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In this paper, I propose to examine Dedekind’s ideal of rigor in the context of some of his mathematical drafts. After a presentation of his ideal of rigor based on statements in his published works, I use drafts from his Nachlass to study his invention of the new concept of Dualgruppe (equivalent to our modern lattice). I question the extent to which these preliminary researches hold up to the same standards of rigor. Focusing on a specific law of Dualgruppe theory, I show that the elaboration of a rigorous work can be the outcome of a process that is not necessarily so. I put forward the trial-and-error and inductive aspects of Dedekind’s research practices. I consider whether the Dedekindian ideal of rigor guided mathematical research in its various phases, and what the consequences were of such an ideal of rigor, if any, on mathematical research.
Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of second-order via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to identifying an initial F-algebra in a certain category, we identify a natural coalgebraic dual to Boolos’s axiom which naturally interprets large parts of Aczel’s non-well-founded set theory via the construction of a certain terminal F-coalgebra, and we suggest a coalgebraic way forward for an abstraction-theoretic axiomatization of the real numbers.
Purity has been recognized as an ideal of proof. In this paper, I consider whether purity continues to have value in contemporary mathematics. The topics (e.g., algebraic topology, algebraic geometry, category theory) and methods of contemporary mathematics often favor unification and generality, values that are more often associated with impurity rather than purity. I will demonstrate this by discussing several examples of methods and proofs that highlight the epistemic significance of unification and generality. First, I discuss the examples of algebraic invariants and of considering a mathematical object from several different perspectives to illustrate that the methods used in contemporary mathematics favor impurity. Then I consider an example from category theory which demonstrates how unification and generality are related to impurity and that impure solutions can be explanatory. In light of this discussion, we see that purity only has marginal value within contemporary mathematics which instead prioritizes the epistemic values associated with impurity.
This paper is divided into two parts, the first being a point of departure for the second. I will begin by discussing a well-known negative argument due to Mark Lange concerning the explanatory role of mathematical induction. In the first part of the paper, I offer yet another response to Lange’s argument and attempt to characterize the sort of explanatory role played by inductive proofs. That account depends on two structural principles about explanatory proof that look like a fragment of a constructive semantics for that concept. The remainder of the paper fills out this semantics and explores its consequences. It will be clear that this framework does not constitute a fully general characterization of the concept of mathematical proof; the question will be whether there is a natural class of proofs that it does characterize. My answer will be that it nicely describes what I shall call grounding explanatory proofs. A proof of this sort explains the sentence proved in terms of the grounds of the fact that it describes. I will conclude by briefly exploring the connections between grounding proofs and the notion of purity.
A formal theory is categorical if any two of its models are isomorphic, that is, there is a structure-preserving bijection between them. It is natural to think that whether or not categoricity is a virtuous feature of a theory depends on whether we have some antecedent belief that the theory in question describes a unique structure. In this paper, I put forth a purpose-relative framework for assessing the virtuousness of a metamathematical property. According to this framework, whether categoricity (or any other property) is virtuous depends more deeply on what we take the purpose or function of formal theories to be. I then develop one important purpose we would like theories to serve which I call the instrumentalist conception of formal theories. I argue that given this instrumentalist conception, categoricity is best understood as having somewhat limited virtue, and that, in contrast, uncountable categoricity is highly virtuous.
In the mid-1970s, Gregory Chaitin proved a novel incompleteness theorem, formulated in terms of Kolmogorov complexity, a measure of complexity that features prominently in algorithmic information theory. Chaitin further claimed that his theorem provides insight into both the source and scope of incompleteness, a claim that has been subject to much criticism. In this article, I consider a new strategy for vindicating Chaitin’s claims, one informed by recent work of Bienvenu, Romashchenko, Shen, Taveneaux, and Vermeeren that extends and refines Chaitin’s incompleteness theorem. As I argue, this strategy, though more promising than previous attempts, fails to vindicate Chaitin’s claims. Lastly, I will suggest an alternative interpretation of Chaitin’s theorem, according to which the theorem indicates a trade-off that comes from working with a sufficiently strong definition of randomness—namely, that we lose the ability to certify randomness.
The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.