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For a ‘Non-mathematical’ Learning of Mathematics.
A Philosophical-Educational Reflection on Philosophical Inquiry and 

Mathematics Classes

Stefano Oliverio

‘A                    , that is, “Let no-one without knowledge of geometry enter:” the 
inscription displayed on the entrance to Plato’s Academy reminds us how close the relationships 

between mathematics1 and philosophy used to be. In this perspective, when we approach the issue of how 
philosophical inquiry can further maths’ teaching/learning, a sort of archaeological attitude (Agamben, 2008) 
is in order, which delves into the layers of a long history, plumbs the recondite depths of Western thought, and 
unearths what remains too often concealed either because it is taken for granted or because we have become 
unable to detect what constitutes the very way in which we think. 

     Accordingly, investigating the role and significance of philosophical inquiry for the learning/teaching of 
maths does not consist simply in reflecting upon the possibility of extending a specific pedagogical model 
(in this present case the Philosophy for Children approach) to another area of teaching or upon the didactic 
strategies necessary to do that (pivotal as they are) but, more radically, it means re-thinking the deep solidarity 
between what we are endeavouring to (re-)harmonize – maths and philosophy. 

     But is that ancient and venerable harmony between mathematics and philosophy the one which we want 
to re-establish in our classrooms? Or is the sense in which we understand philosophy when we appeal to 
its mobilization in math classes profoundly different from that which resonates in the Academy emblem? 
One thing should attract our attention: the warning on the Academy entrance intimated that math is a pre-
requisite for accessing philosophy, a visa to enter the domain of the philosophical. In contrast, by invoking a 
recourse to philosophical inquiry in math classes, we reverse the order and suggest that through the former we 
can approach the latter in more profitable, if not more effective, ways. What is entailed in such a change of 
perspective?

     Actually, things are much more complicated; the Academy admonition says not that math is the door 
through which we can enter philosophy, as if, by studying math, we could gain access to philosophy, but rather 
that only those who already have a knowledge of mathematics can pass the door which opens onto philosophy. 
In other words, no one without knowledge of math can hope to be admitted into a philosophical community, 
but this does not imply that all those who know math will be admitted; math is a necessary but not sufficient 
condition for philosophy. 

     On the contrary, by proposing that through philosophical inquiry, following the model of Lipman’s 
community of inquiry (Lipman, 1988, 1991, 2003; Sharp, 1987, 1996; Splitter & Sharp, 1995; D. Kennedy, 
1995, 1997, 2004a, 2004b, 2012), we can familiarize ourselves with math, we are turning the community of 
philosophy into a (possible) ‘door,’ into a way of access. What does it imply for philosophy? Apparently, it is no 
longer the Academy philosophy, that ‘after-the-door,’ but another kind of philosophy, but which one?
 



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     By questioning the relationships between mathematics and philosophy, within a learning/teaching 
situation, we are compelled to raise the question of what is philosophy when it ceases to be what is ‘after 
the door’ and becomes the ‘door’ itself, so that the community of philosophical inquiry (CPI) could be 
characterized, with a grain of irony, as that group which is at the door (of maths?). And, more interestingly, 
we should ask why this new status of philosophy reveals itself in a particularly clear way precisely when we 
endeavour to understand how philosophy can be mobilized to learn/teach maths. 

      Indeed, the texture of this present reflection holds together three topics which include philosophy, 
mathematics and learning/teaching. Even better, it attempts to bring their unity to light. As I will attempt 
to show, investigating math means constitutively inquiring into learning/teaching and this, in turn, appeals 
to an exploration of what philosophy is and what it can be. If Plato’s Academy is something more than an 
experience in Western pedagogy and has become the prototype of every higher education institution, it is 
because it articulated in an epoch-making way the relationships between mathematics, learning/teaching, and 
philosophy. And it did it with the force of a commandment (it is noteworthy that the warning on the Academy 
entrance is in the imperative mode). Questioning this imperative, reversing the order--in any meaning of the 
word--and making maths and philosophy not two steps in a sort of evolutionary sequence but the allies in an 
educational project which tries to re-define what learning and teaching are, all this is ultimately what the whole 
reflection on CPI and maths classes amounts to.

1. Away from the ‘Mathematical’ Despotism: The Need for a Recovery of Philosophical Inquiry

     Why did people have to know mathematics in order to enter the Academy? To capture the meaning of the 
admonition we have to understand to what end an institution such as the Academy was founded. To put it 
in a nutshell, the Academy was Plato’s response to the scandal of Socrates’ death, and it is connected with an 
overturning of that idea of philosophy of which Socrates had been the embodiment.

     In his later studies on Socrates, Gregory Vlastos emphasized that the elenchus (the Socratic method based 
on questions-answers) is an inquiry which, although it aims at truth, does not envision the same truth as that of 
mathematical reasoning. On the contrary, the ‘mathematical truth,’ and the specific method connected with 
it – the ‘hypothetical’ method which is spoken of in Meno 86e (ex hypothéseos skopeîsthai, we read there) –, is what 
Plato will substitute for the Socratic method in order to obviate the shortcomings of the latter (we will see later 
what they were). The procedure of Socrates, as Vlastos reconstructs it, is clearly marked in its stages:

1) The interlocutor asserts a thesis p, which Socrates considers false and targets for refutation. 
2) Socrates secures agreement to further premises, say q and r […]. The agreement is ad hoc: 
Socrates argues from {q, r}, not to them. Socrates then argues, and the interlocutor agrees, that 
q and r entail not-p. 4) Socrates then claims that he has shown that not-p is true, p false. (Ibid., 
p. 11)

     In other words, Socrates operates on the beliefs of his interlocutors, and aims at showing their 
contradictions. What is most significant in the Socratic method is that it is a real inquiry because the 
philosopher has no guarantee of success and no absolute ground on which to found his argumentation. As a 
matter of fact, all steps are negotiated with his interlocutors. This notwithstanding, Socrates is sure that he will 
always be able to prove the fallacy of the belief p, to stick to the aforementioned example. According to Vlastos, 
Socrates harbors this conviction not because he believes that he owns a godly wisdom but, on the contrary, 
because he has only a human wisdom (anthropíne sophía, we read in Apology 20d-20e), that comes from his 
previous inquiries, which had borne out the consistency of his own system of beliefs, while “[a]ll others, when 
tested for consistency, have failed”(Vlastos, 1994a, p. 27). 

