-transgressing loci of reality,
fictionalist arguments are of the same genus with “as if” philosophies familiar from the
tradition of cartesian scepticism. It is startling that there has been nothing in the discussions
fictionalist philosophies of mathematics have engendered on this fact. Yet what I have aimed
at showing is that at the same time, fictionalism falls short of analysing the semantics of
tropes the existence of which they nevertheless need to take for granted.
References
Balaguer, Mark (2008). “Fictionalism, Theft, and the Story of Mathematics”, Philosophia
Mathematica, advance access.
Brouwer, L.E.J. (1975). Collected Works 1. Philosophy and Foundations of Mathematics, A.
Heyting (ed.), Amsterdam: North-Holland.
Burgess, John (2004). “Mathematics and Bleak House”, Philosophia Mathematica 12, 18-36.
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