On Naturalism in Mathematics

On Naturalism in
Mathematics
Alfred Lundberg
Bachelor’s Thesis, Spring 2007
Supervison: Christian Bennet
Department of Philosophy
Göteborg University
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Contents
Contents.......................................................................................................................................2
Introduction................................................................................................................................ 3
Naïve Questions.............................................................................................................................................. 3
Contemporary concerns................................................................................................................................. 4
Naïve answers of naïve questions ................................................................................................................ 6
Traditional Schools of Mathematical Philosophy ...................................................................................... 7
Point of Departure.......................................................................................................................................... 9
Quine’s philosophy.................................................................................................................. 10
On Ontology.................................................................................................................................................. 10
An Empirical Dogma.................................................................................................................................... 11
Naturalism...................................................................................................................................................... 12
Meaning according to Quine....................................................................................................................... 13
Quine on mathematics.................................................................................................................................. 13
Maddy’s philosophy................................................................................................................. 15
Criticism: Quine........................................................................................................................17
Mathematics in the web................................................................................................................................ 18
Meaning.......................................................................................................................................................... 18
Ontology......................................................................................................................................................... 19
Cartesian doubt is not the way to begin..................................................................................................... 20
Criticism: Maddy.......................................................................................................................22
Conclusion.................................................................................................................................25
References..................................................................................................................................26
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Introduction
Philosophy has aimed to ask and answer clever questions on a wide variety of subjects
throughout the ages. These can be summarized in a single one:
–
Or, spelled out a bit:
– What is its nature
Turning towards mathematics, this can be differentiated and specified in several ways,
each way resulting in one of the traditional philosophical questions of mathematics. These
may, as in other philosophical fields, take simple or more technical form, but they originate,
at least historically, in common sense questions of the nature of mathematics. Therefore,
a common sense starting point for this exploration seems appropriate. Also, for people
not familiar with the discipline this is supposedly the best approach. We will concentrate
mainly on a philosopher who urged the leaving behind of far-fetched philosophical
theorizing in favour of common sense questions in a scientific context. Thus, this approach
will not oppose his manners, but will on the contrary meet central standards of
consequence.
Naïve Questions
So, what is mathematics What does it do What does it tell us Mathematics is a modern
scientific field of research, but with roots of thousands of years. It is definitely among the
oldest subjects found at universities of today. Taking a closer look, many questions arise:
What is the nature of mathematics This rather general question is seldom asked as it
stands. It splits up into a variety of more precise questions.
What does mathematics talk about Mathematics seems to be about something. Mathematicians
talk as if they talked about something. This, any grammatical analysis would consent
to. Well, the answer seems quite simple: Numbers. Mathematics is about numbers, and
their relations and their properties. It is also about functions and matrices, of course, and
sets and points and surfaces and so on. But several of these entities are of a somewhat
strange nature. In what ways are these entities What are they, and where The ontological
status of mathematical objects has been the subject for discussion throughout the ages.
These objects seem to be, in a way, independent of our thoughts and wishes, yet they have
a rather clear mental status, which connects them to the mind, rather than the outer physical
world.
What is mathematical knowledge Mathematics seems to deliver knowledge in some sense.
First of all, mathematical studies have the result that the student, insofar as the studies are
successful, feels that he knows more afterwards. Mathematics has been rigorously important
in the development of the natural sciences. And science tells us something about our
environment. Thus, we have all the usual philosophical questions connected to knowl-
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edge, in their mathematical form: How do we acquire mathematical knowledge Mathematical
knowledge seems more secure than other kinds of knowledge, as it stands in no dependence
upon empirical knowledge, or does it Anyhow, if so, how can that be And if
mathematical knowledge is independent of sensory input, how is it possible to explain the
remarkable success of mathematics in natural science Epistemology has traditionally been
a central concern of the philosophy of mathematics.
Are mathematical statements meaningful, despite their scent of triviality How is it possible for
mathematical statements to be meaningful, as they only state what seems not to could
have been otherwise, what seems to be trivially true
Ever since Pythagoras until our days, mathematics has been worshipped in strong romantic
terms. The field is claimed to offer an unparalleled rational training, to exhibit artand
music-like beauty. It has been claimed to provide knowledge of the most secure form
possible and it has inspired religious tendencies ranging from Pythagorean number mysticism,
via so-called sacred numbers appearing in almost all major world religions, to cabbalistic
permutation and the like. What is the real ground for these reactions What eternal
virtues of mathematics does this witnessing really tell of
Is our mathematical theory the only (possible) one Might another culture, another species,
take a radically different line of mathematical development Thus, in practice we might
ask whether there are other axiomatisations that are equally good but that differs essentially
from our present one. If so, we would consider there to be two possible mathematics,
thus rejecting uniqueness.
Contemporary concerns
The connections of mathematics to logic seem obvious. The fields share a range of attributes
due to their common axiomatic and deductive method, their security of results, etc. It
is hence not a surprise that efforts were made in the late 1800’s to establish mathematics
on the grounds of logic alone. The notion of a collection or a class of objects was already
in the air, and mathematician Gottlob Frege used the concept in trying to provide a
foundation for arithmetic. His aim was to establish the philosophical status of mathematics
through giving a complete account for the natural numbers, and these he defined as
classes of equinumerous sets. Unfortunately, his logical system was shown inconsistent 1 by
Bertrand Russell not long after. To save the project one had to skip what was called the
Axiom of Comprehension, and to rebuild the theory through specifying a number of new
axioms. This was done in order to weaken the system, to avoid contradiction, while retaining
most of its expressive powers, those necessary in expressing and proving the results of
mathematics. This work was mainly accomplished by Richard Dedekind, Abraham Fraenkel,
Ernst Zermelo and Thoralf Skolem, and the resulting theory is commonly known as
Set Theory, or ZFC 2 .
1
A system is inconsistent if it proves a contradiction, consistent otherwise.
2
ZFC denotes the Zermelo-Fraenkel axiomatisation of set theory, together with the Axiom of Choice. This
system is widely accepted as the foundation of all of mathematics, though not the only set theory.
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The notion of equinumerous sets was also the basis for Georg Cantor’s theory of cardinal
numbers. In it, Cantor established some nontrivial facts on the number of elements
of sets, applicable on sets of numbers through their interpretation in ZFC. Now, one soon
ran into further worries, as one of Cantor’s hypotheses proved to be independent of the
axioms of ZFC 3 , and more examples have later been found. Thus, Frege’s initial theory
had proved erroneous, in allowing contradictions to be deduced, and so being to strong.
