Structuralists in philosophy of mathematics can learn from the current heritage of the ancient ar... more Structuralists in philosophy of mathematics can learn from the current heritage of the ancient arithmetician Diophantus. A list of polynomial equations defines a kind of geometric space called a scheme. By one definition these schemes are countable sets built from integers in very much the way that Leopold Kronecker approached pure arithmetic. In another version every scheme is a functor as big as the universe of sets. The two versions are often mixed together because they give precisely the same structural relations between schemes. The practice was vividly put by André Joyal in conversation: "There is no ontology here." Mathematicians work rigorously with relations among schemes without choosing between the definitions. The tools which enable this in principle and require it in practice grew from topology. The three great projects for 20th century mathematics were to absorb Richard Dedekind's and David Hilbert's algebra, to absorb Henri Poincaré's and Luitzen Brouwer's topology, and to create functional analysis. 1 Algebra and topology made explosive advances when Emmy Noether initiated a series of ever-deeper structural unifications. Her group theory became the method of homology of spaces. When abstract algebra spread from advanced number theory into basic topology it became a contender for organizing all of mathematics, as Noether intended. Her protégé Bartel van der Waerden advanced the new hegemony in his Moderne Algebra (1930). Bourbaki based their encyclopedic work on van der Waerden's text (Corry, 1996, pp. 309ff.). This "structural" mathematics was the research norm by the 1950s and the textbook norm by the 1960s. Homology itself continues expanding and linking Dedekind and Poincaré to the latest Fields Medals. 2 This case study looks at schemes, which arose largely in pursuit of a single problem, namely the Weil conjectures, a series of elegant conjectures on counting the solutions to certain arithmetic equations (Weil, 1949). 3 Weil sketched a fascinating strategy for a proof if only the Lefschetz fixed point theorem from topology would apply in arithmetic. It was a brilliant idea with repercussions all across mathematics but only a handful of leading mathematicians thought it could possibly work.
Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic ... more Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind's approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, and a monograph on the Navier-Stokes equation to show structural methods like Dedekind's have long been central to differential equations, and have philosophically respectable ontology and epistemology. There are many things one would say about [the Dedekind cut corresponding to an irrational number] such as that it is a set of infinitely many things . . . that one would certainly be most reluctant to impose as a burden on the number itself. (Dedekind letter to Weber 24 January 1888, quoted by Stein (1988, p. 248).) The field of differential equations has never been transformed in a profound way by the intrusion of structuralist methods.
Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce), Dec 28, 2020
This note describes Saunders Mac Lane as a philosopher, and indeed as a paragon naturalist philos... more This note describes Saunders Mac Lane as a philosopher, and indeed as a paragon naturalist philosopher. He approaches philosophy as a mathematician. But, more than that, he learned philosophy from David Hilbert's lectures on it, and by discussing it with Hermann Weyl, as much as he did by studying it with the mathematically informed Göttingen Philosophy professor Moritz Geiger.
Structuralists in philosophy of mathematics can learn from the current heritage of the ancient ar... more Structuralists in philosophy of mathematics can learn from the current heritage of the ancient arithmetician Diophantus. A list of polynomial equations defines a kind of geometric space called a scheme. By one definition these schemes are countable sets built from integers in very much the way that Leopold Kronecker approached pure arithmetic. In another version every scheme is a functor as big as the universe of sets. The two versions are often mixed together because they give precisely the same structural relations between schemes. The practice was vividly put by André Joyal in conversation: "There is no ontology here." Mathematicians work rigorously with relations among schemes without choosing between the definitions. The tools which enable this in principle and require it in practice grew from topology. The three great projects for 20th century mathematics were to absorb Richard Dedekind's and David Hilbert's algebra, to absorb Henri Poincaré's and Luitzen Brouwer's topology, and to create functional analysis. 1 Algebra and topology made explosive advances when Emmy Noether initiated a series of ever-deeper structural unifications. Her group theory became the method of homology of spaces. When abstract algebra spread from advanced number theory into basic topology it became a contender for organizing all of mathematics, as Noether intended. Her protégé Bartel van der Waerden advanced the new hegemony in his Moderne Algebra (1930). Bourbaki based their encyclopedic work on van der Waerden's text (Corry, 1996, pp. 309ff.). This "structural" mathematics was the research norm by the 1950s and the textbook norm by the 1960s. Homology itself continues expanding and linking Dedekind and Poincaré to the latest Fields Medals. 2 This case study looks at schemes, which arose largely in pursuit of a single problem, namely the Weil conjectures, a series of elegant conjectures on counting the solutions to certain arithmetic equations (Weil, 1949). 3 Weil sketched a fascinating strategy for a proof if only the Lefschetz fixed point theorem from topology would apply in arithmetic. It was a brilliant idea with repercussions all across mathematics but only a handful of leading mathematicians thought it could possibly work.
