- A theorem of Gilmer and the canonical universal splitting ring,
We give a constructive proof of Gilmer's theorem that if every
nonzero polynomial over a field k has a root in some fixed extension
field E, then each polynomial in k[X] splits in E[X]. Using a slight
generalization of this theorem, we construct, in a functorial way, a
commutative, discrete, von Neumann regular k-algebra A so that each
polynomial in k[X] splits in A[X].
(7 pages, 4 August 2012)
- Walker's cancellation theorem,
(with Robert S. Lubarsky)
Let B and C be abelian groups and Z the additive group of integers.
Walker's cancellation theorem says that if the direct sum of Z with B
is isomorphic to the direct sum of Z with C, then B is isomorphic to C.
We construct an example in a diagram category of abelian groups where
the theorem fails. As a consequence, the original theorem does not
have a constructive proof even if B and C are subgroups of the free
abelian group on two generators.
(9 pages, 5 August 2012)
- A constructive theory of minimal zero-dimensional extensions,
In this paper we prove a constructive version of Chiorescu's theorem which
gives a complete set of invariants for minimal zero-dimensional extensions
of a commutative ring with dimension at most one, primary zero-ideal, and
Noetherian spectrum. This is done, in its full generality, without reference
to prime ideals and without the hypothesis of Noetherian spectrum.
(20 pages, 2 December 2010)
- Algebraic functions, calculus style,
A look at algebraic functions according to their definition in calculus texts.
(16 pages, 22 July 2010)
- Zero sets of univariate polynomials,
(with Robert S. Lubarsky)
Let L be the zero set of a nonconstant monic polynomial with complex
coefficients. In the context of constructive mathematics without countable
choice, it may not be possible to construct an element of L. In this paper
we introduce a notion of distance from a point to a subset, more general
than the usual one, that allows us to measure distances to subsets like L.
To verify the correctness of this notion, we show that the zero set of a
polynomial cannot be empty---a weak fundamental theorem of algebra. We also
show that the zero sets of two polynomials are a positive distance from each
other if and only if the polynomials are comaximal. Finally, the zero set of
a polynomial is used to construct a separable Riesz space, in which every
element is normable, that has no Riesz homomorphism into the real numbers.
(15 pages, 22 April 2009)
- Intuitionistic notions of boundedness in N,
We consider notions of boundedness of subsets of the natural numbers
N that occur when doing mathematics in the context of
intuitionistic logic. We obtain a new characterization of the notion of a
pseudobounded subset and formulate the closely related notion of a detachably
finite subset. We establish metric equivalents for a subset of N to
be detachably finite and to satisfy the ascending chain condition. Following
Ishihara, we spell out the relationship between detachable finiteness and
sequential continuity. Most of the results do not require countable choice.
(10 pages, 14 January 2008)
- Real numbers and other completions,
A notion of completeness and completion suitable for use in the absence of countable
choice is developed. This encompasses the construction of the real numbers as well as
the completion of an arbitrary metric space. The real numbers are characterized as a
complete archimedean Heyting field, a terminal object in the category of archimedean
(19 pages, 11 March 2007)
- Transient limits,
(with Katarzyina Winkowska-Nowak)
Let A be a Markov matrix depending on a small parameter s,
and Cn the average of the first n powers
The stationary distributions of A are the rows of S, the limit
of Cn as n goes to infinity.
The limiting stationary distributions are the rows of
the limit of S as s goes to zero. We
investigate transient limits of the sequence Cn.
These idempotent Markov matrices come up implicitly in an algorithm to
compute limiting stationary distributions. They represent the
intermediate-term behavior of the Markov chain at different time scales.
(16 pages, 10 March 2006)
- Subrings of zero-dimensional rings,
(with Jim Brewer)
When Sarah Glaz, Bill Heinzer and the junior author of this article
approached Robert Gilmer with the idea of editing a book dedicated to his work, we
asked him to give us a list of his work and to comment on it to the extent
he felt comfortable. As usual, he was extremely thorough in his response.
When the authors of this article began to consider what topic we wanted to
write about, we were impressed by Robert's comment that he was particularly
pleased with his series of papers with Bill on the embeddability of a ring
in a zero-dimensional ring. So we decided to write about that.
(16 pages, 12 December 2005)
- Near convexity, metric convexity, and convexity,
Scientific WorkPlace file,
It is shown that a subset of a uniformly convex normed space is nearly convex
if and only if its closure is convex. Also, a normed space satisfying a mild
completeness property is strictly convex if and only if every metrically
convex subset is convex.