And its replacing theory, ZFC, now proved to be too weak, as it was not able to settle natural
questions appearing in the theory. This is the picture still today. Hence, the question
arises, what are we to do about such results For every such question, is it to be answered
in the affirmative or in the negative, when the theory cannot advise Or should we simply
accept their independence If mathematics is about real, somehow existing, objects, what
moral is to be drawn from such a situation And how will mathematics proceed
But more is to be said on logic and mathematics, and even more intriguing questions
arise. Among the most important logical results to this day are the so called Incompleteness
Theorems that were proved by Kurt Gödel in 1930. These results came as a mere
chock to the contemporary mathematico-logical society. David Hilbert had in 1899
showed how the Euclidean geometry could be formalized in a first order logic, and Alfred
Tarski proved some years later that Hilbert’s theory was complete and decidable 4 . Now,
the hope was to repeat the feat for number theory. Through the works of Richard Dedekind,
Giuseppe Peano and others, the Peano Arithmetic, PA, was formulated as a logical theory
of first order, and the idea was now to provide rigorous ground for that also PA was
complete and decidable. This project was completely overthrown by Gödels results. His
first theorem shows that in any axiomatizable first order theory T that is consistent and includes
PA, we can construct true sentences, in the language of T, that do not have a proof
or a disproof. That is, any such theory, PA included, is not complete. In his second theorem,
Gödel showed that this result has consequences for what can be said about consistency.
Any logical theory aiming to found mathematics must account for number theory.
In fact, with number theory at hand, most of classical mathematics can be constructed. So
mathematics without number theory is simply not imaginable. Now, the second incompleteness
theorem says that if we are to formalize mathematics in an axiomatizable theory
T that includes PA, and we do this in first order logic, then T cannot prove its own consistency.
Thus, we may not, in first order logic at least, account for the consistency of
mathematics, using elementary mathematical methods. Now, the reason we would want to
do this in first order logic, is that, while second order logic can account completely for
number theory 5 , it has no complete deductive system. On the other hand, any logic with
such a system, first order or not, would again be subject to Gödels theorem. It appears
thus that we cannot, as far as mathematical methods can guide, be certain of the consistency
of mathematics. And also, it appears as number theory inevitably contains statements
3
The Continuum Hypothesis, that any set of real numbers either is equinumerous to the set of all real numbers,
or to the set of natural numbers, or is finite, was proved independent of Set Theory by Paul Cohen in
1963.
4
A theory is complete if every sentence in the language of the theory has a proof or a disproof from the axioms.
It is further decidable if there is an algorithm for determining for any such sentence that it is a theorem, i.e.
has a proof, or not.
5
Dedekind formalized number theory in second order logic in 1880, and subsequently showed it to have exactly
all true statements of number theory as theorems.
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that can be proved neither true nor false. This is of course a fact contradictory to what
might be called an advanced naïve view on mathematics, as held by mathematicians
throughout mathematical history.
Thus, at a first philosophical glance at mathematics, the following themes of questioning
arise:
~ What is its nature
~ The ontology of mathematics
~ Applicability
~ Mathematical epistemology
~ Meaning and mathematics
~ Eternal virtues
~ Uniqueness
~ Incompleteness of the axioms of set theory
~ Incompleteness in the sense of Gödels theorems
Surely, there is no limit to the complexity of philosophical questions on mathematics,
this aims just to account for the major areas of interest.
Naïve answers of naïve questions 6
Given a multitude of ‘naïve’ philosophical questions on the nature of mathematics, we
might as well try to answer first in a completely commonsensical spirit, just to get oriented,
just to establish a common view point of reference in order to prevent any unnecessary
taking off into too philosophical abstractions, or at least to make any such divergence
evident and explicit. Little would indeed be gained in insight if we came up with
fancy theories, but were unable to relate them to common sense and the every day meaning
of words involved. Briefly, then the following is taken to be widely accepted first impressions
of the science.
In its everyday use, ‘mathematics’ may well be used in the following senses: ‘Mathematics
is an activity’, just as any other scientific field. ‘Mathematics is a language’, the mathematical
theories have communicative uses, much seems to be possible to formulate in the
language of mathematics. Further, ‘Mathematics is symbols on papers’, and ‘Mathematics
exists in some way in our minds’. But while mathematics ‘occurs’ in our minds, the mathematical
results seem remarkably independent of this mind.
It seems indeed to speak of objects of some kind, but no one has ever seen any, nor
are we very well advised on the nature of their existence. Somehow, is the immediate conclusion,
the nature of existence seems not very central to mathematics. Seemingly, it has
nothing to do with the essence of mathematics.
If mathematics is to take place, in equal extent, in the head, on a sheet of paper, and
in computers, then one would assume some similar characteristics of these media and the
6
These statements are mainly drawn from [Bennet].
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mind. If the scientific understanding of the mind is the brain, then from what we know of
the brain and the workings of computers, this would be a structural likeness. This seems
perhaps more reasonable, looking from the common sense view point, than an existence
and a related perception of unseen objects.
Mathematics seems appropriate when revealing relations we had not foreseen. But
once discovered, the results seem undeniable, and often trivial. So we experience a meaning
through laying bare, formerly hidden, but trivial, and necessary results.
The eternal virtues of mathematics seem to be the effect of its nonnegotiable results,
its strict rule-governed behaviour. Its beauty due to the succulence of final insight of trivial
truths, the art-likeness due to its pattern-modelling, its unlimited complexity potential,
and so on.
Finally the possibility of a similar realm of thoughts, of equal use, but of a distinguishable
character seems unbelievable, but admittedly on vague grounds.
Traditional Schools of Mathematical Philosophy 7
The traditional schools of philosophy of mathematics all arose, in their modern versions,
in the late nineteenth century, but carry obvious residue from their philosophical ancestors.
In addition the terms Logicism and Fictionalism will be explained:
Realism – This view considers the field of mathematics to be a science of a matter of
fact, attributed to the world. For the realist distinguishes truth from knowledge. The
mathematician becomes an explorer, rather than an inventor, for the mathematical landscape
is ‘already there’, waiting for us to discover it. Therefore, independent questions of
set theory are assumed to have their answers despite their logical independence of the axioms.
We need just take a look in the realm of sets. Therefore, the axioms of the theory
must be judged incomplete in some sense, they do not completely account for the universe
of sets. It follows that the foundational problems concern the actual logical theories
rather than the mathematical questions, or mathematics ‘itself’. Some philosophers (Gödel
e.g.) claim that we acquire mathematical insight through some special faculty of mathematical
perception. As there is no such imaginable thing as the simultaneous being and nonbeing
of an eternal object, or the having and non-having of a given property, of a certain
number, realism must be taken to imply the consistency of arithmetic and set theory, if
not of the actual logical systems of arithmetic and set theory. As the mathematical theory
models the relations in the one mathematical realm, uniqueness must follow too. The
eternal virtues of mathematics are, it is proposed, best understood in connection to their
mystical metaphysical status, and the related perceiving of such entities. The meaning of a
statement of mathematics is something parallel to meaning in ordinary scientific or commonsensical
context. Mathematics talks of a matter of fact as much as does science.
Intuitionism was founded by L. E. J. Brouwer in an attempt to reconstruct mathematics
in accordance with epistemological idealism and Kantian metaphysics. The natural numbers
are the primary objects of mathematical knowledge. These spring from immediate
7
This section originates from, in addition to Maddy, the encyclopaedia articles listed under ‘References’.
