دانلود کتاب Naturalizing Logico-Mathematical Knowledge: Approaches from Philosophy, Psychology and Cognitive Science

This book is meant as a part of the larger contemporary philosophical project of naturalizing logico-mathematical knowledge, and addresses the key question that motivates most of the work in this field: What is philosophically relevant about the nature of logico-mathematical knowledge in recent research in psychology and cognitive science? The question about this distinctive kind of knowledge is rooted in Plato’s dialogues, and virtually all major philosophers have expressed interest in it. The essays in this collection tackle this important philosophical query from the perspective of the modern sciences of cognition, namely cognitive psychology and neuroscience. Naturalizing Logico-Mathematical Knowledge contributes to consolidating a new, emerging direction in the philosophy of mathematics, which, while keeping the traditional concerns of this sub-discipline in sight, aims to engage with them in a scientifically-informed manner. A subsequent aim is to signal the philosophers’ willingness to enter into a fruitful dialogue with the community of cognitive scientists and psychologists by examining their methods and interpretive strategies.

"Themes and Motifs
The truths of mathematics and logic are special in several well-known
respects: they are seemingly impossible to challenge on empirical
grounds—hence they are traditionally called ‘a priori’; there is also a
sense in which they are considered to be ‘necessary’. Yet, while stressing
their specialness, we should not lose sight of the obvious fact that these
propositions are, first and foremost, beliefs that we, human beings, often
assert. As such, important questions about them arise immediately—e.g.,
how did we acquire them? (Or are they, or some of them, innate ? If so,
what does this mean?); What actually deters us from challenging them?
What makes a proof of such a proposition convincing ? What should we
do when no proof is available? Or, what does it mean for such a proposition
to be self-evident ? And so on and so forth.
For all their naturalness, these kinds of queries were dismissed by Gotlob
Frege (1848–1925), the most important logician since Aristotle. He argued
that they are completely misguided, since what drives them—an interest in the
psychological underpinnings of logico-mathematical thinking—is prone to
engender confusion: one should not focus on how humans operate within
the logico-mathematical realm, but rather on how they ought to do it. As
part of this crusade to uphold (this kind of) normativity, Frege insisted
that the only efforts worth undertaking consist in extracting, from the
morass of the ordinary ways of speaking, the network of objective relations
holding—whether or not individual people realize it—between the contents
encapsulated into the logico-mathematical assertions. This line of thinking,
unsurprisingly dubbed ‘anti-psychologism’, expelled a whole family of questions
from the agenda of the philosophers of logic and mathematics. 2
The sharp separation of the ‘logical’ from the ‘psychological’ became
enormously influential in analytic philosophy; it still remains so, although
it has constantly been challenged in various ways. 3 However, despite the
name of this orientation, the intention behind it was not to dismiss psychology
per se as an empirical science aiming to reveal, among other things,
contingent truths about how people actually (learn to) reason, calculate,
construct, or become convinced by proofs. The intrinsic legitimacy of this
kind of research was not contested, only its relevance—for the normative
questions about how we ought to reason. Thus, perhaps not even aware of
the Fregean attitude, entire branches of psychology and cognitive science
have developed and thrived for more than a century now, 4 investigating
precisely the kinds of questions Frege took to be immaterial for genuinely
understanding what mathematics and logic are about.
With rare but notable exceptions, the mainstream work in the epistemology
of logic and mathematics has until recently barely intersected the
trajectories taken by the flourishing cognitive sciences. 5 Yet this is not the
case with epistemology in general, and this discrepancy is not that surprising
given that mathematics and logic are traditionally believed to be the most
resistant to naturalization. It will soon be almost half a century since W. v.
O. Quine, in his famous programmatic “Epistemology Naturalized” ( 1969 )
asked philosophers to recognize that the traditional Cartesian “quest for
certainty” is “a lost cause” and thus urged epistemologists “to settle for psychology”.