(9 pages, 22 March 2005)
- Van der Waerden's construction of a splitting field,
Scientific WorkPlace file,
In his classic book, Modern Algebra, van der Waerden gave a
procedure for factoring polynomials over a finite-dimensional, separable,
simple extension field. I believe that there is a nonconstructive component
to his proof, and I will indicate where it comes in and why. Although I'm
sure that this component could be avoided while staying within the framework
that he set down, it is simpler to get around the problem by working with a
splitting algebra, which is easily constructed for any polynomial and
base field. The existence of a splitting field then follows from van der
(8 pages, 10 February 2005)
- Enabling conditions for interpolated rings,
Scientific WorkPlace file,
Given a ring B, a subring A, and a proposition P,
we can construct a ring C between A and B so that
C = B if P and C = A if not P.
This construction is
often used to obtain Brouwerian counterexamples. We investigate what conditions
have to be put on the inclusion of A in B in order that C
have some property (like being a unique factorization domain) for all P.
(9 pages, 23 January 2005)
- The ascending tree condition,
A strengthening of the ascending chain condition allows a choice-free
constructive development of the theory of Noetherian modules. Related topics
in the theory of PID's and elementary divisor rings are also explored.
(10 pages, 23 December 2001)
- A division algorithm,
A divisibility test of Arend Heyting, for polynomials over a field
in an intuitionistic setting, may be thought of as a kind of division
algorithm. We show that such a division algorithm holds for
divisibility by polynomials of content 1 over any commutative ring
in which nilpotent elements are zero. In addition, for an arbitary
commutative ring R, we characterize those
polynomials g such that the R-module endomorphism of
R[X] given by
multiplication by g has a left inverse.
(9 pages, 18 December 2001)
- Pre-abelian clan categories,
Categories of representations of clans without special loops, and with a
linear ordering at each vertex, are studied with an eye toward identifying
those that have kernels and cokernels. A complete characterization is given
for simple graphs whose vertices have degree at most two.
(12 pages, 18 July 2001)
- Spreads and choice in constructive mathematics,
An approach to choice-free mathematics using spreads:
If constructing a point satisfying property P requires choice,
replace this problem by that of constructing a nonempty set of
elements satisfying P. Then construct a spread, without choice,
whose elements satisfy P. The theory is developed
and several examples are given.
(11 pages, 21 June 2001)
- Equivalence of syllogisms,
Studies of categorical syllogisms typically focus on the valid ones:
which syllogisms are valid, why they are valid, how the valid ones
are classified, how to derive valid ones from other valid ones.
As Lear put it, "Our principal interest in invalid inferences is
to discard them." Here we are interested in the invalid syllogisms
too. The traditional methods for transforming one valid syllogism
into another also transform any syllogism, valid or not, into an
(22 pages, 20 December 2000)
- Weak Markov's principle, strong extensionality, and countable choice,
Ishihara showed, using countable choice, that weak Markov's principle
is equivalent to all real functions on a complete metric space being
strongly extensional. In this note we show that weak countable choice
suffices, and that the theorem fails in sheaf models of the real
(5 pages, 8 August 2000)
- Omniscience principles and functions of bounded variation,
A very weak omniscience principle is formulated, related omniscience
principles are considered, and the theorem that a function of bounded
variation is the difference of two increasing functions is shown to be
equivalent to the omniscience principle WLPO. It is also shown that an
arbitrary function (not necessarily strongly extensional) with located
variation on an interval is the difference of two increasing functions.
(10 pages, 31 July 2000)
- Weakly integrally closed domains: minimum polynomials of matrices,
(with James Brewer)
Must the coefficients of the minimum polynomial of a matrix over a domain
lie in that domain? This question leads to the notion of a weakly integrally
closed domain, over which the answer is "yes" for 3-by-3 matrices. It is
shown that certain subalgebras of k[t] are weakly integrally
closed, as are rings consisting of quadratic algebraic numbers.
(15 pages, 17 November 1999)
- Constructive mathematics without choice,
What becomes of constructive mathematics without the axiom of
(countable) choice? Using illustrations from a variety of areas,
it is argued that it becomes better.
(7 pages, 28 July 1999)
- Gleason's theorem has a constructive proof (revised),
Two recent papers have dealt with the possibility of a
constructive proof of Gleason's theorem. In the first,
Geoffrey Hellman claims to give an example showing that this is impossible
even in three-dimensional Euclidean space. In the second,
Helen Billinge suggests that some reformulation of Gleason's
theorem in three-space may have a
constructive proof. Douglas Bridges has noted that Hellman's example
leaves open the problem of finding a constructive
substitute---a theorem with a constructive proof that is easily seen
to be classically equivalent to Gleason's theorem.