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temporal experience. Therefore, the objects of mathematics stand in ultimate dependence
on an experiencing mind. In this sense, the mathematician invents his mathematical objects.
Thus mathematics is not an attribute of the world, but rather of our experience. This
accounts for the applicability of mathematics on the experienced world. According to
Brouwer, there are no unexperienced truths, therefore mathematical results must be based
on intuitive clarity rather then logical proof, and so a mathematical claim is true if one intuitively
understands every stage of the proof process. In that, mathematical knowledge is
a priori and intuitive. Also, we may read out a theory of meaning where constructive proof
replaces the ordinary correspondence notion of truth. The mathematical knowledge is
thus profoundly linked to our experience of the world, and this attributes to it its glossy
guise. Brouwer accepted the formalisation of intuitionistic logic, and this would suggest
that mathematics describes an aspect common to many, and not completely subjective.
Therefore mathematics must be the only of its kind.
Formalism is a mathematical relative of the much older doctrine of nominalism. The
nominalist view is that concepts at large do not have separate being in any abstract way.
They are but symbols. So, the formalist rejects any type of abstract mathematical entities,
mathematics is no more than a rule-governed game of insignificant notation. The game
might well be of great practical utility, but this implies in no way a significance of the literal
linguistic kind. David Hilbert, the central advocate of this view, held that finitary mathematics
describes indubitable facts about real objects, but that one introduces the ‘ideal’
objects that feature elsewhere just in order to facilitate research about the former. The reference
to real objects establishes uniqueness of arithmetic, and thereby of all of mathematics.
This relation to the physical world would explain some of the virtues attributed to
mathematics. Mathematical knowledge amounts to knowledge of the behaviour of a game
of symbols, and of some aspects of the physical world.
Logicism was the attempt to found all of mathematics on logic, and thus to provide
both ontological and epistemological clarity. Gottlob Frege was the pioneer through his
attempts in constructing mathematical objects from logical concepts alone. Bertrand Russell
further continued the efforts in his Type Theory, so to avoid the paradoxes discovered
in Frege’s system, but he was forced to use ad hoc axioms to be able to account for the
real numbers. The Hilbert Program aimed to secure mathematics through reducing mathematical
theory to formal systems, and then prove the consistency of these. As we’ve seen,
these efforts have not been successful so far. One first thought that this was possible using
elementary, mathematical methods, involving number theory, but this project was
dealt a lethal blow from the results of Kurt Gödel. It is still up to discussion whether or
not his results put an end to Logicism.
Fictionalism, or instrumentalism, is the view that rather than aiming at literal truth, sentences
or the use of objects in science or mathematics is to be seen as a fiction, a way of
speech, which may have various pragmatic advantages. Thus statements are not true or
false, but convenient or inconvenient. In mathematics, we talk as if numbers exist, because
doing so permits us to construct various things in technology, take various associative
steps in science or reasoning, but by saying ‘2’ we do not succeed in actually referring to
anything in particular.
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Point of Departure
Given the questions introduced, the aim is now to approach the philosophy of Willard
Van Orman Quine, and take a closer look at its consequences in the philosophy of mathematics.
Penelope Maddy offers an account that has clear mathematical sympathies and
her book, [Maddy], is thus a major source of the account at hand.
There, she sets out to investigate the possibilities of a modern, mathematical method
of research, firmly based on an aimed-for, accurate view of what have been basic conditions
for the massive mathematical development over the last two hundred years. Her interests
are chiefly methodological, but she grounds her conclusions in a variation of the
naturalistic outcomes of Quinean thought. One that takes actual mathematics as central,
that “extends the same respect to mathematical practice that the Quinean naturalist extends
to scientific practice” 8 . Despite methodological concerns, her investigation also
provides an account of the metaphysical results of the stance.
These pages, though, will overlook methodological intents. The objective is to ask the
ordinary questions, and to answer them in a Quinean spirit, noting in what ways Maddy’s
variation of his thoughts brings her to differing conclusions.
Further, a criticizing part will scrutinize their ideas, establishing the relations of dependence,
looking for misconceptions and overly optimistic conclusions, and ask questions
on their success in answering the naïve questions of the philosophy of mathematics
in a satisfying way.
Quine is chosen on the basis of the still central role of his philosophy in the contemporary
debate. His thoughts dissolve, or so aim to, the traditional dispute on the metaphysics
of mathematics, introduced above, and offer a modern, sound and sane view of
the role of science, and, thereby, of mathematics. This commonsensical spirit motivates
our outset in naïve questions on mathematics.
8
[Maddy], p185 (henceforth referred to just by ‘p185’)
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Quine’s philosophy
The philosophy of Quine emerges from a deep conviction to empiricism, a conviction
that scientific evidence is what evidence there is for any given statement about the world.
Quine’s interactions with Rudolph Carnap, who stressed the importance of the linguistic
framework of statements for their meaningfulness, are present through his understanding
of scientific research of all branches as parts of an over-all scientific project, the unique
scientific framework. Although influenced to a certain degree, Quine would not follow
Carnap in his thus implied views on mathematical meaning and ontology.
A few themes will here be accounted for, that are central to Quine’s view on mathematics,
and which appear in the version given by Penelope Maddy.
On Ontology
Questions gain their intelligibility from the linguistic framework in which they occur. That
is Carnap’s point. Carnap took ontological questions to be external to any linguistic framework,
he took the acceptance of them to be part of the very adoption of such a framework,
and thus to be a pragmatic, rather then a theoretical question. Posed outside the
framework, ontological questions were but pseudo-questions, with nothing to provide
standards by which they were to be evaluated – inside the framework they were trivial. For
instance, in discussing number theory the existence of 2 is a trivial, since presupposed,
matter. If, on the other hand, we take a step back and ask whether 2 really exists, then we
are outside the explanatory framework that specifies what is meant by ‘2’ so that our question
becomes unintelligible, meaningless, as a theoretical question. Thus, the issue of the
‘real’ existence of 2 is settled indirectly through our adoption of a framework that includes
‘2 exists’ as a part. And whether to adopt or not a given framework is a practical, rather
than theoretical, matter.
With the over-all scientific theory in view, also Quine would take ontological questions
for meaningless outside this scientific context, but, opposing Carnap, he considers
these to have meaningful counterparts inside science 9 . These outer questions should be
adopted on pragmatic grounds, such as simplicity, economy, efficiency, just as Carnap argued.
But Quine notes that this is really the ground for adoption of any theory, or any theoretic
part, of science. Thus in the framework of overall science, pragmatic standards are
an integral part of the very framework. Thus ontological questions, those who would be
settled on pragmatic grounds, have their intelligible, internal counterparts in the scientific
context.
So, how are we thus to understand a scientific statement of existence There is nothing
about ‘real existence’ for Quine; electrons don’t ‘really exist’. But the assertion of existence
of electrons is central to the scientific field. It is an important component without
9
Rather, Quine would say that ontological questions are internal to science, while Carnap would consider
them external, and therefore meaningless.
