Mathematics

The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Marcus Giaquinto (Oxford University Press) (Hardcover) The author has performed an impressive balancing act. He manages to treat details precisely without being pedantic. He does not shortchange history, but he also does not permit the pursuit of historical authenticity to interfere with clarity of exposition.

In the early decades of the twentieth century, mathematicians showed an unprecedented concern for the foundations of their subject, not just in expressions of disquiet but also in attempts to find a secure basis. This search for certainty and the crisis that sparked it off is the central subject of this book. First Giaquinto shows mathematical setting of this story to see how the foundational accomplishments grew out of the nineteenth-century quest for clarity and rigor in mathematics.

The clarification of basic properties and relations of analysis set the tone for the search for certainity. The objects of analysis—real numbers and more generally points, classes of points, and functions on classes of points—were taken for granted. But in the later decades of the nineteenth century, according to Giaquinto, mathematicians came to feel that an explicit account of real numbers was needed. also an account of the transfinite numbers discovered—or, some would say, invented by Cantor is explored.

Giaquinto first explains the two best-known accounts of real numbers. Next, he presents a sketch of the way in which the ideas for the transfinite ordinal and cardinal number systems grew out of the study of classes of points, and the rudiments of those number systems are presented. After which he looks at accounts of the natural numbers.

Towards the end of the nineteenth century, the drive for clarity and rigor seemed to be reaching a successful conclusion. Among its fruits were precise accounts of the real and natural numbers, the first general theory of transfinite classes and numbers, and a first account of quantifier logic—no meager harvest. But celebrations had barely begun when certain paradoxes were found in the general theory of classes, which was the basis for all supposedly rigorous accounts of the number systems. This defeat in the hour of triumph made foundational research a major area of concern for mathematicians.

Deeper excavation was needed, and the younger mathematicians who took up the task intended to reach bedrock. So the drive to find sure foundations for mathematics issued largely from problems internal to mathematics, together with the conviction that, if certainty is to be found anywhere, it is to be found in mathematics. In this way, the mathematical concern was tied to a philosophical one: how can we be certain that the theorems of mathematics are trustworthy? The bulk this book examines the attempts to meet this challenge.

The central concern of this book is the epistemic status of non-finitary mathematics. Epistemology is not the only concern in foundational studies, though it has been dominant. The nature and intrinsic organization of mathematics has also been a major concern. Later developments in mathematics show that set theory is not the only basis for this kind of inquiry. Of course, those who think that true mathematics must be constructed will reject not only classical set theory but also the nineteenth-century mathematics out of which it grew. In this regard the development of constructive analysis can be regarded as partial fulfillment of an alternative foundational program. Giaquinto is not able to evaluate the success and significance of this program, and perhaps we are too close to see all of what needs to be seen. For those who accept classical mathematics, category theory has been offered as an alternative to set theory for its catholic reach. Mathematics is definitely not just logic, not just higher-order logic, not just set theory. The old picture of a single fundamental theory to which all else must be reduced has faded. If pure mathematics is the study of abstract structures, set theory is just one framework among others for thinking about that subject matter, and it may not be the best. Universes of sets are themselves structures, and these may be instances of something more general, as is suggested by topos theory. In addition, topos theory sheds new light on the intrinsic organization of mathematics, revealing a surprising unity between apparently disparate fields, topology and algebraic geometry on the one hand, and logic and set theory on the other.

The initial impulse for foundational study was the need to clarify our understanding of the continuum and the basis of infinitesimal calculus. The standard set-theoretic account is an explication that has served well — witness the use made of it in classic textbooks on analysis. But now there are other explications of the basic intuitions. Robinson's non-standard analysis rehabilitated infinitesimals. Non-classical accounts include intuitionistic analysis and Bishop's constructive analysis. The development of synthetic differential geometry gives yet another perspective on the continuum and a novel theory of infinitesimals. Thus we now have a plurality of mathematical ways of refining and abstracting from what are originally spatial intuitions. This too is a way in which foundational study has spread out and away from the monolithic view.