     By drawing upon such an interpretation, and with a certain hermeneutical bending, we could venture the 



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idea that the trust in the consistency of his own system of beliefs is a presupposition of Socrates’ inquiry only 
insofar as it is the outcome of previous inquiries, without being anchored to any warranty going beyond the inquiry 
itself. In any discussion Socrates bets on the robustness of his system of beliefs, tests it and exposes it to the 
risk of failure but, simultaneously, by doing that, he corroborates it whenever he shows the inconsistency of 
beliefs contradictory with his. 

     The educational implications of such a procedure are that Socrates’ position is not absolute in principle 
(despite his charisma risking overshadowing this fact): as far as I understand him, appropriating Vlastos’ 
reflections, Socrates is not a doctrinaire teacher but rather one engaged in a real dialogue, and, accordingly, 
a common inquiry can occur, in which Socrates is also a learner (insofar as he learns that actually his beliefs 
are the only ones to avoid contradiction). As he does not have any godly wisdom but only a human one, his 
philosophizing is always a co-philosophizing (symphilosopheîn), which is a communal philosophical inquiry 
animated by a co-educative tension. In this procedure, knowing and educating each other are two sides of the 
same coin. And such a wisdom is human because it does not pretend to be absolute, it is an inquiry not a 
quest for certainty, to use a Deweyan (1984[1929]; 1986[1933]; 1986[1938]) vocabulary. But “how could it have 
happened that each and every one of Socrates’ interlocutors did have those true beliefs he need[ed] to refute 
[from] all of their false ones?” (Vlastos, 1994a, p. 29): this is a question – Vlastos remarks – which never came 
from Socrates, but Plato attempted to answer through “[t]hat wildest of Plato’s metaphysical flights, that ultra-
speculative theory that all learning is ‘recollection’” (Ibidem). 

     What is most interesting in this context is, however, the reason why Plato felt the need to lay a metaphysical 
foundation for Socrates’ method. The scandal of the trial and of the death of his master convinced Plato that 
the inquiring method of Socrates, destitute of any metaphysical anchoring, was constantly exposed to the 
danger of succumbing to its limitations. Plato endeavored to find a kind of ‘thinking’ which could be a rival in 
strength with the effective and actual force of the leaders, of the Athenian democracy, to which he had been 
a witness during the trial. He identified such a ‘forceful’ kind of thinking in mathematics. Indeed, as Hannah 
Arendt (1965) remarked, in a completely different context, Plato started the tradition of insisting on:

the compelling nature of axiomatic or self-evident truth, whose paradigmatic example […] has 
been the kind of statements with which we are confronted in mathematics. Le Mercier de 
la Rivière was perfectly right when he wrote: “Euclide est un véritable despote et les vérités 
géométriques qu’il nous a transmises sont des lois véritablement despotiques. Leur despotisme 
légal et le despotisme personnel de ce Législateur n’en font qu’un, celui de la force irresistible 
de l’évidence;” […]. In our context it is important to note that only mathematical laws were 
thought to be sufficiently irresistible to check the power of despots. (Arendt, 1965, p. 193) 

     As a consequence of the scandal of the trial and death of Socrates, Plato operated an epoch-making shift 
and what his master had refused (that is, the ‘mathematical’ research of the Meno) is mobilized to give strength 
to Socrates’ thinking. From that point on no one without any knowledge of mathematics would be allowed to 
enter the Academy. Mathematics became a sort of fortified outpost which protected philosophy from lapsing 
into the ultimately losing confrontation with the dóxai, the opinions of the city. Philosophy is no longer an 
inquiry investigating the beliefs of subjects but a kind of mathematical thinking, which derives consequences 
from first principles which are self-evident. 

     In this new thematic constellation, at least as I am interpreting it, the alliance between mathematics and 
philosophy occurs at the expense of inquiry understood as a kind of research which has no guarantee, no 
axiomatic certainty, no absolutely firm ground. It is important to bear in mind this fact because here we find 
both the reasons for the mistrust that the proposal of using P4C for maths classes can encounter and those 
for a possible failure of such an educational proposal, when it is not clear to which idea of philosophy we 
are referring. Indeed, on the one hand, to the extent that math continues to be considered the paradigm of  
‘compelling reasoning,’ any mobilization of P4C could appear to be inappropriate, if not detrimental, insofar 



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as it risks ‘infecting’ with discussion and dialogue what should be intrinsically monological because absolutely 
certain. On the contrary, what will be argued here is that there is – in a quasi-Deweyan sense (1980[1917]) – 
a need for a recovery of mathematics which is parallel and even interwoven with the need for a recovery of 
philosophy, to which Lipman much more than Dewey gave educational expression (Oliverio, 2012b). 

     On the other hand, this recovery is possible only if we are clear about what kind of philosophy we are 
speaking of and about the radicalism of Lipman’s proposal of the community of philosophical inquiry, which 
represents a counter-movement in comparison with the Academic one. To put it in a slogan: through P4C and 
CPI Lipman realizes, at an educational level, a ‘back to Socrates’ (Lipman, 1988, p. 12) movement, Socrates 
being the name for the search for a ‘human wisdom’ as opposed to the ex hypothéseos skopeîsthai characterizing 
mathematics.
 
     As the Socratic method, as far as I have presented it in Vlastos’ wake, predominantly addressed ethical 
issues, it could be suggested that P4C and CPI can and should be used in math classes only when we are 
interested in exploring the moral and political dimensions of mathematics. While it is plausible that this 
represents one of the opportunities offered by P4C applied to maths classes (N.S. Kennedy, 2012a), it does 
not exhaust the range of possibilities. We should understand in what sense Socrates’ inquiry is an ethical one. 
As far as I construe his undertaking, Socrates attempts to bring his interlocutors to ethics passing through an 
examination of morals. ‘Moral’ comes from Latin mos and refers to the mores of a society, to that set of rules 
which are codified and are often taken for granted and complied with without any personal commitment. 
In contrast, ‘ethics’ comes from the Greek éthos, which originally means ‘the appropriate place.’ The ‘ethical’ 
inquiry properly understood does not deal (only and primarily) with morals but with how the individual 
positions him/herself in relationship to the world, and it concerns, consequently, the whole being-in-the-world 
of individuals, their existence and, more particularly, the ways they find something existentially meaningful. 
‘Ethical inquiry’ is a search for meaning and, from this perspective, the entire Lipman and Sharp enterprise, 
insofar as it is directed to meaning (Lipman et al., 1980), is ‘ethical’ and therefore ‘Socratic.’ Accordingly, 
engaging with mathematics through ethical-Socratic inquiry (= Lipman philosophical inquiry) adds up to 
more than investigating the moral or political dimensions of math. It rather concerns the question of to what 
extent math is meaningful, how it can be and how individuals make sense of it. It is a search for meaning while 
learning math: in this perspective it is ‘ethical.’ It allows individuals to find ‘the appropriate place’ for math in 
their existence. And this happens not only at a strictly moral but also at an ‘epistemological’ level (see below § 
3). 