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which science would definitely not be the same. The observation is thus, that science, as it
is, includes the existence of electrons as a façon de parler, an assertion that has meaning in
relation to all other assertions made by science, and this extends to all scientific claims:
“Scientific knowledge is a man-made fabric which touches perceptions only along the
edges.” 10 Quine holds that this is the only meaning ontological questions can acquire.
Now Quine performs a logical trick: In forming our over-all science, we can judge its
ontology by looking at what it claims there is. So, by adopting a given theory, we commit
ourselves to its ontology, that is, to the things that exist according to the theory. This is
Quine’s Ontological Commitment.
An Empirical Dogma
Opposing Carnaps attempt to distinguish the mathematical from the physical portion of
the over-all scientific framework, of which they agree in large 11 , Quine directs a severe attack
on an old, as he says, “metaphysical article of faith” 12 . His attack concerns the
concept of analyticity.
This notion has been around in philosophical debate for quite a while. Kant considered,
as an important part of his metaphysics, the possibility of synthetic truths known
a priori. The whole idea is that an analytic statement is a true statement which is so solely
in virtue of the meanings of the words used. That is, it is a trivial truth in the sense that if
you properly understand the concepts involved, if you know their meaning, then you cannot
deny what is said. Quine’s point is that the concept of analyticity is ill-founded. His argument
is found in his “Two Dogmas of Empiricism” 13 and runs along the following
lines:
First, analytic statements have been defined as statements whose denials are self-contradictory.
Any similar definition lacks explanatory power. These two notions are “the two
sides of a single dubious coin” 14 .
In a similar vein, Quine proceeds to scrutinize a variety of attempts made to clarify
the notion. He considers Kant’s metaphorical notion of ‘containment’, Frege’s work on
the relation of meaning and reference. He traces the explanatory ground for meaning to
that of ‘synonymy’, but, again, he ends up with an explanation owing to an understanding
of some other, equally ill-founded, concept. The way through ‘definition’ seems appealing,
but adds nothing to our understanding. All such, accept the postulating kind, “hinges on
prior relationships of synonymy” 15 . So we’re back at where we started.
Further, Quine has a go at ‘interchangeability salva veritate’. He considers the predicate
‘necessarily’, but points out that such a predicate does indeed presuppose a satisfactory
10
p102, this is closely linked to the Quinean view, that stems from Duhem, and which is called the Duhem
Thesis, that physical theories are accepted or rejected as a whole, and not statement by statement [Cambridge
Dictionary of Philosophy].
11
p177
12
[Quine], II.5, p37.
13
[Quine], I.
14
[Quine], II.1, p20.
15
[Quine], II.3, p27.
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understanding of ‘analyticity’. Such an understanding can be obtained in an external language,
but troubles appear here as well: Extensional agreement, what synonymy becomes
in this context, is far from the type of cognitive synonymy that is needed for the elucidation
of ‘analyticity’. In an intentional language, analyticity is presupposed.
Quine’s last attempt is about semantic rules. Obviously, a semantic rule, itself containing
‘analyticity’, is of no help. We know very well which instances of the language we
might apply the predicate of ‘analyticity’ to, but we know nothing more of what this says
about the considered sentences. We might then consider, Quine proposes, differentiating
among true statements; any analytic statement is of course among these. We might say
that sentences true by virtue of specific semantic rules are to be labelled ‘analytic’. But
even here, nothing is gained. For the reference to ‘semantic rule’ is as dubious and unclear
as is the notion of analyticity itself.
Quine’s conclusion is thus that any attempt to clarify what is meant in the use of ‘analytic’
is condemned to fail. In his own words, “for all its a priori reasonableness, a boundary
between analytic and synthetic statements simply has not been drawn” 16 .
Naturalism
As we have seen, Quine claims that ontological questions are to be understood in, and
only in, the framework of over-all science. But these are not the only philosophical questions
that have dim evidential standards. As Quine only finds one legitimate ground for
understanding statements about ‘how it is’, philosophical questions are intelligible just in
so far as they have counterparts in this context. That is, questions on epistemology, for example,
can only be properly understood in their scientific versions, as questions of how
human beings, as understood by science, acquire information of the physical world, as understood
by science. There is no other understanding of these questions. The traditional
philosophical goal of epistemology has been to found science on some secure, extra-scientific
cornerstone, but we must give up the Cartesian dream, Quine urges.
In sum then, this abandonment of traditional philosophical goals, due to lack of intelligibility,
Quine calls ‘naturalism’. This is “the recognition that it is within science itself,
and not in some prior philosophy, that reality is to be identified and described” 17 . Thus,
science appears as “an inquiry into reality, fallible and corrigible but not answerable to any
supra-scientific tribunal, and not in need of any justification beyond observation and the
hypothetico-deductive method” 18 .
The abandonment of the goals of a first philosophy means not that all questions
asked by philosophers are left behind. Those which are possible to ask inside the scientific
framework are as good as any scientific such, but these must now be understood on scientific
grounds, and answered within the theory. In this way the part of philosophy that
endures Quine’s treatment becomes ‘continuous with science’, as Maddy puts it.
16
[Quine], II.5, p37. Also cited in [Maddy], p100.
17
Quine, cited on p180.
18
Quine, cited on p180.
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We already noted the holistic character of science. Its only principles are observation
and the hypothetico-deductive method, and scientific claims are meaningful only in relation
to other scientific claims and true only insofar as they explain our observational experiences.
The naturalistic philosopher is thus no different than the scientist. He “begins
his reasoning within the inherited world as a going concern” 19 . Without any extra-scientific
ground to hold on to, he must reconstruct science from within, “while staying afloat in
it” 20 . This is the picture of Neurath’s Boat 21 .
Meaning according to Quine
So given this naturalistic view of an ‘unavoidable’ scientific context, or framework, what
has Quine to say on the notion of meaning This is important in view of the naïve questions
raised in the introduction.
Just like Carnap, Quine holds that meaning for a statement arises in relation to a given
linguistic framework. This framework is to supply evidential rules which explain how the
statement is to interact with the other statements of the framework. These rules should
account for what inferences are to be drawn from a given statement, or what is to be deduced
from a certain perceptual impression. For a sentence, this interplay with other sentences
of the framework, and with observation, is all there is to its meaning. That is how
Quine should be understood.
Spelling this out, the meaning of a statement, or a question for that matter, is its relation
to all other internal statements in terms of what is to be inferred from it, and what it
can be inferred from. This is exactly the ‘façon de parler’ earlier mentioned in connection
to ontology.
This lack of meaning in its more metaphysical sense is easily understood from Quine’s
attack on analyticity. A clear interdependence is identified between the two concepts, and
as analyticity is judged as an empty notion, so must also any concept of meaning be considered
outside the framework. Thus meaning outside science lacks explanatory ground,
for the very same reasons as does analyticity.