Consequently, “Epistemology, or something like it, simply falls
into place as a chapter of psychology and hence of natural science” ( 1969 ,
82). 6 Such provocative statements may have been useful to reorient philosophical
agendas 50 years ago, but nowadays, very few philosophers take
them literally. It is quite clear that this radical ‘replacement’ naturalism , as
Kornblith (1985, 3) calls it, is not the best option for a naturalistically bent
philosopher, especially one of logic and mathematics (and perhaps not even
for the scientists themselves). A better alternative seems to be a moderate
view, sometimes called ‘cooperative’ naturalism, which, as the name indicates,
encourages the use of the findings of the sciences of cognition in solving
philosophical problems. 7 Yet what I take to be an even better approach
is to understand ‘cooperation’ in a more extensive fashion, as promoting
interactions that go in both directions; it is a reasonable thought that the
scientists too may profit from philosophical reflection. Thus, fostering
such a dialogue is the primary aim of the present project. The way to
achieve it here is by displaying, for the benefit of both the philosophical
and scientific audiences, a sketch of the landscape of the current research
gathered under the heading ‘naturalized epistemology of mathematics
and logic ’. 8
Before I briefly present the contents of the chapters, it may be useful to
set the reader’s expectations right. Perhaps the first point to make is that,
although traditionally it was the concept of knowledge that took pride of
place in the writings dealing with the epistemology of these two disciplines,
in what follows, this centrality is challenged. In a naturalist spirit, many
contributors here can be described as shifting their attention to the very phenomenon
of knowledge 9 —that is, the remarkable natural fact that human
beings, of all ages and cultures, are able to navigate successfully within the
realm of abstraction. In this type of analysis, it is not so much the symbolism
itself that is being investigated, nor how a generic mind relates to
abstraction, but rather the way in which the (presumably) abstract content
is assimilated and manipulated by concrete epistemic agents in local contexts.
Indeed, at least when it comes specifically to mathematics, there is
no better way to summarize the issues investigated here than by citing the
felicitous title of Warren McCulloch’s (1961) paper, “What is a number, that
a man may know it, and a man, that he may know a number?” Thus, it is
causal stories, sensory perception, material signs and intuition, testimony,
learning, neural activity, and other notions of the same ilk that now hold
center stage in most of the chapters.
Consequently, the elements of the logico-mathematical practice under
examination here no longer retain the purity and perfection traditionally
associated with these two fields. Few, if any, of the perennial (and perennially
frustrating) in principle questions are asked or debated. As expected,
of major interest here is to probe to what extent a robust sense of normativity
can be disentangled from an enormously complicated network of
causal connections involving nonidealized epistemic agents ratiocinating in,
and about, a material world. A central question is not only how but also
whether normativity is possible in practice , or despite all the imperfections,
approximations, and errors people are so prone to. 10 Both the friends and
the foes of these naturalistic approaches will recognize the pivotal issue as
being the following: does revealing the cognitive basis of mathematics and
logic affect (threatens? supports?) the putative objectivity of mathematical
and logical knowledge?
Another aspect worth pointing out is that the collection has not been
conceived to promote a specific philosophical position, hiding, so to speak,
behind the avowed naturalist attitude ‘let us first look and see —what is the
evidence’. 11 Thus, both the empiricistically inclined philosophers/scientists
and their opponents are, I believe, represented; there are chapters inclining
toward what is traditionally labeled as mathematical ‘realism’, while
others display a preference for different metaphysical camps. There is also
variety in terms of the methodological assumptions and conclusions among
the scientifically oriented contributions. Moreover, it is my hope that the
collection as a whole manages to avoid being biased in either of the two
usual ways. It was not meant to provide empirical evidence that certain
philosophical theories are true (or false), nor was it meant to provide reasons
of a philosophical-conceptual nature that certain research programs in
psychology and cognitive science are misguided. Importantly, however, note
that acknowledging this is consistent with some individual chapters having
such goals—although in most cases, a firm dichotomy empirical/conceptual
is implicitly questioned. After all, not only should philosophers look at the
empirical evidence first but also, as the scientists are often aware, what counts
as evidence (i.e., which findings they are justified to present as evidence) may
be influenced by deep commitments of a philosophical-conceptual nature.
To sum up, a reader motivated primarily by philosophical interests is
invited to reflect on a (meta-)question, which I take to be both fundamental
and insufficiently explored: what, if anything, is relevant about the nature
of logico-mathematical knowledge in the recent research in psychology and
cognitive science? Naturally, a corresponding question can be formulated
for the more scientifically inclined reader: what, if anything, offers valuable
insight into the nature of logico-mathematical knowledge in the philosophical
work in this field? Although I regard these two questions to be equally
urgent, and in fact entangled, the potential reader should be advised that
the majority of the contributions here are philosophically oriented, as the
table of contents and the brief presentations of the chapters that follows
show. Moreover, most of the work deals with mathematics (and of a rather
elementary kind), so logic per se receives less coverage than would be ideal."