     Before investigating in more detail the educational and pedagogical implications of this shift, I want to 
explore what the philosophical-educational characteristics of the Academy are so that the peculiarities of the 
CPI approach can stand out more clearly.

2. The Overthrow of the Academic Model and the ‘Pragmatic’ Learning

     In describing the Academy, I will focus not on the historical debates occurring there (which still represent 
a paradigm for intellectually open discussion [Berti, 2010]) but rather on the quasi-archetypical image of it 
which has congealed in the Western tradition. The exploration of this image will allow us, on the one hand, to 
identify the differences of Lipman’s (and more generally a pragmatist) understanding of philosophical inquiry 
in comparison with the traditional educational model and, on the other, to highlight how the predominance 
of  ‘mathematical’ despotism affected the Academic view of philosophy. 

     In the Academy, the reality of philosophy consists primarily and essentially in the philosopher’s vision of 
the principles, the absolutely metaphysical ideas, that constitute the very core of the world. This vision is not 
the outcome of a common inquiry but of the ‘intimacy’ of the philosopher, in his solitude, with the object 
of his theory. The Seventh Letter (341b-341c) bears witness to such a view of philosophy. Plato opens up a 



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gulf between the ‘theoretical activity’ of the philosopher and teaching; the former cannot be taught stricto 
sensu, teaching will always be not only a derivative activity but one that cannot convey what is the innermost 
nature of theory. Consequently, also any cooperation is excluded: inquiry is solitary, it is hardly inquiry in an 
appropriate sense; as a matter of fact, it is vision (theoria, in Greek). 

     Against this backdrop, the imperative on the entrance of Academy is comprehensible: the paradigm for 
such a theoretical activity is the mathematician isolated from the world of worldly appearances and immersed 
in the universe of numbers (purely ‘abstract’ entities, the intercourse with which is an isolated mental matter 
and excludes any inter-human intercourse). 

     Hierarchically subordinate to this first reality is what Paul Landesberg (1923) calls “the second reality 
of philosophy,” which is constituted by the disciples of philosophers and propagates in the successors of 
philosophers during the ensuing centuries. The writings of philosophers are one of the means of spreading 
this second reality but they “are in no way a reality of philosophy but precisely only printed or written paper” 
(Landesberg, 1923, p. 95). The genuine life of philosophy is realized in the vision, which belongs to the 
philosopher and is then (partly) communicated to disciples and successors. In the Academy device, while the 
first level reality can exist without the second level (= the innermost core of philosophy – the vision of the 
thinker – does not need a community of co-inquirers), the second is nothing without the first. 

     It is important to note two things: first, in this perspective, the circle of disciples is only a bridge between 
philosophy in its real essence and its socialization and dissemination. Students are not a part of the very reality 
of philosophy, and do not participate in the production of thoughts but only in their communication. And, 
second, their function is, though, superior to that of the writings because “philosophy is real only when it is 
realized and taught: in philosophizing” (Ibidem). To put it schematically:

- Level 1 = The isolated philosopher’s vision/theory = The innermost reality of philosophy;
- Level 2 = The circle of disciples = The communicated reality of philosophy;
- Level 3 = The writings = The means of communication, therefore destitute of any real bonds 
with the reality of philosophy (indeed, they are connected with Level 2, but as the connection 
is only instrumental, they do not affect in any way the reality).

     With this model in mind, we can assess the novelty of the idea of a classroom turned into a community of 
philosophical inquiry. In CPI, philosophy does not exist prior to and outside the circle of students who co-
philosophize. And students are not disciples of a philosopher, whose vision they rehearse or re-cite, but they 
are the producers of thoughts within a plural setting (Level 2 > Level 1). Philosophy is here not the outcome of 
an isolated soul/mind immersed in the contemplation of ideal entities but of a distributed thinking occurring 
in a space-temporal context. And written texts are not the soulless ‘materialization’ of philosophy but, on the 
one hand, under the form, for instance, of ‘philosophical novels,’ they are what triggers the philosophical 
inquiry, without which no philosophy would exist, and, on the other hand, – under the form of the agenda on 
the paper-board – they are what embodies the distributed thinking of the CPI (and, consequently, they are not 
only written paper but the ‘objective correlate’ (in T.S. Eliot’s phrase) of the philosophical activity). Thus, the 
Academic schema is completely reversed. 

     It is crucial, however, to highlight that, thanks to the CPI approach, it is not a mechanical reversal which 
we have to do with here, but rather a real deconstruction, that is, in a quasi Derridean vein, something that 
activates and operates on what remained unsaid and concealed in the Academy model. This is the case in at 
least two respects: first, in the Academy model, as disciples occupy a middle position (between the vision of the 
thinker and its social dissemination) and as they constitute what ‘actualizes’ philosophy because philosophy is 
humanly actualized only when it is taught (see Landesberg, 1923, p. 95), they are a sort of intermediary and 
have, then, precisely the mediating position of the philosopher, such as Pierre Hadot (2002) magnificently 



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depicted it by comparing Socrates and Eros. The real philosophy in the ‘Socratic’ sense resides not in the 
self-secluded soul/mind of a philosopher en-visioning first principles but in the co-educative dynamics of a 
communal inquiry. There is no genuine philosophy if not within a co-philosophizing as educational relation. 

     Secondly, when speaking of writings and condemning them in comparison with the first reality (the 
spark in the psyche of the thinker), Plato uses an interesting word: súggramma, that is, the semantic root for 
writing (-gramma) and the prefix sun- (=with). The Platonic textuality suggests that the writing and the co-
operative dimension of philosophy as an inquiry are intimately interwoven. And if, as we can state through a 
hermeneutical twist of Landesberg’s text, writings – insofar as they are re-actualized in that wordly realization 
of philosophy that its teaching – are “the lasting power of [the] birth” of philosophy (Landesberg, 1923, p. 
95), then it is within a session of philosophical inquiry, set off by a philosophical novel and – possibly – 
culminating in another written text (the written agenda), that philosophy comes into the world over and over 
again. Better, philosophy is nothing but a continuing re-birth within the context of a communal inquiry made 
possible by writings and animated by an educational tension. The static ‘mathematical’ contemplation is 
replaced, then, with a movement of re-birth. 

