# Semantic Realism: Why Mathematicians Mean What They Say*From: **http://www.bu.edu/wcp/Papers/Math/MathLand.htm*
**ABSTRACT:** I argue that if we distinguish between
ontological realism and semantic realism, then we no longer have to choose
between platonism and formalism. If we take category theory as the language
of mathematics, then a linguistic analysis of the content and structure of
what we say in and about mathematical theories allows us to justify the inclusion
of mathematical concepts and theories as legitimate objects of philosophical
study. Insofar as this analysis relies on a distinction between ontological
and semantic realism, it relies also on an implicit distinction between mathematics
as a descriptive science and mathematics as a descriptive discourse. It is
this
latter distinction which gives rise to the tension between the mathematician *qua* philosopher.
In conclusion, I argue that the tensions between formalism and platonism, indeed
between mathematician and philosopher, arise because of an assumption that there
is an analogy between mathematical talk and
talk in the physical sciences.
In this paper I argue that if we distinguish between ontological realism (the
claim that mathematical objects exist independently of their linguistic
expression) and semantic realism (the claim that mathematical statements which
talk about mathematical objects are meaningful), then we no longer have to
choose between platonism and formalism. If we take category theory as the
*language *of mathematics, then a linguistic analysis of the content and
structure of what we say in, and about, mathematical theories allows us to
justify the inclusion of mathematical concepts and theories as legitimate
objects of philosophical study. Insofar as this analysis relies on a distinction
between ontological and semantic realism, it relies also on an implicit
distinction between mathematics as a descriptive science (the view that
mathematics *is about* mathematical objects) and mathematics as a
descriptive discourse (the view that mathematics *talks about* mathematical
objects). It is this latter distinction, I argue, which gives rise to the
tension between the mathematician *qua* mathematician and the mathematician
*qua *philosopher. When the mathematician claims that a mathematical object
exists he intends to make a semantic claim. On the other hand, when the physical
scientist claims that an object exists he intends to make an ontological claim.
Thus, when the philosopher comes to analyze "existence" claims, he must be
careful to distinguish between these intentions. In conclusion, I argue that the
tensions between formalism and platonism, indeed between mathematician and
philosopher, arise because of an assumption that there is an analogy between
mathematical talk and talk in the physical sciences. Dieudonné characterizes the mathematician as follows: we believe in the reality of mathematics, but of course when
philosophers attack us with their paradoxes we rush to hide behind formalism and
say "mathematics is just a combination of meaningless symbols" . . . Finally we
are left in peace to go back to our mathematics, with the feeling each
mathematician has that he is working on something real. (Dieudonné in Davis and
Hersh, [1981], p. 321)
Given this description, one may ask: Do mathematicians by nature have
multiple-personalities or do philosophers make them crazy? Davis and Hersh would
have us believe the former: the typical mathematician is a [realist] on weekdays and a formalist
on Sundays. That is, when he is doing mathematics he is convinced that he is
dealing with an objective reality . . . But then, when challenged to give a
philosophical account of this reality, he finds it easiest to pretend that he
does not believe in it after all. (Davis and Hersh, [1981],
p.321)
What explains the mathematician's double stance, according to Davis and
Hersh, is his double roles. The mathematician *qua* mathematician is a
realist, the mathematician *qua* philosopher is a formalist. Benacerraf, on
the other hand, holds that the mathematician is driven to this personality shift
by two conflicting philosophical demands, namely: (1) the concern for having a homogeneous semantical theory in which
semantics for the propositions of mathematics parallel the semantics for the
rest of language, and (2) the concern that the account of mathematical truth
mesh with a reasonable epistemology. (Benacerraf in Hart, [1996], p.
14)
Thus, for Benaceraff it is not the double role of the mathematician that
accounts for his differing views. Rather it is the philosophers' double demand
for a homogeneous semantics and a reasonable (causal) epistemology. In this paper I argue that if we distinguish between ontological realism (the
claim that mathematical objects exist independently of their linguistic
expression) and semantic realism (the claim that mathematical statements which
talk about mathematical objects are meaningful), then we no longer have to chose
between platonism and formalism, or worry about whether our semantics and
epistemology matches that of physical discourse. If we construe category theory
as the language of mathematics, then a linguistic analysis of the content and
structure of *what we say* allows us to justify the inclusion of
mathematical concepts and theories as legitimate objects of philosophical study.