If new developments within mathematics advance our understanding of the nature and intrinsic organization of mathematics, epistemological advances are likely to come from developments outside. In the period covered by Giaquinto in this book, the central epistemological concern has been to justify a body of mathematics. Another concern is to explain how it is possible for an individual to have mathematical knowledge and understanding. This inquiry needs fine-grained information about how we actually acquire our mathematical beliefs and abilities; then we can investigate how best to evaluate those modes of cognitive growth in epistemic terms. The empirical input must come primarily from cognitive sciences. Investigations of simple numerical abilities have already proved fruitful, aided by a recent confluence of evidence from different sources: experiments on healthy adults, children, and even infants, clinical tests on brain-damaged patients, brain imaging studies, and animal studies. There is still a long way to go. The history of mathematics is another source of information.

The
Applicability of Mathematics As a Philosophical Problem by Mark
Steiner
(Harvard University Press) analyzes the different ways mathematics is
applicable in the physical sciences, and presents a startling thesis - the
success of mathematical physics appears to assign the human mind a special place
in the cosmos.

This book has two separate objectives. The first is to examine in what ways
mathematics can be said to be applicable in the natural sciences or to the
empirical world. Mathematics is applicable in many senses, and this ambiguity
has bred confusion and error--even among "analytic" philosophers: because there
are many senses of "application" and "applicability," there are many questions
about the application of mathematics that ought to be, but have not been,
distinguished by philosophers. As a result, we do not always know what problem
they are dealing with.

For example, we often use pure mathematics in reasoning
about the empirical world. This raises two questions: what is the logical form
of such reasoning, and what are its metaphysical presuppositions?

The problem of logical form arises because number words
function in mathematics as proper nouns (names of numbers, numerals), while in
empirical descriptions ("three green leaves") they often function as adjectives.
This is an equivocation, which appears to make it impossible to reason using
mathematics in an empirical situation. Thus, some writers have felt themselves
forced to distinguish between "pure mathematics" and "empirical" or "applied"
mathematics. Steiner points out that Frege solved the problem, making this
distinction entirely superfluous, by showing how number words can function as
names, not adjectives, everywhere.

Names of what? Names, apparently, of numbers, thought of as
abstract or nonphysical objects. And this brings up the metaphysical question
concerning application: how can facts about number be relevant to the empirical
world? Frege had a keen answer to this too: they are not. Numbers are related,
not to empirical objects, but to empirical concepts! It is the empirical
concepts that are used to describe the world; numbers are used to characterize
those very descriptions. (Of course, there are other objections to Platonism
than its alleged inability to account for applications, but this is not a book
about Platonism.)

Thus, two frequently asked questions about the application
o mathematics were answered definitively over a hundred years ago. Some
philosophers sometimes write as if this were not so, the reason maybe because
the different concepts of applicability have not been made clear.

To the extent that "naturalism" rejects any anthropocentric
point of view (and Steiner thinks all forms of naturalism do)--then this book
challenges naturalism. This makes the book consistent with natural theology, but
there are many positions available that are neither naturalist nor theological.
Steiner’s research has been inspired in great measure by Maimonides' view that
no philosophy, and in particular no religious philosophy, can be complete
without careful examination of our best physical theories (this is a much harder
task today than it was in the Middle Ages) and that the study of science (and
philosophy) itself can be a religious act . Of course Maimonides would
disapprove of Steiner’s "anthropocentric" conclusions. Considering another
Maimonidean truth also informs
the conclusions in this book: the importance of the enterprise of
scientific inquiry from a religious point of view. Perhaps this book can
contribute to the dialogue between the sciences and the humanities. Steiner
shows how painstaking attention to what seem to be technical details of
mathematical formalism can yield insight into the human mind and its place in
nature--a major goal of the humanities. But this insight cannot be obtained,
either, without the peculiar skills of philosophical analysis.

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