     A second dimension of ‘mathematic,’ one which is related directly to learning, needs to be investigated. In 
order to define ‘mathematics’ Heidegger (1987) refers to tà mathémata, the things insofar as they are learnable, 
from which he distinguishes tà prágmata, that is, the things insofar as they are that with which we have to do, 
to deal with, something related to práxis understood as any human doing. If tà mathémata are the learnable 
things, what is learning? According to Heidegger, learning is a kind of acquisition, a ‘taking’: 

The mathémata are the things insofar as we take cognizance of them as what we already know in 
advance […]. Such a proper learning is thereby an extremely remarkable taking, a taking where 
who takes takes only that which fundamentally he already has. (Heidegger, 1987[1936], p. 56)

       What the student already has and is, therefore, learnable, ‘the mathematical’ in Heidegger’s sense as I am 
idiosyncratically reading him, is what is alien to the existential level and belongs to the ideal realm, that is, to 
that domain accessible only to a disembodied, self-secluded reason. 

     What happens when such a kind of learning, which treats things ‘mathematically,’ that is, as learnable 
things (=ultimately known in advance, in the sense that they are cogently valid and are to be demonstrated 
and not discovered through an inquiry), is replaced with a ‘pragmatic’ learning, that treats its contents (even 
the specifically mathematical ones, that is, numbers, etc.) as something which we have to do with? If we 
can construe the entire Lipmanian undertaking ‘pragmatically,’ that is, as a way of putting philosophy into 
practice, of educating for doing philosophy (Lipman, 1988, p. 12), the issue we should finally investigate is 
what happens when this way of understanding philosophy is used for math classes, for teaching precisely that 
subject-matter that seems the least susceptible to any ‘pragmatic’ learning, and so much so that it is the very 
paradigm – in the Academy – of the flight from existence into the domain of the abstract. 

     In § 1, I have spoken of an ‘ethical’ approach to mathematics, one that explores, through philosophical 
inquiry, the meaning of mathematics within one’s own existence; in this paragraph, the idea of a pragmatic 
learning was put forward. It is time, now, to outline how the deployment of philosophical inquiry can act on 
math classes in an ‘ethical-pragmatic’ perspective.

3. The Tears of Philippa: Philosophical Dialogue and Math Classes

     In his obituary for Rorty, the Italian mathematician Giorgio Bagni (2007) pointed out the significance of 
Rorty’s philosophy for math education by insisting on his anti-Platonism:



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As a matter of fact, a Platonic approach cannot be stated uncritically in educational practice 
[…] The connection between knowledge and social practice is really a crucial issue from the 
educational point of view […] Richard Rorty strongly underlined the crucial importance of the 
community as source of epistemic authority […]. (p. 2)

     Bagni’s remarks are interesting because they allow us to outline the specificity of a Lipmanian approach to 
math. To say it with a formula, we could state that Lipman occupies a middle ground between Plato and Rorty, 
avoiding the shortcomings of both approaches. While it can be maintained – and this has been the topic 
of the previous paragraphs – that a non-Platonic stance towards philosophy and mathematics can represent 
a major gain for math education, it is moot whether a typically Rortyan emphasis on solidarity instead of 
objectivity (Rorty, 1991) would represent real progress in math education. 

     In other words, while Bagni appears to be enthusiastic, in his few lines praising Rorty’s philosophy, about 
the prospects that the latter opens up to math education, I would like to highlight that Rorty’s stress on 
community is profoundly different from Lipman’s and that, while the latter can add to the meaning of math 
learning/teaching, the former risks dissolving it. Indeed, due to his misgivings with the notion of inquiry, it is 
debatable that Rorty would accept serenely the idea of a community of philosophical inquiry. He would have 
probably found it still too mortgaged by an ‘objectivity-oriented’ attitude, as if the value and the raison d’être 
of the community were external to it and to the solidarity-principle. As was sagaciously remarked (Silva, 2010), 
Rorty and Lipman represent two misreadings, in a Bloomian (1973, 1975) sense, of Dewey’s legacy and I would 
tend to believe that, while Lipman can play a major role in reconstructing the practices of math education, 
Rorty can provide us with helpful suggestions for the pars destruens but can have little import on the pars 
construens.

     Indeed, the core of the Rortyan insistence on solidarity is that only dialogue counts and not also what the 
dialogue is directed to. The peril I see in this perspective is that it represents only the antipodean opposition 
to the Academy model I have depicted. If in the Academy community counted for nothing, as far as the 
theoretical searching for truth was concerned, in Rorty’s device the search for truth counts for nothing. The 
notion of “conversation” captures beautifully this change of perspective. If it is obviously possible that within 
a solidarity-oriented community a continuous adding to the repertoires of meanings occurs (indeed, it is the 
mark of a flourishing community), this is not the outcome of an inquiry understood as a search for truth. 

     On the contrary, as Susan Gardner has magnificently argued, “progress toward truth is vital to the practice 
of inquiry and […] if such progress is not made, the term ‘Community of Inquiry’ becomes a misnomer” 
(Gardner, 1995/1996, p. 102).

     The pedagogical repercussion of these distinctions (conversation ≠ dialogue; Rortyan community ≠ 
Lipmanian community) is that in a conversational approach looking over the development of the inquiry 
during a class (or a P4C session) can be overlooked – let the pun be allowed – because the mere progress 
of conversation is enough and can contribute to the strengthening of the solidarity of the community. In 
contrast, in the CPI it is crucial that the glue of the community, that is what holds the latter together, is 
the commitment to a search for truth (leaving, for the moment, aside a more clear characterization of this 
truth). Ignoring this search is calamitous for a P4C session (many of us are painfully aware of how what goes 
on in classrooms is too often not a P4C session but a nice conversation starting from Lipmanian texts and 
sticking to some Lipmanian procedures). And this would be even more the case, if we wanted to mobilize this 
conversationally-weakened kind of CPI for math education. 

     The search for truth, without being a capitulation to a ‘Platonic objectivity,’ supplies the community 
with an inquiring horizon, which should structure the relationships occurring in it and give the dialogues a 
direction (what Gadamer (1960) would call the logos of the dialogue). The teacher oversees such a direction 



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by contributing to the harmonizing of the different strands of the dialogue (corresponding to the different 
positions of the students, which are not to be reduced to uniformity) in a dialogic-unity-out-of-differences.

     I am insisting on this idea of a dialogue directed to (a non-Platonically-objective) truth to prepare a 
conceptual platform for the discussion of the role of philosophical inquiry within math classes, by putting 
in relief both the ways in which this approach is linked with other experiences in math education and its 
peculiarities, which, in my opinion, allow it to eschew some drawbacks.