Quine on mathematics
The consequence of Quine’s philosophy for mathematics is immediate: Mathematics plays
a major role in the over-all scientific theory, it is claimed to be indispensable for science 22 ,
and as we adopt the framework of over-all science, Quine argues, then we must accept the
ontology of that framework, and we must accept, as part of its ontology, what mathemat-
19
Quine, cited on p180.
20
Quine, cited on p180.
21
Neurath compared the scientific project to a ship on the sea that was continuously reconstructed, but in
such a way that it could stay afloat.
22
H. Putnam, cited on p104.
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ics claims there is 23 . This is how a realism concerning mathematical objects develops out
of Quinean thought.
His view also answers to the question of meaning and meaningfulness of mathematics
raised above. As all other scientific statements are to be understood, the mathematical assertion
of the existence of numbers means nothing more than that in scientific practice we
talk and behave as if there were numbers. That is all the meaning that can be assigned to
statements of mathematics. On the other hand, this is all there is to meaning of any statement.
So, a mathematical claim is just as meaningful any other scientific claim. But even
more so. Quine explains the applicability of mathematics and its epistemologically secure
character by saying that mathematical statements are central in the web of scientific knowledge,
and in this sense mathematical claims are even more meaningful than the average
scientific claim.
Now, Quinean thought does not leave mathematics as it is, but retains only a minor
part. If mathematical claims gain their intelligibility and mathematical existential claims
their justification solely through their role in science, not much of pure mathematics could
be spared. Or this is at least Quine’s conclusion 24 .
We have seen how Quine rejects the concept of analyticity, since there can be no clarifying
explanation of its meaning, or of what is an analytic truth, and what is not. The
Carnapian solution to epistemological questions on mathematics would be to say that
mathematical claims are analytical claims, and so they are secure, for they are trivially true.
Quine rejects this solution, because, as noted above, analyticity has not yet been explained
in a satisfying way.
23
This is, together with Quine’s ontological commitment, commonly known as the Indispensability Argument
for mathematical realism.
24
p105
14
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Maddy’s philosophy
In view of Maddys overriding concern to investigate the possibility of settling undecided
issues in set theory, it is easy to understand a temptation for a naturalistic approach similar
to that of Quine on natural science. She reads the spirit of naturalism as synonymous for
“philosophical modesty” towards methodological considerations. Maddy takes her position
to be a ‘variation’ of Quine’s naturalism, but not a philosophy of mathematics, rather
a “position on the proper relations between the philosophy of mathematics and the practice
of mathematics” 25 . We need to know what the philosophical result of such methodological
interests is.
Maddy starts out with Quinean naturalism; actual scientific methods are fundamental.
Not far gone into investigations of the physical world, two facts become obvious. Mathematics
is central to our scientific study of the world. And, the methods of mathematics
“differ markedly from those of natural science” 26 . The first observation underlies arguments
of indispensability for mathematical realism. Maddy takes history and actual fact as
a guide to take a step away from Quine; she offers to mathematics a similar, self-supporting,
framework of its own. This is clearly in opposition to Quine, but in accordance with
the naturalistic spirit of his philosophy. Following Quine’s parallel, Maddy thus takes
mathematics to be “not answerable to any extra-mathematical tribunal and not in need of
any justification beyond proof and the axiomatic method” 27 . So far the methodological
concerns are central. But what are the philosophical effects
In a Quinean fashion, Maddy asks herself what mathematical method has to say on
ontology. If we are to investigate philosophical questions on mathematics, then it would
be interesting to know, as in the case of natural science, which ones might be taken as
continuous with mathematics. Knowing that, we would know which of the questions can
be asked inside the mathematical framework, and thus which can be asked “with some
real hope of finding answers” 28 . This was how Quine succeeded in saving at least some
philosophical questions. In short, she finds not much. Firstly, Maddy notes that “common
sense is more forthcoming on physical than mathematical ontology” 29 . Moving to the level
of mathematical theories, we find existential claims in abundance, but not the slightest
hint of the nature of this existence. Maddy does find some hope for ontological guidance
at the level of mathematical discussion, the level at which, for instance, discussions of justifications
of CH and similar, takes place. But this raises the question of whether these discussions
are continuous with mathematics. The outcome is daunting, and she ends the
section, “The methods of natural sciences – […] – also tell us that ordinary physical objects
and many unobservables exist, […]; but the methods of mathematics – […] – tells us
no more than that certain mathematical objects exist.” 30
25
p161
26
p183
27
p184
28
p181
29
p185
30
p192
15
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The outcome of this investigation, then, is nought. Maddy writes, “our mathematical
naturalist relinquishes her last hope of ontological guidance from the practice of mathematics”
31 . Moving to other parts of traditional philosophy, the picture is the same. She
writes, “it seems no traditional epistemological questions about mathematics and only the
barest ontological questions about mathematics can be naturalized as mathematical questions”.
So, we are to infer that ontological questions are not mathematical questions. The
result is that “naturalism is not a metaphysical rival to realism” 32 . Thus the shift for a
proper mathematical framework is mainly of methodological nature. A scientific study of
mathematics in a naturalistic spirit evaluates mathematical progress on its own terms, but
what are the consequences for the philosophical questions on mathematics All such are
considered first philosophy, and are hence, in Carnaps sense, pseudo-questions. Just one
philosophical question remains, and that is ‘What exists’. Its intra-mathematical counterpart
has a trivial answer: – Numbers, sets, functions, etc. All things that appears directly
after an ‘∃’ in mathematical literature. And the matter ends here.
31
p191
32
p205
16
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Criticism: Quine
The Quinean account has many virtues, indeed, a whole bunch of insights is woven into
the fabric, but they are not always as apparent and explicitly presented as one might
please. Besides, there are a number of questions to be asked.
At first sight, Quine is not very successful in responding to the naïve questions of the
philosophy. Although mathematicians throughout the ages have reported on eternal virtues
of the field, Quine rejects most of these. Mathematics is – in so far as it is intelligible
at all, that is, is applied in science – no more than a scientific theory amongst others, and
is, therefore, as fallible and corrigible as the surrounding, supporting fabric of theory, that
is over-all science. Thus, mathematics provides no knowledge of a more secure kind than
does science in general. This is most certainly opposed to the general naïve philosophical
view. Further, mathematics has no meaning in this far-reaching sense; an existential claim
of mathematics is a way of speech, no more, no less. Pure mathematics is not meaningful
at all. As a part of the actual scientific framework, mathematics is in no way the only possible
field of this sort. Quine holds that we cannot consider our science to be the only
possible science 33 , and unfortunately mathematics goes with it.
Before plunging into specific worries about the Quinean view, a necessary remark
must be made. The naturalistic view in no way does away with philosophy according to itself,
but changes, so to say, the labelling of the issues involved. Where the classical philosopher
discusses meaning in a first-philosophy sense, in the sense of meaning ‘in itself’,
the natural philosopher hears just nothing. What is meaning to the naturalist is merely
how the issue is related to all other issues of knowledge. What is meaning for a term is just
the way we use that term in ordinary speech. This means that if the classical philosopher
were to describe the naturalistic view, then he would say that the naturalist denies any
meaning, or existence for that matter, ‘in itself’, and that the question is now reduced to
that of how the parlance goes. The naturalist, on the other hand, would say that the classical
philosopher is mistaken in his thoughts of meaning, as he considers there to be a
meaning of ‘meaning’ which goes beyond the usage of the word. Uttering “there is
something more to meaning”, would result in nothing more than what goes well with naturalism.