It turns out that Gleason's formulation admits a constructive proof as it
stands. In this paper we discuss this seemingly anomalous situation.
Gleason's theorem itself is somewhat peripheral to the discussion.
What is interesting is the relationship of classical mathematics to
constructive mathematics that is highlighted by this misunderstanding.
(7 pages, 1 July 1999)
- Computing limiting stationary distributions of small noisy
networks, (with Katarzyna Winkowska-Nowak)
The dynamics of opinion transformation is modeled by a neural network
with a nonnegative matrix of connections. Noise is introduced at each
site, and the limit of the stationary distributions of the resulting
Markov chains as the noise goes to zero is taken as an indication of
what configurations will be seen. An algorithm for computing this
limit is given, and a number of examples are worked out. Some of the
mathematical ideas developed, such as visible states, time scales,
and a calculus of indexed probabilities, are of independent interest.
(35 pages, 21 June 1999)
- Pointwise differentiability,
What can be done with pointwise properties as opposed
to uniform properties on compact intervals?
(4 pages, 7 June 1999)
- Linear independence without choice,
(with Douglas Bridges and Peter Schuster)
The notions of linear and metric independence are investigated in relation
to the property: if U is a set of n+1 independent vectors,
and X is a set of n independent vectors, then adjoining some
vector in U to X results in a set of n+1 independent
vectors. It is shown that this property holds in
any normed linear space. A stronger property---that finite-dimensional
subspaces are proximinal---is established for strictly convex normed spaces
over the real or complex numbers. It follows that metric independence and
linear independence are equivalent in such spaces. Proofs are carried out in
the context of intuitionistic logic without the axiom of countable choice.
(9 pages, 12 December 1998)
- Trace-class operators,
(with Douglas Bridges and Peter Schuster)
In this paper we define trace-class and Hilbert-Schmidt operators---the
von Neumann-Schatten classes C1 and C2---without assuming the existence
of an adjoint or even an absolute value. In fact, an operator is in the
class Cp, for p in [1,¥), exactly when a certain supremum
exists. We prove that such operators are compact, hence have adjoints.
The theory is developed without appeal to separability, or to the
existence of an orthonormal basis, and without using countable choice.
We construct the singular values of compact operators, and characterize
them, and the classes Cp, in terms of their singular values.
(22 pages, 29 November 1998)
- The fundamental theorem of algebra: a constructive development
Can constructive mathematics be developed in a reasonable manner without the
axiom of countable choice? Serious schools of constructive mathematics all
assume it one way or another, but the arguments for it are not compelling.
Here it is shown how the fundamental theorem of algebra can be restated and
proved without using countable choice, and it is argued that this is really
the right way to look at it. A notion of a complete metric space, suitable
for a choiceless environment, is also developed. (21 pages, 16 October 1998)
- Nontransitivity of locatedness for subspaces of a Banach space,
Given subsets A contained in B of a metric space X,
such that A is located in B and B is located in X,
does it follow that A is located in X?
It does if A and B are subspaces of a Hilbert space X,
but not if X is just a Banach space. (5 pages, 27 August 1998)
- Subgroups of p5-bounded groups
(with Elbert A. Walker)
Each valuated module B with B(5) = 0 is a direct sum
of simply presented valuated modules and copies of two valuated modules
which come from (finite) hung trees. There are infinite-rank
indecomposable valuated modules B with B(6) = 0.
(21 pages, 8 July 1998)
- Adjoints and the image of the ball,
A bounded operator between Hilbert spaces has an adjoint if
and only if the image of the unit ball is located.
(7 pages, 29 July 1998)
- Is 0.999... = 1?,
A skeptical look at this well-known equation. Copyright
the Mathematical Association of America 1999. All rights reserved.
(8 pages, 15 May 1998)
- A weak countable choice principle,
(with Douglas Bridges and Peter Schuster)
A weak choice principle is introduced that is implied both by countable
choice and by the law of excluded middle. This principle suffices to prove
that metric independence is the same as linear independence in an arbitrary
normed space over a locally compact field, and to prove the fundamental
theorem of algebra.
(5 pages, 19 January 1998)
- Generalized real numbers in constructive mathematics,
Two extensions of the real number system, one given by uppercuts
the other by lowercuts, are developed within a constructive
framework. The first includes distances to arbitrary subsets,
the second includes norms of arbitrary bounded linear operators.
The intuitive meaning of comparing such quantities to ordinary
real numbers is preserved. Difficulties with encompassing both kinds
of numbers in a single system are considered.