     Paul Ernest (1994) focused on the dialogic nature of mathematics itself, showing how it “sits at the 
crossroads of two major currents of modern thought, the recent fallibilist tradition in the philosophy of 
mathematics, and the multidisciplinary use of the conversation as a basic underlying metaphor for human 
knowing and interaction” (p. 33). As to the first current, its main feature consists in the rejection of the 
following four theses: 

1. There is a secure and fixed basis of truth on which mathematical knowledge is founded;
2. There are wholly reliable logical deductions of mathematical theorems from explicit 
premises;
3. Absolute mathematical knowledge based on impeccable proofs is an ideal which is 
attainable;
4. The logical properties of mathematical proof alone suffice to establish mathematical 
knowledge without reference to human agency or the social domain.
- These theses underpin the traditional absolutist views of mathematical knowledge and 
establish its monological character. They are also central assumptions of Cartesian rationalism 
and the modernism based on it. (Ibid., p. 35)

     It is true that these theses belong to modern rationalism but, as I have tried to illustrate above, they have 
their most ancient forebear in the Academy model.

     As to the second current mentioned by Ernest, he summons different thinkers such as Rorty, Wittgenstein 
and Gadamer. In order to launch an alternative to the monological tradition in math education it could be 
appropriate to display such a panoply of philosophies, yet I would tend to consider the ‘conversational’ strand 
of this new dialogic tradition less promising for math classes. Even if we take leave of the absolutist view, it is 
important to bear in mind that inquiry, also within mathematics, demands a direction to truth and, therefore, 
that some ‘dialogic philosophies’ are more suitable than others in order to underpin a renewal of math 
learning/teaching. 

     Ernest shows also how the dialogic nature of mathematics “encompasses its textual basis, some of its 
concepts, the origins and nature of proof, and the social processes whereby mathematical knowledge is 
created, warrented and learnt” (Ibid., pp. 44-46). Apart from the misgivings about the idea of conversation, I 
agree with the main point made by Ernest. But if his argumentations permit us to find, within the domain of 
mathematics education, a respondency to the (philosophical-educational) reflections conducted in this present 
paper, they do not constitute per se the ‘proof’ that something like philosophical inquiry/dialogue is beneficial 
for math classes. 

     To put it in the P4C vocabulary, my discussion plan will be finally the following: if maths education should 
always have a dialogic nature (in keeping with the dialogic nature of mathematics itself), when, why, to what 
end and in what forms is a typically philosophical dialogue helpful? 

     Over the last years Nadia Stoyanova Kennedy has been investigating such issues and has spoken of 
“interruption,” by drawing upon a Biesta (2006; 2010) notion. What is interrupted is the “normal order” 
ruling over the pedagogical praxis in classrooms and the interruption “may be as straightforward as prompting 



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students to question their own understandings of a concept under discussion, to reflect on what they know 
and what they do not know, to question their peers’ understanding of the concept” (N.S. Kennedy, 2012a, p. 
261).

     So understood, interruption is a quasi-Socratic, torpedo-like move, which prevents students from falling 
into a ‘disciplinary’ slumber, that is, into the risk of taking for granted the concepts of a discipline (as Socrates’ 
fellow-citizens took for granted the mores of their city). And it permits individuals to reconnect the learning of 
the discipline to the broader context of their own being-in-the-world (the ethical dimension of learning math, 
in the sense spelled out above). 

     It is important to appreciate the peculiar note of the notion of “interruption:” it does not refer to a 
breaking-in from without, but to something that, by operating on the interstices of the bodies of disciplines, 
breaks the spell of the monological closure – always looming over every discipline – and opens up a space 
for discussion and dialogue. This kind of inter-ruption is the condition for the ‘interest,’ understood 
etymologically as the being in-between, at the door, to use once again the metaphor I have started with. 

     If it opens up a space for dialogue, which reconnects math in its dialogic nature to human transactions with 
the world, philosophical inquiry represents also a radicalization of the inquiring nature of mathematics itself. 
John Mason (2002, p. 109) has beautifully spoken of a “conjecturing atmosphere” which sustains learning of 
mathematics. And in an analogous vein, Derek Holton (1997), by building upon Legrand’s idea of the débat 
scientifique, remarks:

Under le débat scientifique, […] students are seen as participants in a scientific community 
whose methods of development include conjectures, proofs and regulations. Scientific debates 
can arise spontaneously, as when a student asks a question, or can be intentionally provoked. 
The guiding principles for scientific debate include:

- disturbance – students must encounter and deal with conflict;
- inclusiveness – everyone should have an opportunity to understand what we try to 
teach; and
- collectivity – collective resolution of issues shows how to work with contradictions and 
to respect the views of others.

Now it may seem strange that what is labelled “scientific” has such a strong social 
underpinning. Maybe this can be explained by noting that the point of the exercise is to allow 
students to engage in “scientific debate.” This requires an atmosphere where conjecturing is 
supported, where students feel free to put forward their ideas, where they are not embarrassed 
to make a mistake, and where they feel that they are able to modify the ideas of others. (p. 4)

     I have quoted at length this passage because it refers to a socio-epistemic dynamics very similar to that in 
the CPI: what is, then, the specificity of the CPI? If the philosophical dimension has to be more than a content-
ingredient, it should represent rather a difference in that dynamics. 

     To capture this point we have to refer back to the very idea of the community of philosophical inquiry. The 
phrase was drawn by Lipman from Peirce (D. Kennedy, 2010, p. 15), but with a major qualification, expressed 
by the adjective ‘philosophical.’ In Peirce, inquiry is stirred by a “genuine doubt,” that is, by something 
that intervenes to disrupt our beliefs. Genuine doubt is some real “indecision, however momentary, in our 
action” (to use the Peircean expression), which is obviously quite different from the kind of interruption 
which philosophical inquiry produces. The whole project of a community of philosophical inquiry would be, 
therefore, ultimately, pace Lipman, non-Peircean, and disconnected from the way a scientific community works 
and unable to have effects on inquiries concerning science (and also math, for that matter). 