Because, were there no other meaning to “there’s something more to meaning”
than the usage of the terms and the grammatical construction, then the sentence would indeed
fail to say anything about something non-existing. The naturalist claims thus that we
simply fail to talk about meaning or existence, because as we say anything on these matters,
then what we say is meaningless accept for the relation of the spoken word to the
rest of human language usage. Comparing with another famous example, if one could say
that “the king of France is bald” fails in making sense because of lack of referent, then the
naturalist would say (according to the classical philosopher) that “the king of France is
bald” fails to make sense (be meaningful), for there is no such thing as reference (or
meaning), while the naturalist himself would say that “the king of France is bald” makes
perfect sense, and that the classical philosopher talks about invented (pseudo-) worries in
33
p181
17
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discussing reference and meaning as such. To avoid a situation where debate is impossible,
simply as one party claims that the other fails to say what he wants to say, I now take position,
for practical reasons, in the classical camp. Thus, I will allow the idea of a possible
real existence, meaning, etc, and say that the naturalist denies this faculty. 34
Mathematics in the web
Mathematical claims seem to be of epistemological excellence. Nothing seems to be more
secure than claims of mathematics. Why this is so, Quine explains by saying that their position
is central in the fabric of the over-all scientific theory. So, Quine offers a reason, why
we do refrain from questioning these theses as long as we have anything else left to put in
doubt first. But, of course, we could now ask why mathematical claims are so central. And
as far as Maddy’s account goes, we find no answers. That mathematics is of high fidelity,
we knew in advance, but what, in mathematics, makes it so secure This could well be
asked even if we considered merely a degree of security as in the case of Quine. However,
the picture Quine gives goes well with the over-all view, and it is easy to understand the
unpleasant implications for the Quinean project of accepting mathematics as a quite different
faculty of knowledge. If mathematical claims are true in a much stronger sense than
scientific claims, then this must be explained, presumably in terms of concepts not far
from the rejected concept of analyticity. For, if a claim is true in a stronger sense than by
virtue of observational data, what might that truth depend on other than meaning Of
course, there is always the possibility of psychologism. We could say that mathematical security
is due to our mental apparatus rather than an external ‘meaning’ in this sense. But,
then we would still be interested in what makes mathematical states of mind more central
to the field of knowledge than scientific states of mind. If there were states of minds, or
theorizing patterns, of greater generality, thus applicable to a wider range of scientific situations,
then this generality, and the reasons for it explained in mental properties would be
interesting indeed in revealing features of scientific progress. Thus the conclusion is still
the same. Quine says that mathematics is indispensable for science, and therefore central
in the web, but he does not explain why.
If mathematics is to be questionable, it seems natural to ask for a fair description of
how this might actually happen. What would it look like to reject a central claim of mathematics
I would be surprised if there were a single example of a change of mathematical
theory because of incompatibility with any other body of claims of the over-all science.
Meaning
Quine argues that outer questions are pseudo-questions, but despite his relativizing manners,
he does not seem to notice that the claim that outside issues are meaningless is itself
an utterance in the context, and therefore relative to this context. The point here is that if
34
Compare for the introductory note in Quine’s ‘On What There Is’, [Quine], I.
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Quine provides a linguistic framework, which gives utterances in the framework meaning,
then he has not yet provided grounds for the rejection of any other possible meaning to
occur. His argument is a positive one, inner statements are intelligible, but his conclusions
are negative, outer statements are not intelligible. If any kind of meaning occurred outside
the framework, then this framework would not recognize it as meaning. For if meaning is
always relative to a framework, as Carnap argues, then meaning A would not qualify as a
meaning in framework B, that is, a meaning B . So, if Quine is to be understood in this relative
sense, that a sentence that gains meaning from a context, cannot gain this meaning
outside the context, then his remark is indeed trivial. And if he is to be understood to say
that meaning is impossible whatsoever outside the context, then he has no supporting argument
to offer. In this sense, Quine’s argument for meaninglessness outside the scientific
context, is itself meaningful inside the context, but then it follows that the argument can
only work in a positive manner, inside, and not in a negative manner, outside, that is, it
can justify the existence of meaning inside the context, but is useless in rejecting any possible
meaning outside this context. If utterances are to be meaningful just ‘in relation, then
that is meaningful just ‘in relation’.
The sentence ’p → p’ is supposedly included in the Quinean fabric, but is it meaningful
He could say, no, it is trivial and empty. Or he could say yes, in relation to other sentences
of science it is meaningful, but then, if we were to reject it, what could science possibly
look like Some sentences seems fundamental to the very project of science, to the
very possibility of a Quinean fabric of belief. Are these of equivalent dignity as ‘ordinary
statements of science’ Further, how are we to judge in the case of sentences of the following
form: ‘CH follows from the existence of certain large cardinals’. This is a result of
set theory, and it is proven with mathematical rigor and may therefore be considered true
beyond any reasonable doubt. It seems it could not end up false however we were to interpret
mathematics, science or logic. It seems meaningful as it claims something nontrivial.
It seems to be about mathematics, but still within mathematics, as a logical utterance,
that is, not a scientific claim. But how is it to be understood I claim that Quine could not
do justice to these kinds of example. And while not accounting properly for the logical
part of knowledge, chance is considerable that he misses out on closely related mathematics.
Ontology
Metaphysics is of tradition a central concern of western philosophy. And therefore it
seems natural that ontological questions have carried over to the philosophy of mathematics.
The Quinean project, as well as Maddy’s prolongation, is heavily concerned with ‘what
there is’. Establishing her proper mathematical realm, Maddy asks for ontological guidance,
but finds none. Her conclusion is that, as mathematics offers little guidance on the
nature of existence of its objects, then questions of existence cannot be naturalized as
proper mathematical questions. But this whole quest for ontological guidance rests, of
course, on the assumption that mathematics actually claims something to exist, and that
seems not to be evident. Surely, a first acquaintance with mathematics would provide
19
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ground for believing that mathematics talks about existence, but a closer look will reveal
that any statement of existence is preceded by a specification of the required universe of
objects which is needed for the development of the theory in question. Thus, a statement
of existence of a certain object is merely a statement of there being, among the already existing
objects, an object of the intended kind. But, scrutinizing naturalism, we see that what
now differentiates between our accounts is no more than a matter of words. For, if mathematics
does talk about existence, then, in the naturalistic interpretation, following Maddy,
this existence is of no more metaphysical connotation than the very usage of the mathematical
existential claims, which is all that may be said of existence at all. Then, in naturalistic
parlance, mathematics does talk about existence in the only way possible. That is, we
may understand mathematics to make existential claims and then adopt the naturalistic
view and thus make the existential claims of mathematics less metaphysically loaded, or
we may say that mathematics does not talk of existence, referring to the higher order existence
of first philosophy, in order to arrive at the same conclusion. So, in that case, and
again, if the same point is really possible to make from the stance of classical philosophy,
why this new labelling of the concepts If this is really the case, then I cannot see why the
naturalistic account becomes anything more than an observation on the limits of human
knowledge. For interpreted in this way, the only arguments for the naturalistic definitions
of the terms, rather than the classical, which are not arguments for limits of human knowledge,
would be arguments in favour of the stance that ‘there is no real existence’, that
‘there is no real meaning’, etc. And those have not been provided.