The finiteness theorem for invariants of a finite group by Emmy Noether in Erlangen (trans. Colin... more The finiteness theorem for invariants of a finite group by Emmy Noether in Erlangen (trans. Colin McLarty)
This paper explores the set theoretic assumptions used in the current published proof of Fermat's... more This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions. Does the proof of Fermat's Last Theorem (FLT) go beyond Zermelo Fraenkel set theory (ZFC)? Or does it merely use Peano Arithmetic (PA) or some weaker fragment of that? The answers depend on what is meant by "proof" and "use," and are not entirely known. This paper surveys the current state of these questions and briefly sketches the methods of cohomological number theory used in the existing proof. The existing proof of FLT is Wiles [1995] plus improvements that do not yet change its character. Far from self-contained it has vast prerequisites merely introduced in the 500 pages of [Cornell et al., 1997]. We will say that the assumptions explicitly used in proofs that Wiles cites as steps in his own are "used in fact in the published proof." It is currently unknown what assumptions are "used in principle" in the sense of being proof-theoretically indispensable to FLT. Certainly much less than ZFC is used in principle, probably nothing beyond PA, and perhaps much less than that. The oddly contentious issue is universes, often called Grothendieck universes. 1 On ZFC foundations a universe is an uncountable transitive set U such that U, ∈ satisfies the ZFC axioms in the nicest way: it contains the powerset of each of its elements, and for any function from an element of U to U the range is also an element of U. This is much stronger than merely saying U, ∈ satisfies the ZFC axioms. We do not merely say the powerset axiom "every set has a powerset" is true with all quantifiers relativized to U. Rather, we require "for every set x ∈ U , the powerset of x is also in U "
Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical f... more Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue. Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics. Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics. Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical. This article assumes knowledge of category theory on the level of Category theory, introduction to §1.
Elementary axioms describe a category of categories. Theorems of category theory follow, includin... more Elementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations between categories and sets, the latter defined as discrete categories, are described, and applications to foundations are discussed.
This note gives a syntactic presentation for partial algebraic theories (see [1] and [3]). The lo... more This note gives a syntactic presentation for partial algebraic theories (see [1] and [3]). The logic, called left exact logic, is interpretable in any category with all finite limits, and it has coherent logic as a conservative extension, which implies a completeness theorem.
We can learn from questions as well as from their answers. This paper urges some things to learn ... more We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman.
A ‘category’, in the mathematical sense, is a universe of structures and transformations. Categor... more A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets. This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard set theory.
Structuralists in philosophy of mathematics can learn from the current heritage of the ancient ar... more Structuralists in philosophy of mathematics can learn from the current heritage of the ancient arithmetician Diophantus. A list of polynomial equations defines a kind of geometric space called a scheme. By one definition these schemes are countable sets built from integers in very much the way that Leopold Kronecker approached pure arithmetic. In another version every scheme is a functor as big as the universe of sets. The two versions are often mixed together because they give precisely the same structural relations between schemes. The practice was vividly put by André Joyal in conversation: "There is no ontology here." Mathematicians work rigorously with relations among schemes without choosing between the definitions. The tools which enable this in principle and require it in practice grew from topology. The three great projects for 20th century mathematics were to absorb Richard Dedekind's and David Hilbert's algebra, to absorb Henri Poincaré's and Luitzen Brouwer's topology, and to create functional analysis. 1 Algebra and topology made explosive advances when Emmy Noether initiated a series of ever-deeper structural unifications. Her group theory became the method of homology of spaces. When abstract algebra spread from advanced number theory into basic topology it became a contender for organizing all of mathematics, as Noether intended. Her protégé Bartel van der Waerden advanced the new hegemony in his Moderne Algebra (1930). Bourbaki based their encyclopedic work on van der Waerden's text (Corry, 1996, pp. 309ff.). This "structural" mathematics was the research norm by the 1950s and the textbook norm by the 1960s. Homology itself continues expanding and linking Dedekind and Poincaré to the latest Fields Medals. 2 This case study looks at schemes, which arose largely in pursuit of a single problem, namely the Weil conjectures, a series of elegant conjectures on counting the solutions to certain arithmetic equations (Weil, 1949). 3 Weil sketched a fascinating strategy for a proof if only the Lefschetz fixed point theorem from topology would apply in arithmetic. It was a brilliant idea with repercussions all across mathematics but only a handful of leading mathematicians thought it could possibly work.
Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic ... more Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind's approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, and a monograph on the Navier-Stokes equation to show structural methods like Dedekind's have long been central to differential equations, and have philosophically respectable ontology and epistemology. There are many things one would say about [the Dedekind cut corresponding to an irrational number] such as that it is a set of infinitely many things . . . that one would certainly be most reluctant to impose as a burden on the number itself. (Dedekind letter to Weber 24 January 1888, quoted by Stein (1988, p. 248).) The field of differential equations has never been transformed in a profound way by the intrusion of structuralist methods.
Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce), Dec 28, 2020
This note describes Saunders Mac Lane as a philosopher, and indeed as a paragon naturalist philos... more This note describes Saunders Mac Lane as a philosopher, and indeed as a paragon naturalist philosopher. He approaches philosophy as a mathematician. But, more than that, he learned philosophy from David Hilbert's lectures on it, and by discussing it with Hermann Weyl, as much as he did by studying it with the mathematically informed Göttingen Philosophy professor Moritz Geiger.
Structuralists in philosophy of mathematics can learn from the current heritage of the ancient ar... more Structuralists in philosophy of mathematics can learn from the current heritage of the ancient arithmetician Diophantus. A list of polynomial equations defines a kind of geometric space called a scheme. By one definition these schemes are countable sets built from integers in very much the way that Leopold Kronecker approached pure arithmetic. In another version every scheme is a functor as big as the universe of sets. The two versions are often mixed together because they give precisely the same structural relations between schemes. The practice was vividly put by André Joyal in conversation: "There is no ontology here." Mathematicians work rigorously with relations among schemes without choosing between the definitions. The tools which enable this in principle and require it in practice grew from topology. The three great projects for 20th century mathematics were to absorb Richard Dedekind's and David Hilbert's algebra, to absorb Henri Poincaré's and Luitzen Brouwer's topology, and to create functional analysis. 1 Algebra and topology made explosive advances when Emmy Noether initiated a series of ever-deeper structural unifications. Her group theory became the method of homology of spaces. When abstract algebra spread from advanced number theory into basic topology it became a contender for organizing all of mathematics, as Noether intended. Her protégé Bartel van der Waerden advanced the new hegemony in his Moderne Algebra (1930). Bourbaki based their encyclopedic work on van der Waerden's text (Corry, 1996, pp. 309ff.). This "structural" mathematics was the research norm by the 1950s and the textbook norm by the 1960s. Homology itself continues expanding and linking Dedekind and Poincaré to the latest Fields Medals. 2 This case study looks at schemes, which arose largely in pursuit of a single problem, namely the Weil conjectures, a series of elegant conjectures on counting the solutions to certain arithmetic equations (Weil, 1949). 3 Weil sketched a fascinating strategy for a proof if only the Lefschetz fixed point theorem from topology would apply in arithmetic. It was a brilliant idea with repercussions all across mathematics but only a handful of leading mathematicians thought it could possibly work.
The finiteness theorem for invariants of a finite group by Emmy Noether in Erlangen (trans. Colin... more The finiteness theorem for invariants of a finite group by Emmy Noether in Erlangen (trans. Colin McLarty)
This paper explores the set theoretic assumptions used in the current published proof of Fermat's... more This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions. Does the proof of Fermat's Last Theorem (FLT) go beyond Zermelo Fraenkel set theory (ZFC)? Or does it merely use Peano Arithmetic (PA) or some weaker fragment of that? The answers depend on what is meant by "proof" and "use," and are not entirely known. This paper surveys the current state of these questions and briefly sketches the methods of cohomological number theory used in the existing proof. The existing proof of FLT is Wiles [1995] plus improvements that do not yet change its character. Far from self-contained it has vast prerequisites merely introduced in the 500 pages of [Cornell et al., 1997]. We will say that the assumptions explicitly used in proofs that Wiles cites as steps in his own are "used in fact in the published proof." It is currently unknown what assumptions are "used in principle" in the sense of being proof-theoretically indispensable to FLT. Certainly much less than ZFC is used in principle, probably nothing beyond PA, and perhaps much less than that. The oddly contentious issue is universes, often called Grothendieck universes. 1 On ZFC foundations a universe is an uncountable transitive set U such that U, ∈ satisfies the ZFC axioms in the nicest way: it contains the powerset of each of its elements, and for any function from an element of U to U the range is also an element of U. This is much stronger than merely saying U, ∈ satisfies the ZFC axioms. We do not merely say the powerset axiom "every set has a powerset" is true with all quantifiers relativized to U. Rather, we require "for every set x ∈ U , the powerset of x is also in U "
Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical f... more Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue. Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics. Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics. Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical. This article assumes knowledge of category theory on the level of Category theory, introduction to §1.
Elementary axioms describe a category of categories. Theorems of category theory follow, includin... more Elementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations between categories and sets, the latter defined as discrete categories, are described, and applications to foundations are discussed.
This note gives a syntactic presentation for partial algebraic theories (see [1] and [3]). The lo... more This note gives a syntactic presentation for partial algebraic theories (see [1] and [3]). The logic, called left exact logic, is interpretable in any category with all finite limits, and it has coherent logic as a conservative extension, which implies a completeness theorem.
We can learn from questions as well as from their answers. This paper urges some things to learn ... more We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman.
A ‘category’, in the mathematical sense, is a universe of structures and transformations. Categor... more A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets. This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard set theory.
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