(15 pages, 1 November 1997)
- Adjoints, absolute value and polar decomposition,
(with Douglas Bridges and Peter Schuster)
Bilinear forms, equivalent conditions for the existence of an
adjoint, general notion of a polar decomposition. Choiceless
Riesz representation theorem. (13 pages, 1 September 1997)
- Simply presented tag modules,
A tag module is a generalization, in any
abelian category, of a simply presented torsion
abelian group. The theory of
such modules is developed, it is shown that countably
generated tag modules are simply presented, and that Ulm's
theorem holds for simply presented tag modules.
Zippin's theorem is stated and proved for countably generated
(22 pages, 9 June 1997)
- Filtered modules over discrete valuation domains,
(with Elbert Walker)
We consider a unified setting for studying local valuated groups and
coset-valuated groups, emphasizing the associated filtrations rather than
the values of elements. Stable exact sequences, projectives and injectives
are identified in the encompassing category, and in the category
corresponding to coset-valuated groups. (27 pages, 22 May 1997)
- A constructive proof of Gleason's theorem,
(with Douglas Bridges)
Gleason's theorem states that any totally additive measure
on the closed subspaces, or projections, of a Hilbert space
of dimension greater than two is given by a positive operator
of trace class. (25 pages, 21 May 1997)
- The regular element property,
The property that an ideal whose annihilator is zero contains a regular
element is examined from the point of view of constructive mathematics. It
is shown that this property holds for finitely presented algebras over
discrete fields, and for coherent, Noetherian, strongly discrete rings that
contain an infinite field. (8 pages, 25 July 1996)
- Growing forests in abelian p-groups,
The purpose of this note is to finish a development of the theory of
simply presented p-groups which exploits tree structure as
much as possible. A proof that summands of simply presented p-groups
are simply presented that is independent of Ulm's theorem is given.
The same techniques are used to show that countable p-groups are
simply presented. Indeed it is shown that a summand of an Axiom-3
p-group is simply presented, thus settling both problems,
and showing that Axiom-3 p-groups are simply presented, at one go.
(6 pages, 10 June 1996) Journal of Algebra, 187(1997)
- Sets, complements and boundaries, (with Douglas Bridges and Wang Yuchuan)
The relations among a set, its complement, and its boundary are
examined constructively. A crucial tool is a theorem that allows the
construction of a point where a segment comes close to the boundary of a set
in a Banach space. Brouwerian examples show that many of the results are the
best possible. (23 pages, 14 May 1996) Indag. Math. 7(1996)
- Interview with a constructive mathematician,
This interview is cobbled together from a series of conversations that took
place on the list, l-math, during the fall of 1994. It concerns the nature
of constructive mathematics, starting with the question of whether the
objects studied by constructive mathematicians are the same as those
studied by classical mathematicians. (26 pages, 4 March 1996)
Modern Logic, 6(1996) 247--271.
Click here for comments by Gabriel Stolzenberg
on this paper.
- Flat dimension, constructivity, and the Hilbert syzygy theorem,
A constructive treatment of flat dimension of modules including a proof of
the Hilbert syzygy theorem. (13 pages, 13 February 1996)
- Confessions of a formalist, Platonist intuitionist,
A bit of automathography, and some musings. (1) Algebraists.
(2) A Platonist in trouble. (3) Constructive mathematics and recursive function
theory. (4) Different subject matter? (5) Intuitionistic logic. (6) Formalism.
(7 pages, 9 April 1994)
- Intuitionism as generalization,
Intuitionism, in its simplest form, is a generalization of
classical mathematics that accomodates both classical and
(5 pages, 1990)
The traditional Hare proportional system has a number of flaws when used
with a small electorate. For certain values of the parameters the algorithm
doesn't even work. In addition, the problem of ties becomes acute, so the
role of chance in determining the outcome is increased. We present here a
system that deals with these flaws while still staying very much in the
spirit of the original system. The main innovations are floating quotas,
fractional quotas, and breaking ties by complete enumeration of possible
(9 pages, late 1980's)
- A measure of consanguinity,
A natural numerical measure of consanguinity is developed that applies to
individuals with arbitrary multiple kinship connections. For simple
relationships the consanguineal distance specializes to the civil degree,
less two if the relationship goes through full siblings. This measure is
deduced from axioms motivated by an heuristic picture of blood mixtures. The
formula suggests a quantum mechanical probability interpretation whose
classical counterpart yields a generalization of the Murdock degree of
consanguinity. (10 pages, 1977)
Last modified 5 August 2012