Cartesian doubt is not the way to begin
Quine traces the failure of the epistemological project to Hume. He did not succeed in his
attempts to found knowledge of bodies through identification with impressions and the
reconstruction of statements as about impressions. His statements “gained no increment
of certainty by being construed as about impressions” 35 . The conclusion is that we cannot
hope to found science on anything more secure, and we are back at Neurat’s boat. Now,
Quine notes that “the very impulse to found science on a secure foundation is driven by a
scepticism that is itself internal to science: the very notion of sensory illusion depends on a
rudimentary science of common sense physical objects” 36 . As far as a scientific-free situation
is imaginable, I cannot see why the possibility of the doubt that ‘what I spontaneously
think of something, need not be the case’, is so tightly linked to science. If ‘science’
means any human theorizing, if science is what thinking is, then of course this seems right,
but trivially so. If the very idea of common sense physical objects is to be the root of
scepticism, I cannot see how this would be contaminated by statements like ‘I feel as one
person, but there might be no way to specify me in a scientific way’. It seems possible
that, as soon as perception arises, there is a possibility of questioning that impression.
That is Descartes’ insight, indeed. But this doubt applies equally well to any other theory
35
Quine cited on p179.
36
p179
20
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of ‘how it is’: A dedicated solipsist might very well be able to doubt his version, or understand
the limiting characteristics of his perceptual ability. There seems to be a possible
knowledge that presupposes no physical objects, merely a ‘truth’, or a ‘world’, and the
doubt of one’s own thoughts seems to survive.
21
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Criticism: Maddy
All in all, the Maddyan account suffers from far too many methodological concerns. This
is a choice. Depending on the understanding of the field of mathematics, one might
choose different paths towards its perfection. One might wonder what Maddy takes mathematical
triumph to be, as this will surely influence the resulting methodological counsel.
As the philosophy of mathematics is equally the philosophy of mathematical triumph –
answering the one question would advise considerably on the other – this is a crucial
choice. Maddy chooses pragmatic standards and methodology, in a Quinean vein. I take
this to be a mistake. My focus, and what ought to be the focus of any philosophy of mathematics,
is mathematics itself. This choice is probably the major source of the various dissatisfactions
over her views, to be presented in the following. This survives naturalization
if one understands ‘mathematics itself’ to be ‘that, whatever it may be, that generates the
mathematical success as well as the reports on mathematical virtue’.
In what way does mathematics care for the methods of research of actual mathematics
in practice This is really the feel of independence that is so central to the realistic account
of mathematics, and which appears in our primitive answers of the naïve questions.
Maddy starts with ‘actual methods of mathematics’, and this may be wise if we are to
counsel on further goals for mathematical research, but in what way, given that mathematics
is understood as the field itself, which emerges from our commonsense standpoint, the
object of research, may the methods of actual mathematics guide us in the search for its
nature Mathematical results seem indeed unconcerned with the direction of approach.
And this puts doubts on the project of answering natural questions of the character of
mathematics in methodological manners. Hence, Maddy’s methodological concerns are
questioned as such. One could perhaps turn the argument around and say that if actual
mathematical methods are central, then there is no mathematics apart from mathematical
activity. This would indeed be to consider mathematics to be nothing more than yet another
example of sociological phenomena in global society. But, the feel of need for special
treatment of mathematics seems as clear for Maddy as for our naïve view, so let’s not
go there.
It is clear that Maddy’s concern is of practical and methodological nature, but still, she
aims to offer a “philosophically useful account” 37 . The mathematical framework that
Maddy proposes is provided in view of methodological concerns, “the methods of mathematics
differ markedly from those of natural science” 38 . But, in addition to its methodological
usefulness, this context has philosophical consequences. So we ask, what is the
nature of this special mathematical framework What is its relation to that of natural science
Is this framework really a framework of mathematics and nothing else, or does it
enclose other disciplines That is, in what way does this framework define what is to be
found in it My point is simply that a proper mathematical framework is suggested, but
not characterized in kind or content. In that, the Quinean picture of science seems much
more complete.
37
p2
38
p183
22
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Further, if mathematics is worthy of special treatment in this sense, we should not be
far from a definition of mathematics as a field of knowledge; if mathematics has a proper
context, then, presumably, all and only what is found in that context is mathematics. But
of this, Maddy says not much. One could of course ask if Maddy’s project requires such
an account. The answer would be that if she has nothing but methodology in sight, no
such is needed. Indeed, the naturalistic spirit advices us to study mathematics to find the
way to mathematical success. But insofar as she seeks to offer a philosophically useful
view of mathematics, then she could not overlook this. Briefly, a complete characterization
of the mathematical framework, I imagine, would define the subject in some sense.
And an account of the relation of the mathematical and the scientific frameworks would,
first of all be facilitated by a characterization of the former, but further also, answer to the
naïve questions of what the reported unique eternal virtues of the mathematical field
might consist of. These are evident questions which lie close to the naturalistic spirit –
common sense is our first Quinean over-all science. So, it is strange that there is no comment
to be found in Maddy’s exposé. This argument evidently rests on the assumption
that the mathematical framework is not only a methodological construction, in the naturalistic
sense, but also a framework in the Carnapian sense. If not, I cannot see why Maddy
needs a new framework at all. If ‘naturalism’ were among the theories of Quinean science,
then mathematics would enjoy the same methodological respect as would natural sciences,
and we would have no use for a proper mathematical framework. Also, that Maddy engages
in the naturalizing of philosophical queries vis-à-vis the mathematical framework
suggests that she takes it to be more than a mere methodological construction. Anyhow,
this is yet another example of how the relationship science~maths is not clearly understood.
On ontology, Maddy finds nothing. Clearly, the philosophical question of ontology of
mathematics cannot be naturalized within mathematics, as is possible in the scientific context.
Mathematics, in all its parts and linguistic levels, seems mum on the nature of the existence
of which it speaks so overwhelmingly. But, if mathematics has nothing to say on
ontology, then on what grounds do we claim that mathematics actually speaks of ontology
at all This is the same point as made against Quine, and the question is left unanswered
also by Maddy, who somehow has been even closer to this thought in arriving at but a
single philosophical statement surviving mathematical naturalization. This is a shame
simply because a philosophically acceptable account of mathematics would surely not
leave unattended such a central issue. And, further, this very point made clear, might well
cause better understanding of the specific framework of mathematics, and thereby of the
relationship of mathematics and science. This relationship is genuinely central also to the
ontological issue: It seems reasonable to me to assume that the very question of mathematical
ontology is in fact an investigation of in what sense the scientific concept of existence
could successfully be taken as parallel to the apparent existence of objects in the field
of mathematics.