Specifically, such an analysis permits us to justify the claim that mathematical
objects exist independently of us but, at the same time, depend on the structure
of a given category and the content of a given theory. Inasmuch as this analysis relies on a distinction between ontological and
semantic realism, it relies also on an implicit distinction between mathematics
as a descriptive science (the view that mathematics *is about* objects) and
mathematics as a descriptive discourse (the view that mathematics *talks
about* objects). It is this latter distinction, I argue, which gives rise to
the tension between that mathematician *qua *mathematician and the
mathematician *qua *philosopher. When the mathematician claims that a
mathematical object exists he intends to make a semantic claim: he intends to
say that the statement in which its concept occurs meaningfully refers. When the
physical scientist claims that a physical object exists he intends to make an
ontological claim: he intends to say that the concept is meaningful *in virtue
of *its reference to an independently existing object. Thus, when the
philosopher comes to analyze "existence" claims, he must be careful to
distinguish between the intentions that motivate such claims.
**1. The Semantic Tradition**
In this section I argue that we have yet to appreciate the significance of
what Coffa, [1991], has termed "the semantic tradition": we continue to assume
that the conditions for meaningful and true assertions must be grounded in
either what we can know or what exists. As a consequence, we continue to read
those philosophical issues that relate to the semantics of mathematical
discourse as strictly epistemological (either naturalized or psychologized) or
strictly ontological (either physicalized or platonized). In contrast to such
readings, the semantic tradition is characterized by the attempt to justify the
inclusion of concepts as legitimate objects of philosophical study by "offering
an analysis of the content and structure of *what we say*, as opposed to
considering the psychological (or causal) acts by which we come to say it". (
Coffa, [1991], p.1) The semantic tradition began as a reaction to the psychologistic
interpretation of Kant. In particular, philosophers and mathematicians belonging
to this tradition sought to maintain the *a priori* character of
mathematics while at the same time avoiding the seemingly necessary appeal to
pure intuitions. As Coffa notes, [t]he semantic tradition consisted of those who believed in the* a
priori* but not in the constitutive powers of the mind . . . the root of all
idealist confusion lay in misunderstandings concerning matters of meaning.
Semanticists are easily detected: They devote an uncommon amount of attention to
concepts, propositions, senses . . . (Coffa, [1991],
p.1)
Bolzano, in contrast to Kant, sought to account for the nature of both
mathematical and philosophical concepts not in terms of the conditions for the
possibility of experience, but in terms of the *conditions for the possibility
of saying*. He sought to make a study "not of the transcendental
considerations but of what we say and its laws. . . of semantics, not
metaphysics or ontology". (Coffa, [1991], p. 23) Bolzano's aim, then, was to show that like philosophical truths,
"mathematical truths can and must be proven from the mere [the analysis of]
concepts". (Bolzano quoted in Coffa, [1991], p.22) He did this by demonstrating
how "mathematical rigor [could be] both an epistemological as well as a semantic
notion". (Coffa, [1991], p. 26) This latter point is crucial for understanding
the relevance of the semantic tradition, for as Coffa tells us [i]t is widely thought that the principle inspiring such
reconstructive efforts was epistemological, that they were basically a search
for certainty. This is a serious error . . . a no less important purpose was the
clarification of what was being said. (Coffa, [1991], p.
26)
Underlying Bolzano's demonstration of the dual character of mathematical
rigor was the distinction between what he termed 'subjective' and 'objective'
representations. The former being the mental states of the soul and the latter
the intersubjective content of the subjective representation. Meaning, then,
relates not to the subjective representation but rather relates to the
intersubjective content and as such is in no need of assistance from intuitions
(either empirical or pure). Thus, what Bolzano did, more that bringing rigor to
mathematics, was effectively extrude mathematical concepts from the mind and
place them firmly on semantic soil. This separation of semantics from psychology likewise enabled a distinction
between both ontological realism and idealism on the one hand and, what I have
termed, semantic realism on the other. Objective representations are not
ontological entities, "they subsist not indeed as something existing, but as a
certain something . . .". (Coffa, [1991], p. 30) Nor are they psychological
entities: [they] are the substance (Stoff) or [intersubjective] content of
subjective representation. There being in no way depends on the existence of
subjective acts, just as the meaningfulness of expressions in no way depends on
anybody's having the appropriate meanings in mind; and like meanings, there is
only one for each linguistic unit. (Coffa, [1991], p.