     But things are, actually, more complicated and by understanding in what sense Lipman is really faithful to 



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one dimension of the Peircean legacy we can appreciate also the contribution which CPI can provide maths 
classes with. As Browning (1991) noted, by 1905:

Peirce recognized a form of philosophically significant inquiry which did not have a starting 
point of genuine doubt. […] The preferred form of philosophical investigation is [...] that 
which serves both to lead towards and encourage genuine doubt and to proceed, once such 
doubt has been so brought about, to its destruction by belief. But on this view the starting 
point of philosophical investigation is no longer genuine doubt, which now occupies a middle 
point in the investigation, but something quite different. This new starting point, though not 
adumbrated in any detail by Peirce, appears to consist in or be instituted by a sort of voluntary 
act in which one “sets himself” to reflect upon and examine certain of his beliefs. (pp. 20-21)

     The ingenuity of Lipman consisted in the fact that, without presumably having any accurate knowledge 
of Peirce’s thinking, he was able to grasp one important element in the latter’s epistemology and to translate 
it into a powerful educational device, that is the CPI. When used for the learning/teaching of disciplines, 
the CPI is the pedagogical approach which, by soliciting philosophical doubts (step 1), can contribute to the 
emergence of ‘genuine doubts’ (step 2), without which no true inquiry (step 3) can be realized. 

     Indeed, it is often insufficient to promote courses of (non-philosophical) inquiry, which may remain 
incapsulated in the matrix of the discipline. It is obviously welcome to use any kind of teaching strategy, which 
prevents students from lapsing into the trap of what Leibniz called psittacism, the parrotism in the re-citation 
of lessons. But not always, despite their merits, can inquiry-based methods obviate adequately this risk. As 
a matter of fact, if not generated by a genuine doubt, the inquiry which develops, brilliant and interesting 
as it may be, can be less fruitful in terms of a real understanding of the topic than expected. The students 
risk playing more the role of Kuhnian (1970) puzzle-solvers than that of Peircean inquirers. Consequently, a 
significant move of the teacher could/should be that of promoting the emergence of a genuine doubt via the 
mobilization of philosophical inquiry, according to a Peircean-Lipmanian model. 

     I will provide a short example of what I mean. In a 4th grade class in Italy pupils had to order the following 
measurements in an ascending order: 7.50 dm; 8.1 dm; 7.8 dm; 7.09 dm; and 8.15 dm. And the following 
dialogue took place (see Sorzio, 2013, pp. 143-144):

 1) Tommaso: “Seven point nine is smaller … because of the zero.”
2) Giulia: “Because the zero does not have any value, you can then say 7.9.”
3) Andrea: “But there is also 7.8.”
4) Tommaso: “That’s true, ‘7 point 9’ because nine is a millimetre.”
5) Teacher: “Who agrees?”
6) All: “Yeahhhhh!”
7) Teacher: “Give me then a good reason.”
8) Silvia: “Because 7 is a decimetre, 0 is a centimetre, 9 is a millimetre.”
9) Teacher: “0 is not a centimetre, it is zero centimetres. And 7.8, what is it?”
10) All: “7 decimetres, 8 centimetres.”
11) Teacher: “Then, why is 7.09 smaller, can you explain it well?”
12) Silvia: “Because centimetres are bigger than millimetres.”
13) Giulia: “No!”
14) Silvia shows centimetres and millimetres on the ruler.
15) Teacher: “Why is it smaller?
16) Tommaso: “Because it has the millimetres ... because the zero does not have any value ... it 
is the zero that indicates …”
17) Teacher: “No, no, it is not because the zero has no value  ... it is because the zero indicates 
...”



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18) All: “The centimetres.”
19) Teacher: “7 refers to the decimetres, the 0 tells us that this measurement is 0 centimetres 
and 9 millimetres, and the other is 8 centimetres, and then ...”
20) Tommaso: “7.9.”
21) Silvia: “7.09.”
22) Teacher: “And then, what is the following number?”
23) Tommaso: “7.8.”
24) Teacher: “Are you sure?”
25) Giulia: “Yes, because we take a small step back.”
26) Teacher: “No, reflect.”
27) Andrea: “7.50.”
28) Teacher: “Why?”
29) Andrea: “Because 5 centimetres is smaller than 8 centimetres.”
30) Teacher: “Who agrees? Who disagrees?”
Many disagree.
31) Tommaso: “5 is smaller than 8 because the 0 doesn’t have any value.”
32) Teacher: “You should not see the zero as smaller but as something that indicates a smaller 
element.”
33) Silvia: “This 0 should be 0 millimetres; 0 millimetres is not on the ruler and therefore 
millimetres are not there because if it is 0 they are not there.” 
34) Teacher: “So, 8.1 and 8.15, which is the smaller?”
35) Tommaso: “8.15 is smaller because here we have the millimetres.”
36) Teacher: “And for this reason is it smaller?”
37) Tommaso is perplexed and some disagree.
38) Andrea: “8.1 and 8.15 would be 8 dm and 1 cm., and 8.15 8 dm, 1 cm and 5 mm [the 
others agree] here it is only one centimetre and here it is in advance by 5 mm.”

     The dialogue here is often on the verge of turning into a sort of puzzle-solving. The ‘logic’ of the 
interruption could suggest, instead, that at move 17) the teacher could have ‘expanded’ her intervention up to 
a philosophical level and opened up the space for a different kind of questioning, instead of almost intimating 
the right answer, by referring to the words of Tommaso at 16). Before going back to the mathematical problem, 
she could have asked: what is the ‘value’ of zero? What do you mean by saying that zero has no value? Are zero 
and nothing the same thing? When we have zero, do we refer to nothing? What is nothing? Is it the same as 
zero?

     It is apparent that one of the issues of the dialogue (although not the only one) concerns the idea of zero 
having a/no value, and that this represents a sort of “epistemological obstacle” (Bachelard, 1938) to fully 
understand how to order the measurements. Staying only within a purely mathematical framework, that is, 
within a purely intra-disciplinary discussion, could lead finally to the right answer but without a full ‘grasp’ of 
the concepts. The philosophical interruption could offer the chance to fine-tune the understanding of some 
concepts and to establish a theoretical platform to go back to the mathematical inquiry with more awareness.

     The New Zealand mathematician and math educator Bill Barton invented an interesting story, included in 
his book The Language of Mathematics, which aims at “argu[ing] that mathematics is a human creation [and at] 
show[ing] that in the origins of mathematics humans had the opportunity to create it differently that (sic!) they 
did” (Barton, 2009, p. 73). He builds on notions with which I would tend to agree, such as, for instance, the 
central role of communication and language (see also Devlin, 2000), the importance of the metaphor and the 
embodiment (see also Lakoff & Nunez, 2000), and the need to reconnect mathematics and experience, but 
he gives these notions a radical ‘spin,’ which, in my opinion, is too affected by some strands of postmodern 
thinking and epistemology, while I would rather suggest situating them within a Deweyan, post-postmodern 
(Hickman, 2007; Oliverio, 2013) framework. But this is a line of discussion which exceeds the scope of this 



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paper. 