Considered as a philosophy of mathematics, Maddy’s story is unusually mute on the
nature of mathematical knowledge, nor do we find comments about the degree of security
of this knowledge. If Maddy accepts the Quinean view of mathematics in general, that of
its central, but still fallible, position in the web of over-all knowledge, then she is the target
23
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of the very same questions concerning mathematical virtues and seeming independence as
is Quine.
In completion, and in view of our introductory naïve questions, we may note that
Maddy in no way specifies any ground for the expected uniqueness of the mathematical
field, except the tautological one: That there is just one mathematical framework, so the
maths in it is unique. Her mathematical framework neither explains the applicability of
mathematics to science, nor, as already dwelled upon, its air of eternal virtues.
Another point that could be made against Maddy is this, and it’s a simple one. If
mathematical methodology is a pragmatic matter, why not leave it as it is. Why would
Maddy want to present a philosophically useful account, if, as she suggests, philosophy, as
well as science, should take a hands-off attitude towards mathematical method
With regard to what Maddy really aims for, rather than my interests, i.e. a methodological
development in mathematical foundational research, we might ask what would happen
if succeeded in the urge for new mathematical axioms. Mathematics has since the
early 1800’s shifted from being a theory of the world, to being a theory of modelling the
world. This is to say that where we used to talk about, for instance, acceleration as the derivative
of velocity, we now would say that if you take the function that describes the
speed, which is something different from the speed ‘itself’, and then differentiate, then
you get a good description of the acceleration, which again is not the acceleration ‘itself’.
It gets more and more evident that there is a gap between our understanding of mathematics,
and what mathematics actually says 39 . So, if mathematics is a field of eternal value, as
we would say, what would happen if one were to discover a new axiom What could possibly
justify this new axiom – that is Maddy’s question. But another question might be:
What would happen to the former mathematical knowledge, that which is to be rejected
through the adoption of a new axiom Here we encounter different faculties of knowledge,
and will those prevail Specifically, will, for instance, CH cease to be the logical consequence
of the existence of certain measurable cardinals I would say not. It seems that a
good deal of knowledge on mathematical relations would survive despite a newly accepted
axiom candidate. There is already evidence in abundance for this: Mathematicians work
with questions of what might be proved without the use of the Axiom of Choice, what the
logical relations amongst hypothesis such as SCH, GCH, CH 40 , the existence of certain
large cardinals, etc, are. All this knowledge, one would think, is of equal security and kind
as classical mathematical knowledge. We must perhaps find a way of understanding this
type of knowledge, and its virtues, all at once. There is no account for the nature of mathematical
knowledge in Maddy’s book, but this questions the very understanding of the
project of completing the axiomatic system of Set Theory, and also our understanding of
the already established axioms.
39
Compare for Wittgenstein’s ‘prose’, p166.
40
These are the Singular Cardinal Hypothesis, the General Continuum Hypothesis, and the Continuum Hypothesis,
respectively.
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Conclusion
The above is an inquiry into the philosophy of W. V. O. Quine and the consequences for
philosophy of mathematics of his thoughts. P. Maddy is discouraged by Quine’s conclusions
on foundational matters in mathematics 41 , so she attempts to understand mathematics
on its own merits, following the naturalistic spirit of leaving first-philosophy behind
when it doesn’t solve matters 42 . Her intents are chiefly methodological in the sense that
her aim is not (perfectly following naturalism) any real being of mathematical virtues, but a
rather practical one: to investigate the possibilities to evaluate new axiom candidates for
Set Theory 43 . Her account of Quine’s philosophy and her variations of his thoughts bring
us to a picture of mathematics somewhat different from the common sense ideas. Mathematics
does not talk of objects in a ‘higher’ sense, since nothing can talk of anything in
this ‘higher’ sense 44 . In the meantime, Quine’s naturalism, and his view on the project of
over-all science, stresses the leaving behind of first philosophy, and re-establishes common
sense as our inherited starting point of any inquiry 45 . As there is evidence, in experience,
in favour of a special treatment of the mathematical faculty of knowledge, as concluded
in the introduction, we identify a failure, of the Quinean view, to account for the
phenomenon of mathematics. A central point of this criticism is that the naturalistic account
escapes attack through an assumed resistance of criticism, which could be said to
make impossible the very arguing. That one is not persuadable who leaves the room as
soon as an argument is launched.
Some common points are made towards the Quinean/Maddyan accounts. One could
ask what brings ontology into mathematics at all. Naturalism provides, we note, a new
labelling of the concepts to be discussed, but one might well wonder what the use of this
new labelling is, if all that is essentially claimed is that human knowledge is limited in
various ways. What is the persuasive argument that there is no such thing as meaning or
existence, beside that we cannot provide a complete account Not that Quine’s argument
is fallible, but that his interpretation is. He seems indeed unaware of the perspective
relativity of his arguing. Nothing can convince an unabashed solipsist, but as soon as he
tries to shift perspective, his arguments won’t hold. He is as unable to reject outer
intelligence, as is Quine to reject outer meaning.
41
”Note the oddity of this conclusion: Quine’s application of indispensability considerations has led him to a
stand (on V=L) quite opposite to that of the set theoretic community”, p106, and “In sum, then, Quinean
realism avoids the difficulty of the Gödelian version – […] – but doing so brings it into conflict with the actual
practice of mathematics”, p107, and hence in conflict with naturalism.
42
p161
43
p2
44
p177-178
45
p180
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References
~ Audi, Robert (ed.), The Cambridge Dictionary of Philosophy, Cambridge University Press,
Cambridge, 1999.
Articles: ‘mathematical intuitionism’, ‘philosophy of mathematics’, ‘formalism’, ‘Duhem, Pierre-Maurice-
Marie’.
~ Bennet, Christian, Första ordningens logik, Studentlitteratur, Lund, 2004.
~ Bunge, Mario, Dictionary of Philosophy, Prometheus Books, New York, 1999.
Article: ‘Mathematics, Philosophy of’.
~ Maddy, Penelope, Naturalism in Mathematics, Oxford University Press Inc., New York,
1997.
~ Mautner, Thomas, A Dictionary of Philosophy, Blackwell Publishers Ltd, Oxford, 1996.
Article: ‘intuitionism’.
~ Quine, Willard Van Orman, ‘Two Dogmas of Empiricism’ and ‘On What There Is’,
both in his From a logical point of view: 9 logico-philosophical essays, Harvard U.P., Cambridge,
Mass., 1961.
~ Saifulin, Murad and Dixon, Richard R. (ed.), Dictionary of Philosophy, Progress Publishers,
1984.
Articles: ‘Logicism’, ‘Intuitionism’.
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