30)
Unfortunately, this latter demand (that there be a isomorphism between the
intersubjective content of subjective representations and the meaning of
objective representations) was thought by Bolzano to be satisfied by the
assumption that there is an isomorphism between words (in natural language) and
objective representations, i.e., by the assumption that "there is only one
objective representation designated by the word." (Bolzano quoted in Coffa,
[1991], p. 30) Frege overcame this problem first, in the* Begriffsschrift*, by creating
a language of concepts, as opposed to relying on natural language to give names
to objective representations. Second, to Bolzano's demand that we "always
separate sharply the psychological from the logical, the subjective from the
objective", he added, in his *Die Grundlagen der Arithmetik*, that we
"never ask for the meaning of a word in isolation, but only in the context of a
proposition". (Frege, [1978], p.x.) Given these proscriptions, arithmetic could
proceed, as philosophy, through the analysis of the content and structure of its
concepts, modulo the additional demand that, when writ in the language of the*
Begriffsschrift*, such an analysis need only consider *what can be said*
of the relation between concepts and objects. Regrettably, while Frege's context
principle solved Bolzano's semantic problem at the level of concepts, what
Russell's paradox shows is that it does not solve the problem at the level of
objects. Basic Law V has an existential content which is conceivably false,
i.e., which cannot be justified from within Frege's language, and therefore he
could not conclude that arithmetic was analytic. It is clear that both Bolzano and Frege exemplify the semantic tradition.
What is not clear, however, is whether the problems they encountered imply the
demise of the semantic approach itself. If we are to take history as our guide,
then it appears as though we ought to conclude that we must supplement our
semantic analysis with the assumption that either logic or reality provides a
basis for what we say. Does this historical conclusion truly represent all of
our options? To answer this question we must first recognize what assumption is
necessary for such a drastic conclusion, viz., that the only way I can use a
statement to *talk about* objects is if the statement *is about*
objects. What this assumption does, however, is conflate the linguistic claim
that meaningful statements *talk about* objects, with the ontological claim
that meaningful statements are* about* objects. It is this conflation, I
argue, that leads to the lack of distinction between ontological realism and
semantic realism. While semantic philosophers realized that sense need not be constructed out
of psychological intuitions, philosophers following them attempted to legitimize
mathematical and/or theoretical concepts by offering a logical analysis not of
what we say but by, in effect, reducing them to logical and/or empirical 'atoms
of meaning'. Indeed it is the conflation of linguistic and ontological claims
that lead philosophers, like Wittgenstein and Quine, to believe that unless we
restrict ourselves to the world of facts or have access to 'real essences', as
opposed to their nominalistic counterparts, then what we say will be so
intimately bound to the conventions we choose, that any semantic analysis of it
will show nothing. Thus, it is here that we must apply the lessons of the
semantic tradition, before we take our cue from the philosophical history. For,
as Coffa warns: [f]ew things have proved more difficult to achieve in the development
of semantics than recognition of the fact that between our subjective
representations and the world of things we talk about, there is a third element,
what we say . . . many of the best philosophical minds . . . were [and are!]
unable to understand that what we say, sense, cannot be constituted either from
psychological content or from real-world [or logical] correlates of our
representation . . . (Coffa, [1991], p.77)
**2. An Analysis of Discourses**
The questions that I want now to consider are: Is conventionalism a necessary
consequence of a linguistic analysis of mathematical discourse? and Why does a
linguistic analysis not suffice for physical discourse? In this section I argue
that if we take category theory as the language of mathematics, then we can not
only justify the inclusion of mathematical concepts as legitimate objects of
philosophical study, we can also justify the inclusion of mathematical theories.