     In the fictional story (Barton, 2009, pp. 73-77) a teacher sets this problem: ‘1/4 + 3/8 = ?’  and receives four 
different answers:

- Johnny: 1/4 + 3/8 = 4/12
- Mere:  1/4 + 3/8 = 5/16
- Tom: 1/4 + 3/8 =  3/32
- Philippa: 1/4 + 3/8 = 5/8

     Barton is brilliant in describing how Johnny, Mere and Tom provide ingenious explanations (ultimately 
rooted in the possibility and legitimacy of other kinds of mathematics), and, consequently, in driving home 
his point that “[t]he four ways of ‘adding’ (that is, combining) fractions are all valid in their contexts” (p. 77), 
because mathematics is language- and culture-sensitive and, for this reason, a teacher “may be more careful 
about using the words ‘right’ and ‘wrong,’ preferring rather to mention conventions more often, or to explain 
the context of mathematical concepts” (Ibidem). 

     As I have already said, I would not subscribe to the radicalism of his proposal, but I am interested here 
in what happens to Philippa, the only pupil who had given the ‘right’ (pace Barton) answer. She is invited to 
explain why she got that result after the explanations of her classmates:

Philippa refused to move. “I’m the only one who got it wrong,” she sobbed. “I thought I had 
learned the correct method, but when I look at it, it makes no sense. I can understand writing one 
quarter as two eights, but there is no reason to add only the top numbers and not the bottom 
ones. That doesn’t seem right. Why would you do that? All the others have got a reason for 
what they did, and I don’t.” Philippa learned the method the teacher had taught the class, but had no 
example to illustrate it, and no rationale for her technique. She was bright enough to understand the 
methods the others had used, and they made sense for her. Her own method made no sense, and 
no matter what the teacher said, Philippa was too deeply embarrassed to be comforted. (Barton, 
2009, pp. 76-77. Italics added)

     While Johnny, Mere and Tom are ready to give reasons, because their answers are ultimately grounded in 
their experiences and in their worlds-of-life, Philippa does not succeed in providing that kind of justification 
and, consequently, she is cast into despair and begins nurturing doubts about her proficiency in maths (finally, 
she thinks she is wrong, although she should know that the calculations she made were right). Opposite to the 
author’s intention, I interpret this story not as a parable on the plurality of the mathematical worlds but rather 
as an illuminating apologue about how much an ‘existential’ understanding of mathematical procedures, 
concepts, methods, etc. is needed, if we want to have an accomplished knowledge of maths. This requires a 
‘non-mathematical’ but ‘pragmatic’ learning of mathematics, to use Heidegger’s vocabulary. There is the need 
to refer mathematics back to the Lebenswelt (Husserl, 1959), in order to contrast the “alienation of the sense” 
of the mathematical formulae (Ibid., § 9f) and their mechanization (Ibid, § 9g). What is – in Husserl – a grand 
narrative about the destiny of Western civilization recurs on a smaller scale in every class. 

     In Barton’s story, the teacher postponed until the ensuing lesson the explanation to the class of why 
Philippa was ‘right’ and “felt the wave of relief as the bell rang, and went off to rethink what it was that she 
was doing in mathematics” (Barton, 2009, p. 77). The teacher was unable to find any word of comfort for 
Philippa. And, surely, drawing upon her pedagogical armory in math teaching risked being ineffective. The 
tears of Philippa denounced not a lack in proficiency in adding up fractions but in making sense of this. And, 
moreover, how could the teacher foster a really transformative learning (in a quasi Mezirowian (1991) sense) 
in Johnny, Mere and Tom and not just confine herself to inflicting a mathematical explanation on them 
without conveying the sense of why they should abandon their theories grounded in their experience? Without 



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promoting philosophical inquiry, the teacher might never be able to stop Philippa’s tears. 

Endnotes

1.  While the motto on the entrance to the Academy refers to geometry, it can be extended to mathemat-
ics too. In this present paper I will consider the expression “´Aγεωμέτρητος” as equivalent to “the one who 
does not know mathematics.” I can not provide here a more specific justification of such a move (see Hus-
serl, 1959).

 
References

Agamben, G. (2008), Signatura rerum. Sul metodo, Torino: Bollati Boringhieri.
Arendt, H. (1965), On Revolution, New York: Viking Press.
Bachelard, G. (1938), La formation de l’ésprit scientifique, Paris: Librairie philosophique J. Vrin.
Bagni, G.T. (2007), Richard Rorty (1931-2007) and his legacy for mathematics educators, Educational Studies  
 in Mathematics, 67(1).
Barton, B. (2009), The Language of Mathematics. Telling Mathematical Tales, Berlin and New York: Springer.
Berti, E. (2010), Sumphilosophein. La vita nell’Accademia di Platone, Roma-Bari: Laterza.
Biesta, G. (2006), Beyond Learning. Democratic Education for a Human Future, Boulder and London:    
 Paradigm Publishers.
Biesta, G. (2010), Good Education in an Age of Measurement. Ethics, Politics, Democracy, Boulder and    
 London: Paradigm Publishers. 
Bloom, H. (1973), The Anxiety of Influence. A Theory of Poetry, New York: Oxford University Press.
Bloom, H. (1973), A Map of Misreading, New York: Oxford University Press.
Browning, D. (1994), The Limits of the Practical in Peirce’s View of Philosophical Inquiry, in E. C. Moore,  
 R. S. Robin (eds.), From Time and Chance to Consciousness: Studies in the Metaphysics of Charles   
 Peirce, Oxford: Berg Publishers
Davis, P.J., Hersh, R. (1986), Descartes’ Dream. The World According to Mathematics, Mineola, NY: Dover   
 Publications.
Devlin, K. (2000), The Math Gene. How Mathematical Thinking Evolved and Why Numbers Are like Gossip,   
 New York: Basic Books.
Dewey, J. (1917), The Need for a Recovery of Philosophy, in The Middle Works, 1925-1953, vol. 10 (1916-  
 1917),  edited by J.A. Boydston, Carbondale: Southern Illinois University Press, 1980.
Dewey, J. (1929), The Quest for Certainty, in The Later Works, 1925-1953, vol. 4 (1929), edited by J.A.    
Boydston, Carbondale: Southern Illinois University Press, 1984.
Dewey, J. (1933), How We Think: A Restatement of the Relation of Reflective Thinking to the Educative    
 Process, in The Later Works, 1925-1953, vol. 8 (1933), edited by J. A. Boydston, Carbondale: Southern  
 Illinois University Press, 1986.
Dewey, J. (1938), Logic: The Theory of Inquiry, in The Later Works, 1925-1953, vol. 12 (1938), edited by J.A. 
Boydston, Carbondale: Southern Illinois University Press, 1986.
Ernest, P. (1994), The Dialogic Nature of Mathematics, in P. Ernest (ed.), Mathematics, Education and   
 Philosophy: An Internationl Perspective, London and Washington D.C.: The Falmer Press.
Gadamer, H.-G., Wahrheit und Methode, Tübingen: J.C.B. Mohr 
Gardner, S. (1995/1996), Inquiry is no Mere Conversation (or Discussion or Dialogue). Facilitation of Inquiry  
 Is Hard Work!, Analytic Teaching, Vol. 16, No. 2.
Hadot, P. (2002), Exercices spirituels et philosophie antique, Paris: Éditions Albin Michel.
Heidegger, M. (1936), Die Frage nach dem Ding, Tübingen: Max Niemeyer Verlag,1987.
Hickman, L.A. (2007), Pragmatism As Post-Postmodernism, New York: Fordham University Press.



