It turns out that many interesting and important mathematical questions are independent of the basic assertions of set theory. One example is Cantor's continuum hypothesis that there are no sets that are strictly p. Neither the continuum hypothesis nor its negation can be proved in the standard axiomatizations of set theory.
What does this independence say about mathematical concepts? Do we have another sort of relativity on offer?
Can we only say that a given set is the size of a certain cardinality relative to an interpretation of set theory? Some philosophers hold that these results indicate an indeterminacy concerning mathematical truth. There is no fact of the matter concerning, say, the continuum hypothesis. The issue here has ramifications concerning the practice of mathematics. Some take this result to refute mechanism, the thesis that the human mind operates like a machine.
On the other hand, Webb  takes the incompleteness results to support mechanism. To some extent, some questions concerning the applications of mathematics are among this group of issues. What can a theorem of mathematics tell us about the natural world studied in science? To what extent can we prove things about knots, bridge stability, chess endgames, and economic trends? There are or were philosophers who take mathematics to be no more than a meaningless game played with symbols chapter 8 in this volume , but everyone else holds that mathematics has some sort of meaning.
What is this meaning, and how does it relate to the meaning of ordinary nonmathematical discourse? Another group of issues consists of attempts to articulate and interpret particular mathematical theories and concepts. One example is the foundational work in arithmetic and analysis. Sometimes, this sort of activity has ramifications for mathematics itself, and thus challenges and blurs the boundary between mathematics and its philosophy. Interesting and powerful research techniques are often suggested by foundational work that forges connections between mathematical fields.
In addition to mathematical logic, consider the embedding of the natural numbers in the complex plane, via analytic number theory. Foundational activity has spawned whole branches of mathematics. Sometimes developments within mathematics lead to unclarity concerning what a certain concept is. The example developed in Lakatos  is a case in point. For another example, work leading to the foundations of analysis led mathematicians to focus on just what a function is, ultimately yielding the modern notion of function as arbitrary correspondence. The questions are at least partly ontological.
This group of issues underscores the interpretive nature of philosophy of mathematics.
We need to figure out what a given mathematical concept is , and what a stretch of mathematical discourse says. It is not clear a priori how this blatantly dynamic discourse is to be understood. What is the logical form of the discourse and what is its logic?
What is its ontology? The history of analysis shows a long and tortuous task of showing just what expressions like this mean. Of course, mathematics can often go on quite well without this interpretive work, and sometimes the interpretive work is premature and is a distraction at best.
In the present context, the question is whether the mathematician must stop mathematics until he has a semantics for his discourse fully worked out. Surely not. Moreover, we are never certain that the interpretive project is accurate and complete, and that other problems are not lurking ahead. I now present sketches of some main positions in the philosophy of mathematics. The list is not exhaustive, nor does the coverage do justice to the subtle and deep work of proponents of each view. Nevertheless, I hope it serves as a useful p. Of course, the reader should not hold the advocates of the views to the particular articulation that I give here, especially if the articulation sounds too implausible to be advocated by any sane thinker.
According to Alberto Coffa  , a major item on the agenda of Western philosophy throughout the nineteenth century was to account for the at least apparent necessity and a priori nature of mathematics and logic, and to account for the applications of mathematics, without invoking anything like Kantian intuition. The main theme—or insight, if you will—was to locate the source of necessity and a priori knowledge in the use of language. Philosophers thus turned their attention to linguistic matters concerning the pursuit of mathematics. What do mathematical assertions mean?
What is their logical form? What is the best semantics for mathematical language? The members of the semantic tradition developed and honed many of the tools and concepts still in use today in mathematical logic, and in Western philosophy generally.sei-sicite.xtage.com.br/trabajos-en-valladolid.php
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Michael Dummett calls this trend in the history of philosophy the linguistic turn. An important program of the semantic tradition was to show that at least some basic principles of mathematics are analytic , in the sense that the propositions are true in virtue of meaning. If the program could be carried out, it would show that mathematical truth is necessary—to the extent that analytic truth, so construed, is necessary.
Given what the words mean, mathematical propositions have to be true, independent of any contingencies in the material world. And mathematical knowledge is a priori—to the extent that knowledge of meanings is a priori. Presumably, speakers of the language know the meanings of words a priori, and thus we know mathematical propositions a priori.
The most articulate version of this program is logicism , the view that at least some mathematical propositions are true in virtue of their logical forms chapter 5 in this volume. According to the logicist, arithmetic truth, for example, is a species of logical truth. The most detailed developments are those of Frege [ , ] and Alfred North Whitehead and Bertrand Russell . Unlike Russell, Frege was a realist in ontology, in that he took the natural numbers to be objects. For any concepts F , G , the number of F 's is identical to the number of G 's if and only if F and G are equinumerous.
Frege showed how to define equinumerosity without invoking natural numbers. This became known as the Caesar problem. It is not that anyone would confuse a natural number with the Roman general Julius Caesar, but the underlying idea is that we have not succeeded in characterizing the natural numbers as objects unless and until we can determine how and why any given natural number is the same as or different from any object whatsoever. The distinctness of numbers and human beings should be a consequence of the theory, and not just a matter of intuition. The number 2, for example, is the extension or collection of all concepts that hold of exactly two elements.
The inconsistency in Frege's theory of extensions, as shown by Russell's paradox, marked a tragic end to Frege's logicist program. Russell and Whitehead  traced the inconsistency in Frege's system to the impredicativity in his theory of extensions and, for that matter, in Hume's principle.
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They sought to develop mathematics on a safer, predicative foundation. Their system proved to be too weak, and ad hoc adjustments were made, greatly reducing the attraction of the program. There is a thriving research program under way to see how much mathematics can be recovered on a predicative basis chapter 19 in this volume. Variations of Frege's original approach are vigorously pursued today in the work of Crispin Wright, beginning with  , and others like Bob Hale  and Neil Tennant [ , ] chapter 6 in this volume.
The idea is to bypass the treatment of extensions and to work with fully impredicative Hume's principle, or something like it, directly. But what is the philosophical point? On the neologicist approach, Hume's principle is taken to p. Hume's principle is akin to an implicit definition.
Indeed, the only essential use that Frege made of extensions was to derive Hume's principle—everything else concerning numbers follows from that. Neologicism is a reconstructive program showing how arithmetic propositions can become known. Without this feature, the derivation of the Peano axioms from Hume's principle would fail. This impredicativity is consonant with the ontological realism adopted by Frege and his neologicist followers. The neologicist project, as developed thus far, only applies basic arithmetic and the natural numbers.
An important item on the agenda is to extend the treatment to cover other areas of mathematics, such as real analysis, functional analysis, geometry, and set theory. The program involves the search for abstraction principles rich enough to characterize more powerful mathematical theories see, e.
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Coffa  provides a brief historical sketch of the semantic tradition, outlining its aims and accomplishments. Many philosophers no longer pay serious attention to notions of meaning, analyticity, and a priori knowledge. To be precise, such notions are not given a primary role in the epistemology of mathematics, or anything else for that matter, by many contemporary philosophers. Quine e. Quine's view is that the linguistic and factual components of a given sentence cannot be sharply distinguished, and thus there is no determinate notion of a sentence being true solely in virtue of language analytic , as opposed to a sentence whose truth depends on the way the world is synthetic.
Then how is mathematics known? Quine is a thoroughgoing empiricist, in the tradition of John Stuart Mill chapter 3 in this volume. His positive view is that all of our beliefs constitute a seamless web answerable to, and only to, sensory stimulation. Moreover, no part of the web is knowable a priori. This picture gives rise to a now common argument for realism.