Finally, I show why a linguistic analysis of the content and structure of what
we say cannot suffice to justify the inclusion of either physical concepts or
physical theories. While semantic philosophers of mathematics have come to rely on the syntax of
logic or model theory to provide justification for the inclusion of mathematical
concepts and theories, the linguistic approach relies on category-theoretic
notions. If we restrict our analysis to what can be said in, and about,
mathematical theories, then we can justify the inclusion of mathematical
concepts and theories by representing talk of their content and structure in
category-theoretic terms. What category theory does, as far as our talk of
mathematical concepts and relations is concerned, is provide a means for
organizing and classifying what we say about 'the structure of the relationship'
between various mathematical concepts in various mathematical theories. We say
that category theory is the language of mathematical concepts and relations
because it allows us to talk about their structure in terms of "objects" and
"arrows", wherein such terms are taken as syntactic assemblages waiting for an
interpretation of the appropriate sort to give them formulas meaning. Likewise, at the level of mathematical theories themselves, our talk of 'the
structure of the relationship' between mathematical theories and their relations
is represented by category-theoretic notions. We say that category theory is the
language of mathematical theories and their relations because it allows us to
talk about their structure in terms of "objects" and "functors", wherein such
terms are, again, taken as syntactic assemblages waiting for an interpretation
of the appropriate sort to give them formulas meaning. Thus, recalling our
semantic aim (to justify the inclusion of concepts by offering an analysis of
the content and structure of what we say), it appears that both mathematical
concepts and mathematical theories can, when writ in the language of category
theory, be taken as legitimate objects of philosophical study. Returning to our initial questions, we note that while the theoretical
axioms, definitions, etc., may be conventions, i.e., may depend on the terms, or
model, chosen, the category itself remains as an objective representation of
both the content and structure of what such axioms and definitions *say
about* the corresponding concepts from *within* a given mathematical
theory. Conventionalism, then, is a consequence of accepting that mathematical
theories themselves are *local *domains of mathematical discourse, or in
Carnapian terms are local "linguistic frameworks". It is not a condition of the
description. The reason why we do not run into the problems that Carnap encountered,
however, is that we can describe the structure of theories themselves in
linguistic terms, i.e., in category theoretic-terms. So when Tait claims that
Carnap is right that 'external questions' of existence have no
prima-facie sense. But ... his notion of a linguistic framework fails [because]
linguistic frameworks are constructed in our everyday language; and it is hard
to see how we can convincingly determine when we have a 'good' framework and
when we do not. (Tait in Hart, [1996], p. 151)
what we must note is that there is a middle ground. If we take mathematical
theories to be linguistically represented by categories, then the
category-theoretic version of a linguistic framework does not fail. We can
objectively determine when we have a 'good' framework and when we do not. There
is no need, therefore, to reduce the content or structure of mathematical
theories to 'atoms of meaning'; we get all the meaning we want or need from
*within *mathematical theories. The question that remains to be considered is: Why is a linguistic analysis
of the content and structure of what we say not sufficient to justify the
inclusion of either physical concepts or physical theories? What we first note
is that our "facts" occur at different levels. Mathematical facts occur
*within theories*; they are fixed by the way things are in a given
mathematical theory. Insofar as mathematical theories and their elements and
relations are themselves linguistic entities, then so are mathematical facts.
Physical facts, on the other hand, occur *in reality*, their constituents
are ontological entities. The reference of a physical concept, i.e., a physical
object, is not a linguistic entity, it is an ontological one. The question of
the reality of physical objects and the truth of physical propositions thus
cannot be settled linguistically: it must crucially depend on some
extra-linguistic process whereby the connection between language and reality is
both established and justified. This is the lacuna that naturalized epistemology
fills. What the naturalized epistemologist and the platonist must realize, however,
is that this demand for a non-linguistic experience (causal or intuitive) does
not apply to mathematical knowledge. Quite simply there is no gap in
mathematical discourse between language and reality. *Mathematical reality in
contained within, as opposed to merely constrained by, the language of
mathematical discourse*. Insofar as the constituents of mathematical
propositions and the referents of mathematical concepts occur within local
domains of mathematical discourse, mathematical facts are in no need of
assistance from either physical or metaphysical facts. There is therefore no
need to appeal to the ways things are in reality (metaphysical or physical) to
justify the inclusion of mathematical concepts. Likewise, then, our knowledge of
mathematical facts is in no need of assistance from any extra-linguistic
experience. This distinction, between mathematics as a descriptive science (the view that
mathematics *is about* mathematical objects) and mathematics as a
descriptive discourse (the view that mathematics *talks about* mathematical
objects), itself implies a difference in experience. To experience a
mathematical fact we require a meet between use, proof and language; whereas, to
experience a physical fact we require a mesh between reality, observation and
language. If, then, in our account of mathematical experience, there is to be
any appeal to mathematical intuition it must be one which distinguishes it from
both empirical and pure intuition in the following manner: What we call "mathematical intuition" . . . is not a criterion for
correct usage. Rather having mastered the usage, we develop feeling, schematic
pictures, etc., which guide us . . . What is objective about the existence of
mathematical objects is that we speak a common language about them and this
includes our agreement about what counts as warrant for what we say. (Tait in
Hart, [1996], p. 147 & 149)
**3. Semantic Realism**
So where does this analysis of discourse leave our mathematician? He is a
modified-formalist, or structuralist, to the extent that he believes that a
linguistic analysis of mathematical structure justifies our accepting
mathematical concepts and theories as legitimate objects of philosophical study.
He is an internal realist to the extent that he believes that a linguistic
analysis of the way things are *within* a given theory justifies our
accepting that mathematical statements meaningfully talk about mathematical
objects. Thus, what the mathematician *qua* semantic realist believes is
that meaning, truth and existence are a linguistic notions, as opposed to a
logical, psychological or metaphysical notions. This is why the mathematician
means what he says. **References**
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