ANALYTIC TEACHING  AND PHILOSOPHICAL PRAXIS   Vol. 34 Issue 1

13 14

Hildebrand, D. (1996), Genuine Doubt and the Community in Peirce’s Theory of Inquiry, Southwest   
 Philosophy Review, Vol. 12, No. 1.
Holton, D. (1997), Personal thoughts on an ICMI study, retrievable on http://www.maths.otago.ac.nz/ICMI_ 
 Study (last access June, 10th 2013).
Husserl, E. (1959), Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie,    
 L’Aja: Martinus Nijhoff’s Boekhandel.
Kennedy, D. (1995), Philosophy for Children and School Reform. Dewey, Lipman and the Community of   
 Inquiry, in J. P. Portelli & R. F. Reed (eds), Children, Philosophy, and Democracy, Calgary, Alberta   
 (Canada): Detselig Enterprise Ltd.
Kennedy, D. (1997), The Five Communities, Inquiry: Critical Thinking across Disciplines, 16(4).
Kennedy, D. (2004a), Communal Philosophical Dialogue and the Intersubject, International Journal of   
 Applied Philosophy, 18(2).
Kennedy, D. (2004b), The Role of a Facilitator in a Community of Philosophical Inquiry, Metaphilosophy,   
 35(4).
Kennedy, D. (2010), Ann Sharp’s Contribution. A Conversation with Matthew Lipman, Childhood &   
 Philosophy, 6(11).
Kennedy, D. (2012), Lipman, Dewey, and the Community of Philosophical Inquiry, Education and Culture,   
 28(2).
Kennedy, N.S. (2012a), Interrogation as Interruption in the Mathematics Classroom, in M. Santi, S. Oliverio  
 (Eds.), Educating for Complex Thinking through Philosophical Inquiry. Models, Advances, and    
 Proposals for the New Millennium, Napoli: Liguori.
Kennedy, N.S. (2012b), Lipman, Dewey, and Philosophical Inquiry in the Mathematics Classroom, Education  
 and Culture, 28(2).
Kuhn, T.S. (1970), The Structure of Scientific Revolutions (Chicago and London: The University of Chicago   
 Press, 1996).
Lakoff, G., Nunez, R. (2000), Where mathematics comes from, New York: Basic Books.
Landsberg, P. (1923), Wesen und Bedeutung der Platonische Akademie, Bonn: Verlag von Friedrich Cohen.
Lipman, M. (1988), Philosophy Goes to School, Philadelphia: Temple University Press.
Lipman, M. (1991), Thinking in Education, Cambridge: Cambridge University Press.
Lipman, M. (2003), Thinking in Education, Cambridge: Cambridge University Press.
Lipman, M. Sharp, A.M., Oscanyan, F.S. (1980), Philosophy in the Classroom, Philadelphia: Temple    
 University Press 
Mason, J. (2002), Generalisation and algebra: Exploring children’s powers, in L. Haggerty (Ed.), Aspects of   
 Teaching Secondary Mathematics: Perspectives on Practice, London: Routledge Falmer.
Mezirow, J. (1991), Transformative Dimensions of Adult Learning, San Francisco: Jossey-Bass Inc.
Oliverio, S. (2012b), Accomplishing Modernity: Dewey’s Inquiry, Childhood and Philosophy, Education and  
 Culture, 28(2).
Oliverio, S. (2013), La costellazione post-postmoderna. Educazione, filosofia e scienza tra Rorty e Neurath   
 (e Dewey), in E. Corbi, S. Oliverio (Eds.), Oltre la Bildung postmoderna? La pedagogia tra istanze   
 costruttiviste e orizzonti post-costruttivisti, Lecce: PensaMultimedia.
Rorty, R. (1991), Solidarity or Objectivity, in Objectivity, Relativism and Truth. Philosophical Papers, Vol. I,   
Cambridge: Cambridge University Press.
Sharp, A.M. (1987), What is a Community of Inquiry?, Journal of Moral Education, 16(1).
Sharp, A.M. (1996), Self-Transformation in the Community of Inquiry, Inquiry: Critical Thinking Across the   
 Disciplines, 16(1).
Silva, H. (2010), Notas sobre a filosofia da educação de Matthew Lipman a partir do pragmatismo de   
 John Dewey e do neopragmatismo de Richard Rorty: um exercício de desleitura, in Anais do V   
 Colóquio International de Filosofia da Educação, UERJ, 7-10 settembre 2010.
Sorzio, P. (2013), Il curricolo come ambiente diincontro tra schemi cognitivi individuali e oggetti simbolici.  
 Un approccio realista al costruttivismo, in E. Corbi, S. Oliverio (Eds.), Realtà tra virgolette? Nuovo   



ANALYTIC TEACHING  AND PHILOSOPHICAL PRAXIS   Vol. 34 Issue 1

15

 realismo e pedagogia, Lecce: PensaMultimedia.
Splitter, L., Sharp, A.M. (1995), Teaching for Better Thinking. The Classroom Community of Inquiry,   
 Melbourne: Australian Council Educational Research (ACER).
Vlastos, G. (1994a), The Socratic elenchus: method is all, in Id., Socratic Studies, New York: Cambridge   
 University Press.
Vlastos, G. (1994b), Socrates’ disavowal of knowledge, in Id., Socratic Studies, New York: Cambridge   
 University Press.

Address Correspondences To: 
Stefano Oliverio
SInAPSi Centre -- University of Naples Federico II
stefano.oliverio@unina.it


